C E R N 8 7 - 0 3

2\ A P R I L

O R G A N I S A T I O N E U R O P É E N N E P O U R LA R E C H E R C H E N U C L É A I R E

CERN E U R O P E A N O R G A N I Z A T I O N F O R N U C L E A R R E S E A R C H

C A S CERN A C C E L E R A T O R SCHOOL

A D V A N C E D A C C E L E R A T O R PHYSICS

The Queen's College, Oxford, England 16-27 September 1985

PROCEEDINGS

Editor: S. Turner

Vol. 11

G L N E V A

1 9 8 ?

t 'URN - Service d'Information scientifique — RD'127- Í500- Avril I1)«?

ABSTRACT

This advanced course on general accelerator physics is the second of the biennial series given by the CERN Accelerator School and follows cn from the first basic course qiven at Gif-sur-Yvette, Paris, in 1984 (CERN Yellow Report 85-19). Stress is placed on the mathematical tools of Hani Kon i an mechanics and the Vlasov and Fokker-Planck equations, which are widely used in accelerator theory. The main topics treated in this present work include: nonlinear resonances, chromât icily, motion in longitudinal phase space, growth and control of longitudinal and transverse bean emittaiice, space-charge effects and polarization. The seminar progranne treats some specific accelerator techniques, devices, projects and future possibilities.

/

CONTENTS

Page NO.

Foi fiwui d . 1

Opening address 1

Local coordinates for the beam and frequently used symbols 3

j.s. 3eil

Ham i I Ionian mechanics 5 Introduction 5 Poisson brackets 8 Stationary and varying action S Poincaré invariant 10 Lagrange invariant 11 Symplectics U Liouvil ie's theorem 14 Conserved quadratic form 15 Charat ler is lk exponents 16 Canonical transformations 19 Point transformation 20 Change of independent, variable 22 Scaling 23 Dynamical evolution as canonical transformation 24 Poincaré invariant 25 Lagrange invariant 25 Liouville invariant 27 One degree of freedom 29 Action and angle variables 30 Smalt deviations from closed orbit 31 Adiabatic invariance of J 35 Small canonical transformation 36 Canonical perturbation theory 37

E.J.N. Wilson

Nonlinear resonances 41 Introduction 41 The general form of the Hamilton i an 43 The magnetic vector rotential for injltipoies 45 Linear dynamics in action angle variables 46 Perturbation theory 52 Effect of nonlinearities far from a resonance 55 Resonances 55 The third-integer resonance 57 The trajectory of a third-integer resonance 59 The effect of m octupole 62 Phase-space topology with amplitude frequency variation 63 Amplitude growth on crossinq a resonance 65 Synchrotron resonances 66 Beam lifetime due to maq.iet imperfections 6B The effect of two dimensions of transverse motion 69 Three dimensions af magnetic f ie ld 12 Conclusions 73

B. W. Montague

Chromatic effects and their first-order correction Introduction Basic ideas Chromatic perturbation equations T*ro dimensions

G. Guignard

Chromaticity: nonlinear aberrations Introduction Description of the nonlinear perturbations Perturbation theory in the canonical variables Dynamic aperture and analytical approach

G, Dôme

Theory of RF acceleration Energy gain and transit time factor Harmonic number Finite difference equations Differential equations for an arhitrary RF voltage Hamiltonian vith reduced variables Small oscillations around the stable fixed point Wjtion m the vicinity o, the fixed points Stationary bucket with a harmonic cavity Formulae for a sinusoidal RF voltaqe Adiabatk damping of phase oscillations Back Lo finite difference equations. Stochasticity Pfcise displacement acceleration Linear accelerators

L.R. h'voni} and J. Carey t,e

Beam-bean effects Introduction The beam-beam force Experimental and numerical data fron-, f+e" machines Experimental data from hadron machines Nonlinear beam-beam resonances Beam disrupt ion Conclusions

A. Piitinaki

Synchro-betatron resonances Introduction Dispersion in a cavity Transverse fields with longitudinal variation Beam-beam interaction with a crossing anqle

C. Guignard

Betatron coupling with radiation Introduction Perturbation treatment of linear betatron couplina /implitude variation due to radiation and acceleration Equilibrium in the case of betali Dn coupling with radiali Application to emittancç control

C. N • La xIimorn-Dav ies

Kinetic theory and the Vlasov equation Introduction The Vlasov equation The effect of binary collisions Fluid models

C. -V. Lashmove-Davies

Haves in plasmas 235 Introduction 235 Waves in a field-free plasma 236 Waves in a magnetized plasma 240 Low-frequency waves in a magnet i2ed plasma 245 Raman scattering 247

H. O. Hereward

Landau damping 255 Spectrum of linear oscillations 255 Longitudinal instability 259 Nonlinear oscillations 261

J.L. Laclare

Bunched beam coherent instabilities 264 Introduction 254 Longitudinal instabititiss 264 Transverse instabilities 3(16 Conclusion 325

I. Hofmcuin

Space charge dominated bean transport 327 Introduction 327 Basic properties 327 Efjiïttance and field energy 329 Application to eniitta.'ice growth 332

L, Palumbo caul V.G. Vaacaro

Halte fields, inpedarxes and Green's function 341 Longitudinal wake potential and impedance 341 Longitudinal impedance and wake potential for simple structures 344 General analysis 357 Circular accelerators 361 Transverse-wake potential and impedance 363 Remarks 368 Conclusions 369

C, Dôme

Diffusion due to RF noise 370 Statistical properties of random variables 370 FokJcer-Planck equation 372 Differential equations for a stationary bucket with amplitude and phase noise 374 Computation of the coefficients A¿, A2 ™ t n e Fokker-Planck equation 377 Case of a sinusoidal RF voltage 382 Hnite difference equations 386 Diffusion equation 388 Contribution of RF noise to the finite bean lifetime in the SPS collider 392

A. Pivinnki

Jntra-beam scattering 402 Introduction 402 Calculation of rise times and damping times 403 Experimental results 411

D. Bous sard

SchotLky noise and bean transfer function diagnostics 415 Schottky signals "17 Beam detectors 426 Observation of Schottky signals fl40 Bean transfer functions 445

D. !-iohl

Stochastic cooling 453 Introduction 453 Simplified theory, Lime-domain picture 455 A more detailed presentation of betatron cooling, frequency domain oicture 47.g Distribution function equations (Fokker-Planck) and momentum scalina 510

//. Polh

Electron cooling 534 introduction 534 What electron cooling is 534 Why electron cooling? 535 Ion-beam and storage-ring properties 535 How electron cooling works in principle 537 Introduction to electron cooling theory (for pedestrians) 53R Experimental realization of electron cooling 543 More on the theory of electron cooh'nq 552 Recombination 555 Electron cooling experiments 557 Simulation of electron cooling in storage rings 561 Electron cooling diagnostics 562 Applications of electron cooling 564 Electron cooling projects 566

J. Jewell

Electron dynamics with radiation and nonlinear wigglers 570 Introduction 570 The dynamics of electrons in a storage ring 571 Normal modes and optical functions 579 Radiation ¡amping 534 Quantum fluctuations and Fokker-Planck equations 537 Nonlinear wigglers 591

J. Le IM ff

Beam break up 610 Experimental evidence 610 Transverse deflection of charge particles in radio-frequency fields 612 Deflecting modes in circular iris loaded waveguides 614 Regenerative beam break up 617 Cumulative beam break U D 621

Beam loading 626 Introduction 626 Sinqle bunch passaqe in a cavity 627 Multiple bunch passaqes 629 Limiting case í>ü = 0 630 The case of a travelling wave structure 633 Transient correction 635 RF drive generation 637

J. iíuatt

P o l a r i z a t i o n i n e l e c t r o n arid p r o t o n beams 647 Introduction 647 Generalities on polarization and spin mot ion 649 Acceleration of polarized protons in synchrotrons 661 Polarization of electrons in storage rings 671

li. Ma~'-iîi 0. Ripken, A. it'yu I i ah caul t'. 5'jisnidt

P a r t i c l e t r a c k i n g 690 Introduction 690 Hamiltonian description of the proton motion 690 Dynamic aperture 693 Particle tracking 694 Qualitative theory of dynamical systems 696 Studies of chaotic behaviour in HERA caused by transverse magnetic multipole fields 699 Summary 704

M, i'ugit-ji

The r a d i o f r e q u e n c y q u a d r u p o l e l i n e a r a c c e l e r a t o r 706 Introduction 706 The accelerating structure 706 Outline of the T-K expansion 709 The vane tips shaping 71? Physical considerations 717 The structure of an RFQ 720 Design and technical considerations

Recent developments 733

//. HcL Fundamental f e a t u r e s o f s u p e r c o n d u c t i n g c a v i t i e s f o r high energy a c c e l e r a t o r s 736

Introduction 7?fi Some cavity fundamentals 737 Superconducting cavities ,'H\ Cavity design 716 Anomalous losses 751 Cavities covered with superconduclinq thin films 761 Current accelerator projects and achievcnenls 76S

J* La Duff

High f i e l d e l e c t r o n l i n a c s 7-V Introduction 7!¿ Extrapol a U u n of present technologies 7 7? RF compression scheme 775 Ultimate acceleratinq qradienls in conventional structures 7M3 A survey of acceleralinq structures 7Hrj RF power source: the laserIron w

0. Ut.it to Li, / I . l.cnicvi (JUti A. "JVI><S

F r e e e l e c t r o n l a s e r s : a s h o r t r e v i e w o f t h e t h e o r y and e x p e r i m e n t s 79? Introduction 79? FEL; theory and desiqn criteria ;qv> FEL storage ring operation SO.! Single passage FEL operation RHU Conclusions

U.A. u't'iiy <i't't '*'.''. /iff-;;

I S I S , t h e a c c e l e r a t o r based n e u t r o n source a t RAL 81' Introduction R(7 Linac and synchrotron B17 Target station Pl9 Experimental facilities $2\ High intensity performance of the ISIS synchrotron 02?

Heavy ions; Lhe present and future*)

Linear colliders versus storage rings*)

Contribution not received

SCHOTTKY M O I S E AMD BEAM T R A H S F E H F U N C T I O N D I f l G H O S T I C S

D B o u s s a r d

C E f l V , G e n e v a . S w i t z e r l a n d

A B S T R A C T

F o l l o w i n g t h e a n a l y s i s o f S c h o t t k y s i g n a l s f o r t h e u n b u n c i - e d a n d t h e b u n c h e d beam c a s e s , a g e n e r a l s t u d y o f e l e c t r o m a g n e t i c d e t e c t o r s i s p r e s e n t e d . H e r e t h e i m a g e - c u r r e n t a p p r o a c h a n d t h e L o r e n t z r e c i p r o c i t y t h e o r e m w i l l b e u s e d t o e v a l u a t e t h e d e t e c t o r ( o r p i c k - u p > p e r f o r m a n c e f o r s e v e r a l t y p i c a l e x a m p l e s . T h e n , s i g n a l - p r o c e s s i r . g t e c h n i q u e s , w h i c h p l a y an i m p o r t a n t r o l e i n t h e S c h o t t k y s i g n a l a n a l y s i s , w i l l b e r e v i e w e d . T h e beam t r a n s f e r f u n c t i o n w h i c h r e l a t e s t h e beam r e s p o n s e t o a n e x t e r n a l e x c i t a t i o n a l s o p r o v i d e s v e r y u s e f u l i n f o r m a t i o n a b o u t t h e a c c e l e r a t o r b e h a v i o u r . I t r e q u i r e s an e l e m e n t t o e x c i t e t h e beam ( k i c k e r ) w h i c h w i l l b e shot.-n t o b e e q u i v a l e n t t o a d e t e c t o r w o r k i n g i n r e v e r s e . W i t h b e a m - t r a n s f e r - f u n c t i o n m e a s u r e m e n t s an a s s e s s m e n t o f beam s t a b i l i t y l i m i t s c a n be m a d e , l e a d i n g t o t h e d e t e r m i n a t i o n o f t h e o v e r a l l r i n g i m p e d a n c e .

T h e n o i s e g e n e r a t e d I n an o l d f a s h i o n e d e l e c t r o n t u b u i s g o v e r n e d b y t h e S c h o t t k y

f o r m u l a w h i c h s i m p l y r e f l e c t s t h e f a c t t h a t t h e a n o d e c u r r e n t I s c o m p o s e d o f i n d i v i d u a l

e l e c t r o n s r a n d o m l y e m i t t e d b y t h e c a t h o d e . V e r y s i m i l a r l y , t h e beam c u r r e n t i n a c i r c u l a r

p a r t i c l e a c c e l e r a t o r , a l s o e x h i b i t s a r a n d o m c o m p o n e n t , c a l l e d t h e S c h o t t k y n o i s e , w h i c h

r e s u l t s f r o m t h e l a r g e , b u t f i n i t e , n u m b e r o f p a r t i c l e s i n t h e b e a m . I n t h e a b s e n c e o f

r a n d o m q u a n t u m e m i s s i o n s ( i . e . f o r h a d r o n m a c h i n e s } t h e a n a l y s i s o f S c h o t t k y n o L s e s i g n a l s

( o r S c h o t t k y s i g n a l s , f o r b r e v i t y ) i s a v e r y p o w e r f u l t o o l t o s t u d y t h e a c c e l e r a t o r

b e h a v i o u r . H i s t o r i c a l l y , S c h o t t k y s i g n a l s h a v e b e e n o b s e r v e d f i r s t on u n b u n c h e d beam

m a c h i n e s ( C E R N I S R ) , 1 ' ^ l e a d i n g t o t h e d e v e l o p m e n t o f t h e v e r y s u c c e s s f u l s t o c h a s t i c

c o o l i n g t e c h n i q u e . F o r b u n c h e d b e a m s , t h e p r e s e n c e o f s t r o n g " m a c r o s c o p i c " beam s i g n a l s

r e n d e r s t h e o b s e r v a t i o n o f t h e t i n y S c h o t t k y s i g n a l s m o r e d i f f i c u l t . H o w e v e r i m p r o v e d

s i g n a l p r o c e s s i n g t e c h n i q u e s h a v e r e c e n t l y made t h e i r o b s e r v a t i o n p o s s i b l e .

F o l l o w i n g t h e a n a l y s i s o f S c h o t t k y s i g n a l s F o r t h e u n b u n c h e d a n d t h e b u n c h e d beam

c a s e s , a g e n e r a l s t u d y o f e l e c t r o m a g n e t i c d e t e c t o r s i s p r e s e n t e d . H e r e t h e i m a g e - c u r r e n t

a p p r o a c h a n d t h e L o r e n t z r e c i p r o c i t y t h e o r e m w i l l b e u s e d t o e v a l u a t e t h e d e t e c t o r ( o r

p i c k u p ) p e r f o r m a n c e f o r s e v e r a l t y p i c a l e x a m p l e s . T h e n , s i g n a l - p r o c e s s i n g t e c h n i q u e s ,

w h i c h p l a y an i m p o r t a n t r o l e i n t h e S c h o t t k y s i g n a l a n a l y s t s , w i l l b e r e v i e w e d .

T h e beam t r a n s f e r f u n c t i o n w h i c h r e l a t e s t h e beam r e s p o n s e l o an e x t e r n a l e x c i t a t i o n

a l s o p r o v i d e s v e r y u s e f u l i n f o r m a t i o n a b o u t t h e a c c e l e r a t o r b e h a v i o u r . I t r e q u i r e s a n

e l e m e n t t o e x c i t e t h e b e a m ( k i c k e r ) w h i c h w i l l b e dtiown t o b e e q u i v a l e n t t o a d e t e c t o r

w o r k i n g i n r e v e r s e . W i t h beam - 1 r a n s f o r - f u n c t i o n m e a s u r e m e n t s an a s s e s s m e n t o f beam

s t a b i l i t y l i m i t s c a n b e m a d e , l e a d i n g t o t h e d e t e r m i n a t i o n o f t h e o v e r a l l r i n g I m p e d a n c e .

1 SCHOTTKY S I G N A L S

U n b u n c h e d b e a m , l o n g i t u d i n a l

F o r a s i n g l o p a r t i c l e c i r c u l a t i n g i n t h e m a c h i n e ( c h a r g e e , r e v o l u t i o n p e r i o d

T . = 1 / f . ) , t h e beam c u r r e n t , a t a g i v e n l o c a t i o n i n t h e r i n g , i s c o m p o s e d o f an i n f i n i t e

t r a i n o f d e l t a p u l s e s C F i g . l a ) s e p a r a t e d i n t i m e b y T . . I n f r e q u e n c y d o m a i n , t h i s

p e r i o d i c w a v e f o r m i s r e p r e s e n t e d b y a l i n e s p e c t r u m ( r i g - l b ) , t h e d i s t a n c e b e t w e e n l i n e s

b e i n g f . = .

i ^ t ) = e f i £ e x p j n u j t

L o o k i n g a t p o s i t i v e f r e q u e n c i e s o n l y :

i j C t ) = e f L + 2 e f i £ COB m ^ t . ( 2 )

n=.l

T h e f i r s t t e r m r e p r e s e n t e t h e DC c o m p o n e n t , t h e o t h e r s a r e c i m p l y t h e s u c c e s s i v e h a r m o n i e s

o f t h e r e v o l u t i o n f r e q u e n c y .

2 e f ,

a} b)

F i g . 1 a ) T i m e d o m a i n á p u l s e s b ) F r e q u e n c y d o m a i n : l i n e s p e c t r u m

' i - c l e s , r a n d o m l y d i s t r i b u t e d i n a z i m u t h a l o n g t h e r i n g c i r c u m f e r e n c e

(debt :•• : - ti c a s e ) a n d h a v i n g s l i g h t l y d i f f e r e n t , e a c h l i n e a t f r e q u e n c y n f ^ w h i c h

i s i n n : i y n a r r o w i n t h e c a s e o f a s i n g l e p a r t i c l e , w i l l b e r e p l a c e d b y a b a n d o f

f r e q u e n c i e s ( S c h o t t k y b a n d ) w h o s e w i d t h i s s i m p l y :

Ä f . i s t h e s p r e a d i n p a r t i c l e ' s r e v o l u t i o n f r e q u e n c i e s r e s u l t i n g f r o m t h e r e l a t i v e

momentum s p r e a d û p / p a n d t h e m a c h i n e p a r a m e t e r fi = ~ l / -lf ) • f f l i s t h e a v e r a g e

r e v o l u t i o n f r e q u e n c y .

When a v e r a g i n g e q u a t i o n ( 2 ) o v e r N p a r t i c l e s , o n l y Lhe DC te rms r e n a i n < i n ( . - M - f

Q '

t h e o t h e r components c a n c e l due t o t h e random a z i m u t h phase f a c t o r . H o w e v e r , t h e r . m . s .

c u r r e n t p e r band w h i c h i s g i v e n by t h e sum:

Ze f ( c o s 0 + cos 1

does n o t v a n i s h b e c a u s e o f t h e cos 6 t e r m s . One o b t a i n s :

The r . m . s . c u r r e n t p e r band f S c h o t t k y c u r r e n t ) i s i n d e p e n d e n t o f n ( h a r m o n i c number )

and p r o p o r t i o n a l t o t h e s q u a r e r o o t o f ' h e number o f p a r t i c l e s N

As i n d i c a t e d on F i g . 2 , t h e - j w e r s p e c t r a l d e n s i t y , p r o p o r t i o n a l t o i > / û f ,

d o n - f a s e s w i t h n u n t i l o v e r l a p o c c u r s f ft f > f ) . Fo r a g i v e n band t h e lue i l l power d e n s i t y

i s o b v i o u s l y p r o p o r t i o n a l t o t h e number o f p a r t i c l e s p e r u n i t f r e q u e n c y . I f t h e ¡ lar- imct e r

n i s known ( n may be f r e q u e n c y d e p e n d e n t ) , t h e measurement, o f t h e power s p e c t r a l

d e n s i t y , i n one o a r t i c u l n r S c h o t t k y band g i v e s d i r e c t l y t h e A p / p d i s t r i b u t i o n o f t h e

beam.

<J> û f n

F i g . 2 Power s p e c t r a l d e n s i t y c-f s c h o t t k y l i n e a w i t h i n c r e a s i n g n

Th i s forais t h e b a s i s of û p / p beam d i s t r i b u t i o n measurement - ir. UC c o a s t i n g

x a c h i n e s , ( c o o l i n g and a c c u m u l a t i o n r i n g s i n p a r t i c u l a r ) .

N o t e t h a t t h e n o i s e s i g n a l s p e r t a i n i n g t o s u c c e s s i v e ¡ í r h c t t k y bands a r e n o t

c o r r e l a t e d b e c a u s e t h e random a z i m u t h a l phase f a c t o r i s m u l t i p l i e d by n i n Eq . ( M -

1 .? U r b u n c h e d beam, t r a n s v e r s e

F o r a s i n g l e p a r t i c l e , t h e beam c u r r e n t > . ( t ) must be r e p l a c e d by t h e d i p o l e

moment: iV ( t ) J a . ( t ) . i ^ i t ) , w h e r e a ^ ( t ) i s t h e t r a n s v e r s e d i s p l a c e m e n t . The i*"*1

p a r t i c l e e x e c u t e s a s i n u s o i d a l b e t a t r o n o s c i l l a t i o n , o f a m p l i t u d e a . , w h i c h c a n be w r i t t e n :

H e r e • . f. i s t h e o b s e r v e d f r e q u e n c y , ? t a f i x e d l o c a t i o n i n t h e r i n g , q b e i n g t h e

non i n t e g e r p a r t o f t h e b e t a t r o n t u n e ( F i g - 3 a )

I n f r e q u e n c y d o m a : n :

d . ( t ) = a . r o s ( q . u , t * v . ) e f

h 1 _ l n * 1 1 l f i

I I J !

0 n ft ( n*l ] f,o

P i g . 3 T i m e ( a ) and F r e q u e n c y ( b ) doma in r e p r e s e n t a t i o n s o f a s i n g l c -p a r t i c i e t r a n s v e r s e o s c i l l a t i o n

d . ( t ) = a i e f Q

R

e j ¿ ] e * P J l t n * q . ) Wjt + ip i > I j . ( 9 )

T h e s p e c t r u m i s a g a i n a s e r i e s o f l i n e s s p a c e d by t h e r e v o l u t i o n f r e q u e n c y o f t h e i * * 1

p a r t i c l e , b u t s h i f t - e d i n f r e q e n c y by f^ . L o o k i n g a t p o s i t i v e f r e q u e n c i e s o n l y

( F i g . 3 b ) o n e o b t a i n s two b e t a t r o n L i n e s p e r r é v o l u t i o n f r e q u e n c y band as i n t h s c a s e o f

an a m p l i t u d e - m o d u l a t e d c a r r i e r w h i c h e x h i b i t s two s y m m e t r i c a l s i d e b a n d s .

F o r M p a r t i c l e s i n t h e b e a m , a g a i n r a n d o m l y d i s t r i b u t d i n a z i m u t h and i n b e t a t r o n

p h a s e s , a v e r a g i n g e q u a t i o n ( 9 ) , f o r a g i v e n V a l u e o f n + q , g i v e s :

A g a i n , t h e t o t a l p o w e r p e r S c h o t t k y bant] i s i n d e p e n d e n t o f i t s l o c a t i o n i n t h e

i r t q u ^ n c y r p e c t r u m ; i t i s p r o p o r t i o n a l t o t h e number o f . a r t i c l e s i n t h e beam and t o t h e

s q u a r e o f t h e r . m s . o s c i l l a t i o n a m p l i t u d e .

- J : O -

E a c h S c h o t t k y b a n d h a s now a f i n i t e w i d t h w h i c h r e s u l t s f r o m t h e s p r e a d o f r e v o l u t i o n

. ' r e q u e n r i e s A f ^ / f ^ = n A p / p a n d f r o m t h e s p r e a d o f b e t a t r o n f r e q u e n c i e s o q ^ - T h e l a t t e r

u s u a l l y c o m e s f r o m t h e m a c h i n e c h r o m a t i c i t y £ : û q . - Q r f i p / p . b u t may a l s o r e s u l t f r o m

s p a c e c h a r g e , b e a m - b e a m o r n o n l i n e a r e f f e c t s .

T h e l i n e w i d t h o f two a d j a c e n t S c h o t t k y b a n d s {n ± q ) i s g i v e n b y :

û f = (n + q> A f . • 2n f û q .

i f o n l y c h r o m a t i c i t y c o n t r i b u t e s t o t h e b e t a t r o n f r e q u e n c y s p r e a d .

E q u a t i o n ( 1 3 ) s h o w s t h a t t h e w i d t h o f t h e t w o S c h o t t k y b a n d s i s n o t t h e s a m e , d u e t o

t h e m a c h i n e c h r o m a t i c i t y . H o w e v e r , b y c o m p a r i n g t h e two b a n d s n * q , o n e can d e t e r m i n e

t h e û q . o f t h e b e a m . E v e n m o r e , i f m e c a n i d e n t i f y s i m i l a r p o i n t s on t h e d i s t r i b u t i o n

( r e s o n a n c e s , f o r i n s t a n c e ) , t h e i r q c a n b e d e t e r m i n e d b y t h e f o r m i j L a :

û f b e i n g t h e m e a s u r e d f r e q u e n c y d i f f e r e n c e b e t w e e n t h e m . T h i s t e c h n i q u e w a s e x t e n s i v e l y

u s e d i n t h e I S B t o m o n i t o r t h e w o r k i n g l i n e o f t h e m a c h i n e d i s t r i b u t i o n i n t r a n s v e r s e ;

t u n e s .

C o m p a r i n g e q u a t i o n s f 1 0 J a n d ( 6 ) g i v e s a d i r e c t m e a s u r e o f t h e r . m . s . b e t a t r o n

amp I i t u d e :

E q u a t i o n ( 1 5 ) c a n b e i-t,ed t o m e a s u r e d i r e c t l y t h e t r a n s v e r s e beam e m i t t a n c e , i f t h e

beam d i s t r i b u t i o n i s k n o w n . T h i s o b v i o u s l y r e q u i r e s w e l l c a l i b r a t e d l o n g i t u d i n a l a n d

t r a n s v e r s e d e t e c t o r s t o m e a s u r e a c c u r a t e l " d and i u n l e s s o n l y r e l a t i v e m e a s u r e m e n t s

rm<* rms a r e s o u g h t ( e v o l u t i o n o f AA t r a n s v e r s e e m i t t a n c e , f o r i n s t a n c e ) .

1 . 3 B u n c h e d b e a m . l o n R J t u d i n a l

I n t h e b u n c h e d beam c a s e , e v e r y i n d i v i d u a l p a r t i c l e e x e c u t e s s y n c h r o t r o n o s c i l l a t i o n s

a t Lhe f r e q u e n c y The t i m e o f p a s s a g e o f t h e p a r t i c l e i n f r o n t o f t h e d e t e c t o r i s

m o d u l a t e d a c c o r d i n g t o :

• ^ ( t ) = T ¿ s i n Cfî^t * H^) ( 1 6 >

i . ( t ) i s t h e t i m e d i f f e r e n c e w i t h r e s p e c t t o t h e s y n c h r o n o u s p a r t i c l e ( f r e q u e n c y f Q )

and T^ i s t h e a m p l i t u d e o f t h e s y n c h r o t r o n o s c i l l a t i o n , assumed t o be l i n e a r . I n t i m e

d o m a i n , t h e beam c u r r e n t i s r e p r e s e n t e d i n F i g . d , as a s e r i e s o f d e l t a p u l s e s , w i t h a

m o d u l a t e d t i m e o f p a s s a g e . I t can be w r i t t e n :

s i n (Í2 t ^ * . ) ) }

< - L ( t l

F i g . 4 T i m e doma in r e p r e s e n t a t i o n o f a s i n g l e p a r t i c l e c u r r e n t i n ¡ b u n c h e d beam.

U s i n g t h e r e l a t i o n :

exp (j iz s i n e)> = £ .1 < z )

w h e r e i s t h e B e s s e l f u n c t i o n o f o r d e r p , o n e c a n e x p a n d t h e n t h h a r m o n i c i n e q u a t i o n

( 1 7 ) and o b t a i n :

T . ) exp j (n t*! t *• p í l t

Each r e v o l u t i o n f r e q u e n c y l i n e ( n f Q > now s p l i t s i n t o a n i n f i n i t y o f s y n c h r o t r o n

s a t e l l i t e s , s p a c e d by t h e a m p l i t u d e s o f w h i c h b e i n g p r o p o r t i o n a l t o t h e B e s s e !

f u n c t i o n s o f a r g u m e n t nio T . a s shown i n F i g . 5-o l

T h e a m p l i t u d e s o f t h e s y n c h r o t r o n s a t e l l i t e s h e t o m e n e g l i g i b l e beyond a c e r t a i n v a l u e

o f p . T h i s i s b e c a u s e J p ( x ) ~ 0 f o r p > x i f x i s l a r & e . T h e r e f o r e , ( h e s y n c h r o t r o n

s a t e l l i t e s a r e , i n p r a c t i c e , c o n f i n e d i n t o a l i m i t e d b a n d w i d t h :

2p í í = 2n w t . f t ( 2 0 )

J .

n u n

^ Significant bandwidth

F i g . 5 D e c o m p o s i t i o n o f each r e v o l u t i o n l i n e i n t o s y n c h r o t r o n s a t e l l i t e s

t h The s p r e a d i n t h e i n s t a n t a n e o u s r e v o l u t i o n f r e q u e n c y o f t h e i p a r t i c l e due t o t h e

s y n c h r o t r o n o s c i l l a t i o n i s s i m p l y :

C o n s e q u e n t ly , f o r l a r g e v a l u e s o f n , t h e s i g n i f i c a n t b a n d w i d t h a r o u n d l i n e n i s t h f

same as t h a t c f a beam o f many p a r t i c l e s h a v i n g t h e same Äu and t h e r e f o r e t h e same

Ä p / p .

C o n s i d e r now t h e case o f many p a r t i c l e s , w i t h randomly d i s t r i b u t e d s y n c h r o t r o n p h a s e s

and r r a n g i n g f r o m 0 t o ( 2 , ,

m

b e i n E t h e t o t a l bunch l e n g t h ) .

F o r a g i v e n n , t h e c e n t r a l l i n e Cp = 0 ) shows t h e same p h a s e f a c t o r ( e x p j n ^ t ) f o r

a l l p a r t i c l e s : t h e c u r r e n t i n t h e c e n t r a i l i n e i s t h e r e f o r e p r o p o r t i o n a l t o N and n o t / N ;

t h i s i s s i m p l y t h e m a c r o s c o p i c RF c u r r e n t o f t h e b u n c h . On t h e c o n t r a r y , t h e s y n c h r o t r o n

s a t e l l i t e s ( p * 0 ) add r . m . s . w i s e b e c a u s e o f t h e random phase f a c t o r

exp j C n ^ t f p t î s + p i f O ( F i g . 6 ) .

Edch l i n e i s i n f i n i t e l y n a r r o w i f t h e s y n c h r o t r o n o s c i l l a t i o n i s p u r e l y l i n e a r

(fï i s t h e same f o r a l l p a r t i c l e s ) and i f t h e m a c h i n e has no i m p e r f e c t i o n s . H o w e v e r ,

magnet and RF f l u c t u a t i o n s b r o a d e n i n p r a c t i c e e a c h i n d i v i d u a l l i n e . I n a d d i t i o n a s p r e a d

i n s y n c h r o t r o n f r e q u e n c y w i t h i n t h e b u n c h ûSî^ t r a n s f o r m s e a c h s a t e l l i t e (p^O) i n t o a band

o f w i d t h P ^ s • F o r l a r g e v a l u e s o f n , o v e r l a p b e t w e e n s u c c e s s i v e s y n c h r o t r o n s a t e l l i t e s

( p i ß > fî ) can o c c u r w i t h i n t h e s i g n i f i c a n t w i d t h ft t h e S c h o t l k y band o f o r d e r n .

( F i g . 6b>

I f we c o n s i d e r two S c h o t t k y bands w i t h d i f f e r e n t v a l u e s of n , t h e i r c o r r e s p o n d i n g

s y n c h r o t r o n s a t e l l i t e s ( o f o r d e r p ) a r e c o r r e l a t e d . T h i s r e s u l t s f rom Eq . ( 1 5 ) , where t h e

random phase f a c t o r p * . i s t h e same, even f o r d i f f e r e n t v a l u e s o f n .

<Z1 )

A n o t h e r way t o l o o k a t t h e c o h e r e n c e b e t w e e n s u c c e s s i v e S c h o t t k y bands i s t o e x a m i n e

t h e bunch s i g n a l i n t i m e domain C F i g . 7 ) . I t i s composed o f a s t e a d y component

J , " I i

1 »v significant fcandwidiu. 2nUan.iT

b- L arg e n

F i g . 6 L o n g i t u d i n a l S c h o t t k y s p e c t r u m o f a bunched beam.

S f * « d y stgntt Z Jg U m T

F l u c t u a t i n g signal Ï I Jp

F i g . 7 T i m e d o m a i n o f r e p r e s e n t a t i o n o f b u n c h e d beam S c h o t t k y s i g n a l .

( m a c r o s c o p i c s i g n a l r e s u l t i n g f r o m t h e t e r m s : ^ J ( n u r , ) ) and a f l u c t u a t i n g S c h o t t k y

E _ n 0 0 L

? . J (nw T , ) ) . T h e f l u c t u a t i n g s i g n a l e x t e n d s i n t i m e o v e r 2 i , and can be D pTu P 0 1 M

F o u r i e r decomposed i n t o components a t m u l t i p l e s o f t h e f u n d a m e n t a l bunch f r e q u e n c y

= 1 /2 - r^ . A l l i n f o r m a t i o n c o n c e r n i n g t h e S c h o t t k y s i g n a l i s c o n t a i n e d i n t o t h o s e

components t i n t h e l i m i t Q g f . <*Q) • î n o t l i e r w c r ú s s i g n i f i . : i n f u r " . a t i o n a b o u t t h e

S c h o t t k y s i g n a l o n l y a p p e a r s e v e r y f^ f r e q u e n c y i n t e r v a l , t h e o t h e r s p e c t r a l l i n e s i n

( 1 9 ) ( e v e r y f Q ) s i m p l y g i v e r e d u n d a n t i n f o r m a t i o n , I . e . , t h e y a r e c o r r e l a t e d .

As a c o n s e q u e n c e , s a m p l i n g o f S c h o t t k y s i g n a l s a t f q , w h i c h f o l d s many n f Q bands

on t o p o f e a c h o t h e r and o n l y g i v e s one S c h o t t k y s i g n a l , does n o t i n t r o d u c e a n y l o s s o f

i n f o r m a t i o n , i f t h e b a n d w i d t h b e f o r e s a m p l i n g i s l i m i t e d t o + f . / 2 .

1 . à Bunched beam, t r a n s v e r s e

H e r e we h a v e t o combine t h e a m p l i t u d e m o d u l a t i o n ( b e t a t r o n o s c i l l a t i o n ) and t h e t i m e

m o d u l a t i o n ( s y n c h r o t r o n o s c i l l a t i o n ) . One o b t a i n s :

d . ( t ) = a . cosCq.td t • u>.) e f R / ) exp j n w ( t •• ^ s i n ( Q t + U ) . ) ) i i ^ i P i o e ) ¿ j r J o i s i

I f q . i s i n d e p e n d e n t o f w . , t h e n sum becomes:

d = e f a . E J Y" J < < n + q ) u i . ) e x p j [ C ( n + q ) u +pfl ) t * p * . *-<p. i f n o i e ( Z _ _ , p - o i o s 1 1 l

A g a i n , e s c h b e t a t r o n l i n e s p l i t s i n t o an i n f i n i t e number o f s y n c h r o t r o n s a t e l l i t e s ( F i g .

8 ) . The s i g n i f i c a n t b a n d w i d t h , a s i n t h e l o n g i t u d i n a l c a s e , a p p r o a c h e s t h a t o f c o a s t i n g

beams w i t h t h e same û p / p , f o r l a r g e v a l u e s o f n . On t h e c o n t r a r y , f o r s m a l l v a l u e s o f

n , most o f t h e e n e r g y i s c o n c e n t r a t e d i n t h e p = 0 l i n e .

F i g . B D e c o m p o s i t i o n o f e a c h b e t a t r o n l i n e i n t o s y n c h r o t r o n s a t e l l i t e s .

F o r a non z e r o c h r o m â t i c i t y , t h e a r g u m e n t o f t h e Besse 1 f u n c t i o n ( n + ^ " o ' t s h o u l d

be r e p l a c e d by [ ( n + q ) - Q l / n J W Q I . . I n t h i s c a s e , t h e r e l a t i v e a m p l i t u d e s o f t h e

s y n c h r o t r o n s a t e l l i t e s a l s o depend on t h e c h r o m a t i c i t y . I n p a r t i c u l a r , f o r t h e c h r o m a t i c

f r e q u e n e y :

o n l y t h e t e r m J i s s i g n i f i c a n t : a l l t h e e n e r g y o f t h e n t h S c h o t t k y band i s c o n c e n t r a t e d

i n t h e c e n t r a l l i n e

W i t h many p a r t i c l e s , we s h o u l d a v e r a g e o v e r t h e two random v a r i a b l e s ip. and

. U n l i k e t h e l o n g i t u d i n a l c a s e , t h e c e n t r a l L i n e s ( p = 0 ) add up r . m . s . w i s e due t o

t h e random b e t a t r o n p h a s e f a c t o r , t h e c o n s e q u e n c e b e i n g t h a t t h e r e i s no t r a n s v e r s e

m a c r o s c o p i c s i g n a l . S u c c e s s i v e b a n d s a r e c o r r e l a t e d as i n t h e l o n g i t u d i n a l c a s e , a g a i n ,

b e c a u s e a l l t h e s i g n a l i s c o n c e n t r a t e d i n t h e t i m e i n t e r v a l and n o t T q = i ' f ^ as i f

t h e beam w e r e u n b u n c h e d .

The w i d t h o f t h e c e n t r a l l i n e i s d e t e r m i n e d by RF and m a g n e t i c f i e l d f l u c t u a t i o n s ,

b u t a l s o by t r a n s v e r s e n o n l i n e a r i t i e s ( t u n e s p r e a d due t o o c t u p o l e f i e l d s , beam beam o r

s p a c e c h a r g e f o r c e s ) . I n a d d i t i o n , t h e s y n c h r o t r o n s a t e l l i t e s a r e b r o a d e n e d by t h e s p r e a d

i n s y n c h r o t r o n f r e q u e n c i e s w i t h i n t h e b u n c h ( w i d t h pûîî as i n F i g . 6 ) .

The t o t a l p o w e r p e r band ( f o r a g i v e n n ) i s g i v e n b y :

!zy "-i"» V i »

W i t h t h e i d e n t i t y :

one o b t a i n s :

The t o t a l power p e r band i s t h e same as in the c o a s t i n g beam c a s e , f o r t h e same t o t a l

number o f p a r t i c l e s and t h e same t r a n s v e r s e o s c i l l a t i o n a m p l i t u d e ( F i g . 9Í-

F i g . 9 H o r i z o n t a l S c h o t t k y s i g n a l s i n t h e SPS. T o p : d e b u n c h e d beam B o t t o m : b u n c h e d beam.

" BEAM DETECTORS

2 . 1 T h e i m & f t e - c u r r e n t a p p r o a c h

C o n s i d e r t h e v e r y s i m p l e g o o m e t r y o f F i g . 1 0 a , w h e r e a r o u n d beam c i r c u l a t e s i n t h e

c e n t e r o f a c y l i n d r i c a l smooth vacuum c h a m b e r . T h i s i s a t w o - d i m e n s i o n a l p r o b l e m , and i t

i s w e l l known t h a t t h e e l e c t r o m a g n e t i c f i e l d s a r e p u r e l y t r a n s v e r s e , as i n a c o a x i a l l i n e .

- 42b -

2

tí

S 0 I l o a d )

a) b)

F i g . 10 T h e beam i s e q u i v a l e n t t o a c u r r e n t s o u r c e f l o w i n g i n t o t h e d e t e c t o r i m p e d a n c e .

i n t h e l i m i t u = c . I t f o l l o w s t h a t f o r a l l f r e q u e n c i e s t h e beam and w a l l c u r r e n t s a r e

o p p o s i t e :

i . = - i (21 b w

E q u a t i o n ( 2 8 ) i s o n l y v a l i d up t o some u p p e r f r e q u e n c y , d e p e n d i n g on t h e p a r t i c l e

r e l a t i v i s t s f a c t o r Y and t h e t r a n s v e r s e d i m e n s i o n s o f t h e vacuum c h a m b e r . H o w e v e r , f o r

most p r a c t i c a l c a s e s ( h i g h e n e r g y s t o r a g e r i n g s ) t h i s i s n o t a l i m i t a t i o n .

I f now we c u t a gap i n t h e c i r c u l a r w a l l we I n t r o d u c e a c o u p l i n g b e t w e e n t h e i n s i d e

and bhe o u t s i d e o f t h e vacuum p i p e . The l a t t e r i s c h a r a c t e r i s e d by t h e i m p e d a n c e Z w h i c h

we can m e a s u r e b e t w e e n t h e two s i d e s o f t h e g a p . As t h e e n e r g y l o s t by t h e beam when

p a s s i n g t h r o u g h t h e d e t e c t o r i s much s m a l l e r t h a n t h e p a r t i c l e ' s e n e r g y , t h e c u r r e n t i ^ ,

and h e n c e i s i n d e p e n d e n t o f t h e gap v o l t a g e : i t means t h a t t h e w a l l c u r r e n t i ^ w h i c h

f l o w s t h r o u g h Z c a n be r e p r e s e n t e d by a p u r e c u r r e n t s o u r c e ( F i g . 1 0 b ) .

T h e d e t e c t o r , w h i c h s e e n f r o m t h e gap a p p e a r s l i k e an impedance 2 , d e l i v e r s i t s

o u t p u t s i g n a l i n t h e l o a d ( F i g - 1 0 a ) . T h e s e n s i t i v i t y o f t h e d e t e c t o r ( l o n g i t u d i n a l

i n t h i s c a s e ) i s d e f i n e d b y :

F o r a l o s s l e s s n e t w o r k b e t w e e n gap and , one can e a s i l y o b t a i n , f r o m p o w e r

o n s i d e r a t i o n s :

T h e f o l l o w i n g e x a m p l e s w i l l i l l u s t r a t e t h e image c u r r e n t a p p r o a c h f o r t h e e v a l u a t i o n

o f beam d e t e c t o r s ( o r beam p i c k - u p s ) .

a ) T h e r e s i s t i v e - g a p p i c k - u p

I n t h i s c a s e t h e l o a d r e s i s t o r R^ i s s i m p l y c o n n e c t e d t o t h e vacuum chamber g a p .

H o w e v e r , t o p r o v i d e a l o w i m p e d a n c e DC r e t u r n p a t h f o r t h e w a l l c u r r e n t , a s h o r t - c i r c u i t e d

c o a x i a l l i n e i s b u i l t a r o u n d t h e vacuum c h a m b e r , as shown on F i g . 1 1 . T h e l i n e i s f i l l e d

w i t h l o s s y m a t e r i a l ( f e r r i t e s ) SUCH t h a t , f o r t h e o p e r a t i n g f r e q u e n c y o f t h e p i c k - u p , i t

a p p e a r s as a t e r m i n a t e d l i n e . T h i s i n t r o d u c e s a l o w - p c s s c h a r a c t e r i s t i c i n t h e d e t e c t o r

r e s p o n s e .

aösorbins msfE'ial at high frequency (ferrirel

F i g . 11 R e s i s t i v e - g a p p i c k - u p

T h e u p p e r f r e q u e n c y l i m i t i s d e t e r m i n e d b y t h e p a r a s i t i c c a p a c i t a n c e a t t h e g a p .

M a k i n g R^ s m a l l ( s e v e r a l p a r a l l e l r e s i s t o r s ) w i l l p u s h t h e u p p e r f r e q u e n c y l i m i t , a t t h e

e x p e n s e o f s e n s i t i v i t y .

T h e SPS w i d e - b a n d l o n g i t u d i n a l d e t e c t o r 3 * u s e s e i g , h t p a r a l l e l 50 fi s t r i p l i n e s

s y m m e t r i c a l l y c o n n e c t e d t o t h e g a p , and a f e r r i t e l o a d e d c o a x i a l l i n e w i t h 25 £3

c h a r a c t e r i s t i c i m p e d a n c e . T h i s a r r a n g e m e n t g i v e s Z = 5 ß . T h e e i g h t gap s i g n a l s a r e

c o m b i n e d i n an e i g h t p o r t p o w e r c o m b i n e r g i v i n g an o v e r a l l s e n s i t i v i t y , i n a SO l o a d :

5^8 ¡ 14 Q

i n s t e a d o f t h e maximum S = yj 6 . 2 5 x 5 0 = 1 7 . 6 « i f no power w o u l d be l o s t I n t h e

f e r r i t e s ( v e r y h i g h i m p e d a n c e c o a x i a l l i n e ) .

T h e b a n d w i d t h e x t e n d s f o r m A MHz t o 4 GHz w i t h a l m o s t no r e s o n a n c e s . To i m p r o v e t h e

l o w - f r e q u e n c y r e s p o n s e t h e i n d u c t a n c e o f t h e s h o r t c i r c u i t e d l i n e can b e i n c r e a s e d by

l o s s l e s s f e r r i t e s , b u t h i g h - f r e q u e n c y r e s o n a n c e s may be d i f f i c u l t t o s u p p r e s s .

b ) T h e d i r e c t i o n a l - c o u p l a r p } c k - u p

Afl shown on Fig. 1 2 a , t h e r e a r e two g a p s I n t h i s d e t e c t o r , j o i n e d t o g e t h e r b y a p i e c e

o f c o u x i a l l i n e o f c h a r a c t e r i s t i c i m p e d a n c e R^, s u r r o u n d i n g t h e vacuum c h a m b e r . U i t h

t h e t>+o l o a d r e s i s t o r s K w h i c h a r o c o n n e c t a i ] t o e a c h ( a p , one c a n d r a w t h e ä q u i v a l e n t

- 4 : s -

( 3 0 )

v a n i s h e s i f v and v a r e e q u a l .

F i g . 12 D i r e c t i o n a l - c o u p l e r p i c k - u p a ) : s c h e m a t i c s , b ) : e q u i v a l e n t c i r c u i t

F o r t h e l o a d R on t h e l e f t , one f i n d s e a s i l y t h e c u r r e n t :

( 3 1 )

and t h e c o r r e s p o n d i n g s e n s i t i v i t y :

c i r c u i t o f F i g , 1 2 b . T h e two beam c u r r e n t s o u r c e s , a t ep.ch g a p , a r e in o p p o s i t e

d i r e c t i o n , and a r e s h i f t e d i n p h a s e b y t h e beam t r a n s i t t i m e .

T h e c u r r e n t f l o w i n g i n t h e l o a d R q on t h e r i g h t i s t h e sura o f t h e c o n t r i b u t i o n s

f r o m t h e two c u r r e n t s o u r c e s '

— e x p ( j u l / v > l e f t s o u r c e ¿ tp

- exp C-jwl / Up) r i g h t s o u r c e

v and v b e i n g t h e wave and beam v e l o c i t i e s and Í t h e d i s t a n c e b e t w e e n g a p s . V P

The t o t a l c u r r e n t :

S = j- ( 1 - e x p < - 2 j w - ) > ( 3 2 )

I f t h i s s y n c h r o n o u s c o n d i t i o n i s f u l f i l l e d , f o r i n s t a n c e i f = c and t h e c o a x i a l

l i n e i s i n v a c u u m , t h i s d e t e c t o r i s d i r e c t i o n a l : t h e s i g n a l o n l y a p p e a r s a t t h e u p s t r e a m

p o r t ( w i t h r e s p e c t t o beam v e l o c i t y ) . W i t h e o u n t e r r o t a t i n g beams ( p and p b a r s f o r

i n s t a n c e ) t h e d i r e c t i o n a l p i c k - u p c a n s e p a r a t e t h e s i g n a l s f r o m t h e two t y p e s o f

p a r t i c l e s . I n p r a c t i c e t h e d i r e c t i v i t y i s o f t h e o r d e r o f 30 t o 35 dB. N o t e t h a t

d i r e c t i v i t y c a n , i n p r i n c i p l e , b e o b t a i n e d a l s o by c o m b i n i n g t h e s i g n a l s o f s e v e r a l

i d e n t i c a l d e t e c t o r s .

T h e s e n s i t i v i t y o f t h e d e t e c t o r , g i v e n by E q . ( 3 2 ) i s f r e q u e n c y d e p e n d e n t ( F i g . 1 3 ) .

I t shows a s u c c e s s i o n o f z e r o s and max ima c o r r e s p o n d i n g t o :

t h e s e n s i t i v i t y b e i n g s i m p l y R a t t h e m a x i m a .

A mplifude

2 I

F i g . 13 T r a n s f e r f u n c t i o n s o f t h e d i r e c t i o n a 1 - c o u p 1 e r p i c k u p .

T h e t r a n s i e n t r e s p o n s e o f t h e d e t e c t o r can be o b t a i n e d by m a k i n g t h e i n v e r s e F o u r i o r

t r a n s f o r m o f E q . ( 3 2 ) , b u t i t i s o b v i o u s f r o m t h e e q u i v a l e n t c i r c u i t o f F i g . 12b t h a t i t

i s composed o f two o p p o s i t e d e l t a p u l s e s s e p a r a t e d i n t i m e b y t w i c e t h e t r a n s i t t i m e

( 2 1 / c ) ( F i g . 1 4 a ) .

S e v e r a l i d e n t i c a l p i c k - u p s can be c o m b i n e d t o i n c r e a s e t h e o v e r a l l s , - n s i t i v i t y . W i t h

p o w e r c o m b i n e r s , t h e o u t p u t s i g n a l s a r e a d d e d p o w e r w i s e g i v i n g a n o v e r a l l s e n s i t i v i t y

f o r i d e n t i c a l d e t e c t o r s , and t h e same f r e q u e n c y r e s p o n s e . One c a n a l s o

- 4 3 0 -

combine s e v e r a l d i r e c t i o n a l c o u p l e r d e t e c t o r s i n c a s c a d e and o b t a i n , w i t h t h e p r o p e r

d e l a y s , a t r a n s i e n t r e s p o n s e as i n F i g . l i b . T h e r e t h e maximum s e n s i t i v i t y i s

p r o p o r t i o n a l t o , b u t t h e f r e q u e n c y r e s p o n s e now shows a s i n f / f c u r v e p e a k e d a t

L = 4 . I n o t h e r w o r d s , t h e h i g h e r s e n s i t i v i t y ( p r o p o r t i o n a l t o f î ) r e s u l t s i n a n a r r o w e r

b a n d w i d t h .

F i g . 14 T r a n s i e n t r e s p o n s e o f d i r e c t i o n a l c o u p l e r a ) : s i n g l e b ) : m u l t i p l e ( w i t h t h e a s s o c i a t e d f r e q u e n c y r e s p o n s e ) .

D i r e c t i o n a l c o u p l e r p i c T t - u p s a r e i n f a c t m o s t l y u s e d as t r a n s v e r s e d e t e c t o r s . W i t h

s e v e r a l s t r i p s s y m m e t r i c a l l y a r r a n g e d i n t h e vacuum c h a m b e r , as i n F i g . 1 5 a , t h e t o t a l

w a l l c u r r e n t l y s h o u l d be r e p l a c e d by ö / 2 w f o r e a c h s t r i p , p r o v i d e d t h e beam i s i n t h e

c e n t e r . F o r a n o n - c e n t e r e d beam t h e p r o b l e m i s t r u l y t h r e e d i m e n s i o n a l n e a r t h e g a p s . By

a p p r o x i m a t e ^ t h e e l e c t r o m a g n e t i c f i e l d by t h a t o f a p u r e TEH wave one can o b t t i n t h e w a l l

c u r r e n t d i s t r i b u t i o n a l o n g thfe vacuum c h a m b e r a z i m u t h w h i c * o b v i o u s l y d e p e n d s on t h e beam

p o s i t i o n . F o r s m a l l beam d i s p l a c e m e n t , fix, t h e d i f f e r e n c e of t h e s i g n a l s o f two

o p p o s i t e s t r i p l i n e s i s p r o p o r t i o n a l t o o x :

Û V * V 2 ' V l = S û L b & x ( 3 3 :

S ¿ b e i n g d e f i n e d by e q u a t i o n ( 3 3 ) as t h e t r a n s v e r s e s e n s i t i v i t y o f t h e d e t e c t o r ( i n

o h m s / m e t e r ) .

F i g . 15 C r o s s s e c t i o n o f t r a n s v e r s e d i r e c t i o n a l - c o u p l e r p i c k - u p a ) : c i r c u l a r b ) : r e c t a n g u l a r

- 4 3 1 -

I n t h e c a s e o f a r e c t a n g u l a r g e o m e t r y , o f t e n used i n w i d e - a p e r t u r e c o o l i n g r i n g s f o r

i n s t a n c e , t h e s e n s i t i v i t y S i s g i v e n b y :

The f o r m f a c t o r t a n h t r r w / h ) s i m p l y r e f l e c t s t h e f a c t t h a t some f r a c t i o n o f t h e w a l l

c u r r e n t f l o w s o u t s i d e t h e s t r i p l i n e g a p s .

T h i s t y p e o f p i c k - u p ( s o m e t i m e s c a l l e d l o o p c o u p l e r ) i s w i d e l y u s e d i n c o o l i n g

s y s t e m s . r t o f f e r s a good c o m p r o m i s e b e t w e e n b a n d w i d t h ( o f t h e o r d e r o f one o c t a v e ) and

s e n s i t i v i t y . The s i g n a l s o f many c o u p l e r s a r e o f t e n a d d e d power w i s e on a c o m b i n e r b o a r d ,

i n s i d e v a c u u m , t o i n c r e a s e t h e o v e r a l l s e n s i t i v i t y . I f o n l y one t y p e o f p a r t i c l e i s

p r e s e n t , t h e d o w n s t r e a m r e s i s t o r R^, w h e r e no c u r r e n t f l o w s , c a n be r e p l a c e d by a s h o r t

c i r c u i t ( h e n c e t h e name o f l o o p c o u p l e r ) , b u t m i c r o w a v e r e s o n a n c e s may be h a r m f u l i n t h i s

c a s e -

c ) T h e e l e c t r o s t a t i c p i c k - u p

I f t h e c o a x i a l l i n e o f F i g 1 2 a I s much s h o r t e r t h a n t h e w a v e l e n g t h ( ï . < < \ ) , i t

c a n be r e p r e s e n t e d by a s : . m p l e c a p a c i t o r C •= l /R^u^ ( F i g 1 6 a ) . F a r a v e r y h i g h t o a d

r e s i s t o r , t h e e q u i v a l e n t c i r c u i t o f F i g , 16b r e p r e s e n t s t h e e l e c t r o s t a t i c d e t e c t o r , w i t h

t h e two c u r r e n t s o u r c e s p h a s e s h i f t e d by .

S , = - 2 ( t a n t , ™ )

2 h h

Í 3 4 )

Out b a

F ¡ S - i t T h e e l e c t r o s t a t i c p i c k - u p

T h e v o l t a g e d e v e l o p e d on t h e l i n e (- t h e e l e c t r o d e ) i s s i m p l y :

( 3 5 )

( 3 6 )

T h e q u a n t i t y * - x

b

/ v ^ * s t n e °eam c h a r g e q c o n t a i n e d i n t h e d e t e c t o r l e n g t h , ( a s s u m i n g a

s l o w l y v a r y i n g c h a r g e d i s t r i b u t i o n w i t h r e s p e c t t o t h e e l e c t r o d e l e n g t h ) . I t f o l l o w s :

V - q / C

a s t h e e l e c t r o s t a t i c t h e o r y w o u l d h a v e g i v e n i m m e d i a t e l y .

F o r v and i n t h e a p p r o x i m a t i o n o f a h i g h l o a d r e s i s t o r , Eq . ( 3 6 ) c o m b i n e d w i t h C = £ / R v l e a d s t o t h e v e r y s i m p l e r e s u l t : o tp r

s = R o

The s e n s i t i v i t y i s i n d e p e n d e n t o f t h e f r e q u e n c y and o f t h e l r . g t h o f t h e d e t e c t o r . Of

c o u r s e t h i s i s o n l y t r u e a t medium f r e q u e n c i e s . The non- i.if m i t e l o a d r e s i s t o r ( u s u a l l y

a n a m p l i f i e r w i t h h i g h i n p u t i m p e d a n c e ) i n t r o d u c e s a l o w f r e q u e n c y c u t o f f w h e r e a s a t h i g h

F r e q u e n c i e s t h e a p p r o x i m a t i o n l < < \ is no l o n g e r v a l i d .

T h e t r a n s v e r s e v e r s i o n o f t h e e l e c t r o s t a t i c p i c k - u p can be o b t a i n e d by s p l i t t i n g t h e

e l e c t r o d e c y l i n d e r i n two h a l v e s a l o n g a l i n e a r c u t . ( F i g . 1 7 ) . E l e c t r o s t a t i c t h e o r y

shows t h a t Uie d . . . U,. i ' • >f ' '

beam d i s p l a c e m e n t . Many v e r s i o n s o f t h e t r a n s v e r s e e l e c t r o s t a t i c p i c k - u p w i t h v a r i o u s 4 )

shapes c o u l d be f o u n d i n t h e l i t e r a t u r e ( c i r c u l a r , r e c t a , ¡ v . i l a r , e l l i p t i c a l ) They a r e

m o s t i y u s e d f o r closed o r b i t m e a s u r e m e n t s i s o m e t i..jes h o r i z o n t a l and v e r t i c a l pickups a r e

c o m b i n e d i n a s i n g l e u n i t ) .

I f t h e l i n e a r i t y r e q u i r e m e n t i s l e s s i m p o r t a n t , t h e l i n e a r c u t c o u l d be a b a n d o n e d ,

f o r i n s t a n c e i n t h e so c a l l e d " b u t t o n s " t o be u s e d i n LEP ( F i g . 1 8 ) . T h e r e , o n l y t h e h i g h -

f r e q u e n c y r e s p o n s e i s i m p o r t a n t , and c o n s e q u e n t l y t h e l o a d r e s i s t o r i s a SO n < :ab le .

T h e l i n e a r i t y can be r e s t o r e d by a p r o p e r a l g o r i t h m a t t h e s i g n a l p r o c e s s i n g l e v e l .

The e l e c t r o s t a t i c d e t e c t o r c a n be made r e s o n a n t , w i t h a c o l l ( o r t r a n s f o r m e r )

c o n n e c t e d t o t h e e l e c t r o d e . A t r a n s v e r s e v e r s i o n i s s k e t c h e d i n F i g . 1 9 a , w i t h t h e

e q u i v a l e n t c i r c u i t of F i g . 1 9 b .

A t r e s o n a n c e t h e v o l t a g e a c r o s s t h e p l a t e s V i s g i v e n b y :

11

F i g . 17 T r a n s v e r s e e l e c t r o s t a t i c p i c k - u p l i n e a r l y c u t

"b " b

( 3 8 )

f o r a l o s s l e s s t r a n s f o r m e r .

- 454 -

T a k i n g i n t o a c c o u n t t h a ohmic l o s s e s o f t h e c o i l (Q - q u a l i t y f a c t o r o f t h e

r e s o n a n t c i r c u i t , Q = l o a d e d q u a l i t y f a c t o r ) , one o b t a i n s :

T h i s t e c h n i q u e has been u s e d i n t h e CERN SPS, f o r a d e d i c a t e d , v e r y s e n s i t i v e

S c h o t t k y d e t e c t o r " * ' [ s e n s i t i v i t y : 75ii/min> .

2 . 2 P i c k - u p e v a l u a t i o n u s i n g t h e r e c i p r o c i t y t h e o r e m

The r e c i p r o c i t y t h e o r e m , w e l l known i n a n t e n n a t h e o r y , r e s u l t s f rom M a x w e l 1 e q u a t i o n s

a p p l i e d t o a l i n e a r , i s o t r o p i c s y s t e m . I f we h a v e two s e t s o f c u r r e n t s o u r c e s i n t h e

s y s t e m J * and J * ' w h i c h p r o d u c e t h e e l e c t r i c f i e l d s E ' a r . i z ' and t h e i r a g n e t i c f i e l d s H*

and h" ' , t h e f o l l o w i n g r e l a t i o n i s v a l i d :

( 4 1 )

w h e r e t h e v o l u m e v i s e n c l o s e d by t h e s u r f a c e s (n i s t h e u n i t y v e c t o r on t h a t s u r f a c e ) .

F o r t h e a p p l i c a t i o n o f t h e r e c i p r o c i t y t h e o r e m , ( F i g . 2 0 ) . we t a k e J " = i f c ,

( i f a i s t h e beam c u r r e n t a l o n g t h e d e t e c t o r a x i s ) , and J ' ' = 1 ^ ( 1 ^ i s a p u r e

c u r r e n t s o u r c e a p p l i e d a c r o s s t h e l o a d r e s i s t o r R ) .

" o

We c o n s i d e r an i n t e g r a t i o n vo lume l i m i t e d by t h e m e t a l l i c e n c l o s u r e o f t h e p i c k u p ,

w h e r e t h e e l e c t r i c f i e l d s a r e n o r m a l t o t h e s u r f a c e , w h i c h makes t h e l e f t s i d e o f Eq . ( 4 1 )

v a n i s h and l e a d s t o :

: .v . ( I o u t f

c i t e d b y i ^ , and i s t h e on a x i s

component o f t h e f i e l d i n t h e p i c k - u p s t r u c t u r e when e x c i t e d by 1 . Fo r a g i v e n g e o m e t r y

and a g i v e n f i e l d c o n f i g u r a t i o n , E^ can be r e l a t e d t o 1 ^ . f r o m power c o n s i d e r a t i o n s . T h e n

a p p l i c a t i o n o f E q . ( 4 3 ) d i r e c t l y g i v e s t h e d e t e c t o r s e n s i t i v i t y s = v

o l l t

/ i b ' f ° r c a s e s w h e r e

t h e image c u r r e n t a p p r o a c h w o u l d f a i l ( e . g . m i c r o w a v e s t r u c t u r e s ) .

V out

W II

F i g . 20 A p p l i c a t i o n o f t h e r e c i p r o c i t y t h e o r e m t o a beam d e t e c t o r

N o t e t h a t t h e r e c i p r o c i t y t h e o r e m , t r a n s p o s e d i n c i r c u i t t h e o r y , s i m p l y s t a t e s t h a t ,

f o r a p a s s i v e q u a d r u p o l e , t h e d e t e r m i n a n t o f i t s t r a n s f e r m a t r i x i s u n i t y .

A p p l i c a t i o n o f t h e r e c i p r o c i t y t h e o r e m w i l l b e i l l u s t r a t e d i n t h e f o l l o w i n g by two

e x a m p l e s : t h e s l o w - w a v e and t h e s l o t - l i n e p i c k - u p s .

The s l o w - w a v e p i c k - u p i s e s s e n t i a l l y a n e l e c t r o m a g n e t i c wave g u i d e i n w h i c h t h e p h a s e

v e l o c i t y h a s b e e n s l o w e d down t o m a t c h t h e v e l o c i t y o f t h e p a r t i c l e s . D i e l e c t r i c s l a b s 6 )

( F i g . 2 1 a ) o r c o r r u g a t i o n s ( F i g . 2 1 b ) h a v e b e e n c o n s i d e r e d f o r t h i s p u r p o s e A

d e s c r i p t i o n c*f t h e f i e l d i n t h e s t r u c t u r e w i l l be g i v e n b y s t a n d a r d wave g u i d e t h e o r y .

W i t h r e s p e c t t o t h e t r a n s v e r s e d i m e n s i o n , t h e f i e l d c o n f i g u r a t i o n i s e i t h e r s y m m e t r i c a l

( e v e n mode) o r a n t i s y m m e t r i c a l ( o d d m o d e ) , l e a d i n g t o a l o n g i t u d i n a l o r a t r a n s v e r s e

d e t e c t o r r e s p e c t i v e l y .

F i g . 21 S l o w - w a v e p i c k - u p s a ) d i e l e c t r i c s l a b b ) c o r r u g a t e d w a l l

I n t h e c a s e o f a p u r e t r a v e l l i n g - w a v e s t r u c t u r e , t e r m i n a t e d a t b o t h e n d s by r e s i s t o r s

ft v i a m a t c h e d t r a n s i t i o n s , t h e p o w e r f l o w P i n t h e w a v e g u i d e i s r e l a t e d t o I b y : o p i

- ASb -

( 4 4 )

N o t e t h a t o n l y I 12 f l o w s t o w a r d s - h e w a v e g u i d e , t h e r e s t b e i n g d i s s i p a t e d i n t h e l o a d

r e s i s t o r R .

T h e s e n s i t i v i t y i s g i v e n b y :

I t i s f o u n d t o be p r o p o r t i o n a l t o t h e t r a n s i t t i m e f a c t o r :

s i n V i r - P 2

1 P 1

p v

k , ß , p : p r o p a g a t i o n c o n s t a n t s i n f r e e s p a c e , w a v e g u i d e and beam r e s p e c t i v e l y , o ip p

The s e n s i t i v i t y i s op t imum f o r ß and ß ( s y n c h r o n i s m c o n d i t i o n ! as e x p e c t e d . F o r a P *P

g i v e n f r e q u e n c y , op t imum d i m e n s i o n s o f t h e w a v e g u i d e s a r e g i v e n by t h e s y n c h r o n i s m

c o n d i t i o n ( a s i n F i g . 1 4 b ) . H a k i n g t h e d e t e c t o r l o n g e r i n c r e a s e s t h e s e n s i t i v i t y

( p r o p o r t i o n a l t o 1 ) b u t r e d u c e s i t ^ b a n d w i d t h a c c o r d i n g t o ( 4 6 ) .

The s l o t - l i n e p i c k - u p ? , B ^ o f f e r s a n o t h e r i n t e r e s t i n g e x a m p l e , i n w h i c h t h e waves

p r o p a g a t e i n a d i r e c t i o n p e r p e n d i c u l a r t o t h a t o f t h e beam ( F i g . 2 2 ) . A t h i n s l o t i n a

meta 1 l i e p l a n e on a d i e l e c t r i c s u b i t r a t e c a n s u p p o r t quas i TEH waves i n t h e u p p e r r e g i o n .

T h e e l e c t r i c f i e l d , n o t too c lose- o t h e s l o t , i s p u r e l y t a n g e n t i a l : i t s a m p l i t u d e i s

g i v e n b y :

( 4 5 )

,(»> ( 4 7 )

F i g . 22 S c h e m a t i c s o f s l o t - l i n e p i c k - u p

H a n k e 1 f u n c t i o n o f f i r s t o r d e r .

The l o n g i t u d i n a l f i e l d E^, a l o n g t h e beam ( a t a d i s t a n c e d f r o n t h e m e t a l l i c p l a n e ) i s

s i m p l y .

c i c

V i s r e l a t e d t o t h e p o w e r f l o w P a l o n g t h e gap by t h e s l o t - l i n e impedance 1:

c V ^ o 7 / 2 * d \

E q u a t i o n ( 5 2 ) can b e shown t o b e v a l i d a l s o e v e n i f X and X' a r t n o t v e r y c l o s e :

V 5ljuuld t h e n be r e p l a c e d by \ ' i n ( 5 2 ) .

I t i s i n t e r e s t i n g t o r e m a r k t h a t i n t h e l i m i t d < < \ , E q . ( 5 2 ) r e d u c e s t o

S - * / R 2/2 w h i c h i s t h e r e s u l t g i v e n b y t h e image c u r r e n t a p p r o a c h , w i t h t h e f °

r e c i p r o c i t y t h e o r e m , t r a n s v e r s e p r o p a g a t i o n w h i c h was p r e v i o u s l y n e g l e c t e d c a n be l a k i ' n

i n t o a c c o u n t .

I f t h e s i g n a l s o f t w o s y m m e t r i c a l p l a t e s w i t h two s l o t s a r e c o m b i n e d , a t r a n s v e r s e

d e t e c t o r c a n b e b u i l t . I t s s e n s i t i v i t y w n u l d b e :

h s í n h ( « h / K ' )

h b e i n g t h e d i s t a n c e b e t w e e n p l a t e s .

- 438 -

S l o t - l i n e p i c k - u p s w o u l d be i n t e r e s t i n g , b e c a u s e t h e y can be e a s i l y p r o d u c e d by

s t a n d a r d p r i n t * > d - c i r c u i t t e c h n i q u e s , even i n t h e m i c r o w a v e r e g i o n . T h e i r b a n d w i d t h i s

o n l y l i m i t e d by t h a t o f t h e s l ^ - t - l i n e t o s t r i p - l i n e t r a n s i t i o n s ( t h e wave on t h e s l o t i s

c o u p l e d t o o u t s i d e v i a a s t r i p l i n e d e p o s i t e d on t h e o p p o s i t e s i d e o f t h e d i e l e c t r i c ) .

Because o f t h e t r a n s v e r s e p r o p a g a t i o n , t h e i n h e r e n t d e l a y o f t h e d e t e c t o r d e p e n d s on t h e

t r a n s v e r s e beam pos i t i o n . T h i s c o u l d be u s e f u l f o r some s t o c h a s t i c c o o l i n g schemes -

2.3 r m p u l s e r e s p o n s e

C o n s i d e r a g a i n a t r a v e l l i n g wave d e t e c t o r l i k e , f o r i n s t a n c e , t h e c o r r u g a t e d w a l l

w a v e g u i d e , w h e r e a number o f c e l l s ( o r i n d i v i d u a l r e c t a n g u l a r b o x e s ) a r e c o u p l e d t o g e t h e r

v i a t h e beam p i p e . Uhen e x c i t e d b y a s h o r t beam p u l s e , t h e r e s p o n s e o f t h e d e t e c t o r i s ,

and d u r a t i o n t. A f t e r f i r s t a p p r o x i m a t i o n , a n RF b u r s t ( F i g . 2 3 ) o f a m p l i t u d e V Q u t .

t h e t i m e T , a l l t h e e n e r g y d e p o s i t e d i n t h e d e t e c t o r h a s b e e n t r a n s p o r t e d w i t h t h e

g r o u p v e l o c i t y u t o t h e end o f t h e s t r u c t u r e and t h e n t o t h e t e r m i n a t i n g r e s i s t o r R

F i g . 23 I m p u l s e r e s p o n s e o f a t r a v e l l i n g - i

o u t What i s t h e r e l a t i o n b e t w e e n t h e d e t e c t o r s e n s i t i v i t y S and i t s o u p u t v o l t a g e

t h i s c a s e ? I f we assume a p e r i o d i c t r a i n o f s h o r t beam p u l s e s ( c h a r g e q ) , s e p a r a t e d i n

me by T , t h e RF component o f t h a beam c u r r e n t Í . a t t h e c e n t r a l f r e q u e n c y o f t h e p i c k - u p

i s s i m p l y i ^ = 2 q / i . O b v i o u s l y t h e o u p u t a m p l i t u d e V

beam i n p u t , w h i c h g i v e s : o u t

i s c o n s t a n t , w i t h t h a t p a r t i c u l a r

f o r t h e l o n g i t u d i n a l ca.^e a n d :

f o r t h e t r a n s v e r s e c a s e .

The e n e r g y W d e p o s i t e d by t h e c h a r g e q i n t h e d e t e c t o r i s r e l a t e d t o t h e g e o m e t r y o f

t h e s t r u c t u r e v i a i t s " l o s s p a r a m e t e r " d e f i n e d b y :

T h e k f a c t o r i s a l s o t h e R/Q o f t h e s t r u c t u r e ( k = M i¿ R / Q ) .

- 439 -

( 5 8 )

and a s i m i l a r e q u a t i o n f o r t h e t r a n s v e r s e c a s e .

The l o s s p a r a m e t e r k d e p e n d s e s s e n t i a l l y u p o n t h e c e l l g e o m e t r y , and c a n be

c a l c u l a t e d a n a l y t i c a l l y i n some s i m p l e c a s e s ( n e g l e c t i n g t h e e f f e c t o f t h e beam h o l e ) o r

e v a l u a t e d b y c o m p u t e r c o d e s l i k e SUPEHFISH f o r i n s t a n c e . On t h e o t h e r hand i

c h a r a c t e r i z e s t h e c e l l - t a - c e l l c o u p l i n g v i a v ^ .

From E q . ( 5 8 ) , t h e maximum s e n s i t i v i t y i s a g a i n p r o p o r t i o n a l t o I ( d e t e c t o r l e n g t h )

as b o t h k and T a r e t h e m s e l v e s p r o p o r t i o n a l t o I . Of c o u r s e t h e b a n d w i d t h d e c r e a s e s

c o r r e s p o n d i n g l y as was shown i n t h e e x a m p l e o f t h e m u l t i p l e d i r e c t i o n a l c o u p l e r ( F i g .

1 4 b ) . H a t e t h a t t h i s m u l t i p l e d i r e c t i o n a l c o u p l e r can be c o n s i d e r e d as a b a c k w a r d -

t r a v e l l i n g wave s t r u c t u r e w i t h = c .

I n t h e f o l l o w i n g e x a m p l e , we s h a l l e v a l u a t e t h e k f a c t o r f o r t h e s i m p l e g e o m e t r y o f

F i g . 2 4 : a c h a i n o f c o u p l e d c y l i n d r i c a l c a v i t i e s . We c o n s i d e r the mode E ^

( t r a n s v e r s e d e t e c t o r ) w h e r e t h e e l e c t r i c f i e l d i s o n l y l o n g i t u d i n a l :

The e n e r g y l o s t by c h a r g e q i s g i v e n b y :

T h e f a c t o r 1 / 2 s i m p l y r e f l e c t s t h e f a c t t h a t t h e c h a r g e q o n l y s e e s o n e h a l f o f i t s own

i n d u c e d v o l t a g e ( f u n d a m e n t a l t h e o r e m o f beam l o a d i n g ) . W i s a l s o o b t a i n e d by i n t e g r a t i n g

E o v e r t h e w h o l e c a v i t y v o l u m e :

E l i m i n a t i n g E b e t w e e n ( 5 9 ) , ( 6 0 ) and ( 6 1 ) f i n a l l y g i v e s :

C o m b i n i n g ( 5 4 ) , ( 5 6 ) and t h e r e l a t i o n :

V

2 R

v a l i d f o r a l o s s l e s s d e t e c t o r one f i n d s :

- 4 4 0 -

Z

L _

U ' " .J F i g . 24 A c h a i n o f c o u p l e d c y l i n d r i c a l c a v i t i e s as t r a n s v e r s e d e t e c t o r

4 * 2 . 3 B \ X ¿ V wL/X /

E q u a t i o n ( 6 2 ) shows . h e i n t e r e s t of h i g h - f r e q u e n c y d e t e c t o r s as f a r as s e n s i t i v i t y i s

c o n c e r n e d ( f a c t o r s to and l / \ ) . B u t t h e i n f l u e n c e o f t h e beam h o l e w h i c h has been

n e g l e c t e d i n t h i s s i m p l i f i e d a n a l y s i s w i l l become more and more i m p o r t a n t . An e x a m p l e o f

t h i s t y p e o f d e t e c t o r , u s i n g t h e f i r s t t r a n s v e r s e mode o f t h e a c c e l e r a t i n g c a v i t i e s i n t h e

CKRH SPS i s g i v e n i n R é f . 9 .

3 OBSERVATION OF SCHOTTKY SIGNALS

3 . 1 S p e c t r a l a n a l y s i s

As a l r e a d y m e n t i o n e d i n s e c t i o n 1 t h e measurement o f t h e p o w e r s p e c t r a l d e n s i t y o f

t h e S c h a t t k y s i g n a l s g i v e s t h e p a r t i c l e d i s t r i b u t i o n i n e i t h e r momentum o r b e t a t r o n t u n e

( o r a c o m b i n a t i o n o f b o t h ) . T h e r e f o r e , s p e c t r a l a n a l y s i s i s t h e n a t u r a l t e c h n i q u e f o r

o b s e r v i n g S c h o t t k y s i g n a l s .

The f r e q u e n c y span o f i n t e r e s t i s of t h e o r d e r o f t h e r e v o l u t i o n f r e q u e n c y , o r e v e n

l e s s , ( i n most c a s e s b e l o w 1 0 0 k H z ) . C o n s e q u e n t l y , t h e F a s t F o u r i e r T r a n s f o r m ( F F T ) o r ,

more p r e c i s e l y , t h e D i g i t a l F o u r i e r T r a n s f o r m (DFT> t e c h n i q u e s w h i c h o p e r a t e a t low

f r e q u e n c i e s , can be used t o e v a l u a t e i n r e a l t i m e t h e s i g n a l s p e c t r u m . The S c h o t t k y band

t o be a n a l y s e d must be t r a n s l a t e d a t low f r e q u e n c y p r i o r t o FFT a n a l y s i s , as i n a

c o n v e n t i o n a l s p e c t r u m a n a l y s e r . T h i s may r e q u i r e a c a r e f u l p r e f i l t e r i n g t o r e j e c t t h e

u n w a n t e d image f r e q u e n c i e s .

I n t h e OFT t e c h n i q u e , t h e s i g n a l i s s a m p l e d and d i g i t i z e d a t f r e q u e n c y f g . Each

d i g i t a l word i s s t o r e d i n a memory w i t h M l o c a t i o n s ( t y p i c a l l y 2 = 1024 l o c a t i o n s ) :

t h e d u r a t i o n o f t h e s i g n a l s a m p l e t o be a n a l y s e d i s t h e n T = H / f s - The f r e q u e n c y

c o n t e n t ( f r e q u e n c y s p a n ) o f t h e s a m p l e d s i g n a l e x t e n d s o n l y up t o f

s ' 2 ( N y q u i s t

t h e o r e m ) , and t h e r e s o l u t i o n o f t h e f r e q u e n c y a n a l y s i s I s o f t h e o r d e r o f 1 / T ( F i g . 2 5 ) .

Resolution ~ ' /

f r e q u e n t / span ft/2

F i g . 25 S p e c t r a l a n a l y s i s ( D F T ) o f S c h o t t k y s i g n a l s

d e p e n d i n g o n t h e c h o i c e o f t h e s i g n a l p r o c e s s i n g " w i n d o w i n g " , t h e r e s o l u t i o n v a r i e s a

l i t t l e : 1 / T f o r t h e r e c t a n g u l a r w i n d o w ; 1 . 4 / T f o r t h e "Hamming w i n d o w " , b e t t e r o p t i m i z e d

f o r n o i s e s i g n a l s .

F o r a g i v e n r e s o l u t i o n o f t h e beam d i s t r i b u t i o n m e a s u r e m e n t ( i n û p / p , o r Ô Q / Q ) , T

i s minimum f o r t h e l a r g e s t w i d t h o f t h e S c h o t t k y b a n d . F o r i n s t a n c e , i n t h e l o n g i t u d i n a l ,

d e b u n c h e d beam c a s e , one w o u l d m i n i m i z e T by l o o k i n g a t t h e h i g h e s t f r e q u e n c y S c h o t t k y

b a n d s ( w i d t h n f l f ) l i m i t e d by e i t h e r f g / 2 « t f i e d e t e c t o r s e n s i t i v i t y , o r t h e o v e r l a p

c o n d i t i o n . T h i s i s o f p a r t i c u l a r i n t e r e s t f o r t h e o b s e r v a t i o n o f " p s e u d o " S c h o t t k y

s i g n a l s i n p u l s e d m a c h i n e s t o m e a s u r e t h e beam momentum s p r e a d d u r i n g d e b u n c h i n g . ( T i s

t h e r e s t r i c t l y l i m i t e d by t h e d u r a t i o n o f t h e m a g n e t i c c y c l e f l a t t o p ) . T h e beam

d e v e l o p s , d u r i n g d e b u n c h i n g a t h i g h i n t e n s i t y , a v e r y c o m p l i c a t e d s t r u c t u r e w h i c h i s more

o r l e s s e q u i v a l e n t t o random n o i s e , b u t o f m a c r o s c o p i c n a t u r e ( " p s e u d o - - S c h o t t k y s i g n a l ) .

I t s s p e c t r u m a n a l y s i s p r o v i d e s an e s t i m a t e o f t h e momentum s p r e a d o f t h e beam d u r i n g

d e b u n c h i n g .

Even i f T c a n b e made v e r y l o n g , t h e r e s u l t o f t h t DFT on a n o i s e s i g n a l does n o t

g i v e a good e s t i m a t e o f i t s s p e c t r a l d e n s i t y . T h i s i s b e c a u s e t h e v a r i a n c e o f t h e p o w e r

m e a s u r e m e n t i s c o m p a r a b l e t o i t s mean v a l u e : i t d o e s no, d e c r e a s e when T i s made l o n g e r .

A b e t t e r " e s t i m a t i o n " o f t h e t r u e p o w e r d e n s i t y i s o b t a i n e d by a v e r a g i n g s e v e r a l

s p e c t r a t a k e n a t d i f f e r e n t t i m e i n t e r v a l s . T h e " d e g r e e o f c o n f i d e n c e " o f t h e m e a s u r e m e n t

i n c r e a s e s w i t h t h e number o f a v e r a g e d s p e c t r a ( F i g . 2 6 ) , a t t h e e x p e n s e o f t h e t o t a l

a n a l y s i s t i m e ( w h i c h may be d i s t r i b u t e d o v e r s e v e r a l m a c h i n e c y c l e s i n t h e t h e p r e v i o u s

e x a m p l e ) .

1 6 16 11 fA 128 2S6 Number of av*rjg*d spectr*

F i g . 26 90% c o n f i d e n c e l e v e l of t h e n o i s e s p e c t r a l d e n s i t y measurement

3.7, P a r a s i t i c s i g n a l s o f t h e S c h o t t k y s p e c t r u m

Due t o t h e v e r y l o w l e v e l o f t h e S c h o t t k y s i g n a l s , many s o u r c e s o f d i s t u r b a n c e c a n

b e h a r m f u l and s h o u l d be e l i m i n a t e d w h e n e v e r p o s s i b l e .

P a r a s i t i c s i g n a l s may come f r o m t h e beam i t s e l f ; i f t h e r e i s a c o h e r e n t e x c i t a t i o n

( i . e . t r a n s v e r s e ) i t w i l l a p p e a r as a b e t a t r o n s i g n a l , b u t w i t h an a m p l i t u d e p r o p o r t i o n a l

t o N and n o t y/TT as f o r t h e S c h o t t k y s i g n a l . T h e l o n g i t u d i n a l l i n e i n a t r a n s v e r s e

S c h o t t k y scan c a n be s u p p r e s s e d b y c a r e f u l c e n t e r i n g o f t h e p i c k - u p on t h e beam a x i s . I n

t h e b u n c h e d beam c a s e , a d d i t i o n a l s h a r p f i l t e r i n g w i t h a c r y s t a l f i l t e r i s n e c e s s a r y 5 ' .

T h e l i n e r e l a t e d components a r e r e d u c e d by a c a r e f u l d e s i g n o f t h e a m p l i f i e r s , power

s u p p l i e s and e a r t h c o n n e c t i o n s . I f t h i s i s n o t s u f f i c i e n t , n a r r o w band s y n c h r o n o u s

f i l t e r i n g ( l o c k e d t o t h e m a i n s f r e q u e n c y ) can a l e o be e m p l o y e d .

Of more f u n d a m e n t a l n a t u r e i s t h e d i s t u r b a n c e due t o t h e t h e r m a l n o i s e o f t h e f i r s t

p r e a m p l i f i e r , a f t e r t h e d e t e c t o r . The a m p l i f i e r i s c h a r a c t e r i s e d by i t s n o i s e f a c t o r F

( e x c e s s n o i s e w i t h r e s p e c t t o a s i m p l e r e s i s t o r R Q f - The a v a i l a b l e n o i s e s p e c t r a l

d e n s i t y r e s u l t i n g f r o m t h e a m p l i f i e r and w h i c h i s fciven b y : F K

Q t o R , ( k Q : B o l t z m a n n

c o n s t . - . t , t o : t e m p e r a t u r e ) must be c o n s i d e r a b l y s m a l l e r t h a n t h e S c h o t t k y n o i s e s p e c t r a l

d e n s i t y . One p o s s i b i l i t y i s t o c o o l t h e p r e a m p l i f i e r and t h e t e r m i n a t i n g r e s i s t o r s o f t h e

- 4 4 3 -

p i c k - u p s ( A C 0 L ) . From F i g . 2 , i t i s c l e a r t h a t t h e b e s t s i g n a l t o n o i s e r a t i o i s o b t a i n e d

a t l o w f r e q u e n c i e s { f o r t h e c a s e o f u n b u n c h e d b e a m s ) . U n f o r t u n a t e l y , o b s e r v a t i o n o f

S c h o t t k y s i g n a l s a t h i g h f r e q u e n c y i s more f a v o u r a b l e as f a r as s e n s i t i v i t y and a n a l y s i s

t i m e a r e c o n c e r n e d .

3 . 3 B u n c h e d - b e a m s s i g n a l p r o c e s s i n g

As a l l S c h c t t k y l i n e s i n a frequency i n t e r v a l a r e correlated (see section 1), it

i s i n t e r e s t i n g t o s a m p l e t h e beam s i g n a l a t t h e r e v o l u t i o n F r e q u e n c y . A l l l i n e s w i l l b e

f o l d e d i n t h e b a s e b a n d g i v i n g a much b e t t e r s i g n a l t o n o i s e r a t i o as w i l l be shown i n t h e

f o l l o w i n g .

C o n s i d e r t h e RF b u r s t a m p l i t u d e V Q u t o f F i g . 23 d e l i v e r e d by a t r a v e l l i n g wave

t r a n s v e r s e p i c k - u p , when e x c i t e d b y a s h o r t b u n c h . F o r a t r a n s v e r s e r . m . s . beam

d i s p l a c e m e n t x , t h e o u t p u t v o l t a g e o f t h e d e t e c t o r and t h e t h e r m a l n o i s e v o l t a g e o f t h e

a m p l i f i e r , r e f e r r e d t o t h e i n p u t , a r e r e s p e c t i v e l y :

- - / F k t w o o

B b e i n g t h e a m p l i f i e r b a n d w i d t h .

T h e p o w e r s i g n a l t o n o i s e r a t i o , d u r i n g t h e t i m e i n t e r v a l T i s t h e r e f o r e :

T h i s i s a l s o t h e s i g n a l t o n o i s e r a t i o a f t e r s a m p l i n g . We c a n s e l e c t B (B = B

o p t i

t o o p t i m i z e 1 / U . B i s t h e minimum b a n d w i d t h f o r w h i c h t h e u s e f u l s i g n a l i s n o t r o p t

r e d u c e d s i g n i f i c a n t l y . T h i s h a p p e n s i f t h e r i s e t i m e o f t h e band l i m i t e d RF b u r s t i s o f

t h e o r d e r o f i t s l e n g t h : 1 / B = T , as i l l u s t r a t e d i n F i g . 2 7 . More p r e c i s e l y B Q p t i s

t h a t o f t h e so c a l l e d " o p t i m u m f i l t e r " ( r a d a r t e r m i n o l o g y ) f o r w h i c h t h e i m p u l s e r e s p o n s e

i s t h e t i m e r e v e r s e d image o f t?.e RF b u r s t . W i t h t h a t c o n d i t i o n ( 6 5 ) b e c o m e s :

w h i c h i s t h e same as f o r t h e d e b u n j h e d beam c a s e , e x c e p t f o r t h f onhancemont f a c t o r . 1 0 )

1 / i f w h i c h c a n be much l a r g e r t h a n u n i t y

F i g , . 27 Optimum f i l t e r i n g o f a n RF b u r s t f rom a beam d e t e c t o r

T h e o v e r a l l s i g n a l p r o c e s s i n g s y s t e m f o r a b u n c h e d beam t - a n s v e r s e S c h o t t k y s i g n a l i s

d i s p l a y e d o n F i g . 2 E . F r e q u e n c y t r a n s l a t i o n down t o t h e base band f r e q u e n c y c a n be done

by peak d e t e c t i o n , as i n d i c a t e d , o r w i t h a s y n c h r o n o u s d e t e c t o r d r i v e n by t h e sum s i g n a l

o f t h e p i c k u p . Tn t h i s c a s e , i t i s i n t e r e s t i n g t o r e m a r k t h a t t h e odd s y n c h r o t r o n

s a t e l l i t e s a r e r e j e c t e d f o r an i n phase d e t e c t i o n ( l i k e f o r a p e a k d e t e c t i o n ) , w h e r e a s "or

a q u a d r a t u r e d e t e c t i o n , i t i s t h e e v e n s y n c h r o t r o n s a t e l l i t e s w h i c h a r e r e j e c t e d . T h i s

f e a t u r e may be u s e F u l i f one w r i t s t o i s o l a t e t h e c e n t r a l J l i n e of t h e S c h o t t k y b a n d .

Pu 1 • I

DFT

Hatched filter

Sample H Q\i

tow Pass filter

F i g . 28 Bunched beam s i g n a l p r o c e s s i n g s y s t e m

A l t h o u g h t h e t h e r m a l n o i s e o f t h e p r e a m p l i f i e r i s o f l e s s i m p o r t a n c e f o r bunched beam

s i g n a l p r o c e s s i n g , t h e e f f e c t o f s p u r i o u s c o h e r e n t e x c i t a t i o n o f t h e beam may be more o f a

p r o b l e m . T h i s i s b e c a u s e , even a low f r e q u e n c y e x c i t a t i o n , n e a r t h e f i r s t b e t a t r o n l i n e ,

a p p e a r s e v e r y w h e r e i n t h e s p e c t r u m , c o n t r a r y t o t h e d e b u n c h e d beam c a s e , and may s p o i l

e v e n a h i g h f r e q u e n c y S c h o t t k y s y s t e m . A s o l u t i o n t o t h a t p r o b l e m i s t o r e j e c t t h a t p a r t 9 )

o f t h e d e t e c t o r s i g n a l w h i c h i s c o h e r e n t f r o m one bunch t o t h e n e x t

à BEAM TRANSFER FUNCTIONS

4 . 1 P r i n c i p l e o f beam t r a n s f e r f u n c t i o n s

The name o f beam t r a n s f e r f u n c t i o n a l m o s t s p e a k s f o r i t s e l f : i t r e l a t e s t h e r e s p o n s e

o f t h e beam ( a m p l i t u d e a n d p h a s e ) t o a known e x c i t a t i o n . I d t h e c a s e o f a t r a n s v e r s e

e x c i t a t i o n b y a d e f l e c t o r ( o r k i c k e r ) , t h e beam r e s p o n s - i s m e a s u r e d b y a t r a n s v e r s e

p i c k u p a s i n d i c t e d on F i g - 2 9 a , w h e r e a s F i g . 29b s h o w s t h e a r r a n g e m e n t f o r t h e

m e a s u r e m e n t o f a l o n g i t u d i n a l t r a n s f e r f u n c t i o n * * * .

1 _ C . — Network

A n a l / s e r

Cavity

exci ration

— h et work

Analyser

aï b) F i g . 29 P r i n r i p l e o f beam t r a n s f e r f u n c t i o n m e a s u r e m e n t

To m i n i m i z e the a n a l y s i s t i m e a n d t h e d i s t u r b a n ' o to t h e b e a m , i t i s i n t e r e s t i n g t o

e x c i t e the beam w i t h a w h i t e n o i s e s p e c t r u m ( a l l f r e q u e n c i e s are Dresent i n t h e b a n d u f

i n t e r e s t ) . T h e r e the o u t p u t w i l l a l s o b e a n o i s e s i g n a l , s i m i l a r t o t h e S c h o t t k y n o i r e ,

and f o r w h i c h s i m i l a r p r o c e s s i n g t e c h n i q u e s c a n b e a p p l i e d . To e x t r a c t t h e p h a s e

i n f o r m a t i o n s p e c t r a l d e n s i t y m e a s u r e m e n t s a r e n o t s u f f i c i e n t and a d u a l c h a n n e l DKT

i n s t r u m e n t i s needed. A g a i n l i v e r a g i f i f rojny t ronsícr f u i t e t i o n s r e d u c e s t h e v a r i a n c e o f t h e

e s t i m a t e ( F i g . 2 6 ) . I n F i g - 29a and b , a new e l e m e n t a p p e a l ' s , n a m e l y t h e k i c k e r ( e i t h e r

t r a n s v e r s e o r l o n g i t u d i n a l ) w h i c h w i l l b e e x a m i n e d m o r e i n d e t a i l i n t h e l o l l o w i r g .

4 . ? K i c k e r s

A l o n g i t u d i n a l k i c k e r i s a f a i r l y s t r a igl' .t f o i

e l e c t r i c f i e l d i s p r o d u c e d . T h e p a r t i c l e gain;

k i c k e r o r c a v i t y ) , w h i c h i s s i m p l y g i v e n b y :

AH - / e E dz ( h / )

J * z

T h e a p p l i c a t i o n o f t h e r e c i p r o c i t y t h e o r e m to a l o n g i t u d i n a l beam d e t e c t u r h a s led vis t o

K q . ( 4 1 , w h i c h comli i n e d w i t h (hi) r e s u l t s i n :

•ward d e v i c e i n w h i c h n l o n g i t u d i n a l

; an e n e r g y aw, when c r o s s i n g t h e

( 6 8 )

- AAb -

s h o w i n g t h a t t h e e n e r g y g a i n o f t h e k i c k e r and t h e s e n s i t i v i t y o f t h e p i c k up a r e s i m p l y

p r o p o r t i o n a l . I n o t h e r words a l o n g i t u d i n a l k i c k e r i s n o t h i n g b u t a l o n g i t u d i n a l d e t e c t o r

w o r k i n g i n r e v e r s e . T h i s i s a l m o s t o b v i o u s f o r c a v i t y l i k e d e t e c t o r s , b u t i s a l s o t r u e

f o r a d i r e c t i o n a l c o u p l e r t y p e o f p i c k - u p f o r i n s t a n c e , w h e r e a q u a s i TEH wave

p r o p a g a t e s . T h e r e , o n l y t h e f i e l d a t t h e ends o f t h e c o u p l e r a r e u s e f u l f o r beam

e x c i t a t i o n .

F i g . 30 A p p l i c a t i o n o f thr> i n d u c t i o n l a w t o t h e e v a l u a t i o n of t h e t r a n s v e r s e f o r c e

C o n s i d e r now t h e c a s e o f a t r a n s v e r s e d e f l e c t i o n p r o d u c e d by t h e L o r e n t z f o r c e :

e {F. *• xi x B) dz

w h i c h p r o j e c t e d on t h e x a x i s ( F i g . 3 0 ) can be w r i t t e n :

e ( E + » .B ; d z

To e v a l u a t e t h e q u a n t i t y E + u . B , we a p p l y t h e i n d u c t i o n IRW t o a s m a l l

c t t f i i g l e i n t h e xOz p l a n e :

0 b e i n g t h e f l u x o f t h e m a g n e t i c f i e l d B on t h e c o n t o u r C. Ona o b t a i n s

dx dz + - — dz d x = - j n ) B d x d z

W i t h :

de dE dt ï s - Í jU) E

x y ju dx

E q u a t i o n ( 7 5 ) shows t h a t o n l y t h e l o n g i t u d i n a l f i e l d E^ ( m o r e p r e c i s e l y d E ^ / d x )

i s i m p o r t a n t f o r t r a n s v e r s e d e f 1 n o n . T h i s i s a w e l l known r e s u l t ( 1 i n a c t h e o r y f o r

i n s t a n c e ) w h i c h h a s a f e w i n t e r e s t i n g c o r o l l a r i e s . F o r i n s t a n c e , one c a n n o t d e f l e c t a

beam n e i t h e r w i t h a p u r e TEH wave n o r w i t h a p u r e H mode i n a c a v i t y i f t h e end e f f e c t s

a r e n e g l e c t e d . A t r a n r ^ e r s e k i c k e r must show a l o n g i t u d i n a l e l e c t r i c f i e l d , i n t h e same

way as a t r a n s v e r s e p i c k u p e x t r a c t s e n e r g y f r o m t h e l o n g i t u d i n a l v e l o c i t y o f t h e

j i r t i c l e s . T ^ ' . e i s c o m p l e t e e q u i v a l e n c e b e t w e e n p i c k ups and k i c k e r s e v e n i n t h e

t r a n s v e r s e p l a n e . T h i s w i l l b e i l l u s t r a t e d i n t h e f o l l o w i n g e x a m p l e .

T h e " T E H " t r a v e l l i n g wave k i c k e r h a s t h e same g e o m e t r y as t h e t r a n s v e r s e d i r e c t i o n a l

c o u p l e r p i c k - u p ( F i g . ' l a ) . T h e f i e l d i s t h a t o f a TEH wave a l o n g t h e two l i n e s , e x c e p t

a t t h e two e n d s w h e r e a l o n g i t u d i n a l component e x i s t s ( F i g . 3 1 b ) . Assume, f o r

s i m p l i c i t y v =v : t h e p a r t i c l e s r e c e i v e s u c c e s s i v e l y two o p p o s i t e t r a n s v e r s e k i c k s a t P *

e i t h e r end o f t h e k i c k e r , t h e r e s u l t b e i n g a z e r o d e f l e c t i o n ( a n o t h e r way o f s a y i n g t h e

same t h i n g i s t h . i t t h e e l e c t r i c and m a g n e t i c d e f l e c t i o n s a l o n g t h e l i n e e x a c t l y c a n c e l

e a c h o t h e r ) . On t h e c o n t r a r y , f o r u = -i> (beam i n t h e o p p o s i t e d i r e c t i o n ) t h e two k i c k s

P f

add e x a c t l y i f t h e y a r e s e p a r a t e d b y h a l f a p e r i o d o f t h e RF wave ( 1 = V / 4 ) . T h i s g i v e s a

v a r i a t i o n o f t h e t y p e s i n 2n H / X . The k i c k e r e f f i c i e n c y K | i s , f r o m E q . ( 7 5 ) p r o p o r t i o n a l

F i g . 31 " T E H " t r a v e l l i n g wave k i c k e r

w h i c h can bp w r i t t e n :

The f i r s t t e r m i s p r o p o r t i o n a l t o t h e DC d e f l e c t i o n ( p r o p o r t i o n a l t o ft) and t h e t e r m

b r a c k e t s g i v e s t h e f o r m f a c t o r w h i c h i s f r e q u e n c y d e p e n d e n t .

A . 3 Debunched beam t r a n s f e r f u n c t i o n

The beam i s composed o f a c o l l e c t i o n o f p a r t i c l e s , e a c h h a v i n g i t s own o s c i l l a t i o n

f r e q u e n c y q . u . , s u b m i t t e d t o a common d r i v i n g f o r c e F ( u ) . The e q u a t i o n of m o t i o n ,

f r j r e a c h i n d i v i d u a l p a r t i c l e i s , i n l i n e a r a p p r o x i m a t i o n :

i t h a f o r - ? d s o l u t i o n o f t h e f o r m :

x . - X . ciip j u t

F i n ) J FCj

The a v e r a g e beam r e s p o n s e < X . > / F ( w ) i s g i v e n by t h e i n t e g r a l :

(BOJ

where p ( q . u ^ ) i s t h e n o r m a l i z e d d i s t r i b u t i o n of t h e b e t a t r o n f r e q u e n c i e s w i t h i n tho beam,

( q ^ ^ and <ij

í->j b e i n g t h e two e x t r e m e í r<jquc-ne L rr . ) .

T h i s i s a s i n g u l a r i n t e g r a 1 , b e c a u s e of ( h e p o l e at q . u . - u Jt c a n bo decomposed

i n t o i t s Cauchy p r i n c i p a l v p i u e , w h i c h i s r e a l , and i t s r p : ; i d u c at t h e p o l e ( i m a g i n a r y ) ;

<* >

F U Ö = 2 ^ t p r i n c - V a l u c ( 8 1 )

We no 1* r e p l a c e < K ; : - by j u < X . > t o o b t a i n a r e a l t r a n s f e r f u n c t i o n B ( u ) when e n e r g y is

a b s o r b e d ( T o r c o and d i s p l a c e m e n t i n q u a d r i t u r p ) and o M a i n :

B ( u ) = ~ ( i r p t u ) + j P r i n t . V a l u e ) ( 8 2 )

The r e a l p a r t o f t h e t r a n s f e r f u n c t i o n g i v e s t h e p a r t i c l e d i s t r i b u t i o n i n t u n e l i k e

t h e s p e c t r a l p o w e r d e n s i t y o f t h e S c h o t t k y s i g n a l O u t r i d e t h e f r e q u e n c y band

( q ^ , t n G r e a l P a r l o f B ' l J ' v a n i s h e s ( p u r e i m a g i n a r y r e s p o n s e ) . T h e f a c t t h a t a

c o l l e c t i o n o f l o s s l e s s o s c i l l a t o r s r e s p o n d s l i k e a damped r e s o n a t u r i s t h e b a s i s o f Landau

damping and i s i l l u s t r a t e d i n F i g . 3 2 -

* Prase

F i g . 32 R e s p o n s e o f a l a r g e number o f l o s s l e s s r e s o n a t o r s - - - i n d i v i d u a l ^ a r t i c l e s

a v e r a g e

The e v a l u a t i o n o f t h e s t a b i l i t y o f t h e beam c e r t a i n l y c o r r e s p o n d s t o t h e most

i n t e r e s t i n g a p p l i c a t i o n o f beam transfer f u n c t i o n m e a s u r e m e n t s . C o l l e c t i v e e f f e c t s ( a n d

i n p a r t i c u l a r beam i n s t a b i l i t i e s ) r e s u l t f r o m t h o p r e s e n c e o f p a r a s i t i c imf -edances i n t h e

m a c h i n e w h i c h g e n e r a t e a d e f l e c t i n g f o r c e ( i n t h e t r a n s v e r s e c a s e ) , when e x c i t e d by a

c o l l e c t i v e d i s p l a c e m e n t o f t h e beam. t n o t h e r words t h e e x c i t a t i o n F ( w ) in Eq- ( 7 8 )

s h o u l d be c o m b i n e d w i t h a t e r m p r o p o r t i o n a l t o t h e beam r e s p o n s e j u < X . > . T h i s l e a d s

t c t h e w e l l known f e e d b a c k l o o p o f F i g . 3 3 , w h e r e H ( j w > i s l i n k e d t o m a c h i n e p a r a m e t e r s

and i s p r o p o r t i o n a l t o t h e impedance o f t h e m a c h i n e Z(ii>) • Fo r i n s t a n c e i n t h e

t r a n s v e r s e c a s e :

eo> i

m i s t h e r e s t mass of t h e p a r t i c l e .

j - < X , >

F i g . 33 F e e d b a c k l o o p due t o t h e m a c h i n e impedance

From F i g . 33 t h e new t r a n s f e i f u n c t i o n becomes:

By p l o t t i n g t h e c u r v e 1 / B ( u ) f o r d i f f e r e n t beam i n t e n s i t i e s i ^ one o b t a i n s a

f a m i l y o f c u r v e s s h i f t e d i n t h e c o m p l e x p l a n e b y t h e q u a n t i t y HCw) ( F i g . 3 4 ) . T h i s

s h i f t b e i n g p r o p o r t i o n a l t o Z ( o ) , t h e m a c h i n e impedance can be d i r e c t l y m e a s u r e d a t any

, i "

Uhen t h e s h i f t e d 1 / B ( u ) c u r v e r e a c h e s t h e complex p l a n e o r i g i n , s t a b i l i t y of t h e

beam i s l o s t ( B i u ) •* •») . t h i s means t h a t t h e d i s t a n c e o f t h e c u r v e t o t h e o r i g i n i s a

m e a s u r e o f beam s t a b i l i t y . I f a f e e d b a c k s y s t e m i s e m p l o y e d t o s t a b i l i z e t h e beam, i t s

e f f e c t w h i c h s h o u l d b e t o s h i f t t <? c u r v e t o w a r d s t h e r i g h t s i d e o f t h e complex p l a n e

c o u l d a l s o be e v a l u a t e d .

Stabilit

B ( u » )

F i g . 34 E v a l u a t i o o f t h e beam s t a b i l i t y w i t h t r a n s f e r f u n c t i o n measureme t s

W i t h v e r y s e n s i t i v e d e t e c t o r s and p r o v i d e d l o n g a n a l y s i s t i m e s a r e a v a i l a b l e (DC

s t o r a g e r i n g s ) , beam t r a n s f e r f u n c t i o n i s a v e r y p o w e r f u l t e c h n i q u e , a l m o s t non d i s t u r b i n g

t o t h e beam; i t c a n a l s o be u s e d i n a s i m i l a r way f o r t h e l o n g i t u d i n a l p l a n e .

ù . 4 B u n c h e d - b e a m t r a n s f e r f u n c t i o n

The m a i n d i f f e r e n c e w i t h r e s p e c t t o t h e u n b u n c h e d beam c a s e i s t h a t a n e x c i t a t i o n o f

t h e beam a t a g i v e n f r e q u e n c y w , n o t o n l y r e s u l t s m a beam r e s p o n s e a t u , b u t a l s c a t

a l l f r e q u e n c i e s nu>Q + w- ( T h i s i s b e c a u s e t h e b u n c h e d beam s a m p l e s t h t u w a v e f o r m

a t t h e r e v o l u t i o n f r e q u e n c y '«» 0 )- The p r o c e s s i s t h e r e f o r e f u n d a m e n t a l l y n o n l i n e a r ,

and as a c o n s e q u e n c e , t h e beam t r a n s f e r f u n c t i o n i s n o t d e f i n e d i n g e n e r a l , u n l e s s

a d d i t i o n a l c o n d i t i o n s a r e i m p o s e d 1 2 * . F o r i n s t a n c e , i f b u n c h t o b u n c h c o u p l i n g c a n be

n e g l e c t e d , o n e c a n d e f i n e u n a m b i g u o u s l y t h e beam t r a n s f e r f u n c t i o n o f a s i n g l e b u n c h , f o r

a g i v e n mode o f o s c i l l a t i o n ( d i p o l e , q u a d r u p o l e e t c . ) , i . e . v i t h i n an f ^ f r e q u e n c y

i n t e r v a l . A n o t h e r i n t e r e s t i n g c a s e i s when t h e b u n c h e d beam b e h a v e s l i k e a n u n b u n c h e d

beam: many e q u a l b u n c h e s , f r e q u e n c y r a n g e f r o m DC up t o and n e g l i g i b l e e f f e c t s

b e y o n d .

I n t h e t r a n s v e r s e p l a n e , t h e m e a s u r e m e n t o f t h e m a c h i n e t u n e i s n o t h i n g b u t a beam

t r a n s f e r f u n c t i o n m e a s u r e m e n t . Many d e s c r i p t i o n s o f t u n e m e a s u r e m e n t s y s t e m s e x i s t i n t h e

l i t e r a t u r e ; e x c i t a t i o n c a n b e s i n u s o i d a l o r random ( b a n d l i m i t e d n o i s e ) n e a r a b e t a t r o n

l i n e , o r p u l s e d ; beam m e a s u r e m e n t c o u l d be a t t h e same o r a t d i f f e r e n t f r e q u e n c y . I n

g e n e r a l t h e m a c h i n e i m p e d a n c e Z ( u ) c a n n o t b e m e a s u r e d d i r e c t l y , as a f u n c t i o n or

f r e q u e n c y ; o n t h e o t h e r hand i f t h e s h a p e o f Z ( . ] i i known ( e . g . r e s i s t i v e w a l l ) one c a n

d e t e r m i n e i t s m a g n i t u d e b y m e a s u r i n g t h e t u n e s h i f t as a f u n c t i o n o f beam i n t e n s i t y .

The RF s y s t e m and i t s a s s o c i a t e d f e e d b a c k l o o p s s t r o n g l y p e r t u r b s t h e l o n g i t u d i n a l

t r a n s f e r f u n c t i o n o f a b u n c h e d beam. T h i s i s p a r t i c u l a r l y t r u e f o r t h e d i p o l e mOile;

f o r t u n a t e l y t h e q u a d r u p o l e mode i s e a s i e r t o a n a l y s e and c a n p r o v i d e m e a n i n g f u l

m e a s u r e m e n t s o f t h e m a c h i n e i m p e d a n c e . A m p l i t u d e m o d u l a t i o n o f t h e RF w a v e f o r m a . a r o u n d

t w i c e t h e s y n c h r o t r o n f r e q u e n c y e x c i t e s t h e q u a d r u p o l e mode o f a s i n g l e b u n c h ; t h ;

q u a d r u p c l e o s c i l l a t i o n can b e o b s e r v e d i n s v e r y s i m p l e way by p e a k d e t e c t i n g t h e b u n c h

s i g n a l f r o m a w i d e band l o n g i t u d i n a l d e t e c t o r .

The m e a s u r e d beam t r a n s f e r f u n c t i o n , a t l o w I n t e n s i t y shows a s h a r p p h a s e

d i s c o n t i n u i t y , ai» t h e b u n c h c e n t e r , w h e r e t h e p a r t i c l e d e n s i t y i s maximum, and a ;:mooth

p h a s e c u r v e n e a r t h e b u n c h edge ( F i g . 3 5 a ) . T h i s c o r r e s p o n d s t o t h e 1 / B ( u ) p l o t .n

F i g . 3 5 b and p r o v i d e s a d i r e c t m e a s u r e m e n t o f t h e c e n t e r s y n c h r o t r o n f r e q u e n c y . A : h i g h e r

i n t e n s i t i e s , t h e i n d u c t i v e w a l l e f f e c t s h i f t s t h e 1 / B ( « ) c u r v e a l o n g t h e i m a g i n a r y a x i s

( r e a l f r e q u e n c y s h i f t ) and t h e p h a s e c u r v e o f F i g . 35a shews a s h a r p e r t r a n s i t i o n . From

t h o s e m e a s u r e m e n t s , t h e m a g n i t u d e o f Z ( u ) / n f o r t h e i n d u c t i v e w a l l c a s e c a n be

d e t e r m i n e d o v e r a f r e q u e n c y i n t e r v a l o f t h e o r d e r o f f . .

REFERENCES

1> J . B o r e r , P. Braraham, H . C . H e r e w a r d , K. H ü h n e r , W. S c h n e i 1 , L . T h o r n d a h l , Non d e s t r u c t i v e d i a g n o s t i c s o f c o a s t i n g beams w i t h S c h o t t k y n o i s e . I X t h I n t . C o n f . on H i g h E n e r g y A c c e l e r a t o r s , SLAC, May 1 9 7 4 , (SLAC, S t a n f o r d , 1 9 7 4 ) .

2 ) H . G . H e r e w a r d , W. S c h n e l l , S t a t i s t i c a l phenomena , P r o c . o f t h e f i r s t c o u r s e o f t h e I n t . S c h o o l o f p a r t i c l e a c c e l e r a t o r s , E r i c e , S i c i l y , 1976 (CERN 7 7- 1 3 , 19 7 71 .

3 ) T . L i n n e c a r . The h i g h f r e q u e n c y l o n g i t u d i n a l and t r a n s v e r s e p i c k ups used i n t h e SPS. CERN S P S / A R F V ) 8 1 7 , ( 1 9 7 8 ) .

4 ) J . B o r e r , R. J u n g , CERN A c c e l e r a t o r S c h o o l , A n t i p r o t o n s f o r c o l l i d i n g beam f a c i l i t i e s , CKRM, 1 9 8 3 . (CERN 84 1 5 , 1 9 8 4 ) .

5 ) T . L i n n e c a r , W. S c a n d a l e . A T r a n s v e r s e S c h o t t k y n o i s e d e t e c t o r f o r bunched p r o t o n beams. IEEE T r a n s . H u c l . S e i - , NS-28 p a g e 2147 ( 1 9 8 1 / .

6 ) Ü. B o u s s a r d , G. D i H a s s a , H i g h f r e q u e n c y s l o w wave p i c k u p s , CERN S P S / 8 6 4 , ( 1 9 8 6 ) .

7) K C a s p e r s . P l a n a r s l o t l i n e p i c k - u p s and k i c k e r s , CERN PS/AA N o t e 8 5 - 4 6 , ( 1 9 8 ^ ) .

8 ) 0 . B o u s s a r d , E v a l u a t i o n o f s l o t l i n e p i c k up s e n s i t i v i t y , CERN SPS/ARF N o t e 8 5 - 9 ( 1 9 8 5 ) .

9 ) D. B o u s s a r d , T . L i n n e c a r , W. S c a n d a l e , Ri c e n t d e v e l o p m e n t s on S c h o t t k y beam d i a g n o s t i c s a t t h e CERN SPS c o l l i d e r , IEEE T r a n s . H u c l . S e i . NS-32 page 1908 ( L 9 8 5 ) .

10 ) 0 . B o u s s a r d , S. C h a t t o p a d h y a y , C. Dôme, T . L i n n e c a r , F e a s a b i l i t y s t u d y o f s t o c h a s t i c c o o l i n g o f b u n c h e s i n t h e SPS, CERN A c c e l e r a t o r S c h o o l , A n t i p r o t o n s f o r c o l l i d i n g beam f a c i l i t i e s , CERN, 1 9 8 3 , (CKHN 84 15 1 9 8 4 ) .

11 ) J . B o r e r e t Í I L . , I n f o r m a t i o n f r o m beam r e s p o n s e s t o l o n g i t u d i n a l and t r a n s v e r s e e x c i t a t i o n , 1 FEE T r a n s . N u c l . S e i . NS^26 p a g e 3 4 0 5 ( 1 9 7 9 ) .

1 2 ) S. C h a t t o p a d h y a y , Some f u i damcr.ta 1 a s p e c t s o f f l u c t u a t i o n s and c o h e r e n c e i n c h a r g e d p a r t i c l e beams i n s t o r a g e r i n g s , CERN 84 11 ( 1 9 8 4 ) .

- 455 -

STOCHASTIC COOLIWS

0 . Hohl

C E R N , Geneva, Switzerland.

ASSfRrtCr

This puper describes the main analytical approaches to s tochast ic coolinq. The f i r s t i s the time domain picture in which the beam - s rapidly sampled and à s t a t i s t i c a l analysis is used to describe Ihe coolinq behaviour. The second is the frequency domain picture , which is part icu lar ly useful s ince the observations made on the beam are rninly in th is dr-iain. This second picture is developed in detai l to assess ingrédients of modern cooi inq theory l ike mixing and siqnal shie ld ing and lo i 1 lus trate some of the diagnostic methods. Final ly the use of a d ' s lr ibut 'on f u nct 'on and the Fokker-Planck equation are discussed, which qwe the most complete description of the beam djring the cool ing .

1 . INTRODUCTION

Beam cool ing aims at reducing the s i z e and the enerqy spread of a par t i c l e heam c ^ c u l a t ' n q *ri a

s'orage ring. This reduction of s i z e should not be accompanied hy bea^ loss ; thus the aoal i s to

increase the par t i c l e dens i ty .

Since the beam s i ze varies with the focus i nq proper fes of the sloraqe • - , ni , i t j s - J S P ' ; ' n

introduce normalized measures of s i z e and density . Such quant i t ies are the (horizontal , vert ica l and

longitudinal) emiltances and the phase-space dens i ty . For aux orpsent purpose they may be regarded

as the (squares of (.fieÍ horizontal and vert ica l heam diameter 1,, the eneroy spread, and the dens ' ty ,

normalized by the focusinq strength and the s i z e of the ring to make thmi independent of the sloraq°

rinq propert ies .

Phase-space density i s then a qeriral figure of merit of a p a r t i c l e heam, and coniinq impinv"S

th i s figure of merit.

The terms beam temperature and beam coolinq have been takm over from (ne \ ' n í , t ' c t V r t r v nf

gases. Imagine a beam of par t i c l e s qo^nq around m a sloraoe n n q . Part>cles wil l o s o ' l a t o Í I M U I K I

the oeait centra in much the sarne way that par t i c l e s of a hot qas ooimce b*cfc and forth h e t w ^ n f i - '

walls of a container. The larger the mean square of the ve loc i ty of these o s c i l l a t i o n s in a heam the

larger the beam s i z e . The mean square ve loc i ty spread is us^vi In define the beam I funeralure in

analogy to tne tanperalure of the qas which is determined by the k inet ic energy 0 .5 W"'^ of the

molecules.

l."hy co we wanl heam coolinq? The resultant incr'ease of bp AT q.i^.'ilv 's very d u r a b l e fo> y,

least three reasons:

i) Accum I at ion of rare p a r t i c l e s

Cooling to make space avai lable so that more heam can be stacked into the same storaq. 1 rinq.

The Antiproton Accumulator (AA) at CERN is ¿n example of th is (see Fiq. 1 ) .

- 4 5<1 -

QUANTITY BEAM IN

STACK GAIN

N 10 7 6 « 1 0 " 6 «ÎO-

lOOn 3.5n [imi-mrad] ÎS

£ 1DO« 2.0n [rom-inradj 50

4p/p 7.5 2.0 4

N

E h - E / p / p 130 1 * 1 0 1 0 3 *10 8

Fig. 1 The CERN Antiproton Accumulator (AA). Sketch and table of performance with nber of part icJes , hori2onta) en i t tance , vert ica l emittance and momentum spread of incoming beam and of stack after 24 h of accumulation {desiqn values) .

i i) Improvement of interaction rate and resolut ion

Cooling to provide sharply co l l iga ted and highly mono-energetic beams for precis ion experiments

with co l l id ing beams or beams interacting with fixed targets . The Low Energy Antiproton Ring

(LEAR) at CERN is an example of th is (see Fig. 2 ) .

i i i ) Preservation of beam quali ty

Cooling to compensate for various mectíánisms tearting to growi/t of beam size ând/or loss of stored beam. Again LEAR is an exanple of th i s appl icat ion.

Several cooling techniques are operative or have been discussed 11; Election beams have a

tendency ID COO) 'by themselves 1 owing to the emission of radiation a1- the orbit is curved. The

energy radiated decreases very strongly with increasing rest mass of the p a r t i c l e s . For ( a n t i - ) -

protons and heavier p a r t i c l e s , radiation damping i s neg l ig ib le at energies currently access ible in

accelerators . ' A r t i f i c i a l ' damping had therefore to be devised, and two such methods have been

success fu l ly p-jt to work during the last decade: i) cooling of heavier ^art ic les by the use of an

electron beam — th i s is the subject of H. Poth's chapter in these proceedings; and Ü ) s tochast ic

cooling by the use of a feedback system, which wi l l be discussed later in th i s chapter.

- 455 -

(a) Momentum cool ing at inject ion in LEAR; /íjN/dp displayed against momentum; 3 * 1 0 9

antiprotons, before and after 3 minutes of cool ing. Ap/p is reduced by a factor 4.

(b) Comparison of the cooled beam extracted from LEAR to the low enerqy antiproton beams

previously obtained in secondary beam l ines

300 MeV/c beam from

production target:

200 antiprotons/s +

several 101* conta­

minants

Beam s i z e 4 x 4 cmJ

Fig. 2 An example of momentum spread cooling and properties of the cooled beam from the CERN Low Energy Antiproton Ring (LEAR) compared to a secondary bean used be f e e 1983

2. SIMPLIFIED THEORY, TIME-DOWIH PICTURE

2.1 The basic set-up

The arrangement for cooling of th*. horizontal beam s i z e is sketched in Fig. 3. Assume, for the

moment, that there i s only one par t i c l e c i r c u l a t i n g , 'unavoidably, it wi l l have heen injected with

some small error in posit ion and angle with respect to the ideal orbit (centre of the vacuum

chamber). As the focusing systsn continuously t r i e s to restore the resultant deviat ion, the par t i c l e

o s c i l l a t e s around the ideal orb i t . Detai ls of these 'betatron o s c i l l a t i o n s ' ^ ) are given by the

focusing structure of the storage ring, namely by the d is tr ibut ion of quadrupoles and qradient mag­

nets (and higher-order 'magnetic l enses ' ) which provide a focusing force proportional to the p a r t i c l e

deviation (and to higher-order powers of the d e v i a t i o n ) .

For the present purpose, we can approximate the betatron o s c i l l a t i o n by a purely sinusoidal

motion. The cooling system is designed to damp th i s o s c i l l a t i o n . A pick-up e lectrode senses the

horizontal posit ion of the part ic le on each traversa l . The error signal - - ideal ly a short pulse

Cooled beam from LEAR.

Several I 0 b ant iprotons/s

Typical beam s i z e 5 x 5 nm2

- Abb -

with a height proportional to the p a r t i c l e ' s deviation at the pick-up — is amplified tn a broad-band amplifier and applied on a kicker which def lects the par t i c l e by an angle proportional to i tb error.

In the simplest case, the pick-up 3 ) cons i s t s of a plate to the le f t of the beam and a p late to

the right of i t . If the par t i c l e passes to the l e f t , the current induced on the le f t plate exceeds

the current on the right one and vice versa. The difference between the two s ignals is a measure of

the posit ion error. The 'kicker' i s , in principle , a similar arrangement of plates on wtvch a

transverse electromagnetic f i e l d i s created which def lects the p a n i c l e 3 ) .

Beam *

Fiq. 3 The principle of (horizontal) s tochast ic cool ing. The pick-up measures horizontal deviation and the kicker corrects anqular error. They are spaced by a quarter of the betatron wavelenqth \^ (plus multi­ples of Xß/2). A posit ion error at the pick-up transforms into an error of anqle at the kicker, which i s corrected.

K.cket

Since the pick-up detects the pos i t ion and the kicker corrects the angle, their separation i s

chcsen to correspond to a quarter of the betatron o s c i l l a t i o n (plus an integer number of half wave­

lengths if more distance is necessary). A par t i c l e passing the pick-up at the crest of i t s o s c i l l a ­

tion wi l l then cross the kicker with zero posit ion error but with an angle which is proportional to

i t s displacement at the pick-up. If the kicker corrects just th i s angle the part ic le wil l from

tfiereon rrove an the nominal orb i t . This is the most favourable s i tuat ion {sketched as Case 1 in

Figs . 4 and 5} . A part ic le not crossing the pick-up at the crest of i t s o s c i l l a t i o n s wi l l receive

only a partial correction (Cases 2 and 3 in Figs. 4 and 5) . As we shall see l a t er , i t wi l l then take

several passages to el iminate the o s c i l l a t i o n .

P U K

A The importance of betatron phase: Part ic le 1 crosses the pick-up with maximum d i s ­placement. Its o s c i l l a t i o n is ( idea l ly ! completely cancel led at the kicker. Part ic 1 e 2 arr i ves at an mtermed i ate phase; i t s osci11 at ion is only partly eliminated. Part ic le 3 arrives with the most unfavourable phase and i s not affected by the system.

At pick-up Aî kicker

Fig. 5 Phase space representation of betatron cool ing. The same as for Fig. 4 except that a 'polar diagram' x' = f (x ) i s used to represent the betatron motion x. = x s in [Q(s/R) + ¡ i , u j , x' = (R/Q) x' = x cos [Q{s/R) + (!.{,]. The undisturbed motion of a par t i c l e i s given by a c i r c l e with the radius equal to the betatron amplitude x. Kicks correspond to a jump of x ' . The cool ing system tr i e s to put par t i c l e s onto smaller c i r c l e s , ^art ic les 1, 2 and 3 are sketched with the mast favourable, the intermediate, and the least favourable i n ; t i a l phase, re spec t ive ly . As the number of o s c i l l a t i o n s per turn is dif ferent from an integer or half-integer , par t i c l e s come back with different phases on subsequent turns and all par t i c l e s wi l l be cooled progress ively .

Another part icu lar i ty of s tochast ic cool ing is e a s i l y understood from the s ing l e p a r t i c l e

model (Fig. 3 ) : the correction signal has to arrive at the kicker at the same time as the t e s t par­

t i c l e . Since the signal i s delayed in the cables and the amplif ier, whereas a hiqh-enerqy par t i c l e

moves at a speed c l o s e to the ve loc i ty of l i g h t , the cooling path has usually to take a short cut

across the ring. Only at low and medium energy (v/c < 0.5} is a paral le l path f e a s i b l e .

We have thus famil iarized ourselves with two constraints on the distance pick-up to kicker:

taken along the beam, th i s distance is f ixed , or rather quantized, owing to the required phase r e l a ­

t ionship of the betatron o s c i l l a t i o n ; taken along the cooling path th is length i s fixed by the

required synchronism between par t i c l e and s ignal . A change of energy (part ic le ve loc i ty} and/or a

change of the betatron wavelength wi l l therefore require special measures. Incidental ly , the f i r s t

of these two conditions i s due to the o s c i l l a t o r y nature of the betatron motion. For momentum spread

cool ing in a coasting beam, where the momentum deviation of a p a r t i c l e is constant rather than o s c i l ­

latory, th is constraint does not come into play and a greater freedom in the choice of pick-up-to-

kicker distance e x i s t s .

It i s now time t o leave the one-part ic le consideration and turn our attention t o a beam of

par t i c l e s which o s c i l l a t e incoherently i . e . with di f ferent amplitudes and with random, i n i t i a l phase.

By beam cooling we shall now mean a reduction with time of the amplitude of each individual par­

t i c l e . To understand stochast ic cool ing , we wi l l next have a c loser look at the response of the

cool ing system. This permits us to discern groups of par t i c l e s - - so-ca l led samples — which will

rece ive the same correcting kick during a passage through the system.

- 4S8 -

SID SID

Fiq. 6 I l lus trat ion of the Küpfmtíller-Nyquist re lat ion: a signal whose Fourier decomposition 5(f) has a bandwidth W, has a typical time duration T, = 1/(2W). I l lustrat ion for a Mow-pass' [case (a)J and a 'band-pass' signal [case ( b ) j .

*) The bandwidth/pulse length relat ion was introduced by Nyquist and independently by Kupfrnul1er in 1928. This theorem is c lose ly-re lated to the more general sampling theorem of communication theory: If a function S(t) contains no frequencies higher than W cycles per second, it i s completely described by i t s value S(ml s) at samplinq points spaced by At = T s = 1/2W ( i . e . taken at the 'Nyquist rate' s e e , for example, J.A. Betts , Signal Processing and Noise [English Univers i t ies Press, London, - w

2.2 Notion of beam sample;.

To be able to analyse the response of the cooling system, let us start with an excursion into

elementary pulse and f i l t e r i n g theory 1*). What we would l ike to take over is a bandwidth/pulse-

length relation known as the Kù'pfmù'l 1er or Nyquist theorem*):

If a signal has a Fourier detonjosition of band-width 6f - then i l s 'typical' tine duration

will be

T s = 1/(ZWJ .

This is i l lus tra ted in Fig. 6, where we sketch the Fourier spectrum of a pulse and the resul t ing

time-domain s ignal . Clearly the two representations are linked by a Fourier transformation, and th is

permits us to check the theorem.

For cur ios i ty , note the difference between a pulse with a low-frequency and a high frequency

spectrum (both cases are sketched in Fig. 6 ) . In sp i te of the different shape of the time-domain

s igna l , the 'typical duration' is in both cases 1/(2W).

A orollary to the theorem i s well known to people who design systems for transmitting short

Uten i short pulse is filtered by a low-pass or band-pass filter of bandwidth M, the resulting pulse has i 'typical' time width (see Fig. 7)

T = 1/(2W). 12.1)

sin LOW •—- PASS

FILTER

v - 1 Sf(f|

Fig. 7 Input and output signal 5 ( t ) of a low-pass system and 'rectangular' approximation to the output pulse S f ( t )

In i is form, the theorem i i d i rec t ly applicable to our cool ing problem, u which we now

return. : issing through the pick-up, an o f f - a x i s par t i e l* induces a short pulse with a length given

by the trans i t time. Owing to the f i n i t e bandwidth (H) of the coolinq system, the corresponding

kicker s - al i s broadened into a pulse of length T s . To simplify considerat ions, we approximate

the kicke' pulse by a rectangular pulse of to ta l length T s (Fig. 8 ; .

A t e ' par t i c l e passing the system at t D w i l l then be affected by the kicks due to a l l par t i c l e s

passing G ing the time interval tu ± T s / 2 . These par t i c l e s are said to belong to the sample of

the t e s t a r t i c l e . In a uniform beam of length T (revolution t ime) , there are = T/T^ = 2WT

equally spaced samples of length T s with

N = N/(2WT) p a r t i c l e s per sample (2-2)

•| Pick-up / Motion of '~"ceritre of

gravity ot sample

Uli)

— T = J -

/ l s JW

n 1 \ „ >

Pulse c t oichup R é p o n s e <\\ Sticker

Fig. 8 F k-up signal of a par t i c l e and corresponding kicker pulse ( idea l i zed) . The t e s t part ic le e eriences the kicks of al l other par t i c l e s passing within time - T s / 2 < i t < T s / 2 of i t s c ival at the kicker. These par t i c l e s are said to belong to the sample of the test p t i c l e . Cooling may be discussed in terms of the centre-of -gravi ty notion of samples.

Table 1

An example of samples corresponding to cooling at injection in LEAR'J

j No. of par t i c l e s in the beam N 10' 1

j Revolution time

T

0 .5 [.s j J Transit time in one pick-up unit

\ 0.1 ns

j Cooling system bandwidth u 250 mz [

j Sample length

T s

2 ns 1 1

j No. of samples per turn \ - T ' T s

250 j J No. of par t i c l e s per sample

1

N s a » 10" j

1

2,3 Coherent and incoherent e f f e c t s

The model of samples has allowed us to subdivide the bean into a larqe number of s l i c e s which

are treated independently of each other by the cooling system. If 'hp band*)din can be marie larqe

enough so that there are no other part ic les in the sample of the t e s t par t i c l e , then the s i n g l e -

par t i c l e analysis is s t i l l val id . However, to account for the r e a l i t y nf some mil l ion par t i c l e s per

sample, we have to go a step further and do some simple algebra. This n i ! l penni JS to discern two

s l i g h t l y different pictures of the coolinq process. In the "test part ic le DKture' we shall v^ew

cooling as the competition between: t) the 'coherent e f f e c t ' of the t e s t part ic le Lvon U s e l f via

the cooling loop; and i i ) the 'incoherent e f f e c t 1 , i . e . the disturbance to the tpst par t i c l e by the

other sample members (see Fig. 9 ) . In the 'sampling picture' we shal l understand stochastic cool ing

as a process where samples are taken from the beam at a rate 2 S per turn. By measur>nq and reduc­

ing the average sample error, the error of each individual part ic le wil l (on the dveraqe) slowly

decrease.

A few simple equations wi l l i l l u s t r a t e these pictures- Let us denote by x the error ;>f thp tpst

part ic le and assume that the the correspontf-inq c o n e c t i o n at the fcicfcer i s proportional to x, say

A.x. With no other par t i c l e s present, the error would he changed from x to a corrected

x = x - x (? .3 )

i . e . the t e s t part ic le receives a correclinq kick,

Ax = -xx . (2.4)

In r e a l i t y the i^icks -Axi of the other sample members have to be addfd, and the corrected prror after one turn and the corresponding kick are

- 461 -

_ i x

T * 1

'rev

Fig. 9 Cooling system s igna l s for the t e s t p a r t i c l e p ic ture . Siqnals at the instant of pa-,saqe of the t e s t part: :1e are sketched. The upper trace gives the coherent correction siqnal d'je to the t e s t - p a r t i c l e i t s e l f . The lower trace sketches the incoherent siqnal due to the other par t i c l e s in the sample. The kick experienced is the sun-, of coherent anJ incoherent e f f e c t s . If I i j amplif ication is not too strong and the sample population is small , the coherent e f f e t which is systematic wi l l predominate over the random heating by the incoherent signa s .

r -L — incoherent ef fect

coherent e f f ec t

Ax = -\x - I \.K. . {2.5) s'

In our rectangular response nodel, \\ = k is the same for a l l sample nember-s. Hence, we can a lso

write

i x = -Ax - K V x . (2 .6) s '

Equations (2.5) and (2.6] c l e a r l y •.hibit the 'coherent' and the 'incoherent' e f fec ts 'petitioned ¿hnve.

The sum labelled s' includes al part ic les in the sample except the test p a r t i c l e . You may want to

rewrite th is sum including the ( st par t i c l e ( th i s sum wil l be labelled s) and interpret ?t m terwis

of the average sample error (the ample centre of gravity if you l i k e ) , which •' *• def in i t ion

Eqjations (2 .5) and (2 .6) then become

(2.Sa)

Ax = - (),N s)<x> s s -g<x> s . (2.8b)

This introduces the second picture . What the coolinq systen does is to measure the average sample

error and to apply a correcting kick, proportional to <x> s to the t e s t part i c l e . Up to now the

sample i s defined with respect to a spec i f i c t e s t p a r t i c l e ; however, to the extent that any beam

s l i c e of length T s has *.he same average error <x>s our considerations apply to any t e s t par­

t i c l e . This is true on a s t a t i s t i c a l b a s i s , as wi l l become clear la ter .

A word about notation. It has become customary to write *.NS = g, and to ca l l g the 'ga in ' .

Remember that this g is proportional to the amplif ication (the e lec tronic gain) of the system and

proportional to >V As from Eqs. ( 2 . 8 ) , -g = ûx/<x> s , a more precise (but longer) name i s

' fract ion of observed sample error corrected per turn' .

How, we can again separate the coherent and incoherent e f f e c t s and rewrite Eq. ( 2 . 6 ) , by using

the above notation:

Clearly, the problem is how to treat the incoherent term. The following approximations wi l l be

discussed:

Firs approximation: Neglect the incoherent term

Seccr approximation: Treat it as a f luctuating random term

Third approximation: Treat i t as a f luctuating random term with some coherence due to imperfect

mi xing

Fourth approximation: Include additional coherence due to 'feedback via the beam'

2 . 3 . First approximation

(2 .9)

coherent term

(cooling) (heating)

incoherent term

Neglecting completely the incoherent term in Eq. (2 .9) we get a best performance estimate

i * = - 2 - x . (2.101

We expect an exponential form, x = x 0 e ~ t / t for the amplitude of the t e s t par t i c l e which gi*es

the damping rate

I I 1 ^ í . - i - J 5 6 1 * t u r n • (2 11) i = " i dt ° i i t " i 1

SutKt i t i l ing into Eq. 12.11) from Eq. (2 .W) g ives

l g

Interpreting g as the fractional correct ion, we i n t u i t i v e l y accept that it i s unhealthy to cor­

rect more than the observed sample error, i . e . we assume g < 1. Let us put q = L to make an estimate

of the upper l imi t .

Final ly i t i s convenient to express \ in terms of the total number o ' p a r t i c l e s , N, in the

beam and by the system's bandwidth W, i . e . N s = N(TS/TJ = N/2VT [ see Eq. (2.2)¡. We then obtain,

a f i r s t useful approximation to the cooling rate:

2W W (2.13)

Amazingly enough, t h i s simple re la t ion overestimates the optimum coolinq rate by only a factor of 2,

However, to gain confidence, we have to j u s t i f y some of our assumptions, e spec ia l l y the r e s t r i c t i o n

of g t 1 and the neglect of the incoherent term. In fac t , an evaluation of th i s term wil l c l a r i f y

both assumptions and provide guidance on how t o include other adverse e f f e c t s such as amplifier

no ise .

2 .3 .2 Towards a better evaluation of the incoherent term

To be able to deal with the incoherent term, we make a detour into s t a t i s t i c s to recal l a few

elementary 'sampling r e l a t i o n s ' 0 ) . Consider the following problem.

Given d beam of N part i c l e s characterized by an avpraae <x> - 0 and a variance - x'' of rms

sime error quantity x, suppose we take a random sample of part ic les and do s t a t i s t i c s m the

sample population - - rather than on the whole beam - - to determine

- 464 -

i] the sample average <x>s;

n the sample variance <*.^\;

n i 1 the square of the sample average •'x> i i J / , i . e . the square of [')•

What are the most probable values [the expectation values, denoted by E ( ' > - s ) , « t c . ; of these

sample character i s t i cs?

For random samples the most probat, le values are:

it sample average + beam averaqe;

i i ) sample variance * bean variance;

i n ) square of sample average * beam variance/sample population.

Or, m more mathematical lanquage,

E«x> ) = <x> = 0 (2.14a)

E(<x 2 > s ) = <x 2> = x ^ (2.14b)

E.' (<X> ) 2 \ = x2 /N . (2.14c) ' s ' ruts 5

Results (?.14a) and (2.14b) are in agreement wnb common sense, which expects that, the sample charac­

t e r i s t i c s are true approximations of the corresponding population c h a r a c t e r i s t i c s . This i s the basis

for sampling procedures. Equation (2.14c) is more subtle as it s p e c i f i e s the error to be expected

when one replaces the population average by the sample average.

or symbolical ly

In other words: the larger the beam variance and the smaller the sample s i z e { \ ) , the more

imprecise is the sampling. (n this form, Eqs. (2.14) are used in s t a t i s t i c s l<~ determine the

required sample s i z e for given accuracy and presupposed values for the beam variance x 2 . rms

A s l i g h t l y different interpretation i s useful in the present context: suppose we repeat the

process of taking beam samples and working out <x> s many times. Although the beam has zero <x>,

the sample average wi l l in general have a f i n i t e (pos i t i ve or negative) <x> s . The sequence o f sam­

ple averages wi l l f luctuate around zero (around ix> in general) with a mean-square deviation x¿ H . rms s

Trrs is the f luctuation (or, if you prefer, the noise) of the sample average due to the f i n i t e

part ic le number.

- J 6 5 -

A simple exanple to "illustrate the sa ip l inq re la t ions and t o fami l iar ize us further w i f ' * 2 \

and ( < x > s ) 2 i s given in Table 2 . It i s a n t i n g to note that in th i s exanple 'the most probable

values* 1/3 and 2/3 respect ive ly [which agree with E^s. (2 .14 ) ] never occur for any of the possible

samples — jus t another instance of s t a t i s t i c s dealing with averages and being unjust to the

individval .

Table 2

An example of the sampling re lat ions

Assuma a d i s c r e t e d i s tr ibut ion such that the values x = - 1 , 0 , 1 occur with equal probaui l i ty .

Hence, beam average: <x> = 0, and beam variance: <x 2> = x 2 , ^ = 1/3 i ( - l ) 2 + 0 2 + l 2 j = 2 / 3 .

Consider samples of s i z e : N s = 2 . To work out the most probable values of the sample character i s ­

t i c s , write down a l l poss ible samples of s i z e = 2, determine <x> s , ( < x > s ) j ! , and < x 2 > s , and

take the average of these averages to find the expectat ions .

Sequence Sanple avera K S

<*>s (<*> s) 2

-1 -1 -1 1 i -1 0 - 0 . 5 0.25 0.5

-1 1 0 0 1

0 -1 - 0 . 5 0.25 0 .5

0 0 0 0 0

0 1 0 .5 0.25 0 .5

1 -1 0 0 1

1 0 0 .5 0.25 0.5

1 1 1 1 1

Expectation = 0 = 1/3 = 2/3 = average cf above values <x> <x2>/2 <x 2>

To conclude our detour, l e t us mention that the sampling re lat ions (2.14) are a consequence of

the more general 'central limit theorem' 6 ) of s t a t i s t i c s . For the present purpose we can quote

th i s theorem as fo l lows:

Ilten a large number of random saiples of size U$ are taken from a population with statistics <x> = O and <x2> = *2rms then the distribution of the sanpíe averages is approximately Gaussian with a nean equal to the population mean and a standard deviation a - xr^/^H^.

2 .3 .3 A better approximation of the cooling rate - second approximation

Returning to Eq. ( 2 . 8 a ) , but re-expressing <x> s in ful l we have,

- Abb -

9 .

In order to profit from the samplinq r e l a t i o n s , i t is more useful to evaluate the chanqe A ( x 2 )

K2 - x2 of the squared error rather than ÙK. Thus we obtain,

A(x 2 ) = -2g J-l x. + ( j ^ i ^ J 2 •

The second term in Eq. (2-16) imnediately gives

where we nave used the sampling relat ion (2.14c) to express the expected variance of the sample

average in terms of the beam variance x 2 , - ^ . To work out the f i r s t term we separate the test

part ic le (once again) from the sum and write

1 .. x 2 X ..

Next we apply the sampling re la t ion (2.14a) to the remaining sum, i . e . we take

s s '

under the assumption that the sample ( labe l led s ' ) without the t e s t part M le is a random sample such

that Eq. (2.14a) a p p l e s . Then

[ ( < T ¡ ' , ) T - (2-18)

Thus the f i r s t term in Eq. (2.16) has non-zero expectation. Clearly th is is due to the fact that the

x at the front "coheres" with the corresponding term inside the sum.

Putting together the terms, the expected change is then

Equation \2 .19) appl'es to any l e s t p a r t i c l e . Taking as typical a par t i c l e with a-> error enjal to

the beam r.m.s . we can write espec ia l ly :

T M**) * -TT (29 - 9 2

*rms * \

This gives the cool.ng rate (per second) for the beam variance:

1 1 1 2W - T x 2 = ]tf ( 2 g - s ' = r { 2 9 - g2) •

T x 2 rms s (2.21)

Clearly the term 2g presents the coherent e f fect already i d e n t i f i e d . The -q*' term represents the

incoherent heating by the other p a r t i c l e s . The inclusion of th i s term i s the improvement obtained

in the s t a t i s t i c a l evaluation of th i s s ec t ion .

It emerges quite natural ly from Eq. (2.21) that g should not be too large! In f a c t , optimum

cooling (maximum of 2g - g 2 ) i s obtained with g = 1, and antidamping occurs if g > 2 (see Fig. 10}.

It should be remembered that Eq. (2.21) gives the cooling rate l/i K2 for x z ; the rate 1/x for

x i s half of t h i s , as can be ver i f i ed by comparing •*} = x 2 exp {-t/T 2) and x 2 = [ x u exp ( - t/1)] 2.

I

Optimum gain I

Gain

Fig. 10 Cooling or heating rate when considering the incoherent term as a random f luctuation

- J 6 3 -

2.3 .4 Alternative derivation

for those who were not pleased with the way in which we separated the test part ic le from i t s

sample and regarded the remainder as a random sample of s i z e - 1, we give yet another derivation

of Eq. (2.21) which i s due to Hereward (unpublished notes 19?6, see a lso R e f . 7).

We restart from Eq. (2 .16 ) , which we write as

ú ( x 2 ) = -2gx • <x> s + g 2 (<x>J 2 . (2.22)

This is the charge for one t e s t p a r t i c l e and one turn. We now take the average of t h i s over the

sample of the t e s t par t i c l e (before, we took the average for one p a r t i c l e over many turns ) .

A s l ight complication ar ises from the fact that s t r i c t l y speaking each part ic le defines i t s own

sample, as sketched in Fig. 11. We can assume, however, that the long-term behaviour of any sample

{ i . e . any beam s l i c e of length T $ ) is the same, so that expectation values are independent of the

choice of the sample.

0

Fig, 11 Sample of the original t e s t - p a r t i c l e (0) and of a part ic le passing earl ier ( i ) . Working out the average <x,0(> s > s of x¡<x> s each p a r t i c l e has to be associated with i t s own sample. To the extent that al l beam samples have the seme s t a t i s t i c a l propert ies , al l King-term averages are the same: <X^<K> S> S + -

Then the only variable on the r . h . s . involved in averaging over the original sample is the x in

the f i r s t term, ard we obtain

C û ( x 2 ) > s * -Zg(<x> s ) 2 + g 2 ( < * > s ) 2 - (2.23)

Next we use the sampling re lat ions (2.14b) and (2 .14c ) . We include the fact that the correction

(2.23) is applied to a l l beam samples once per turn. Thus,

< A ( X 2 ) > S - A x ^ ,

(<x> ) 2 •> x 2 /N , s rms s

and the expected correction of beam variance per turn i s

i . e . exact ly as assumed in EQ - ( 2 .20 ) .

Tnis leads to the same cooling rate as that given by the previous approach, but the derivation

lends i t s e l f to the following formulation of the "sampling p i c ture ' .

Take a random beam sample of N $ p a r t i c l e s . Measure and correct i t s average error <x"-s by

giving a kick -g<x> s to al l p a r t i c l e s . Owing to the f i n i t e part ic le number, the beam variance

appears as a f luctuation with 'noise ' ( < x > s ) 2 + X

r m s / N

s °f the centre of qravity < x ^ . By correct ­

ing <x> s to (1-g) of i t s value ( i . e . to zero for fu l l g = 1 ) , one reduces the sample variance (on

the average) by 1/N S ( 2 g - g 2 ) . Repeat N/N s times per turn to reduce the beam variance by the sane

amount. Repeat for rtany turns.

Table 3

'Simulation' of a one-turn correction (with g = 1) using the example of Table 2. We note down ail poss ible samples of s i ze = 2 and reduce the sample errors to zero by applying the same correc­t ion t o both sample members. This reduces the beam variance from 2/3 to 1/3 , : . e . û x 2 fx2 = m = 1/2.

rms nns s

Before correct ion After correct nn

Sequence Sample Sequence Sarrple

Average

< V s Variance

<« 2 >s

Average Variance

-1 -1 -1 1 0 0 0 0

-1 0 -0 .5 0.5 -0.5 0.6 0 0.25

-1 1 0 1 -1 1 0 1

0 -1 -0 .5 0.5 0.5 -0 .5 0 0.2S

0 0 0 0 0 0 0 0

0 1 0.5 0.5 -0 .5 0.5 0 0.25

1 -1 0 1 1 -1 0 1

1 0 0 .5 0 .5 0.5 -0 .5 0 0.25

1 1 1 1 0 0 0 0

'8eam variance' (average of al 1 sample variances)

2/3 1/3

Thus, rather than treating s ing l e p a r t i c l e s , one measures and corrects the centres of gravity of

beam samples. It is amusing (but not too surprising) to note that the total number of measurements,

namely the number of turns n = N s required for reasonable cooling multiplied by the number £ s =

N/Ns of samples ptr turn, is N, as if we treated the N part ic les individually.

It i s easy to t e s t th is sampling prescription for simple d i s tr ibut ions ; in Table 5 we use the

previous example {Table 2) to verify that the fu l l correction (g = 1) reduces the variance by 1/S S

per turn. More general ly , the sampling recipe can e a s i l y be simulaLed on a des-; computer using a

random number generator.

In the next two sect ions we wi l l use the t e s t par t i c l e and the samolinq picture a l ternate ly to

introduce two final ingredients, namely e lectronic noise of the amplifier and mixing of the samples

due to the spread in revolution time.

2 3.5 f. refinement to include system noise

A large amplification of the error s ignals detected by the pick-up is necessary to qive the

required kicks to tne beam. Electronic noise in the preamplifiers then becomes important. In Table 4

we ant ic ipate some typical numbers pertaining to transverse coolinq of 1 0 9 antiprotons in LEAH. This

example should convince us of the necess i ty to rewrite the basic equations to include noise. It i s

convenient 7 ) to represent noise by an equivalent sample error (denoted by x n ) as observed at the

pick-up. He then regard the system sketched in Fig. 12 and write

x c = x - g<x> s - g x n . (2 .24)

Table 4

Signal, no i se , and amplification of a cooling system; orders of magnitude for 10 par t i c l e s and 50 s cooling time

Pick-up signai 50 nA

Preamplifier noise current 150 nA

Kicker voltage per turn 1 V

Corresponding current ( into 50 u) 20 mA

Power amplification - 2 * 1 0 1 0

Parht le orbit

Fig. 12 Cooling loop including system noise . The noise i s represented as an equivalent sample error x n ( t ) as observed at the pick-up.

Going once again through our basic procedure, taking random noise uncorrected with the par t i c l e s we

obtain the expected cooling rate

1 ZW , , — = r [2g - g 2 ( l . U ) j

where U = E(x 2 ) /E<x> 2 ) i s the ra t io of the expected noise to the expected signal power*), ca l led the

'no i se - to - s igna l power r a t i o ' or no i se - to - s igna l ra t io for brevi ty .

This introduces the noise into our p ic tures: i t increases the incoherent term by [1H1). System

noise and the disturbance caused by the other par t i c l e s enter in much the same way; the lat ter i s

therefore also ca l l ed par t i c l e no ise .

Several things can be observed from Eq. ( 2 . 2 5 ) . Coolinq remains poss ible despi te very poor

s i g n a l - t o - n o i s e r a t i o s (1/U « i ) . AU we have to do i s to choose g small enough (g < = 1/(1+11} -1/U), which unavoidably means slow cooling (T > NU/2W). In other words, we have to be patient and

give the system a chance to d i s t i l a s ignal out of the no ise .

In the i n i t i a l cooling experiment (ICE) 0 ) with 200 c i rcu la t ing antiprotons the system worked

with s i gna l - to -no i s e ra t ios as low as 1 0 - 6 .

Secondly, U has a tendency to increase as cooling proceeds: namely the noise tends to remain

the same, whereas the signal decreases as the heam shrinks. This is the case unless the pick-up

plates are mechanically moved to stay c lo se to the beam edge — as wi l l be done in the new antiproton

c o l l e c t o r ACOL9) to be bu i l t at CERN.

With changing U, cooling i s no longer exponential . Equation (2.25) qives a sort of instantane­

ous ra te , and cool ing stops completely ( l / i + 0) when U has increased such that (1+U) = 2/g. M this

s i t u a t i o n , equilibrium i s reached between heating by noise and the damping e f fect of the system. To

avoid t h i s 'saturation' i t is sometimes advantageous to decrease g during coolinq in order to work

always c lo se to the optimum gain [maximum of Eq. (2.25)1 g 0 = l( {I + U).

In al l cases i t is important to obtain a good s iqna l - to -no i se r a t ' o . Frequently, th i s means

having a large number of pick-ups as c lo se as possible to the beam, as well as high qual i ty , low-

noise preamplifiers often working at cryogenic temperatures.

2.4 Mixing - third approximation

So f a r , oi l our considerations have been based on the assumption of random samples. This is a

good hypothesis for an undisturbed beam. However, the cooling system is designed to correct the

*) Stochastic cooling of heavy ions is becominq very important so we should note that U •+ WfV where Z i s the charge number of the par t i c l e and U the noise to signal rat io calculated for s ing ly charged p a r t i c l e s .

- 472 -

AT Ap

where the off-momentum function 2 ) r\ = y-¿ - y - 2 is giyen by the disLance of the working energy (y)

from trans i t ion [Y^ I -

If mixing is fast so that complete re-randomization has occured on the way from kicker to

pick-up then the assumption of random samples made in the previous sections is val id . If however,

mixing is incomplete, cooling is slower. In fac t , if i t takes M turns for a part ic le of typical

momentum error to move by one sample length with respect to the nominal part ic le ( ip/p = 0 ) , then

i n t u i t i v e l y one expects an M times slower cooling rate .

A s l i g h t l y different way of looking at imperfect re-randomization sugqests i t s e l f in the frame

of the t e s t par t i c l e picture: bad mining means that a p a r t i c l e stays too long — namely M rather

than i turns — together with the same noisy neighbours. This increases the incoherent heatinq by

the other part ic les by a factor M.

We thus generalize the basic Eq. (2.25) (a rigorous derivation w-ll be given later)

1 2A (2.Z7)

and cal l M * 1 the mixing factor.

M is defined as the numbe" of turns for a part ic le with one standard deviation in momentum to

migrate by one sample length T v

Equation (2.27) has the optimum,

g * ft, = 1/(M + U) ,

N

This underlines the importance of having good mixing - - M •* 1 — on the way from correction to the

next observation, but . . .

s t a t i s t i c a l error of the samples. Just after correction, samples wi l l no longer be random. For ful l

correction the centre of gravity <x>c w i l l be zero rather than J*2 /M as experLed for random 3 r i s S

condit ions. Cooling will then stop as no error signal is observable.

Fortunately, owing to momentum spread, part ic les in a storage ring go round at s l i g h t l y d i f fer ­

ent speeds, and the faster ones continuously overtake the slower ones. Because of th is mining, the

sample population changes and the sample error reappears, unti l ideal ly al l par t i c l e s have zero

error. The dispersion of revolution time with momentum is governed by

What about mixing between observation and correction? Surely if the sample i s observed is very di f ferent from the sample as corrected, then adverse e f f e c t s can happen. Let u, aqain resort to the t e s t part ic le description and try to imagine how the coherent and the incoherent e f f ec t s change. As to the l a t t e r , we expect that it is to f i r s t order not affected. 'We C - i just assume that the per­turbing kicks are due to a new sample which has the same s t i t i s t i c a l properties as the original bean ' s l i c e ' .

The coherent e f f ec t w i l l , however, change because the system wi l l be adjusted in such a way that

the correction pulse wil l be synchronous with the nominal part ic le (¿p/p = 0 ) . Part ic les that are

too slow or too fast on the way from pick-up t o kiclcer wi l l therefore s l i p with respect to their

self- induced correction (Fig. 13) . In f a c t , in the rectangular response model used above, the

coherent e f fect wi l l be completely zero i f the part ic le s l i p s by more than half the samóle length

(JliTpKI > T

s / 2 ) . M this s tage , i t is more r e a l i s t i c to use a parabolic response model of the

form 1 - ( ¿ T / T c ) 2 , where T c , the useful width of the correction pulse, is about equal to the

sample length T& for a low-pass system. But T c i s shorter than T for a high-frequency band­

pass system with f m i n > W, with a response as sketched -n Fig. 6 (b ) ; ùJ i s the t ime-of - f l ight

error of the par t i c l e between pick-up and kicker. Introducing the typical error ¿Tp^ and ca l l ing

iJpK/T c = we can modify the coherent term g + g [ l - K2] t o account for unwanted mixi"

between observation and correct ion. In a regular l a t t i c e t i e f l i g h t time from pick-up to kicker i 3 u

f ixed fraction of the time from kicker to picfc-up, and the two mixing factors M and M are propor­

tional to each other, M = a N, with a being the rat io of the corresponding distances — hence the

interes t in having a shcrt beam path from pick-i , . to kicker.

' 0

SJ \

V. f

I ¡ ; 2TC ;

Fig. 13 Synchronism between part i c l e s and their correcting pulse on their way from pick-up to kicker. The response cf the cooling system to a par t i c l e (the 'coherent e f f e c t ' ) i s approximated by a 'parabola' s ( t ) = 1 - ( A t / T c ) J of width tT c instead of the 'rectangle ' used in Figs. 7 and 8. A nominal par t i c l e (0) arrives at the kicker simultaneously with the correction kick. The par t i c l e f i s much too fast and advances i t s correction pulse . The par t i c l e s i s s l i g h t l y too slow. Thus, the three par t i c l e s rece ive fu l l correct ion, no correct ion, or partial correct ion, respect ive ly .

The reduced correction becomes,

This wi l l give us a s l i g h t l y different form for the basic eauati.-n.

f - 2 = f ^ [ 2 g ( l - M - 2 ) - g 2 (H«j)j - (2-3Q1

Coherent Incoherent (cooling) (heating)

By a clever choice of the bending and focusing properties of the s'orage ring i t is possible* in

pr inc ip le , to make ATpj( •* 0 independent of momentum, and ATKP large to approach the desired

s i tuat ion of M - 2 = 0 and M = 1. But th i s complicates the storage ring l a t t i c e . The compromise

adopted in e x i s t i n g designs is to s a c r i f i c e some of the desired re-randomisation in order to avoid

too much unwanted mixing.

rol lowing conven'-ion, we now return to the cool ing rate for x rather than x¿ (using 1/x = 1/2

1/t i). Including both mixing e f f ec t s as well as amplifier noise , we write

with M > 1, 'J > 0, M~ < 1.

2 ) - g2(M+U)j (2.31)

Equation (2.31) i s the main resu l t of our simple analys i s . It exhibi ts some of the fundamental

l imitat ions of s tochast ic cool ing. We f i r s t note that 1/T has a maximum characterized by

1 - M-2

9« • T T ? • (:-32»

1 w ,(1 - «H) 2

As an example of r e l a t i v e j y straightforward technology, we take W = 250 MHz. Then, in tht jest of

a i l cases (M * 1, U + 0, H- 2 + 0) th i s gives

1 /T = W/N = 2.5 x lOVN [sec- 1 J Í2.34)

i . e . x = t s at 2.5 « 10a p or \ - 1 day at L0i3 p.

To include mixing, we assume that the t ime-of - f l ight dispersion between pick-up and kicker and

between kicker and pick-up and the system response are such that the unwanted mixing M ¡s one half of

the wanted mixing, i.<i. we put U s an example) H-L = 0.5 M" 1 . We further assume that the s e n s i t i v i t y

and the number of pick-ups are such that u = 1 ( l i t t l e is gained in this example in qoing to more

pick-ups, such that U « 1 ) . Then the best cool ing, obtained with M - 1.5, is

1 -. - 0.32 W/N

Tirs i j bout three times slower than the r a t P '2 .34) with K~¿ * 0, U - 0. We reta'n that ove' a

wide range of parameters l / i - a W/N.

From Fig. 14 we conclude that ex i s t ing cool- q systems follow a 'workinq l ine ' with 1/t - 0.1 to

Û.3 W/N, i . e . a,j * 0 . 1 - 0 . 3 . A bandwidth of 250 : i ¿00 is (more or less ) standard; 2 to 4 GHz

will be used in the CERN-ACOL and the Fermilab ai t i proton sources. Bands of 4 to 8 GHz or higher

have been contemplated for sources accumulat ;ng : V1 antiprotons in a few hours'"), as desirable

for fftjltt-TeV c o l l i d e r s {see T a b l e 5 ) .

Tahle 5

Parameters of present, future and 'ultimate' too' inq systems. The quantity a u , defined by 1/t = a 0 W/N describes the e f f i c i e n c y of solving the noise and mixing problems.

Machine Date U

GHz an /N

Achieved ICE 1976 0.1 0.03 1/3 x 10 b

AA 1980 0.25 0.1 1/2 x 10'

(precool ing)

Future Fermilab 1986 2 0.25 1/5 « 10"

ACOt 1987 2 0.25 1/5 » 10 s

Ultimate 15 0.5 1/7 x 10 3

2.5 Practical d e t a i l s

So far we have, in a general way, discussed a system for correcting 'some error x ' .

In practice cool ing i s used to "educe the horizontal and/or ver t i ca l betatron o s c i l l a t i o n and

the momentum spread of the beam. Table 6 gives a summary of the corresponding hardware.

The simple time-domain approach can be d i r e c t l y applied to momentum cooling by the Palmer-

liereward method. This wi l l be discussed in the next subsection. A discussion of the other momentum

cooling methods and of betatron o s c i l l a t i o n cooling wi l l be deferred to later s ec t ions .

- 4 " t , -

1 1 • ) 1 1 1 1 (- -KJ* TO7 « • M f l 1 0 1 0 1 0 " 1 0 * 2 t o ' 3

Fig. 14 Normalized cooling time versus intens i ty . The inclined lines represent the mixing IrniU. For low intensi ty the cooling time levels off because of noise . The points represent i n i t i a l cooling in various machines. These points roughly follow a l ine with t - 10 N/w. During coo l ing , noise and/or mixing tend to become more important and the cool ing time ionger. Note that the vert ical scale is normalized for 100 KHz bandwidth. Futu--p systems [ACDL, Fermi lab) _m at 2-5 GHz bandwidth.

Table 6

Type Pick-up Corrector

Betatron cool ing , horizontal or vert ical

Difference pick-up Transverse kicker

Mjmentjm cool ing , Palmer-Hereward type

Horizontal di f ference pick-up

RF gap (acce lerat ion/ deceleration)

Moment JT I coo l ing , f i l t e r method

Longitudinal (sum) pick-up +• comb f i l t e r

RF gap

Momentum cool ing , t rans i t time method

Longitudinal pick-up + d i f f erent ia tor or two longitudinal pick-ups

RF gap

2 .6 Palmer cool ing

A horizontal pos i t ion pick-up i s used to detect the horizontal orbit displacement x = D<Ap/p>5

concurrent with the momentum error of the sample; D (a l so denoted by up or xp) is the value of

the orbit 'dispersion function' at the pick-up as determined by the focusing properties of the

storage r ing. In addition to the momentum dependent displacement there are further contributions to

the pos i t ion error , e spec ia l l y the betatron o s c i l l a t i o n (xp) of the p a r t i c l e s . We shall neglect

th i s contribution, assuming that the pick-up i s placed in a region of large dispersion so that

<D(Ap/p)> s dominates over <xß\. We are then in a s i tuat ion where fnomentum cool ing as

envisaged by R. Palmer (private communication to L. Thorndahl and H.6. Kereward in 1975} is

poss ib l e . At the RF' gap the p a r t i c l e receives a 'kick' of momentum and hence a change of x propor­

t ional to the detected error.

The basic one-passage equation (including noise) i s written as

x c = x - g[£Kap/p> s + x n J .

This i s completely equivalent to Eq. ( 2 . 1 5 ) , thus leading to the cooling rate (for x 2 and Ap', i . e .

for the mean square of the momentum dev ia t ion) :

1 2W — — = ï r [ 2 g ( l - M"2) - g2(H+U)J , (2 .36)

CAp 2 1 1

where U = E(x2

n)/E[_(CKip/p> J2\ is the no ise - to -s ignal r a t i o ; x n is the system noise expressed as

the equivalent pick-up signal D{Ap/p), and E(x 2 ) the expectation ( i . e . the long-term average) of x 2 . n n

Above we assumed that the orbit dispersion x = D(Ap/p) dominates at the pick-up so that the betatron o s c i l l a t i o n is neg l ig ib le there. He also implied that at the kicker the dispersion function 0^,

as wel 1 s" i t s der ivat ive , DV, are zero. Otherwise the momentum correction leads to

Stochastic cooiing systems in use or proposed

an exc i tat ion of horizontal betatron osci'• lat ions. The e f fect is t -at the -ncnent-ri lock -niroduces

an abrupt change of the equi1ibrium orbit and the par t i c l e s tar t s to o s c i l l a t e around th i s new

displaced orb i t .

The more r e a l i s t i c case where both x and x 0 are present at the p'ck-up and whçr^ z\ is

non-zero at the kicker was analysed by Hereward. He showed - 1 ) tne l u t j a ! heating anj at the sare

time the p o s s i b i l i t y of using the Palmer system for simultaneous longitudinal and horizontal coolinq

by ä sui table choice of the p^ck-up to kicker distance.

3. A HOft- DETAILED PRESENTATION OF BETATRON COOLING, FREQUENCY DOW IN PICT'J*E

3.1 Betatron equation

Before entering into d e t a i l s , i t .s worth trying to e s tab l i sh a simple picture of betatron coo l -

inq in which the various phenomena can be ident i f i ed .

Consider f i r s t the smooth sinusoidal approximation for the betatron not ion' ) of a s ing le

p a - t i c l e (subscript i ) in a storage ring, with forcing terms on the right-hand s ide ar is ing from t ts

pnper motion, the motion of other part ic les (subscript j ) and system noise .

x (t ) + u ^ O / x . tU = 6 . . x. ( t - l ) + i G.. x . ( t - t ) + 'system noise ' ( 3 .1 ) i v n ) p' J 1.) j l p '

Coherent ef fect Incoherent e f f ec t Additional incoherent e f fect

- Mixing PU * K - Mixinq K + PU - Enhancement of cooling - Betatron phase - Signal shieldinq

errors

We interpret the left-hand s ide as the nrjtion on entering the coolinq kicker (K) and the forcing

terms on the right-hand side as being derived from the motion seen ear l i er ( i . e . at t - tp î in the

pick-up (PU). The character i s t i c s oF the pick-up, amplif ier, transmission system and kicker enter

inlo both t..„- coe f f i c i en t s G-j-j and the "system noise". 'J is the revolution frequency and Q the

t n e of tne storage ring.

3.? Simplified coherent e f f e c t

if we neglect the incoherent terms in Eq, (3.1) and make = constant* we obtain a s inq le -

par t i c l e cooling equation,

x(t ) + ^ x ( t ) = x ( t - t p ) (3.2)

putting = J).

For a weak perturbation term, we can expect a solution of the form:

Substituting into Eq, Ci.2) qwt-..

This is the expected response of a feedback system. The 'eai uart of \~ is Lhe frea-wy s i - r t ,

oí the perturbed o s c i l l a t i o n and the imaginary part of J J q)ves the .lamo'nq heat'''a' t h ­

o s e ; l a t i o i .

1 G n i{. . - n/7)

- = I Ü H » = Re e : . {3.4)

Equation (3 .4) would be exact if the observation and feedback on !)ean were coït'niioA, ani

Gi, c instant , wh'.ch are manifestly not the case . We "ust now, therefor?, invest ig ; te the e f f e c t ;

of periodic observation and correct ion.

3.3 Orbit equation for a constant local ized kick

The orbit in a storage ring with constant kicks can be regarded as a betatron o s c i l l a t i o n wh'ch

c loses onto i t s e l f by virtue of the angular d i s c o n t i n u i t i e s at the kicks . The closed cvbU is <rven

for any dis tr ibut ion of kicks by the well-knnwn e q u a t i o n 2 ) ,

:ä ( s ! E (O „ _ z

OgrJo Pi'

where E x i s the transverse e l e c t r i c f i e ld jV/mj

¿Bz is the transverse magnetic f i e ld LTJ error (vert ical f i e ld for horizontal oHiit informa­

tion and vice versa)

B0o l J is tne magnetic r i g i d i t y t T-nj = 3.3356'10~ y p ^eV/cj

p is the part ic le monenLim :eV/V and P = v/c

s = ;ict is the distance along the orb i t .

The Eq. (3 .5) is good for numerical ca l cu la t ions , but is inconvenient For analytir. i! wit '•., -since

it is defined piecewise in s .is one proceeds around t.ir1 machine through the various c louant s.

¡Jsing the wel 1 known Co irait and Snyder transformât ioní) the eq...i! mn c a.n h-> * - i w 1 : r J-: I-, i

"driven har:ronic o sc i l l a tor" with f u e d frequency Q rather than with azuiMna! ly vjryitin i;(s' is • ••

Eq- ( 3 . 5 ) :

n(t) + Qán = q'^/¿(s)F(s) 13.h)

- -iíSO -

where r] = x f s j ß j ' 1 ' ' 2 { s ) i s the normalised displacement, d$ = ds/Q3„ oefines the Courant and

Snyder angle which increase by Zu per turn, 0 X i s the betatron function of the storage ring and '

now indicate 1- d i f ferent ia t ion with respect to

Equation (3 .6 ) is quite general, but we are e s p e c i a l l y interested by a s ing le narrow kick, which

we can represent by a del ta function, or rather by a periodic delta function (c) with the revolution

frequency (see f i g . 1 j ) .

As

-6TIR

- 3 T - i r r R

- 2 T

- 2 n R

- T

2 n R

T A n ft

2T &T!R s 3 T t

Fig. 15 For a s i n g l e narrow kick in a storage ring the p a r t i c l e sees a periodic influence

The kick as seen by a c ircu la t ing p a r t i c l e is represented by,

F(s) = (FuAsJMs-nZrô) = (F 0As) I 6{s-n2nR) (3 .7)

(accepting that high frequency components, i . e . f > c/ûs are not required).

Equation (3 .7 ) i s a good representation of a short kicker with a constant kick, hut ana ly t i ca l l y

i t would be eas ier to manipulate if we could replace the discontinuous delta functions with continu-

which appears on the r . h . s . of Eq. ( 3 . 6 ) .

^ / 2 ( s ) F ( s ) = I f ^ U ) * (3 .8)

where,

1 2n - U $ 1 2n , n -iJi$(s) ds

Since F is a delta function the integral (3.8) simply leads to Fourier coe f f i c i en t s which are al l

equal. (This is the advantage of the complex formulation, which avoids the f 0 / 2 coef f i c i ent of the

real expansion.)

f" i 2 nq P

K 2„q P

K

(3 .9)

- 4 8 ] -

where ^ is. the value of the beta function at the Kicker (s=0) and

i B 2 E

0 = ! B ¡ ^ * ^ ) "

i s the kick strength.

We can now rewrite Eq. (3 .6) in terms of continuous functions

y J\ » Í¿¿

n{ç) + O n = Q e Ï e (3-lCJ

This i s in fact a l l we rea l ly need, but we can make two variable changes, which wi l l make the

equation more fami l iar . F i r s t l y , s ince we prefer to think in terms of the time, we can introduce a

' t ime- l ike ' variable i/Q i . e . the Courant and Synder normalised phase. î , scaled by the revolution

angular frequency, ß . Rather loose ly we wi l l s t i l l refer to th i s variable as t . In f a c t , t h i s lack

of rigour i s not too ser ious , s ince t wi l l coincide with the true time at l eas t once every revolution

at the kicker (s = 0 ) , which i s the one point where true time is important. In any case , in most

l a t t i c e s i>/ß wi l l not s tray far from true time at any point in the ring. Secondly, we l ike to think

in the transverse deviation x, so we undo the normalisation of the variable n, but again we are only

r e a l l y interested in true deviat ions at the pickup, so we define a quasi -posi t ion variable

r,ßl/2 which gives true pos i t ion once per turn at the pick-up and again we loosely c a l l i t x PU

: Tip l ¿ = true posit ion . (3.11)

Using (3.11) in ( 3 . 1 0 ) , we f ind,

OV rCT 2 ( t ) + 0 V x = ^ a i e ' * * , (3 ,12)

Thus the betatron motion i s driven by an in f in i te se t of Fourier harmonics of equal amplitude

and separated in frequency, one from the other, by the revolution frequency a ( see Fig. 16).

Fourier amplitude

¿ • ^ . 2 * Frequency T

Fig. 16 Fourier spectrum of the s ing le constant kick of Fig. 15

3.4 Transverse RF knockout

In the previous sect ion we a n a l y s e the equation of betatron motion with a constant kick. If we

now modulate th is exc i ta t ion we can s t i l ' use the expansion of the k»ck ¡ 3 . 7 ) . Vte s'-o^y hav«3

to make the kick strength I of Eq. (3.9) a function of time. Just man* ne c ! s ) in Eq. !3.7) n>j 1 r i -

plied by some tmdulation factor, < ay e T ^ . You can keeo this factor separate, expanrf the rest as

before and then multiply with the modulation factor. The result i s that all Fourier coe f f i c i ents

f£ (Eg. 3.9} are modulated by the same factor. Th's leads to the phen-menon knowi as transverse

3F knockout. Let us rewrite Eq. (3.12) as:

i v.t. » i i s t x ( t ) * ¿ . t * ) l u e 1 e ¡3.13)

3 i = -

wr.ere

% being the amplitude of the kick, o - e i u l t .

Note that Vu e , a j t uses true time i . e . the time at the kicker. Equation (3 .13; may also be

written as

( U • w ? x = V u l e

i ( í í í t u ! ) t . (3.14)

Equation (3.14) is expressed with negative and pos i t ive frequencies, which correspond to the

slow and fast waves set up by the d's'urbinq kick. Those nut familiar with complex voltages and

currents as used by e l ec tr i ca l engineers may wonder about the s iqn'f icance of neqative frequencies-

In fact above we assumed a complex exci tat ion 5 = <JL, e-'-*'1 of the kicker as th<s qreatly s impl i f i e s

the algebra. In real l i f e we deal with cos ine- rather than e 1 , J l t - t V D e of kicker f i e l d s . U is easy

to go from the complex to the real world by taking the real parts of Eq. (3 .14) . Then the r . h . s .

contains terms which can be written in the form cosfm sit + <¿t), cos (m, -A - ..a) mû - if •-> 1 »; a lso

cos ( u t - n¿ -A) - with pos i t ive frequencies only (m: any inteqer > 0, m. any inteqer > u/£ and vv¿

any integer 0 < m < <o/u).

Thus the par t i c l e ' sees ' the frequencies mi i u, i . e . two sidebands spaced by the kicker

exci tat ion frequency ^ left and right of each revolution harmonic. "ï.;. This is i l lus tra ted in Fiq. 17

where the spectrum of the compfex exc i tat ion (r .r t . s . of Eq. (3.14\¡ and t(s r e j e c t i o n into the res'i

world are sketched. Taking the real part simply corresponds to ret'lectinq the neqative frequencies

into the pos i t ive f-plane.

The revolution sidebands at nu t W are very similar to the sidebands at ... * -o . of nsc mod

an amplitjde modulated o s c i l l a t o r . For th is simpler example the complex and the real analysis are

once again summarized in Table \H.

Ampli tude

Frequency

Spectrum of exc i ta t ion

Negat ive frequencies Positive frequencies

- 3 n . \ - 2 i l \ - A \ 0 / D. / 2 f t / 3 ( l Frequency

Spectrjn seen by tre beam using negative and pos i t ive frequencies (complex notation)

! Slow waves

I F a s t w a v e s

A - w i j

ft 2n 3f t Frequency

Reflection of negative frequencies onto pos i t ive frequency side gives the true spectrum seen by the

beam

Fig. 1? Simple harmonie exc i tat ion on a short kicker, spectrum of exci tat ion waveform and of waves seen by a part ic le

Table 18

The spectrum of an amplitude modulated o s c i l l a t o r (u f i f. > )

Complex notation 'Real world'

Modulated amplitude a = A e l w m l a = A cos u t m

Carrier c . 0 . 5 ( e ^ F t . e " ^ ' ) C = COS i^pt

Posit ion of spectral

l ines of modulated <ÜF * "m>

and ^ F * "m

and

signal a * c "Si- - I n »

Returning to Eq.i3.lflJ, c l ear ly i f the driving term has a harmonic at the natural frecuency of response, wp, then the beam wil l behave resonantly.

For resonance, (ftätu)2 =

i . e .

M + = sa i * ß = {i t | | Q | | ± a)a (3.15)

71

u± = (t» r q)Q (3.16)

where q i s the fractional part and j | Q j | the integer part of the tune. Hence n = i ± j | q | | i s also

an integer .

Equation (3.16) shows that the beam wil l respond resonantly at the "betatron sidebands" (n*q)ü

centred on the revolution frequency harmon-'cs. Take as an example LEAR at 600 MeV/c with f 0 = 1 MHz,

q = 0 . 3 . Resonant beam response ("RF - knockout") wi l i occur when the kicker is excited at:

0.3 Miz [= (0 + q ) f 0 J , or 0.7 rtH2 [= (1 - q ) f 0 ] or 1.3 MHz . . . .

Figure 13 sketches how two voltages of different frequency {w and UJH3 respect ive ly) on a short

kicker can produce the same ser ies of kicks as seen by a p a r t i c l e . Note the analogy to an RF

accelerating cav i ty which can in pr inc iple work at any revolution harmonic.

Having established the beam response to a kicker we shal l next analyse the reciprocal problem of

the signal response of the pick-up to the betatron o s c i l l a t i o n of a par t i c l e .

F ig . 18 Example of how a beam can be driven in the same way by di f ferent frequencies applied from a short kicker. The bars are the kicker voltages at the moment the par t i c l e passes , i . e . the kick experienced. The low frequency (u) and the hiqher frequency (w + *Q) exc i ta t ion produce the same apparent kick.

3.5 Signals from a c irculat ing par t i c l e

A c irculat ing p a r t i c l e passes once per turn through the pick-up and induce:* a short pulse . This

can be represented by a periodic de l ta function and we have t h i s , now rsther familiar, picture in

Fig. 19-

- .e 'it

I(t)=eS(t-tp.nT)

n -3T -2T -T O T 2T 3T

Fiq. 19 Representation of the pick-¡ J O siqnal from a S ' i q l e c ircu lat ing o a r t i c l e

Time

The induced signal is given by,

accepting that tvgh frequency Ctfnponents, i . e . f ^ 1 / i t , are not required. H P ^ P e is tne charqe per

turn ) I ( t ) dt passing the pick-up, i . e . the charge of the p a r t i c l e .

As before we Fourier analyse the signal in order to replace the discontinuous delta functions hy

continuous functions

inv'(t-t O . l S l

The Fourier harmonic amplitudes are constant. (In practice as the harnon-c freq,i^ncy approaches

1/At the amplitudes wi l l decrease. )

The pick-up we use for a betatron coolinq system wil l need to be s ens i t i ve tn the transverse

beam pos i t ion . This is achieved by placinq e lectrodes on either s ide of the bean. Each plate wil l

have a signal induced of the form of Eg. (3 . IS) and i t s amplitude wi l l he proportional lo the d i s ­

tance of the beam par t i c l e from the p ía te . From the . inference signal between (he two plates the

p a r t i c l e ' s transverse posit ion (x) is obtained. We wr ¡ t e the d i f f e r e n t s ' q m ' as I ( t í :

I s*Sp*x/h, where Ii,(t) is the sum signal Eq. -J 3 .13 ) , h is the half aperture of the pick-uo .vvA

5p a factor of the order of unity. This factor as well as the iss'imcd l inear i ty don^nd vpry -iich

on the construction of the pick-up. For a par t i c l e performinq hetalron o s c i l l a t i o n s we obtain an

induced signal modulated by the transverse betatron motinn

(3.11\

- 4 8 6 -

This signal i s :

I ( t í = I S ¡ 7 ) I s* ' _± p l h ' m ilTû(t-t )

8_JL iQQt im3(t-t )-iu 0

Since only the real part of the betatron motion ( 3 . 1 9 ) and hence of the current ( 3 . 2 0 ) in teres t s u s ,

we may write Eq. ( 3 . 2 0 ) as

i (m±Q)at-imQt * v u

I ( t ) » S e r : Î e ' v

s P 2 T l h m = 0 (3.21)

Looking at the exponent we find aga ;n that the p a r t i c l e induces s ignals at the sidebands,

u , p u = (m t | | q | | î q)fi = (n t q)o . ( 3 . 2 2 )

These are the sars as the beam response frequencies ( 3 . 1 6 ) . Thus the beam "respoi and "talks" at

the same frequencies.

Figure 20 shows the time and frequency domain picture of the pick-up s ial of a s ingle par­

t i c l e . This signal [Eq. ( 3 . 2 0 ) ] wi l l be used in calculat ing the coherent ef fect of the t e s t part ic le

upon i t s e l f . The incoherent e f fect due to the other part ic les can be obt- ; ied by adding up the ir

currents ( 3 . 2 0 ) with a proper d is tr ibut ion in ampl i tude and phase.

In a coasting beam of N part i c l e s with random i n i t i a l betatron phase and random time of arrival

these induced currents add in square to qive noise l ike s iqnals at the f- luencies ( 3 . 2 2 ) .

M M

III)

—it—; |— " r L < — a f0 • + » — — a f0 *\ ' ' I

1 I frequency, •

(m-Q)f 0 mf0 (m.Q)f 0

Fig. 20 Time and frequency domain signal of a partie j performing a betatron o s c i l l a t i o n . A posi ­t ion s e n s i t i v e pick-up records a short pulse t each traversal modulated in amplitude by the betatron o s c i l l a t i o n . The frequency spettn- contains l ines at the two sideband frequencies (mtQlfj of each revolution harmonic mfQ.

As different par t i c l e s have s l i g h t l y di f ferent revolution and betatron frequencies, these

s ignals occur in bands with a spectral power density

¿I2 t x sc 1 . e rms, dN

1 7 = 2 Í S P T — ) S ( 3 - ? 3 )

where dN/du is the fraction of par t i c l e s with sideband frequencies in a range of w.dth oV, ¿rounü .M =

(n i q)ü. These are the Schottky noise bands discussed in 0. Boussrrd's chapter in these proceed­

ings. With a dispersion of revolution and betatron frequencies (AS and Aq respect ive ly) the width of

the hand at (n t q]Q i s (n ± q)Aû i Ûûq. A spectrum analyser usual ly records current, i . e . the

square root of the signal ( 3 . 2 3 ) . A pract ical example i s aiven in Fiq. 21 , where the spectrum

analyser picture of the signal from a horizontal pick-uo is shown. The frequency band of th is

'Schottky scan' i s centered around a revolution Harmonic nfL. and contains the two sidebands

( " - Q KJ - * t e that the height of these sidebands [root of Eq. (3 .23) j is propprt ;onal io the

r .m.s . betatron amplitude x r r tis and thus decreases during cool ing .

All pick-uo currents discussed here are the 'induced currents'. To obtain the true output

s ignal one has to include the response functions of the pick-up structure and the jcnuis i t ion

system. Usually one aims at making these response functions as f la t as poss ib l e .

Fig. 21 Exanple of a horizontal Schottky scan in LEAR at 600 MeV/c, Th? central hand, the harmonic n = 100 of the revolution frequency, i s v i s i b l e as the beam is not completely centred at ihr posit ion pick-up. The right and le f t bands are Ihe sidebands (98*0)f u and ( 103-1) )f L when> Q - 2.3. During omittance cool ing the sidebands decrease. The difference between iht basi? l ine of the trace and the bottom l ine (rero s ignal ) is qiven by the noise of Ibe pick-.jp system. The span covers (approximately) half a revolution interval f u . Uurinq horizontal cool ing the height of the sidebands decreases .

- 4b3 -

3.6 Coherent e f fect

W» Câfl now hdvÇ ä ff&sn loo'' at the motion Of part ic le l. retaining for the trnnVTlL Only iti, "se If-terms". We takp a kicker voltaqe on the r . h . s . of Eq. (3.13) which is proportional to th^

pick-up signai (Eq. (3 .20) ; of the c irculat ing par t i c l e . As before we reft>r to ¿11 i iqnals at thou

time of arrival at the kicker.

The part ic le takes a nominal time tp and the cooling siqnal a time t r to travel from rjic<-up

tu kicket . The electronic delay t c is in general frequency dependent. WP <nclun> th's by a p"dsc?

factor '-tní ^ trul^t.-^n) in the development (3.20) of the picic-iip s t a i a 1 . The fr CQUCK conta'ied if- I i i s signal are the (n - q)i= betatron sidebands {3-??i- In our c.'nrj'f-' n.-aatio-

(*nc 1 'jdinq pos i t ive and neqative frequencies) they simply appear A S

•J s fa + Q)w ; m = -« to » . f l . ? 4 ;

The p'î * fi t to retain is thai al l siqnal transmission occurs at tV- ' > ^>ir-:.t_:- , ' 3 , / i ; .

In addition to the i n i t i a l phase ,i0 [Eq. (3 .19)J , we hav*1 tn inr.lu-1-

-„ -p (if the betatron ose 111 at ion of the part ic 'e en ! ts way from pick-' , \ • 1:,

exponential factor 'n Eq. n . 20) ie f erreri to the kicker is written as ,

p ' . ^ t - i ^ + f{'^rU{\ - tp)J .

To complete the driving term in the betatron equation Í3.13) we 'ntrnducp a transfer function

G ( ^ T , ; . It has to include the pick-up response Zpfa) [ i . e . the voltage output for the induced

current (3 .21)J , the transfer function of the coolmq loop between pick-up and kicker (with cables ,

ampli f iers , f i l t e r s e t c . taken into account) as well as the kicker response. Let us for s impl ic i ty

a lso absorb the constant factors

<fr / — i 2u " V P U D U

of Eq. (2 .13) and the factor

<J I S p e UT h

of Eq. (3.20) into th is transfer fu:-.I ion hut keep <\ = xi pi (i"pt+n,_, ) separate. Hence we

rewrite Eq. (3.14) as

, i , - \ -ip+imut + i*(oi J-imutr, \ ixut ,-, o c v x + w x = x I ,u )*e ^ r i m' p* L e v3.25) 1 P 1 1 ne.* 1 £ =

Me lake GJ , (HH ) as ent ire ly real and include al l phase s h ' f l s in « ( ^ ) . Note that the second

sum in Eq. (3.25) is the 'samplinq term' - appearinq already in Eq. (3.131 - due to the fact that the

part ic le passes the short kicker once per turn. The f i r s t sum c lear ly is due t i thp localised nature

of the pick-up.

Equation (3.25) ib almost the sane as ïq. [3.2) exc°pt that we •r.c'ud* 1 f requency dependent

'gain' G u f w J - e 1 * ^ ) and localised p ' c k - j p and k'cker. The p iO iMcl of thp two su^s frin rpad ' l y

be conver ted into a double sum

îroa îiL't i {m*¡M , , e • ¿ o : ; ¿ c , (3.26}

netinq that in qeneral

Equation (3.25) may now be interpreted as an o s c i l l a t o r with a frequpney shift that vat 'es >n

'IIT°. An aoprox lmatc solution to such equations is obtained by lak'hu. the time avet aq* of t.,e f. e-

quency sh i f t only, i . e . if we retain terms with £ = -m in Eq. (3.26) and drop Llio rapidly o s c i l l a t i n g

frequency s h i f t s . Using th is approximation (3.25) becomes:

i; i i L '. i v m'

This defines a change of betatron frequency:

1 .. - 1

As v/e a s s i e d x = x e'^j'- the dampinq rate

l / i = Im (A* ) = Im 1 G (u ) • e - i f (^l-M'-miAp . ^ '''[•i m ^ ^ ^

Optimum cooling i s given by

1 / t = ^ í |GiiK)| { 3- ? n A ]

Jy m—-

and obta ned for all m, if the phase factor iL properly chosen si;c!i tha t -!m(p

requires

(3.29)

Usually Eq. (3.29) is s a t i s f ' by puttina the kicker at the 'procer' betatron phase advancp from tho

pick-up, i . e .

- 49f i -

r. 3 5 1 1 = 2 ' ^ 2 * W U h s i q n a inversion, or - -

and designing f f ^ ) to be as clos<=> as possible to mßt p:

Ideally th i s s q u i r e s a signal delay of the coolinq loop, t ^ t ^ ) , equal to l D , the part ic le

trave l l ing time pick-up to kicker, independent of frequency. If the optimum spacing pick-up to

ktcfcer = K/2 mod. u) is not poss ible one can in principle include f i l t e r s fas f i r s t proposed by

Thorndahl) with a time delay character i s t i c t c ( w ) such that Eq. (3.29) i s s l i U s a t i s f i e d . This

requires however "steep" f i l t e r s with a phase delay v ( ' - J ) varying by ?(M.-H /2) from the n+q to the n-q

betatron band. In Tact writinq Eq. (3.29) in the form

*( W J - n ß t c = (a - ~) = <V

and noting that for any network ^>(-u) = -ip{u) we need for pos i t ive m (correspondmq to the n + q

sidebands):

- mat = b»

and for negative m (corresponding to the n-q bands)

• w - M * p = -<»>

thus requirinq a phase difference 2óu oetween neiqhbourinq bands.

Other e f f ec t s can be identif ied from Eq. (3 .28) :

- The t ime-of - f l ight error Atp of a par t i c l e ('mixinq between pick-up and kicker') as well as

improper delay At c of the cooling loop or improper pick-up to kicker betatron phase advance <v =

u - n/2 appear as a phase factor in Eq. (3 .28) which may be rewritten as

where btm = # t ^ ) - nctp = m i U c - t p ) .

- The f a l l - o f f of the pick-up-current spectrum at hiqh frequency can be included in the expansion

(3.21) and absorbed into the transfer function. The similar ef fect of the f i n i t e kicker length

can be included in much the same manner v ; a the expansion on the r . h . s . of Eq. (3 .13) . Final ly we

remark that Eq. (3.28) can be written in various other forms involving sums over pos i t ive m only

which c l ear ly reveal the (mq) bands. This is left as an exercise to those interested.

Having establ ished the interaction of the par f ' c l e with i t s e î f we iext include the noise due to

the other par t i c l e s and the e lectronic system.

3.7 Noise

Noise wil l be treated in detai l in G. Dome's chapter in these proceedings. For convenience we

repeat the e s s e n t i a l s here, which are useful to include the incoherent e f fect in the frequency-domain

analysis of s tochast ic cool ing.

Look at an osc i l loscope picture l ike Fig. 22 which displays a pick-up signal u( t ) when 'no beam

is m the machine', i . e . the e lec tronic noise of the system. It is customary to represent the mean

square {averaged over a long enough time T) of such noisy voltages by a pseudo Fourier transformation

The 'spectral power density function' 4>(u) i s c l o s e l y related to a Fourier development of u ( t ) . In

al l practical applications the noisy voltage has been 'switched on' at some time t = 0 and we reqard

i t up to t = T. Outside th i s range the waveform i s irre levant , so , for the purpose of computation we

can per iodica l ly continue i t (Fig. 23) . We then deal with a periodic function u(t ± nT) = u{t) which

we can Fourier-expand in the usual way

u 2 ( t ) = J O(LI) du -

Fig. 22 Noise signal on a pick-up

u(t ) = )' u e

i m J l [ , t • diu = 2-K/J . (3.31)

U ( t )

- T ¡0 ¡1 2 T

Periodic i R a n g e of continuationi interest for

uCO Periodic • continuation1

Fig. 23 A noisy voltage u(t) observed from time t = 0 to t = T and i t s periodic continuation outside th i s range to permit a Fourier development

The Fourier anpl i t jdes:

- I T -imw,.t u m = j j u ( t ) e dt H .3? )

u

are in general complex hut for real u( t ) u ^ i s the conjugate of u ,. The «çan sq-jare of

Eq. (3.31) over the observation time T i s by def in i t ion

u ¿ ( t ) \ I u 2 U ) dt (3.33)

which y ie lds after some calculat ion (transforming the square of the Sum into a double sum similar to

(3.26) and noting that averaged over a period all ~^u>j terms the analysis in conjunction with Eq

vanish except for k = 0)

> K\'¿ (3.34)

This i s known as Perseval 's equation :n the theory of Fourier s e r i e s ; i t applies any Fourier

development! Equation (3.34) presents the 'average noise power' u 2 ( t ) as the sum of i t s sopctral

contributions at frequencies - n2u/T. Analysed over shorter or lonqer time T the spectra are as

sketched in Fig. 21.

um°< T

As T increases üí, decreases

t^=2TT/T T increased fourfold -r-

Fig. 24 The power spectrun ju^t^lj of a noisy voHaqe which is observed for a time -'ntfrval T ind oer iod ica l ly continued outside this interval , lncreasinq T the heiqht of the spectral l ines decrease proportionally to their spacing <JL = 2r./J so that the quantity ]u"'(-<>}| remains the same. In the limit T • ~ one has a continuous spectrum where u''(a>}/:. > l{u_>) is the spectral power density function of Eq. (3 .30) .

As the SIJTI of ttje rays

(for larqe T), i . e . u^-f/uj :

is u ' ( t )

canst.

n both cases their height sca les proportional to the<r sparinq

For very large observation ti'ne T * ~ the spectrum is pract ica l ly continuous and the sum

Eg. (3.34) -s approximated by an integral of the Form of Eq. (3 .30) :

Hence we ident i fy for T *

This interpretation permits us to ca lculate (at least in simple casesi and lo e s t sb l - sh

following important theorem:

When noise with a spectral power density ^(u) is transmitted through a linear system with

(complex) transfer function H(u) then the power spectrum at t lw output is

5 2 M = | H M | 2 * I > Î

This follows immediately from the preced'nq notina that each of the cotinipnt', - r the >-.h.s

(3.31) when transmitted through the network transforms according to Eq. (3.361:

An example of the theorem (3.37) is the transformation of 'broadband no 1 se ' into han.1 !-n«t

noise by a band pass f i l t e r (Fig. 25) .

0,(1*/)

Ideal

b a n d

p a s s

filter

ZJS H (XLT) xxr

Fig. 25 An example of the theoren [Eq. (3 .37) ] : broadband noise transmitted through ,m idea! band pass f i l t e r is unvested 'iilo a h.iid-1 in - tn i n<\>^

This is as much as we need aho.it no'se fo- the purpnsp nf this chapter. Mi-w

facts wil l be discussed in Georges Dome's presentation.

Beam response to a noisy _k_icker

Consider" a linear o s c i l l a t o r driven hy a no'sy exc i ta t ion u ' t ) «ith piwe*' d f ^ ' t y

This problen1 was treated ( m a Türe general context) a quarter of a century ago -n a c l a s i c a ' oaper

by Hereward and Johnsen1"').

Their r e s u l t , the "Hereward-Johnsen theorem" may - m our present case - be s i a l y l as f ^ ' i w s :

For A particle injected at t = 0, the square of the amplitude x of x, Eq. (3.38) expected (Fig. 26) at time t (t large) is

, 2K

In words: l i e amplitude gro^s in a diffusion ' i<e manner [x <• / t j at a "-ate which is ^r-terrne:

the spectral density of the noise at the resonance frequency

Time t Fig, 2b Amplitude "ï of betatron o s c i l l a t i o n of a p a r t i c l e driven by a noisy kicker. The

expectation value ( i . e . the averaqe) of x' grows l inearly in time at a rate qiven by the spectral density of the noise at the resonance frequency. In ñddifnn to th is averaqe growth there i s a f luctuating motion which is of l i t t l e importance for the long term behaviour.

Equation (3.39) is for a simple harmonic o s c i l l a t o r . If we inject «o 'se into the c m ! i n ¡ ) loop (or

d i r e c t l y onto trie kicker) we have aqain to include the "sampling factor" e l W t of Eq. (3.13)

because we use a short localized kicker. Thus we use

x • ujjit = u( l ) > e 1 * " ^ " ^ ' - (3.40)

The pf fee l of ?ach i. *np:ineit, c' ( is to 1 shif t ' thp freqi.pncy conipnt of the dr w'nq f o-ce

•J * u + L<. In th is sense we may interpret the r . h . s . of Eq, (3.40) as a sum nf noisv di w n n forces

with frequenc u",

He can apply the Hereward-Johnsen result to each of these bands ¡"oíinq mat in work^nq out »" ( t ,

cross-terms between bands average to zero) , hence the response of La. f'î.û'V to a no-se is:

x 2 = 4- t : <t> U , + U) . r 3.411 t = —

Equivalenlly if you prefer to work witn posU'v? frequencies rr'y y nay i ! . e vising ,»--S s

; ( - ) , , Eq. (3.41) as

Í3.4?}

This c l ear ly presents the amplitude growth in ter*rs of th>> spectral ¿ens-ty •-' '.r,-. '•-•se at trie r-eta-Iron sidebands.

In working out the long term average of x*( l ) leacing to Eq. (3.^1) snl Fq. (3.4?) we hav-*

assumed that cross terms between different bands a averaqe to zro a*i.i also that :f-..,•) = :(-•). A j u s t i f i c a t i o n [which can be carried through, e .g . usinq an e^pans^n c>f t >v type (3.311 for u''_) anl

the def in i t ion (3.36) of Q[U)\ is left as an exercise to those interested.

3.9 Back to beam Schottky noise and amplifier noise

Upturning to a t e s t - p a r t i c l e : apart from i t s self-term it will r>xn-v > ( T H e the 'beam Schottky

noise ' due to the presence of the other part ic les and th*1 e lectronic n j i s e of the preamplifier Ptc.

The spectral density $( ÍU) = d l ^ / d u j mside a Schottky hand is Jeterrmied fty ta . (3 .23) an<1 mjy be rewritten here {see D. Boussard's chapter on SchotUy no i se ) :

» s c ( " ) - í(j (3.43)

U | = (t 1 q)-J , ( = 1,2,3 . . .

N e'« >.' 5'

I "ITS p

cL d-N—„ - J L if al l bands Are separated, i . e . rV, 2qi.' 2N d u t Z,V,(

1 for complete band overlap, i . e . ,V( ' :J.

Hete / I l i r is the -nean-square-betalrnn arnpliturie of part ic les with a s'doband frequency near

ilti/faip is the number of part ic les with frequency in a -amje iif witltii ck' near (li¡,.

2« : 77 1 > (! • q)ul

The dimensionless quantity M(.j) is c l o se ly related to mi*inq as wil l become clear later-. It i s

customary to appro*imate ON/cLi by a recUnqular d i s l ' ibut or r;/,'... of total wdlh

where tju(i) is the width of the sidebands at the <-th harmonic. This leads to fïp approximat'ons foi M(ux) md'caled under Eq. (3.43) and sketched *n Fiq. 27.

Separated bands

A-u/(l)<2q

i -q

Partial overlap

il>iu/(l)>2q

A u d i

Full overtao

0 ( w ) = 0o

M, =1

Fiq. 27 Transverse hearr. Schottky noise for separated bands (low frequency), partial overlap (intermediate frequency) and complete band overlap

For hiqh frequencies, bands completely overlap and the noise has a continuous power sppctruni

with density p^o) = iv as given by Eq. (3.43) with M = 1. Note that in th is limit Eq. (3.43) is just

the c l a s s i c a l Schottky formula for the nmse of a "DC-current" 10- Defininq the noise dpns'ty j- = dl^/duj with respect to angular frequency u> as we cons i s t ent ly do in the present chapter, this formula

writes as:

* M -- P. \ J n . (3.4e,!

In our case the c irculat ing current is I L

: Nrti/2* and CI.5 ( S 0 x r m s / h ) ' enters as we take the

betatron o s c i l l a t i o n signal from a difference pick-up.

The noise is transmitted through the cooling loop in the same way as the 'sel f -s igna! " of the

t e s t - p a r t i c l e . Hence the transfer function is the same except for phase factors due to the different

arrival tunes nf part ic les and due to different betatron phase. We denote th is transfer function,

which has l i e same nudulus as , (u) and G,J(UJ) . by G(^i).

By virtue of Eq. f3.37) tne noise density on the kicker i s therefore

• p M = \G{u)\¿ i ' ( u )H( w ) (3 .46)

u ¿ rms

Here a.'L is the same as $ u , Eq. (3.43) except for the factor (S p &i/2îih) 2 which was absorb into

G ¿{LJ) as before, see the discussion preceding Eq ( 3 . 2 5 ) .

We now turn to the e lectronic noise and assume that — referred to at the ent> e of the coo l ­

inq loop (exit of the pick-up) i t has a power spectrum ç a ( u ) . Let the t- ^fer function from

this point to the kicker be H(w). Clearly th i s H(w) i s the same as GU' cepl for the pick-up

response function. The noise seen by the t e s t par t i c l e ( j ) (Fiq. 28) is t 1 i

= I G U j l l V M I . j ) . | H ( U J ) | 2 . > 0 ( 3 i 4 7 )

Schottky nc ise amplifier noise seen by seen by p a r t i c l e p a r t i c l e

Here ^ = ^ /(Speü/2nh)' f is the e lec tronic noise reduced by the same factor as for consistency.

To recover previous re su l t s i t i s useful to rewrite Eq. (3.47) as

*(u) = |G(W)|V[MM + U(o.)j (3.47a)

Clear)? Ufa) is the rat io of amplifie- to beam noise (the latter in the hiqh frequency limit where

bands overlap) . Usinq Eq. (3.41} we can write down the expectation value for the amplitude of the

t e s t par t i c l e as driven by the noise Eq. (3.47)

x 2j(t) 4 t o ; I | G ( U j ( ) | 2 Í M ( U j [ ) 4 UUpl (3.49)

with as given under Eq. (3 .46 ) ; M(u} as given under Eq. (3.43) and U(HJJ) as given under Eq.

(3.47a) and u( ~ ( i + qj)uj i s now the sideband frequency of par t i c l e j .

Equation (3.48) determines the 'incoherent e f f e c t ' as experienced by the test p a r t i c l e .

S c h o t t k y b a n d s C o n t i n u o u s

Low f r e q u e n c y High frequency bad mixing M ^ l good mixing M= 1

Fig. 28 The noise seen by a test part ic le is the Schottky noise due to the other part ic les and the electronic noise of the amplifier etc- At low frequency tbe Schottky noise occurs in bands with a density M times higher than in the s i tuat ion of complete over U p . Thii mo-ease of noise density corresponds to enhanced 'heating of the t e s t part ic le due to bad mixing'.

3.10 Cooling rate

We can now calculate the expected amplitude x 2 j { l ) of the t e s t part ic le fj) by addinq up the

coherent and the incoherent e f f e c t s . To use Eq- (3.28) for dU'Vdt we note that in qengraI

1 dV]_ 2 d x 2 "x7 dt = x dt =

~ T" •

Hence we have from Eq. (3.28) for the coherent (damping) e f f ec t :

x * - I Re [G(uO e i 6 * , u i J t ! " ^ j (3.49) J w ß A»™ * _____

I 2

For the incoherent (heating) ef fect we rewrite Eq, (3.48) suhstitutinq from Eq. (3.46)-

a « = -

3 4

The resu lUnt cooling equation is:

0(x

( 3 . 5 1 1

- 499 -

Equations (3.49) to (3 .51) represent cool ing as a sum of the contributions at the sideband frequency ul = (* + QjJûj °f the p a r t i c l e . In th i s form a l l frequency character i s t ics of the coolinq

loop can readi ly be included. This i s e spec ia l l y handy for those who l ike to measure and ca lcu la te

in the frequency domain. In addition we can rediscover and re- interpret the e f f e c t s discussed before.

1 The influence of imperfect synchronisation of p a r t i c l e and cool ing signal ('mixing pick-up to

k i c k e r ' ) , bff * 0 . It enters as a phase error in the coherent term.

2 The influence of betatron phase errors (imperfect spacing) pick-up to kicker, 6^ * 0. It enters

as another phase error in the coherent term.

3 Imperfect mixing on the way kicker to pick- - expressed here as enhancement (M > 1) of the

heating by Schottky noise which is concentrated in bands and hence increased in density . Good

mixing [M = 1) corresponds to overlap of Schottky bands.

4 Amplifier (and other e l ec tron ic ) no i se , U > 0.

Equation (3 .49) so far i s for any t e s t p a r t i c l e . To obtain the damping rate for the mean square

amplitude x ^ ^ we have to averane Eq. (3.49) and Eq. (3 .50) over the frequency d is tr ibut ion of the

beam p a r t i c l e s . In the simple e i s e of perfect 6<p = 0 , on = 0, M = 1 and constant 6(u) = G, U(w) = U

ins ide the passband you can rediscover the familiar

- = -12g - gMl + U)j (3-52)

by c a l l i n g

G*N (3.531

To work out the sums over i , note that with a passband of width Af = W in the pos i t ive frequency

plane the number of betatron l ines contributing is 2-2TI W/Q as sketched in Fiq. 29, namely 2n W/ü for

p o s i t i v e i (the n+q bands) and the same number again for negative i (the n-q bands).

w / f D - b e t a t r o n bands w i t h negat ive I

GCfJ

w/ f 0 - b e t a t r o n b a n d s wi th posit ive 1

p a s s b a n d

^ W i

p a s s b a n d

w

p a s s b a n d

^ W i

Negat ive frequency Positive frequency, f

29 Passband of cooling system in the f = -~ to f = - frequency plane. There are 2n U/a = w/f g

l ines u)j¡_ = ( ï + g)u in the passband at negative freguencies (.legative I ) and the same number in the passband at pos i t ive frequency (pos i t ive A).

You Can generalise Eq. (3 ;2) to include mixing factors , betatron phase errors and frequency

dependence in G and U by int- preting the sums in Eq. (3.49) as averaqes over the passband. With

th i s interpretation you may * i t e Eq. Í3-521 in various different forms useful far comparison with

previous r e s u l t s , for instance:

<Síx¿

dt • 3e[g(u,) e ( i * -<v ) , .

passband m s |q ' (uO|[MH • Ufa) ] ; (3.52a)

g M

a/2u -< " "Vassband 2w

3.1L Feedback via the beam and signal shielding

We shall now attempt to introduce a final ingredient of cool ing theory known as 1 feedback via

the Deam' or 'signal sh ie ld ing ' . Althouqh th is refinement wil l channe our pmvo. i s resul ts ny at

most a factor- of 2, the change of the beam Schottky s ignals when the rnolinq loop 's closed hj«,

become an important diagnost ics tno l 'M.

Where did we miss out th is e f fect in our treatment so far?

part ic le equation

In fact consider i rio Ihn t e s t ­

e t e t oF p a r t i c l e ijpor. i t s e l f : c o ­

herent tenu

O.K.

effect of other jar 11-" l e s : Schottky r.oise, f luctuating lei"Ti with zero aver aq-1

Not O.K.

(3.54) amp!iflPr nol i" P I C . f '-irtuat inq with ¿ero av»i oqp

We h-ive oesenbeo* the effect of the other par t i c l e s - , G, y ;

turbed beam, i . e . as a fluctuating tenu with zero time avpiaqp.

correct . F. Sachere: 1 1*) has pointed out that - in i hp case o?

• Vi Sr.hnttky I Q I S Í ? nf an vmdi*a-

This avsumpt ion is not general 1 y

'jtinr •. • mc - , Gi Í X I dl^S lead

to a coherent o s c i l l a t i o n with f i n i t e a v é r a i amol'tude. Th» fluctuation x o n s . v u N i d this av.-r au->

amplitude and not around zero as it would he the case in an undisturtiM h"am. Thp is that

pari of the "modulation" imposed al the kicker is s t i l l present at thp pick-jp ami I P - . ntt'is \h" loop

as sketched in Fiq. 30. Thus the noise on a b e n s-jhiect to rnnhnn 's different f i :*n i he f r p p beam

noise . The feedback of the cooling s ignals via the beam chanqps alt mqri?dnanls of the analys i s ,

namely heam noise as well as the influence of the coherent term and the amplif'er nms"-

- 5 0 1 -

PU KICKED

f i g . 30 Cooling system includinq the coherent beam modulation xh imposed at the kicker and p a r t i a l l y preserved up to the pick-up due to imperfect mixing. The lower diagram shows Sacherer's equivalent feedback loop. Anplif ier noise ( x n f and Schottky noise ( x 5 ) are random noises whereas the coherent modulation i s fed back via the beam from kicker to the pick-up. This feedback changes the open loop response to x n + x s by a complex transfer function T(u) which depends on the amplification (cooling strength) and the degree of mixing between kicker A I I J pick-up.

Fortunately F. Sacherer has a lso shown the road to rescue our previous r e s u l t s . The way out he

ides i s a beautiful piece of accelerator theory.

*s a pre-exercise: consider a system of N o s c i l l a t o r s with a harmonic drivinq force and a

i t i v e force* oroporticnal to the average displacement of the o s c i l l a t o r s . Take for the q-th

ic le - sorry o s c i l l a t o r

harmonic c o l l e c t i v e force driving force

Here the term (Gy x^) may be interpreted as the weighted contnbot ion of part ic le k to the

average

A 'mechanical' and an ' e l e c t r i c a l ' analog of Eq. (3.55) are sketched in Figs . 31 and 32.

Fig. 31 A 'mechanical analog' of Eq. (3 .55 ) . Person V t r i e s to exc i t e a system of osc i i ators (masses on springs) by shaking their point of suspension. Person G t r i e s to dare the motion by observing the average displacement <x> of the o s c i l l a t o r s and appl> iq a damping force G<x>.

A i

- e - I i i Array of resonators

J

Fig. 32 An ' , ' ec tr ica l analog of Eq. ( 3 . 5 5 ) . A qroup of LC-resonators i s driven by a voltage V e i l 0 t . The sum I = i Ik of the currents through the resonators is fed back through an amplifier with gain G to add an input voltage G ¿I¡<.

503

To solve Eq. (3.55) we inserí a tr ia l solution

J - j T4T7 ¡ V M < ¿ X >] (3.56)

multiply with Gj and average bath s i d e s . Call:

and so lve for the average

< - T T 7 7 > = S M (3.57)

S(U) -<G.x t^ = R • V(U) . (3.58! * k 1 - S

Thus we do have a f i n i t e coherent amplitude <x>. We can now use Eq. (3 .58) to el iminate the

' c o l l e c t i v e force' term from Eq. (3 .55) . We find

x , * a2* = V(OO) e i u , t [ 1 + — ~ — ^ \ . (3.55a) J J J 1 - S M

shie lding factor

Instead of treat ing the orig inal Eq. (3.55) with a

r . h . s . = [driving force] + [weiqhted average displacementJ

we can therefore treat the same equation with the more convenient-.

r . h . s . = [driving forceJ*[shielding factor , .

This is the essence of 'Sacherer's t r i c k ' . In the cool'nq equation we shall want to replace for each

of the betatron bands involved:

r . h . s . = [coherent termj + N»[weiqhted average displacement] + [Schottky noise] + [amplifier noisej

by

r . h . s . = [coherent term + SchotU.y noise + amplifier no i se j - , sh i e ld ing factor] .

A quantity of key importance i s the 'dispersion function' S(u) entering into the shielding factor

h«;-*h + - ~ i-I-1;] 1 - S 1 - s

(3.59)

For large N rte have

G ( U j ) n ( U j ) j G ( a j ) n ( U j ) (3 .60)

Here n(wj)duj is the fraction of par t i c l e s with eigenfrequencies in â band of width duj near

Dispersion integrals of the type 13.60) are treated in H.G. Hereward's chapter on Landaj damp-

i n g l i l ) . For convenience some features are repeated in Appendix 2. Due to Lhe pole, the integral

has an imaginary part even if G(wj) is r e a l . Details depend on the distr ibut ion n(wj) of e igen­

frequencies and on G(w). A typical behaviour of S(w) is sketched in Fiq. 33.

Part ic le eigenfrequency

S(W)/5

-Re(is/G)

Driving frequency

- Im( iS /G)

Fig. 33 Frequency dis tr ibut ion and typical behaviour of the dispersion function Eq. (3.60) for a given function G(w) which i s constant (or slowly varying) near the beam response frequency wp. This behaviour of G(w) i s required for betatron cool ing.

- sos -

A useful approximation is

< 0 t'i

i - for jto-u^j < W 2

or a l l other values (3.61)

m. = <w.> = r: f u i . averaqe eigenfrequency p J n i J

To go one s tep further we now analyse a problem which i s of some practical importance namely

beam exc i ta t ion by a s ing l e harmonic driving force on a kicker when the cooling loop i s c losed. We

write the equation of motion of par t i c l e j as

x. +«2.K. - [l {GM I e ^ - V - n x, eK^ + V e W l J . \ e - ^ j Î ^ r V - (3 .6?) J J J k m

Here the f i r s t sum (k) i s over the N beam p a r t i c l e s , the sum over m i s the 'samplinq term' due to the

local ized pick-up and the sum over i represents the harmciiics of the local ized kick, t^ i s the

arrival time of par t i c l e k at the pick-up, x(_ + xk e _ 1 ^k presents the transformation of i t s

o s c i l l a t i o n from pick-up to kicker, « m {u) = mût^u) i s the signal delay of the cooling loop, t p

i s the t rave l l ing time of par t i c l e j from pick-up to kicker, hence i t s arrival time at the kicker i s

t j + t p , V e ^ t is the external driving force , the term proportional to G(w) i s the correspond­

ing 'driving force' given by the response of the cooling loop to the beam o s c i l l a t i o n .

Once again we drop a l l rapidly varying 'frequency s h i f t s ' , i . e . we only take harmonics with

m = I in the f i r s t term on the r . h . s . of Eq. ( 3 . 6 2 ) .

In the second term we only retain frequencies w i ifl - op - WJ c lo se to resonance. We assume

that a l l bands are well separated so that only one i leads to resonance. Thus we s impl i fy Eq. {3.62)

... . A . - U «.SM e ^ V " 1 " ! « e 1*'"' . V e1"1] • e ^ j ^ Y V • (3.62a)

J J J I. *•

As response to the driving term V e ' w t we expect a, so lut ion of the farm

i ( u - i û . ) t + ' t f l . ( t . H ) x j = x j J J J P

for any p a r t i c l e j .

- 506 -

where we define

\ M = B( U ) e ^ k ^ ' k

The quantity = - îflfctp i s the synchronisation error between part-c ie k and the coolinq

s igna l . Let us denote the resonant driving frequency - as introduced aVeîdy in Eq. (3 .24) - by

J J J J *

and use w 2 j - (w-if l) 2 - 2DJ [ U J - (u-Jlû)J.

From the preceding analysis we can now define a shielding factor for the present s i tuat ion

. , S[",W»,I

Up = <u)j> : beam averaqe of betatron frequency.

Using th i s shielding factor we can rewrite Eq. (3.62a) as

Xj = T ^ H - e ^ * 1 " 1 " * (3.62b)

where we use o¿ = - î û j ( t j H p ) to denote the phase factor due t o the arrival time of p a r t i c l e

j at the kicker. Thus when the ceo ling loop is c losed, the response of Eq. (3-62) to V e^l

changes by T(u). In th is way T(w) can be observed and G(w) can be deduced from i t . Usually these

measurements are done using a network analyser t o display the beam response to a swept s ine wave

(beam transfer fundion measurement) as sketched in Fig. 34. This permits us to adjust the

character i s t ics of the cooling loop band by band.

Tc complete o-r analysis we return to Eq. (3 .62) hut now assume a general drivinq force repre­

sented by a Fourier series (or a Fourier integral) with a spectral density function V(w). We invoke

superposition ara resonant behaviour of the betatron equation at the frequencies Thus we

rewrite Eq. (3.62b) as

\j - ? - d m / * * V 1 ( 3 - 6 3 )

I: Upon subst i tut ion [using the corresponding expression for xk on the r . h . s . of Eq. (3 .62a)] we

KZ

Cooling loop

--cm—" - B e a m

Network analyzer

Output Input

j Pickup

/Ampl i f ier

Fig. 34 Arrangement to measure beam transfer function. The frequency sweep of the network analyser i s set to cover one or several betatron sidebands. The difference in beam resporse with cool ing loop open and closed can be used to optimise the loop gain.

ï TK)VU> ) e (3.62b)

which presents the e f fect as the sun of the interaction at the sidebands = (Jt+Q)Q. As a conse­

quence of the beam feedback each band now has i t s proper shie lding factor T ( W J l ) , Eq. (3.64) (well

separated bands, i . e . poor mixing assumed). The e f f e c t of the shie ld ing factor i s fu l l y equivalent

to introducing a transfer function T(u) between the driver and the kicker.

We can now generalize the cooling rate in Eqs. (3.49) and Eq. (3 .50) to include sh ie ld ing . We

can interpret V(t)» Eq. (3.65) as the cooling s ignals discussed before (namely the s e l f - e f f e c t of the

t e s t p a r t i c l e , the Schottky noise due to the other par t i c l e s and the amplifier n o i s e ) . Since the

beam feedback acts l ike a transfer function we simply include th i s into Eqs. (3 .49) and Eq. (3.50) by

subst i tut ing

(3.65)

A typical behaviour of the shie ld ing function i s sketched in Fig. 35. (Vote that for small 'gain'

(N.G small) and small S(u) the shie lding factor i s c l o s e to 1.

T(TJ)

Im(T[bJ])

Fig. 35 Typical behaviour of the shie lding factor T(w) near a resonance frequency of the beam

- 508 -

To gain further insight we only look at part ic les near the centre o ' the d is tr iL ion (W£ -

<u>j + iQj> = ap + iß ) and assume perfect betatron phase and perfect signal delay pick-up to

kicker (ô> = O . î y ^ J = 0 ) . Then the gain function G(«A) = |G(o>A)| e " 1 ^ 2 - V + a f *

becomes purely imaginary and S(IN¿) and T ^ ) r¿al in the centre of the band. Let us introduce

the 'reduced gain'

|6<VI'

p

in analogy to Eq. (3.53} and recal l the def in i t ion of the mixing factor M¿ - i/2ùua for well

separated bends ( see under Eq. ( 3 . 4 3 ) ) ,

UVng the s impl i f icat ion Eq. (3.61) for the dispersion integral and Eq. (3.64) for Tf.jj)

we have

sw - i

- g , H / 2 (3.66)

The cooling rate equation for any part ic le i s obtained from the expressions of sect ion 3.10 by replacing G(u) + T(w)G(u), Eq. ( 3 .65 ) . We obtain in the present case:

1 B/2% " 2 g* 9 i

• ' T 1 i ' l l > a A / 2 ) - [ l M A f l ) ' ( 3 ' 6 7 )

passband

This i s formally the same as Eq. (3 .52) i f we subst i tute

h * V i = 1 . hHt/2

Optimum cool ing i s obtained from Eq. (3.67) when:

i . e . when for al1 bands

1 2 9Jt = M 9/2 + U, h + M ( 3 - 6 8 1

- 5 0 9 -

The l imiting case (+} is for neg l ig ib l e amplifier no i se , U¡. « Mp/2. The optimum shielding factor

corresponding to Eq. (3.68) i s :

1 J .

V 1 <- Y I M , • 2U,) * 2 i 3 M }

and the optimum damping rate

1 3/2« ,. 1 7 = T Ñ ~ ( 3 ' 7 0 1

Thus in the s i tuat ion of neg l i g ib l e amplifier no i se , optimum cool ing is obtained when the qain

(at all bands involved) leads to signal reduction by a factor of about ?.. 8y comparinq open and

closed loop s ignals (e i ther Schottky noise or driven-beam response) the gain can thus be optimized

band by band. An example o* Schottky signal shie lding of a band i s given in F^g. 36. Note that the

optimum gain Eq. (3.68) for U = 0 is twice the optimum Eq. (2.28) calculated without bean feedhack.

When the amplifier noise becomes important (U » M) then Eq. (3.68) and Ea. (3.69) y i e ld the optimum

g + 1/U, T * 1 as in the case without sh ie ld ing .

[. Cool ing loop open

Closed

Fig. 36 Reduction of a Schottky noise band when the cooling loop is c losed, w>tn n e q h q i b l e ampli-f i er noise and well separated hands o p t i o n gain of the cool ing loop correspunds to a signal amplitude reduction by about 2 in the centre of the hands.

Thus the inclusion of beam shie lding (which was done in an approximate manner here) leads to an

improved expression for the cooling rate and - more importantly - to an adjustment cr i ter ion for the

cool ing system.

The analysis done here for betatron cool ing can be repeated for rronpntiim spread damp-nq where

similar gain adjustment c r i t e r i a apply.

4. DISTRIBUTION FUNCTION EQUATIONS (FOKKER-PLANCK) AND MOMENTUM SCALING

4.1 Distribution functions and par t i c l e f lux

To follow the de ta i l s of the cooling process , we (may) want to know more than the evaluation of

the mean-square beam s i z e and the r .m.s . momentum spread — the only quantit ies used up to now to

characterize cool ing. In fac t , a beam prof i l e monitor records the par t i c l e d is tr ibut ion with respect

to transverse posit ion (see Fiq. 37 as an example), and a longitudinal Schottky scan such as Fig. 2

gives the (square root cf the) momentum dis tr ibut ion . These pictures are rich in fine information on

peak d e n s i t i e s , dens i t i e s in the t a i l s , asymmetries, and other practical d e t a i l s which are overlooked

if only the r .m.s . is regarded.

b) Fig. 37 Evolution of beam prof i l e (number of part ic les vs. vert ica l pos i t ion) during stochast ic

cool ing test in "ICE". The scans were obtained with a prof i le monitor which records the pos i t ion of e lectrons liberated by beam part ic les through c o l l i s i o n s with the residual gas. a) Before cool ing; b) after 4 min of cool ing.

It is therefore challenging to find an equation which describes al l that can be observed and

that is of practical importance. Such an equation does in fact e x i s t !

- 511 -

For s tochast ic cooling toe problem was ( to my knowledge) f i r s t tackled by Thorndahl 1") who

already in 1976 worked with a Fokker-Planck type of equation for the par t i c l e density . This l ine was

followed by v i r t u a l l y all subsequent w o r k e r s 1 7 ) , and compute" codes for solving the dis tr ibut ion

function equations are extens ive ly used in ihe desiqn of s tochast ic cooling and stacking systems-

The basic ideas behind th i s 'd is tr ibut ion function ana lys i s ' are simple, so that a l s? the

beginner can get — h o r e f u l l y without too much pain — some f i r s t degree of fami l iar i ty with th i s

powerful tool of cooling theory. I w»ll f i r s t give the recipe and then try to j u s t i f y i t .

Let <V(K) (Fig. 39) be the p a r t i c l e d is tr ibut ion with respect to the error x ( e . g . x = ûp/p) .

Define 4-(x) = dN/dx so that 4.{x) dx gives the number of par t i c l e s with an error in the range x to x +

dx. During cooling we find different d is tr ibut ions + (x ) , taking snapshots at different times (see

Fig. 2 as an example). We characterize th i s by le t t ing < = ^ ( x , t ) be a function of ti-ne a l s o . The

part ial d i f ferent ia l equation which describes the dynamics of J<(x,t) can be written in the followinq

form:

(4 .1)

x

Fig. 38 A p a r t i c l e d i s tr ibut ion function •*-(*) defining the ntinber of par t i c l e s ¡JN = <i-(x)dx with an error in the interval from x to x + dx

The cooling process i s completely characterized by the two c o e f f i c i e n t s F and f) (wh'ch desrnbp ihn

cool ing system) and the i n i t i a l conditions +(x , I = 01 {which describe the d 's t^h- i t inn it thn

start ) . Part i c l e loss due to wal1 s or influx during stack inq can he included vi J appropri.it o

boundary conditions * { x i ) = 0, ( iV / JxKxj ) = cons t . , e t c . Two represent a t i v 5 exinpl^s of r p s i l t s

obtainable with Ëq. (4 .1 ) are given in Fig. 39, taken from Ref. 13, anil ^-g . 40 fn>n a»»*. 1 ' .

To analyse a given system we have to f :nd *.'- coef f ic -enls F and VI. ^hes'1 quant u i^s -vi*

c l o s e l y related to t¡ie coherent and incoherent e f fpc t , re spec t ive ly , which WP hav»s -di-it I ' H M before. In fact

[ '•; Momentum coolinq :-• j"] • ; ( lonqitudmal density \m ; " ; ;; versus Ap/p)

i : ' ; '.: :

i

i: t ..r .." .;(...,. » j \[mii ":••!""••, '{' mtmt I

Fig. 39 Momentum cool ing at 600 Mr>V/c in tEAR computed using Eq. ( 8 . 1 ) . (Curves taken from Ref. 18.)

' 0 1 * ( ê i "I Ve, Ej • 75 He*

Fig. 4TJ Evolution of the stack in the AA during stochast ic accumulation, turves computed using the d is tr ibut ion function equation with the boundary condition of constant par t i c l e influx simulating the new p added every 2.6 s e c . {Curves taken from Ref. 19) .

i s the expectation value (long-term average) of the coherent change £.x per turn of the error, and

2D/f 0 = C(AX) 2:, (4 .3)

i s the expectation of the square of th i s change. The quantit ies F and 2D alone are corresponding

average changes per second. Note the di f ference between ( A X } 2 = { x c - x ) 2 used here, and è[x*) =

- x 2 as frequently used before!

The important thing is that a d is tr ibut ion function Eq. (4.1) — similar to the Fokker-Planck

equation used in a variety of f i e l d s - - e x i s t s and that r e l a t i v e l y simple prescript ions (4.2) and

(4 .3 ) permit us to e s tab l i sh the two c o e f f i c i e n t s F and D for any given s tochast ic cooling system.

Incidentally* an equation similar J " Eq. (4 .1 ) had long been used (before 1976!) by the Novosibirsk

Group to study the dynamics of electron cool ing . Also the kinetic equations in plasma physics

c l o s e l y resemble our d is tr ibut ion equation.

Let us now try to fol low a simpU- derivation of Eqs. ( 4 . 1 ) - ( 4 . 3 ) . This derivation is due to

Thorndahl 1 6 ) , It proceeds along f e l ines used in textbooks to derive tne diffusion - - or heat

transfer - - equations which reser J c Eq. ( 1 . 1 ) . Imagine a distr ibut ion funrtion <i.(x) and ca l cu la te ,

for a part icular value of x, the number of par t i c l e s per turn which are transferred from x-values

below xi t o values above X[ (Fig, 16) . If the correction per turn at the kicker K Ax, then par­

t i c l e s with an error between < ( and x 0 = xj - A- (cross-hatched area in Fig. 41) pass throuqh %i.

Their number i s

* l

UN = j 4,(x) dx . (4.1) H

1 i |*-~-ÛX M

Fig. 41 A look at the dis tr ibut ion function Fig. 38 through a magnifying g l a s s . When the er'or for par t i c l e s with a value near xj i s changed uy Ax, p a n i c l e s m the dark shaded area have the error value changed from values belnw to values above X j , Eq. (4 .6 ) expresses th i s area as the difference between the ^eclanqle and the tr iangle sketched in the f igure.

Expanding ^ at x l t

the integration y ie lds

AN = * ( X J ) * ¿X - j . (4 .6)

The f i r s t and second terms can be interpreted as the area of the rectangle and the tr iang le , respec­

t i v e l y , sketched in Fig. 41 .

We now define the (average) p a r t i c l e f lux

* = "u<ûN>t

as the expected number of par t i c l e s per second passing a given errar value. Clearly, then, from

Eq. ( 4 . 6 ) , the instantaneous flux i s :

• (x ) = f 0 <ûx> l * (x) - ~ <(Ax) 2 > t — .

This g ives the flux in terms of F and 0 as defined by Eas. (4.2) and ( 4 . 3 ) . The assumption has

t a c i t l y been made that the change Ax per tarn at the kicker is small and 4-(x) smooth, so that hiqher

expansion terms in Eq. (4.5) can be neglected.

Having found the flux we can immediately obtain Eq. (4 .1) from the continuity equation

a* 04.

s * « - 0 - ( 4 - " It s tates that the change per second of the density is given by the 'gradient'-a<i/öx of the f lux .

This i s s imilar to cont inuity considerations in other f i e lds l i k e , fr instance, the charge conserva­

tion law of electrodynamics:

a i 9p

5T+5t-0 • '4-8> relat ing current density j and charge density p.

Like .r;' continuity equations, Eq. (4.7) can be obtained by looking at the flux going into and

coming out of an element of width dx in 4., x-space (Fig. 42):

Incoming flux per second :

Outgoing flux per second : $ 2 " *i * T~ dx

Surplus per second : A* - «t>j. - <t*2 = - ^ dx .

Tiie resul t ing density increase (per second) in the element i s thus

dx " " ox '

and conservation of the p a r t i c l e number requires a ôcji/at equal to t h i s .

<l«

<t>1

dx *- »

<t>2 *

• Xj X 2 X

f i g . 42 The flux into and out of a narrow element of width dx in 4<-x space. An excess of incoming over outgoing flux leads to an increase with time of the density <\, = iN/dx of par t i c l e s in the element.

This completes the der ivat ion. The resu l t ing equation (4 .1 ) agrees with observations made in

the ISR and a l l subsequent machines using s tochast ic cool ing . The reader who might have had some

d i f f i c u l t y in appreciating the derivation may now be pleased to learn that the exact form of

Eq. (4 .1) has been a subject of discussion for quite some time. Looking at the derivation of the

Fokker-Planck equation in t e x t b o o k s 2 0 / , one is tempted to put the coe f f i c i ent D under the second

derivat ive as i s correct for a variety of other s tochast ic processes. In 1977 a machine

experiment 2 1 ) was performed at the ISR to clear up th i s question for cooling and diffusion problems

in storage rings. The experiment c lear ly indicated that in the present case the diffusion term

should be ô /ôx L D ( ^ / ô x ) j a', in Eq. (4 .1) and not ( o 2 / n x 2 ) ( 0 v ) .

4 .2 Example of asymptotic d is tr ibut ions and Palmer cooling

We may conclude from the preceding sect ions that i t is r e l a t i v e l y e imple to determine the

d is tr ibut ion equation pertaining to a given cool ing problem. It is usually much more d i f f i c u l t to

solve the equation. This is because in general the c o e f f i c i e n t s F and D are functions of x, t , and +

i t s e l f . Analytical so lut ions have therefore only been obtained in a few simple cases .

- Sib -

As an etAiple , let us br i e f l y look at Palmer cooling with tire following simplifying assumption: No unwanted mixing, and Schottky noise neal iq ibte cor^ared with amplif1er no i se . Denoting x = (Ap/p), the correction per turn i s

¿x = -g[<x> + x i y L s n J

as given by Eq. (2.24) in Section 2. In analogy to Eq. (2.18) in Section 2.3.3, we assume that the

long-term average of <x> s = (1 /N 5 ) I x-j i s zero exceot for the contibution x/N s of the test

part ic le upon i t s e l f . The noise has zero average. Hence

x 2W

In a similar way (using the assumption that < x ^ t * < < x > s 2 > t ^ ' e ' a r n P ' ^ i e r n o i s e dominating over Schottky noise)

<(Ax) z>. = g2<x2\ = g*V = const. 1 ' t 3 n t ' n.rris

Hence in th i s simple case F = F yx and 0 = D 0 , where F¡j = (2W/N)g and 20 = f t t q 2 x 2

n r m s are

constants . In th i s case , Eq. (4.1) i s amenable to an analyt ic so lut ion . Try

i . e . a Gaussian with a changing in time. Upon subst i tut ion , one obtains an ordinary d i f ferent ia l

equation for the width, o , of the Gaussian:

à/a = -Fy + Du/a 2 .

Special cases:

D 0 = 0: o 2 = o 2 e ~ ^ ü ^ (continuous c o d i n g ) ,

F' = 0: o 2 = o 2 + ZDut (dif fusion) . u u

General so lut ion:

o2 = ¿ e - ^ + DO/F; .

This describes cool ing towards an asymptotic (Gaussian) d i s tr ibut ion with o«, = **(>(,/F0Ï In th is

s i tuat ion an equilibrium between heating and cooling is reached. A similar result is arrived at from

the simple cooling equations [ e . g . Eq. (25), Section 2[ which suggest 1/t * 0 when the signal

( < x > s ) 2 has decreased so much that gU = g | x 2 / ( < x > s ) 2 j * 2. The new information obtained from

Eq. (4.1) is that the asymptotic 4- is Gaussian in the simple case considered.

The e x i s t e n c e of asyfnptot ic e q u i l i b r i u m d i s t r i b u t i o n s is a comnon f e a t j r e a 1 so :n m r e c o m p l i ­

ca ted cases o f Eq. ( 4 . 1 ) . The f i n a l d i s t r i b u t i o n ^ can o b t a i n e d p j t t i n q v / u =• 0 , w h ^ n

conver ts Eq. ( 4 . 1 1 i n t o a s i m p l e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n :

d ^

-FuJ^ + 0 — = c o n s t . 4 . 9 )

The constant is f r e q u e n t l y zero ( e . g . when F ( x ) - 0 and ty/a* = 0 f o r x = 0 as can o f t e n be i n f e r r e d

f rom the symmetry o f the p r o b l e m ) . Equat ion ( 4 . 9 ) i s impor tant as i t i n d i c a t e s the l i m i t i n g d e n s i t y

which can be r e a c h e d .

4 . 3 Momentum c o o l i n g by f i l t e r and t r a n s i t t i m e Tiethods

These -nethods measure the r e v o l u t i o n f requency o f p a r t i c l e s or t h e t ime e f f l i g h t between p i c k ­

up and k i c k e r in order to d e t e c t the momentum e r r o r . The f i l t e r method o f C a n o n and T h o r n d a h l ' O

( F i g . 43} uses a notch f i l t e r between the p r e a m p l i f i e r and the powe" a m p l i f i e r , w i t h notches at a l l

r e v o l u t i o n harmonics in the passband ( F i g . 4 4 ) . In the s i m p l e s t case the f i l t e r is a t r a n s m i s s i o n

l i n e shor ted a t t h e f a r end ( F i g . 4 4 ) , w i t h a length cor respond ing t o h a l f the r e v o l u t i o n t ime of the

p a r t i c l e s i n t h e s t o r a g e r i n g . The notches are produced by \(2 resonances , where i d e a l l y the input

impedance is z e r o and tVe phase changes s i g n . Because of these phase and a m p l i t u d e c h a r a c t e r i s t i c s ,

p a r t i c l e s w i t h a wrong r e v o l u t i o n f requency a r e a c c e l e r a t e d or d e c e l e r a t e d u n t i l i d e a l l y a l l have

' f a l l e n i n t o t h e n o t c h e s ' . The f i l t e r method i s impor tant f o r the c o o l i n g of l o w - i n t e n s i t y beams,

and in f a c t the whole a n t i p r o t o n complex at CERN would p r o b a b l y not have worker! w i t h s t o c h a s t i c

c o o l i n g had t h i s techn ique not heen invented in due t i m e . Sum p 'ck -ups a r e used , and these produce a

much l a r g e r s i g n a l than the d i f f e r e n c e d e v ' c e s t h a t a re necessary w i t h o t h e r T e t h o a V The f i l t e r

reduces not o n l y the p a r t i c l e s i g n a l s but a l s o the p r e a n p l i f i e r no ise at the c r i t i c s ! f r e q u e n c i e s .

Th is f e a t u r e i s impor tant f o r f a s t c o o l i n g at low i n t e n s i t y . The p r i c e t o pay for U r s is I hat a l l

t h e S c h o t t k y bands used have t o be w e l l s e p a r a t e d , so t h a t p a r t i c l e s 'know' t h e notches in to which

t h e y have t o f a l l . Th is means u n a v o i d a b l y imper fec t m i x i n g . However, t h i s s l i g h t d i s a d v a n t a g e could

p r o b a b l y be c i rcumvented by us ing the s i g n a l f rom a second p i c k - u p - - r a t h e r than the r e f l e c t i o n of

t h e p r e v i o u s t u r n p u l s e v i a a c a b l e — t o cance l s i g n a l s o f a p a r t i c l e w i t h the c o r r e c t t ime o f

f l i g h t between the two p i c k - u p s and to a c c e l e r a t e / d e c e l e r a t e o t h e r s ^ 3 ) .

F i g , 43 The bas ic set up f o r momentum c o o l i n g by t h e f i l t e r method. An ortvanlaqe y>f t h i s method is t h a t a sum p i c k - u p is used which is s e n s i t i v e even t o sTial1 (wan s i g n a l s . Secondly , Schot tky and p r e a m p l i f i e r n o i s e a r e reduced by t h e f i l t e r .

- M.s -

Vet another t ime-of - f l ight method has been discussed at Fermi 1 ah J l*J. Essent ia l ly , the idea :s

to d i f f erent ia te the pick-up pulse and arply th is signal on the kicker with a delay so that par t i c l e s

with the correct time of f l ight between pick-up and kicker are not affected, whereas slow or fast

ones get a correct ion.

Fig. 4 4 A simple periodic notch f i l t e r namely a half-wave low-loss transmission l ine (used as a stub resonator) . The ( ideal ized) gain and phase character i s t i c s ¿re qiven by the haîf-wave resonances at the multiples of the revolution frequency. Additional elements are usually added to reduce the ga :n between the harmonics.

Both variants of the f i l t e r -netliod are less e f f i c i e n t ;n noise Suppression and hsve, therefore,

not found applications so f<jr.

We shall return to the time domain for a short moment to sugqest s l i ^ t l y different explanations

of the f i l t e r method: the pulse sent into the coolinq systen by a part ic le of nominal frequency wi l l

be cancelled by i t s pulse from the previous revolution ref lected at the end of the l ine . For part i ­

c l e s that are too slow or too f a s t , the cancel lat ion is imperfect and .y.celeration or deceleration

«MI r e s u l t .

The f i l t e r method is usually analysed using the dis tr ibut ion Eq. ( 4 . 1 ) . The coe f f i c i en t s F anri

3 can be worted D J I theore t i ca l ly and/or by mea sûrement, s on the system. Usually, measurements snd

calculat ions are done harmonic by harmonic, int'udmg various ingredients suç> as imperfect mixing

and signal suppression. All we want to do here is to write down the qeneral form of the relevant

coe f f i c i en t s F and 0 which, expanding up to second order in the error -luantity x = ¿E/E take the

following form:

F = -Gox

D = G2 o-tV t K ) + &¿ [¡.{í2 * K-) , i ¿

where x is the re la t ive energy error; Gg (proportional to the 'gain' q) , G'¡(« g ¿ ) , and ß*>(« q')

are given by the ideal f i l t e r , * 0 and by the l o s se s ; and •,< re la tes to the amplifier no ise . The

f i r s t term of D (which is proportional to the density <i0 gives the Schottky noise f i l t e red by the

notches, and the second term the f i l t e r e d preamplifier noise . For more d e t a i l s , the reader should

consult the spec ia l ized l i t erature .

REFERENCES

1) See for example Bibliography. 1977:1, 1978:3 and 12, 1980:1, 1991:1, 3 and 12, 1982:1 a n d t , 1983:7, 1 9 8 4 : 1 , 2 and s, 1 9 8 6 : 1 1 .

H. Poth, Electron cool ing, these proceedings. K. Hùbner, Radiation damp'ng, Proc. CERN Accelerator Scnool, Genera'' accelerator D h y s i c s ,

Gi f - sur - lve t t e , 1984, CERN 85-19 (1985). K. Hubner, Synchrotron radiat ion, ibid, p. 239.

2) H. Bruck, Accélérateurs c i rcu la ires de p a r t i c u l e s . Bibliothèque des Sciences Nucléaires, Paris (1966).

M. Sands, SLAC report 121 (1970).

3) See Bibliography, 1981:12, 1984:4, 1984:7.

0. Boussard, Schottky noise and beam transfer function diagnost ics , these proceedings.

4) w. Meyer Eppler, Grundlagen und Anwendungen der Informationstheorie, Springer Verlag (1969).

5) See Bibliography, 1980:9 and 10. 6 ) M.R. Spiegel , S t a t i s t i c s (Schaum's Outline Series) (McGraw-Hi1I, New York, 1972), Chapter 8.

L. Maisei, Probabil i ty , s t a t i s t i c s and random processes (Simon and Schuster Inc . , Mew York, 1971) Chapter 6 .

7) See Bibliography, 1977:1.

8) M. Bregman et a ) . , Measurement of antiproton l i fe t ime u ' . 'nq ihv ICE storaqt: r ing, Phys. Lett . 78B (1978) 174.

See a lso Bibliography, 1980:1.

9) See Bibliography, 1983:19 and 28. 1985:3 and 5.

10) See Bibliography, 1983:2. 1984:9. 10. 11 and 12.

11) See Bibliography, 1977:1, 1980:1.

12) H.6. Hereward and K. Johnsen, The effect of radio frequent? no i se , CERN W - M (1960).

13) See Bibliography, 1980:1 and 1982:18.

14) See Bibliography, 1978:3.

15) rl.G. Hereward, Landau damping, these proceedings, and:The elementary theory of Landau ninpinq, CERN 65-20 (1965).

16) See Bibliography, 1977:8.

17) See Bibliography, L977:10, 1978:3, 1979:1, 2 and 3 , 1980:3 and U , 1981:1. S. van der Meer and L. Thorndahl, Computer programs for solving Eq. ( 4 . 1 ) , private

communication.

18) See Bibliography, 1980:)O.

19) See Bibliography, 1978:15.

20) S. Chandrasekar, Stochastic problems in Physics and Astronomy, i n : Noise and stochast ic processes , ed. N. Wok (Dover Press, New York, 1954). G. Isbimaru, Basic pr inc iples of plasma physics (U.A. Benjamin Inc . , Reading, Mass., 1973).

21) L. Thorndahl et a l . . Diffusion in -irjmentim caused by f i l t e r e d noise , CERN internal report

ISR-RF-TH Machine Performance report, 19 Auqust 1977.

22) See Bibliography, 1978:2.

23) See Bibliography, 1984:3.

24) See Bibliography, 1980:4, p. 777.

- .S J O -

BIBLIOGRAPHY

Chronological l i s t of internal reports , conference presentat ions , and other publications

related to s tochast ic cooling

1838

T. 0. L iouv i l l e , J. Math. Pures et Appl. 3, 348.

1918 TÎ U. Schottky, über spontane Schwankungserscheinungen in verschiedenen E lektr i z i tä t sha lb le i t ern ,

Ann. Phys. 57, 541. 1956 T D.B. Lichtenberg, P. Stahle and K.R. Symon, e d i f i c a t i o n of L iouv i l i e ' s theorem required by the

presence of d i s s i p a t i v e forces , Midwestern Univ. Research Assoc. report MURA-08L-PS-KRS1.

1958

T. F. Mills and A.M. Sess ler , L iouv i l i e ' s theorem for a continuous medium report MURA-433.

1972 "H S. Van der Meer, Stochastic damping of betatron o s c i l l a t i o n s , internal report CERN/1SR PO/72-31. 2 . rt. Schnell , About the f e a s i b i l i t y of s tochast ic damping in the ISR, internal report CERN/ISR

RF/72-45.

1973

T. R.B. Palmer, Stochastic cool ing , Brookhaven Nat.- Lab. report 8NL 18395.

19^4 T. 0. Borer, P. Bramham, H.G. Hereward, K. Húbner, W. Schnell and L. Thorndahl, Non-aestructive

diagnost ics of coasting beams with Schottky no i se , Proc. 9th Int. Conf. on High Energy Accelerators. Stanford (USAEC Conf. 740522, Washington, 1974), p. 53.

1975 T. P. Bramham, G. Carrón, H.G. Hereward, K. Hübner, W. Schnell and L. Thorndahl, Stochastic cooling

of a stored protron beam, Nucl. Instrum. Methods 125, 201.

2. L. Thorndahl, Stochastic cooling of momentum spread and betatron o s c i l l a t i o n s for low-intensity s tacks , internal report CERN/ISR-RF/75-55.

1976 T. P. S tro l in , L. Thorndahl and D. Mohl, Stochastic cooling of antiprotons for ISR physics , internal

report CERN/EP 76-05.

2 . D. Cl ine, P. Mclntyre, F. Mills and C. Rubbia, Collecting antiprotons in the Fermilab booster and very high energy proton-antiproton interact ions , Fermilab internal report TM 689.

3 . C. Rubbia, P. Mclntyre and 0. Cline, Producing massive neutral intermediate vector bosons with e x i s t i rig acce lerators , Proc. Int. Neutrino Conf., Aachen, 1976 (Vieweg Verlag, Braunschweig, 1977), p. 683.

4 . K. Hübner, D. Mohl, L. Thorndahl and P. Strol in , Estimates of ISR luminosities with cooled beams, CERN/PS/DL Note 76-27.

5. G. Carrón, L. Fal t in , W. Schnell and L. Thorndahl, Stochastic cooling of betatron o s c u l a t i o n s and momentum spread, Proc. Vth All-Union Part ic le Accelerator Conf., Dubna (USSR Acad. S e i . , Moscow, 1977), p. 241.

6. G. Carrón and L. Thorndahl, Stochastic coolinq of vert ical betatron o s c i l l a t i o n s in the frequency range 80-340 MHz, CERN ISR Perf. Report, ISR-RF/LTS/ps (RUN 775).

7. W. Hardt, Augmentation of damping rate for s tochast ic coolino by the additional use of non-linear elements, CERN/PS/DL 76-10.

8. E.O. Courant, Workshop on phase-space cool ing, in Proc. ISABELLE Workshop, Brookhaven (Report BNL 50611, Upton, 1976), p. 241.

lT~Hi.G. Hereward, S t a t i s t i c a l phenomena — Theory, Proc. 1st Course Int. School of Par t i c l e Accelerators , Erice (Report CERN 77-13, Geneva, 1977), p. 281.

2 . W. Schnell , S t a t i s t i c a l phenomena — Experimental r e s u l t s , i b i d . , p. 290.

3 . L. Fa l t in , RF f i e l d s due to Schottky noise in a coasting par t i c l e beam, Nucl. Instrum. Methods 145, 261.

4. L. Thorndahl, Stochastic cooling of betatron o s c i l l a t i o n s in ICE, CERN/ISR Technical Note I S R / R W / p s .

5. G. Carrón, L. Fa l t in , H. Schnell and L. Thorndahl, Experiments with s tochast ic cool ing in the ISR, Proc. Part ic le Accelerator Conf., Chkaqo, 1977 [IEEE Trans. Nucl. Se i . NS-24 ( 3 ) , 1 9 7 7 J , p. 1402.

6. G. Carrón, L. Fa l t in , H. Schnell and L. Thorndahl, Recent re su l t s with s tochast ic cool ing in the ISR, Proc. 10th Int. Conf. on High-Energy Accelerators, Serpukhov (IHEP, Serpukhov, 1977), vo l . I, p. 523.

7. S. van der Meer, Influence of bad mixing on s tochast ic accelerat ion, internal note CERN/S?S/DI/pp 77-8.

8. I . Thorndahl, A d i f f erent ia l equation for s tochast ic coolinq of momentum spread with the f i l t e r method, technical note ISR-RF/LT/ps.

9. S. van der Heer, normalized solut ion for linear momentum coolinq of a square d i s t r ibut ion , unpublished document.

10. L. Jackson Las l e t t , Evolution of the amplitude dis tr ibut ion function for a beam subjected to s tochast ic cool ing , Berkeley report LBL-6469.

T. ICE Team, In i t ia l Cooling Experiment progress reports Nos. 1 and 2, CERN-EP Div.

2. G. Carrón and L. Thorndahl, Stochastic coolinq of momentum spread with f i l t e r techniques,

internal report CERN/1SR-RF/78-12 and ISR-RF/Note LT/ps.

3. F. Sacherer, Stochastic cooling theory, internal report CERN ISR-TH 78-11.

4 . G. Carrón et a l . , Stochastic cooling t e s t s in ICE; Phys. Lett . 77B, 353.

5. Design Study Team, Design study of a proton-antiproton co l l id ing beam f a c i l i t y , internal report CERN/PS/AA 78-3.

6. F, Bonaudi et a l . . Antiprotons in the SPS, internal report CERN DG 2. ?. S. van der Meer, Stochastic stacking in the Antiproton Accumulator, internal report CERN/PS/AA

78-22.

8. S. van der Meer, Precooling in the Antiproton Accumulator, internal report CERN/PS/AA 78-26.

9. D. Mühl, Stochastic cool ing, Proc. Workshop in P o s s i b i l i t i e s and Limitations of Accelerators and Detectors , Batavia (FNAL, Batavia, 1978), p. 145.

10. L. Fa l t in , Slot-type pick-up and kicker for stochast ic beam cool ing , Nucl. Instrum. Metnods 148,

449.

11. A.G. Ruggiero, Are we beating L i o u v i l i e ' s theorem, in Proc. Workshop on Producing High

Luminosity high Energy Proton-Antiproton Co l l i s ions , Berkeley (LBL, Berkeley, 1978).

12. A.G. Ruggiero, Stochastic cool ing with noise and good mixing, ib id .

13. S. van der Meer, Stochastic cooling theory and devices , ib id . 14. H. Herr and D. Mühl, Bunched beam stochast ic cooling, Proc. Workshop on the Cooling of High

Energy Beams, Madison, 1978 (Univ. Wisconsin, Madison, 1979) and internal report CERN/PS/DL/Note 79-3 (1979) and CERN/EP/Note 79-34 (1979).

15. Design study of a proton-antiproton co l l id ing beam f a c i l i t y , internal report CERH/PS/AA 78-3.

1979 T! J. Bisognano, Transverse s tochast ic cool ing , Berkeley (181) internal report BECON-9.

2 . J. Bisognano, Vertical transverse s tochast ic cool ing , Berkeley (LBL) internal report BECON-10, LBIB-119.

3. J. Bisognano, Kinetic equations for longitudinal stochastic cool ing, Berkeley (LBL) internal

report BECDN-11, LBD-140.

4. Ya.S. Derbenev and S.A. Kncifets , On stochast ic cool ing , Part ic le Accelerators, 9 , 237.

5. Ya.S. Derbenev and S.A. Kheifets , Daiping of incoherent .-notion by d ' s s ipat ive elements in a

storage ring, Soviet Phys. Techn. Phys. 24(2) , 203.

6. Ya.S. Derbenev and S.A. Kheifets, Stochastic coolinq, Sov. Phys. Tech. Phys. 24(2) , 209.

7. G. Carrón e t a l . . Experiments on stochast ic cooling in ICE, Proc. Part ic le Accelerator Conf., San Francisco, 1979 [IEEE Trans. Nucl. Se i . NS-26 ( 3 ) , 1979], p. 3¿56.

8. S. van der .teer, Deounched p-p operation of the SPS, internal report CERN/PS/AA 79-42.

9. D. Höhl, P o s s i b i l i t i e s and l imits with cooling in LEAR, Proc. Joint CEKRN-KfK '.Workshop on Physics with Coaled Low-Energy Antiprotons, Karlsruhe (KfK report 2836, Karlsruhe, 1979), p. 27.

10. 3. Leskovar and C.C. Lo, Low-noise wide-bano" amplifier system for stochastic beam cool ing experiments, Proc. Nuclear Science Symposium, San Francisco, 1979 | IEEE Nucl. Se i . N5-27 ( 1 ) , 1980J, p. 292 (preprint L8L 9841).

1980 T. [). Mcihl, 6. Petrucci , L. Thorndahl and S. van der Meer, Physics and technique of s tochast ic

cool ing , Phys. Rep. 58, 75.

2 . S. van der Heer, A different f o r m a t i o n ° f the longitudinal and transverse beam response, internal report CERN/PS/AA 80-4.

3 . J. Bisognano, Kinetic equations for longitudinal s tochast ic coolinq, 11th !nt. Conf. on High

Energy Accelerators, Geneva (Birkhäuser, 3 a s e l , 1930), p. 772.

4. W. Ke l l s , F-ilterless fast momentum cool inq, ib id . , p. 777.

5. G. Lambertson et a l . , Stochastic cooling of 200 MeV protons, i b i d . , p. 794.

6. E.N. Demet'ev et a l . , Measurement of the thermal noise of a proton heam in the NAP-M storage r ing, Sov. Phys. Tech. Phys. 25(8) , 1001.

7. F. Krienen, I n i t i a ' cooling experiments (ICE) at CERN, 11th Int. Conf. on High Enerqy

Accelerators, Geneva (Birkhäuser, Basel 1980), p. 781.

8. D.E. Young, Progress on beam coolinq at Fer-r.ilab, i b i d . , p. 800.

9. P. Lefevre et a l . , The CERN low enerqy antiproton ring (LEAR) project , i b i d . , p. 019.

10. Design study of a f a c i l i t y for experiments with low enerqy antiprotons (LE*»R), internal report CERN/PS/DL 80-1

11. Y.V. Parkhomchuk and D.V. Pestrikov, Thermal noise in an intense beai) in a storage ring, Sov. Phys. Tech- Phys. 25(7) , 818.

1981 T. r .T. Cole and E.E. Mi l l s , Increasinq the phase-space density of high-energy part ic le beams, Ann.

Rev. P.'ucl. Part. Sei . 31, 295. 2. J. Bisognano and S. Chattopadhyay, Bunched beam stochast ic coolinq, Berkeley ( iBfi internal

report BEC0N-18.

- 5>S -

3 . A.N. Skrinski and V.V. Parkhomchuk, Methods r ' cooling beams of charaed p a r t i c l e s , Sov. J. part. Nucl. 12 ( 3 ) , 223.

4 . 5- van der Meer, Stochastic cool ing in the CZM Antiproton Accumulator, Proc. Part ic le Accelerator Conference, Washington, DC I IEEE Trans. Nucl. Se i . NS-28 ( 3 ) , 1984J, p. 1994.

5. J. Hardek et a l . , ANL stochast ic cooling experiments using the FNAL 200 MeV cooling ring, i b i d . ,

p. 2455.

6. i. Bisognano and S. Chattcphadhyay, Stochastic cool ing of bunched beams, i b i d . , p. 2462.

7. G, Lamberton et a l . , Experiments on s tochas t i c cooling of 200 MeV protons, i b i d . , n. 2471.

8. T. Linnecar and W. Scandale, A transverse Schottky noise detector for bunched proton beams, i b i d . , p. 2147.

9. A.G. Ruggiero, Pickup loop ana lys i s , Fermilab internal p note 148.

10- A.G. Ruggiero, Stochastic cooling - A comparison with bandwidth and l a t t i c e funct ions, Fermilab internal p note 171.

11. E.N. Dement'ev et a l . , Experimental study of s tochast ic cooling of protons in NAP-M, .Novosibirsk preprint 81-57 [t .-anslated at CERN as internal report CERN/PS/AA 32-3 (1982) ] .

12. J. Bisognano and C. Leeman, Stochastic coo l ing , in The physics of hiqh energy accelerators (Proc. Batavia Summer School on High Energy Part ic le Accelerators) (AIP Conf. Proc. No. 87, New York, 1982), p. 583.

1982

1. 0 . Höhl, Phase-spice cool ing techniques and their combination in LEAR, IVoc. Workshop on Physics at LEAR with Low-Energy Cooled Antiprotons, Erice (Plenum Press, London, 1983), p. 27.

2. A.G. Kuggiero, Theory of signal suppression for s tochast ic cooling with multiple systems, Fermilab internal p note 193.

3 . C. Kim, Design options for the fas t betatron precooling systems in the debuncher or on the inject ion o r b i t , Berkeley (LBL) internal report BEC0N-25.

4. S. Chattopadhyay, Stochastic cooling of bunched beams from f luctuat ion and kinet ic theory, Thesis Univ. of Berkeley, Calif , internal report LBL 14826.

5. C.S. Taylor, Stochastic cooling measurements — The CERN AA, Proc. Beam Cooling Workshop,

Stoughton (Univ. Wisconsin, Madison, 1982).

6. S. Kramer, LßL/ANL Cooling Experiments, tbid.

7. L. Tecchio, Comparison of e lectron and s tochast ic cooling for intermediate enerqy ranqe, ib id .

8. F. Voelker, Electrodes for s tochast ic cooling of the FNAL antiproton source, ib id .

9. 8. Autin, Fast betatron cool ing in an antiproton c o l l e c t o r , ib id ,

10. 8. Leskovar and C.C. Lo, Low-noise wide-band amplifiers for s tochast ic beam cooling experiments,

ibid.

11. K. Takayama, Effects of non-linear electrode on transverse cool ing , ibid.

12. A.G. Ruggiero, 2-4 GHz t a i l s tochast ic cooling system with f i l t e r s , ibid.

13. J. Harriner, Stochastic stacking without f i l t e r s and accumulator gain p r o f i l e , ib id .

14. J. McCarthy, Superconducting f i l t e r s for s tochast ic systems performance measurements, ib id .

15. C.R. Holt, A normal mode analysis of coupled betatron o s c i l l a t i o n s including cool ing , ib id .

16. D. Höhl and K. Ki l ian, Phase-soace cooling of ion beams, Proc. Symposium on Detectors in Heavy-Ion Reactions, Berlin (Vol. 178 of Lecture Notes in Physics, Springer Verlag, Berl in, 1982), p. 220.

17. A. Ando and K. "ikayama, RF stacking and ta i l cool ing in the antiproton accumulator, Fermilab internal report TM 1103.

18. S. van der Meer, Gain adjustment cr i ter ion for betatron cooling in the presence of amplifier n o i s e , internal report ŒRN/PS/AA Note 82-2.

19. ?. Leskovar, R e l i a b i l i t y considerations at travelling-wave tube and gal Ilium-arsenide f i e ld e f f ec t transistor amplif iers, internal report LBL 1381.

1. G. Brfanti, Experience with the CERN pp complex, Proc. Part ic le Accelerator Conf,, Santa Fe, 1983 I IEEE Trans. Nucl. Se i . HS-30 ' 4 ) , 1983] , p. 1950.

2. G.R. Lambertson and C.W. Leeman, Intense antiproton source for a 20 TeV c o l l i d e r , i b i d . , p.

2025.

3. G.R. Lambertson, K.J. Kin and F.V. Völker, The s lo t ted coax as a bean e lectrooe , i b i d . , p. 2158.

4. 8. Leskovar and C.C. Lo, Low-noise gallium arsenide f i e l d - e f f e c t transistor preamplifiers, i b i d . , p. 2259 (preprint LBL 15122).

5. F. Volker, T. Henderson and J. Johnson, An array of 1-2 GHz electrodes for s tochast ic coolinbq, i b i d . , p. 2262.

6. S. Chattopadhyay, On stochast ic cooling of bunches in the c o l l i d i n g beam mode in h'gh-enerqy pp storage r ings , i b i d . , p. 2334.

7. J. Bisognano, Stochastic cool ing: recent theoretical d irec t ions , i b i d . , p, 2393.

8. E. Peschardt and M. Studer, Stochastic cooling in the CERN ISR during pp co l l id ing beam physics, i b i d . , p. 2584.

9. G. Carrcn, R. Johnson, S. van der Meer, C. Taylor and L. Thorndahl, Recent experience with antiproton cool ing, ib id . , p. 2587.

10. W. Ke l l s , S t a b i l i t y and signal suppression of Schottky s ignals from s tochas t i ca l ly cooled beams, i b i d . , p. 2590.

11. 6, Autin, J. Marriner, A. Ruggiero and K. Takayanw, Fast betatron cooling in the dehuncher rinq for the Fermila Tevatron I project , i b i d . , p. 2593.

12. A. Ruggiero and J. Simpson, Momentum precooling in the dehuncher ring for the Fermilab Tevatron I project , i b i d . , p. 2596.

13. A. Ruggiero, Signal suppression analysis for momentum stochastic cooling with a w l t i p i e system, i b i d . , p. 2599.

14. A. Ando and K. Takayama, Effects of rf-stacking on t a i l cooling in the Fermi I ab antiproton accumulator, ib id . , p. 2601.

15. S. Chattopadhyay, Vlasov theory of signal suppression F or bunched beams interacting with a

stochast ic cooling feedback loop, i b i d . , p. 2646.

16. S. Chattopadhyay, Theory of bunched beam s tochast ic cooling, i b i d . , p. 2649.

17. 5. Chattopadhyay, A formulation of transversely coupled betatron stochastic cooling of coasting beams, i b i d . , p. 26^2.

18. 8. Leskovar and C.C. Lo, Travelling-wave tube amplifier character i s t i c s study for stochast ic beam too l ing experiments, i t n d . , p. 342 3 , oreprint LBL 14142.

19. R. B i lhnge and E. Jones, The CERN antiproton source, Proc. 12th Int. Conf. on High Energy Accelerators, Batavia (FNAL, Batavia* 1984), p. 14.

20. B. Autin, Technical developments fur an an.'ioroton co l l ec tor =i "EKN, 'h! ' ) . , o. 39 3.

21. S.L. Kramer, J. Simpson, R. Konecny and ' j . Suddeth, R e l a l w i s t i c bean pickuo te s t *"aci'Uy, i b i d . , p. 258.

22. N. Tokuda, H. Yonehara, T. Hattori , T. Katayama, A. Nöda and M. ^osVjaws, Stochastic cooli

7 HeV protons at TARN, i b i d . , p. 336.

23. J. ¡*arriner, The Fermi tab antiproton stack t a i î system, ' b i d . , D. 5-79.

24. R.E. Shafer, The Fermi lab antiproton dehuncher betatron coolinq system, ib id . , p. 531.

25. R.J. Pasquine l l i , Superctwiducting notch f i l t e r s • r.-rmilab antiproton source, ' b i d . , 584.

26. S. van der Meer, Optimum gain and phase for stocha. ic coolinq systems, internal report

CERN/PS/AA 33-48.

27. Design report Tevatron I project , FNAL internal report (1983).

28. Design study of an antiproton c o l l e c t o r for the Antiproton Accumulator, renurt TERN 83-10.

1984 I - "5. van der Meer, Stochastic coolinq coolinq and the accumulation of antiprotons [Nob^I lect

1984). CERN preprint PS 84-32 (AA) published in Rev. Mod. Phys. 1985, p. 689. 2. S. van der '-leer, An introduction lo s tochast ic cool ing , lecture at l'i" ]9'tf us Su-nner "ir.hon

High Energy Accelerators, CERN preprint PS 84-33 (AA). i . 1). Mohl, Stochast ic cooling for beq i n n e n , in Proc. CEWH i\cxeW>r,^nr Vlv.w", " î*. î-.vv.t ,v.s '

co l l id inq beam f a c i l i t i e s " , CERN 84-15 I1934Í, p. 9?.

<1. C S . Taylor, Stochastic cool'.nq hardware, i b i d . , p. 161.

5. S. van der Meer, Opt i ism qa>n anil phase f o r s tochast ic cool ing , ¡hi-l . , n. 183.

6. 0. Br .ssard et a l - , F e a s i b i l i t y study of s tochast ic coa Win ^ f hi.rir'v^. n the S^S, ih - ¡ ; . , -i

7. J. \iorrr and R. Jung, Di ag-.osl i c i , i b i d . , o. 385.

8. P. 3ryant, Antiprotons in the CERN intersect ing stor.iqe (-''mis, ' lud , , r». W.

9. Ii. Autin, Die CERN antiproton c o l l e c t o r , i b i d . , p. 525.

10. R. Autin, Antiproton betatron cooling for the SSt., in Proc. Í1FP wurkshop on pn options f ir H

supercoll ider (J .E. P\chl*r and Ri White, erf.) Chioino Í9íM. p. i l l .

11. I). Mtihl, Fast s tochast ic cooling of batches of lf)'J p a r t i c l e s , i l n d . , p. US.

I?. R.P. Johnseon, Proton coolinq for the SSC, i b i d . , p. 3?1.

¡3. R.P. Johnson jrtd J. Srnpson, pbar source yoftn siwinarv, ih îd . , p . 101.

14. I). Möhl, A comparison between electron coolinq and stochast ic f in i u m , "roi;, if At^'t'Sv ip m electron cool ing , ECOOL 84, Karlsruhe Sepl. 1984, KFK roport t8-16 (H. Polo, H . i o. ""H (preprint PS-LEA 85-8) .

15. Ç.C. Lo ,ind 8. Leskovar, Cr yogóme a 1 ly cooled Iwojd b-)nd :îvV> fii'M .^tfoit I f IDS ,t

preamplif iers, IEEE Trans. Suc I. S c . NS-Jl (1984), .1. ' . '5 n-.l IT,

16. H.r,. Jackson and T.Í . Sh im/u , A low - i d ' i p 2-1 d'-z pre vr i I i i 1 , \.'¿\. ]'•'.[[.

19% T."""R. Bi Hinge , Introduction t.i :".ERNS a i t ' o m h v i f a c ' i t ' . ' s f ••- th.« '.')>in, :•• ¡!t ,r. *..| ;

Wirkiiiop, '^hySlCS W i t h CnoV '1 low t w - . } y n t rjrot wis, T h i n g s , N'i'i . ; !. i n:' i i v - , Gif-sur-Yvett=, l ' )HS) , | ) . 11, p n ^ ' n i í'UN '.'SOI ' M .

2. G. Carrón et a l . , Status and future p o s s i b i l i t i e s of the s tochast ic cooling system for LEAR, i b i d . , p. 27, preprint CERH/PS/LEA 85-12.

3. E. Jones, Progress on ACOL, i b i d . , p. 8, preprint CERN/PS/AA/85-14 (1985).

4. G. Carrón et a l . , Development of power amplifier modules for the ACOL stochast ic coolinq sys tans ,

CERN 85-01 (1985).

5. B. Autin, The new generation i f antiproton sources CERN/PS/AA/85-43 (1985).

6. E. Brambilla, A microwave CerenW pick-up for s tochast ic cooling CERN/PS/AA/35-45 (1985).

7. N. Tokuda e t a l . , Stochastic cooling of a low eneray beam at TARN, IEEE Trans. Nucl. Se i . NS-32 S

(1985), p. 2415. "

8 . F. Caspers, Planar s lo t l ine pick-ups for stochast ic cool ing, CERN/PS/AA/B5-48 (1985).

9. C.C. Lo, Stochastic beam cooling amplifier system frontend components charac ter i s t i c s , ; n Proc. 1985 Part. Acc. Conf. Vancouver 1985 (IEEE Trans. Nucl. Sei. 39 1985), p. 2174.

10. D.A. Goldberg et a l - . Measurement of frequency response of LBL coolinq arrays for TeV-1 storaqe r ings , i b i d . , p. 2168.

11. J.K. Johnson and R. Nemetz, Power combiners/dividers for loop pickup and kicker array for FNAL stochast ic cooling ring, i b i d . , p. 2171.

12. W.C. Barry, Suppression of propagatinq TE modes in the FNAL antiproton source s tochast ic beam cooling system, ibvd. , p. 2424.

13. P. Lebrun, S. Miîner and *. Poncet, Cryogenic desinn of the coolinq pickups for the CERN antiproton co l l ec tor (ACOL), Adv. cryog. eng. 31 (1986) 543, preprint CERN/LEP/MA 85-29.

14. H.G. Jackson et a l . . Preliminary measurements of qamna ray e f f e c t s on character i s t i c s of GaAs f i e ld e f fect transistor preamplifiers, LBL Berkeley report PUB-5141, LBL-21546.

T. 0. Boussard and G. öi Massa, High frequency slow wave pick-ups, CERN/SPS/ARF7B6-4 (1966).

2. 0. Boussard, Schottky noise and beam transfer function d iagnost ics , CERN Accelerator School,

Oxfcvd, 1985 [present proceedings). Preprint CERN/SPS/ARF/86-11 (1986).

3. N. Tokuda, A Helix coupler for a pick-up of a low ve loc i ty beam, CERN/PS/LEft/Note 86-5 (1991).

4 . 0. Höhl, Perspectives of ion cooling rings CERN-PS/LEA 86-32, to be published in Proc. 1986 international cyclotron conference, Tokyo Oct. 1986.

5. O. Hohl, Principle and technology of beam cool ing , CERN-P5/LEA 36-51, to he published in Proc. of

the RNCP-Kickuchi sunnier school on accelerator technology, Osaka Oct. 1986.

6. E.J.N. Wilson, Antiproton production and accumulation, CERN/PS/AA 86-22 (19B6).

7. N. Beverini et a l . , Stochastic coolinq of charged part ic les in a Penninq trap, Europhys. Lett. 1, p. 435 (l9R$f.

8. S. Mtingvta, Stochastic cnoling of antiprotons at tne Tevatron, Fermilah internal report Pbar Note 445 (January 1986).

9 . F.E. Mi l l s , Cooling of stored beams, Fermi I ab internal report, Pbar Note 463 (10 June 1986). 10. J. Marriner, Stochastic coolinq at Fermilab, preprint Fermilah-Conf-86/124 (August 1986)

(Presented at the XIII International Conference on Hiqh Enerqy Accelerators, Novosibirsk, USSR, 7-11 August 1936}.

11. V.V. Parkhomchuk, The cool ing of heavy p a r t i c i e s , presented at the XÍII International Conference on High Energy Accelerators Novosibirsk, USSR, 7-11 Auqust 1986.

Workshops on beam cooling

Workshop on Phase-Space Cooling, Brookhaven, 1976. (Summary report edited by E. Courant, puolished in Proc. 1976 ISABELLE Workshops, Brookhaven Nat. Lab. report BNL 50611, 1976, p. 241).

Workshop on Producing High Luminosity High Energy Proton-Ar tiproton C o l ' i s i o r s , Berkeley, 1978. (Copies of transparencies and reports presented, published by LBL, Berkeley, 1978).

Workshop on Cooling of High Energy Beams, Madison, 1978. (Proc. edited by D. Cline, published by the University of Wisconsin, Madison, 1979).

Workshop on pp in the SPS, Geneva, 1980. (Summary report on Higr t'nergy Beam Coolinq edited by F. M i l l s , Proc. published as report CERN/SPS-pp-lf Gene-a, 1980).

Beam Cooling Workshop, Stoughton, 1982- (Proc. edited by D. Cline and f. Mi l l s , published hy University of Wisconsin, Madison, 1982.)

Design reports of f a c i l i t i e s using s tochast ic cool ing

AA design report , 1978:15. LEAR deisgn report, 1980:10. TEVATRON design • w> ACOL design study, -9B3:23.

Summary reports and reviews Numbers refer to the Biblioaraphy

1977: 1 1 9 8 1 : 1 , 3 and 1 2 1978: 3 and 1 J 1982: 1 , 7 1980 : 1 1983: 7

1984: 1, 2 , 3 and 4 1986: 5

Special problems of coolinç of heavy-ion beams are discussed in

1981: 3 1982: 16 1986: 4

Append y 1

For à li>nq timp the pr 1 ne i p le of stochastic coolinq was reaardpd as \oo f ar-f et rhori lo tie p ' i r -

1 •. » ' û f t - r pr-r '-non* .a ' -¡f«nni,( î- a* ! ni- w a l 1 T- ' w*. on 1 V V'VC1" yr- a' ' ^ f ' 1 ' ' :- ' ' ' • . .

y^ars af te1* tne f • r St p j b ' ic al on of t h e '.dea (Table A i } . The mvpntor. S. van df*r Mei.-r, anrt the

--•arly workers fiad 'tiainlyf emittanr.p coolmq of h íqh- ¡ntens i t v b,".a^ m m'nd with a v'«w (r, Krmrnv'nq

thy luminosity m the r.ERN ISR.

A new era began >n 1975 when P. St rol in, comi nq hack from a v i s i t to Novas ibirsk, and

L. Thor nd ah i rea • ' zed the interest of stochast >c coo 1 ;no - - for hot h T i U a n r n and nvyr^nt — of

low-intensity p beätis for the purcose of stacking. Stochastic coolinq at low intensi ty is d' f terent

fron (hr. c r i c r i s ! van ripr Meer rnolinq. The e^tens^on of the theory f i r s t done M rpviarrt And

L. Thorndahi, and the design of the momentum coolinq hardware ( I . ThorndahI, G. Carrón pt a l . ) , are

oerhaps as fundamental as the oriqmal invention and the earl ier f e a s i b i l i t y s t u p e s by S. van der

Meer and W. Schnei 1.

Follow!-^ upon th is broadeninq of the f i e ld of application, in 19/5 P. Strol in p\ al . worked out

p co l l ec t ion schemes for the ISR using stackinq in momentum 'Dace, an^ C, RubtVa et a l . irade their

f i r s t proposals of the pp scheme for the SPS usinq similar techniques of s tochast ic coolinq and

accumulation. This work gave new l i f e to the idea at a time when the ISR was routinely stackinq such

high currents that proton beam-coolinq became unnecessary — or even impossible. Further milestones

between 19?b and 1980 wp«-e the invention of the f i l t e r method nf momentum cool ino, Ihp refinement of

the s tochast ic stackinq schemes, the resu l t s of the i n i t i a l cooling experiment (ICE), and l a s t , but

not l eas t , the success of the AA. The ICt ring was used tn make a r.^rnful compa-^SiVi nf cool i no

theory with r e a l i t y , including bunched beam coolinq. By the middle of 19/fi a l l systems worked so

well that bean l i fet imes at 2 GeV/c of the order of a week were reached, compared with l i fetimes of a

few hours without cool inq. This permitted a measurement of the s t a h i l i t y of the antiproton, and th i s

experiment improved the lower limit in one qo from 1?D ,is to 30 h.

One essent ia l inqredient in this experiment was the techn'que developed at ICE lo observe as few

as 50 c i rcu lat ing par t i c l e s in a non-destrucr.ive manner. This was made possible hy stnchasM'c

cooling which reduced the momentum spread to 1Q-'J su that a resonant Schottky noise pick-up with the

corresponding qual i ty factor could be used.

P,unmnq-m of the AA started in tne summer of 1980, and s ince 198L/B?, stacks Qf several 1011 p

a'e routinely accumulated from hatches of a few 10b p per second. The AA uses a tntal of seven coo l ­

ing systems for longitudinal ccohnq of different 'reqions' of the beam and for horizontal and v e r t i ­

cal emittance cool ing. Time constants are of the order of a second at up to 5 * i d ' ' p or 30 minutes

for 5'101! p, thus nearing desiqn spec i f i ca t ions . The AA is at the heart of CERN1s antiproton pro­

gramme, which culminated in the observation of the Intermediate Vector Bosons predicted hy the unify­

ing electroweak theory. As you know these neu1 par t i c l e s were produced in the c o l l i s i o n s of dense

proton- and antiproton bunches m the SPS as proposed hy C. Rubhia and co-workers.

[ri the ISR stochast ic 'post-coo!inq' o f antiprotons fron tne AA was used (a-noiost other app lua-

f n n s ) to improve the beam l i fet ime and the résolut ion in conjunction w'th an ^nte^-na' hydroqpn iet

target . In th i s way charmoniun s tates formed m proton antiproton >nteractions could be observed

with high oret i^i^n. This was another <mportant achevèrent of the ISR iust pr'or tn u s f'na'. shut­

down. In fact the very last ISR bean was sucn an antiproton bejm c r c - l a t i n g ring 2- '.x was

f i n a l l y dumoed at 6.00 h on ?S J-jnp 19R4 thus ¡ir-nuinn to a déf in i t» pnd t^p a'amrr-nus career of a

unique machine. The use of un interna' target in conjunction witn phase-space coo 1 'no of the c r c u -

1 atinq beam - - as proposed by the Novosibirsk group many years before — had thus been put to work

for the f i r s t time. This arrangement has stimulated much interest as an option for LEAR and as onp

of the basic techniques for the ion coolinq rings now under riesiqn or c o n s t r u c t s .

The low-energy antiproton ring LEAR, which after ISR and SPS became the third customer of the

AA, has given high qual i ty p beams to i t s 17 user qroups on an operational basis s ince 1983. One

part icu lar i ty of LEAR is an "adjustable" system which allows s tochast ic cooling at many different

energies . In 1985 alone, beam was extracted at 17 different momenta between 105 MeV/c (correspond­

ing to 5.8 Met/ kinet ic energy} and 1.7 GeV/c (~ 1 GeV). Relat ive ly fast coolinq witn time constants

of 2 tc 5 minutes for up to 4 x p works at 3 or 4 s tra teg ic energies; slow coolinq with time

constants of 10-20 minutes can be used at prac t i ca l l y al l momenta to keep the heam in shape riurinq

the one-hour extract ion. In conclusion s tochast ic cool ing has wade a unique proqranroe of antiproton

physics poss ib le at CERN.

From about 1978 onwards, other laboratories , e s p e c i a l l y the Novosibirsk nmun, who had pioneered

electron cooling before, an ANL-LBL-Fermilab col laboration and, ncre recent ly , a qroup at the INS

Tokyo, have done work (boih experimental and theore t i ca l ) on s tochast ic coohnq . This work has

placed emphasis on various important aspects such as low-noisp cryogenic ampli f iers , very high

frequency systems, cooling of heavy ions, or coolinq bunches.

In 1903 the Tevatron-1 project at Fermi'ab vid the p co l l ec tor AC0L to he added to the AA at

CERN were approved. Both systems aim at s tochast ic cooling and stackinq of antiprotons at a rate of

several 10 p per seccnd. Some of the new features are cryoqenical ly cooled components on the low-

level s'd-. (ampli f iers , terminating r e s i s t o r s and to some extent even cables and the pick-up plates

themselves) to improve the s igna l - to -no i se ra t io and bandwidths in the Gigaherz ranqe. The Fermilab

antiproton source is now (Dec. 1986) in i t s f inal runninq-in phase; coolinq and stackinq rates c lo se

to design performance have already been obtained, at least in t e s t s with protons. As in the CERN

case the Fermilab source is e n t i r e l y based on s tochast ic coolinq: the ñ GeV debuncher rinq uses fast

anittance cooling (x a 2 sec at 7 x 10 y par t i c l e s ) of the p-pu'se from the production tarqet . The

7.9 GeV accumulator ring combines s tochast ic coolinq for stackinq in momentum space with emittance

cool ing to improve the transverse density of the stack. In the accumulator alone there i s a total of

s ix spec i f ic coo/inq systems ail working the I to 2 or 2 to 4 GH2 rana?, each of them ccvnbmma ¿

large number of pick-up and kicker loops. The system for momentum coulinq nr the stack t a i ' — which

i s the largest system — uses more than 100 pick-up and kicker u n i t s .

The ACOL ring at CERN, under construction s ince September 1986, is planned to come into opération in

late summer 1987. Together with the AA, which i s being modified to work in cascade with ACOL, it

should improve the p flux by an order of maqnitude compared to tnat avai lable with the AA alcne. The

number and the complexity of the cool'ng systems of AA-AC0L is as 'mprpss've as in thp Fermilab case.

- S30 -

There has thus been a rapid development of s tochast ic cooling over the last decade and roughly

one order of magnitude has been gained every four years in the cooling power, i . e . in the number of

par t i c l e s which can be cooled with a time constant of 1 s . This has become possible by making larger

and larger bandwidths avai lable . Probably an 'absolute' limit of the cooling power in the range of

10 9 to 1 0 1 J par t i c l e s per second will be reached unless bandwidths and frequencies nuch above 10 GH2 can be used where (most) vacuum chambers transmit waveguide modes and where the beam s i z e becomes

comparable to the RF-wavelength.

Since about 1980, interest in cooling of heavy ion beams developed rapidly, and the combination

of s tochas t i c 'pre-cool ing' with post-cool ing by electron* looks a t trac t ive for some appl icat ions- A

number of ion cooling rings with some resemblance to LEAR are being planned in the USA, Japan and

Europe. All of them foresee electron coolinq and many plan to use both e lectron- and s tochas t i c

damping. Three ion coolers: TARN II at the INS, Tokyo; the IUCF-cooler at Bloonnngton, Indiana and

CELSIUS at Uppsala should come into operation in 1987 or early 1988. Six others, TSR at the 1*1

Heidelberg, ESR at GSI, Oarmstadt, ASTRIO at Aarhus, CRYRING at Stockholm, COSY at KfK Jül ich, and

the RNCF-cooler at Osaka are authorized or at l eas t partly authorized projects . Other ion cooling

rings are being planned at Oak Ridge, Berkeley and Brockhaven National Laboratory. Thus in the

coming years phase space coo l '" i — both by electrons and by the s tochast ic method — wil l be used t o

a very large extent at low and medium energy.

For the very highest energies , ideas on bunched beam cool ing are being followed up, and a

thorough study on s tochast ic cooling of bunches in the SPS c o l l i d e r has been carried out. It i s

being complemented by the study of some of the components needed such as the pick-ups and kickers as

well as the system to transmit cooling s ignals over long dis tances .

Table Al

History

LÎDUVi 1 le

Schot iky MURA group (Lichtenberg, M i l l s , Sess l er , Stahle, Symon)

van der Meer

ISR s ta f f (Borer, Bramham, Hereward, Hiibner, Schnell Thorndahl)

van der Meer

Schnei 1

Hereward

Bramham, Carrón, Hereward, Hübner, Schnei 1, Thorndahl

Palmer [BNLJ, Thorndahl

S tro l in , Thorndahl

Rubbia

Thorndahl, Carrón

Thorndahl, Carrón

Sacherer, Thorndah1, van der Meer

ICE team

Herr

AA team

LEAR team

Kilian

Novosibirsk group

ANL-LBL-Fermilab group

TARN group at INS Tokoyo

Fermi lab group

AA team

Berkeley group

SPS p team

Fermi 1 ab group

AA-ACOL team

Groups in several d i f ferent labs

1B33

1938

1956-58

1953

197?

1972

1972

1972-74

1975

1975

1975

1975

1976

1977

19/ / -78

1978

1978

1981-82

h I

1980/82

1979/83

1983/84

1983

Prehistory

Invariance of phase space area

Noise in o.e. e lectron beams

L iouv i l i e ' s theorem applied to part ic le storaqe rings

History Idea of s tochast ic coolinq

Observation of proton beam Schottky noise in the ISR

19-

19 36

19- -87

1 9 ^ -

Theory of emittance coolinq

Engineering s tudies

Refined theory, low-intensity cooling

First experimental demonstration of emittance cooling

Idea of low-intensi ty momentum cooling

p accumulation, schemes for the ISR usinq s tochast ic coolinc

p accumulation, schemes for the SPS

Experimental demonstration of momentum cool ing

F i l t er method of momentum cooling

Refinement of theory; imperfect mixing, Fokker-Planck equations

Detailed experimental ver i f i ca t ion

Demonstration of bunched beam cooling

Accumulation of several 1 0 1 1 p from batches of several 1 0 b , coolinq times c l o s e to desiqn spec i f i ca t ions

Stochastic cooling to permit loss - free deceleration of p. Interest in combining s tochast ic and electron cool inq.

Stochastic cooling of heavy ions (proposal)

Stochastic coolinq experiments, work on coolinq theory

Theoretical and hardware s t u d i e s , s tochast ic cooling experiments at FNAL

Stochastic coolinq experiments in the TARN rinq

Design report of the Tevatrcn-1 project usinq a debuncher ring and an accumulator ring for fast sU.:hast ic coolinq and stackinq of 4 * 10 7 p per second

Oesiqn report of the p co l l ec tor ACOL usinq fast momentum

and emittance cooling and stacking of 4 * Ï 0 7 p per second

p co l l ec t ion scheme for 20 TeV col l ide*- ' . Collection of

5 * 10 B p per second

F e a s i b i l i t y study of bunched beam cooling in the SPS col l i der

Construction and running in of p source

Construction of ftCOL

Construction of (heavy) ion cooling r*nqs

Appendix 2

DISPERSION INTEGRALS

In th i s appendix we wish lo have a brief look at the dispersion ; nteqral iq. •' 3. f5D ) as req.i'r°d

for the 'signal shielding' ca lculat ions . A more general discussion is given iiy H.5: he>eward I j ) in

the context of Landau damping.

To deal with the s ingular i ty of the integrand we assume that the eiqenfrequfncy of the te=it

part ic le has a small imaoinary part, I . e . we take WJ •* LUJ + u such that the free o s c i l l a t i o n

A . E I U J J T corresponding to Eq. {3.55} is damped. Later we go for the limit i •* 0.

With a complex eigenfrequency Eq. {3.60) becomes

S(U») = T " / J ^ tb. .

we are e spec ia l l y interested in the contribution due to the damping term a:

The main contribution to this integral comes from the range o j = m * 'a few t ines near the

p-;l n . For' small * we can usually assume that G(-i)j) and n ( ^ ) are constant in this ranqe and thus

take the weighting function G(-jjj)n(uj) = G(^)n(u>) out of thf 'nlenral. Inteqt at i rig 1 he rest From

a minimum : J J •'. u to a maximum eigenfrequency > -U

and in Lhe 1imit <

- G(.,)n(„) ; atan

• ^ G ^ ) n ( ^ ) • u

Clea; 1/ th i s is the residuum 'Jue to the p ; l e of the interjrand. %>' to the :)hj«,ic*. nf ¡>IP .T ;ihI'.*» l.hc

value - n has to be retained.

'tie 'unaininc part jf the integral >s the pi->nopal value. It can be oxp- P S S 1 ^ in '.PÏTIS yf the

Hilpert transform ( a e e Eide'lyi et a l . , tables of intpgral transforms Vol. 2, MacGr.iw Hill N.Y. 1954)

defintj'l by

H[f(x)j = g(y) - ^ - d «

This transform has been tabulated for a large c o l l e c t i o n of functions. In terms of the Hilpert

transform the principal value of Eq. (3.60) may be written as

Further d e t a i l s depend on n(wj] and G(iuj). For betatron cool ing G(JJJ ) i s ideal ly constant (and

purely imaginary and negative) whereas n(uj) is "bell shaped" around the average betatron frequency

(or the centre of the Schottky band). It i s therefore convenient to work in terms of the devia­

tion from uß denoting x = u>j - t^, y - u> - up. Then

S(y) = ^ [ - i G ( y ) M i ( y ) + H ( G ( x ) - n ( x ) J .

Two d is tr ibut ions can serve as models and permit the construction of approximations tike Eq. (3 .61)

above:

1) The semi-circular dis tr ibut ion (which models a d i s tr ibut ion with sharp cut o f f l

For constant G one obtains

G r, T S(y] = - —TI (y + i / a - - y ) .

P

2) The Lorentzian d is tr ibut ion (which models a d i s tr ibut ion with important t a i l s ) :

which for constant G y i e l d s

- 53-1 -

ELECTRON COOLING

H . P o t h * 1

K e r n f o r s c h u n g s z e n t r u m K a r l s r u h e , I n s t i t u t f ü r K e r n p h y s i k , K a r l s r u h e , F e d . R e p . Germany .

ABSTRACT

A c o m p r e h e n s i v e i n t r o d u c t i o n t o e l e c t r o n c o o l i n g i s g i v e n . A f t e r a b r i e f e x p l a n a t i o n o f e l e c t r o n c o o l i n g a n d i t s a p p l i c a t i o n s , t h e r e a d e r i s i n t r o d u c e d t o t h e o r y by a s i m p l e a p p r o a c h . N e x t , e x p e r i m e n t a l a s p e c t s o f a n e l e c t r o n c o o l i n g d e v i c e a r e d i s c u s s e d . T h e n , t h e t h e o r y i s d i s c u s s e d i n more d e t a i l . T h i s i s f o l l o w e d by a summary o f t h e e l e c t r o n c o o l i n g e x p e r i m e n t s a n d a c o m p a r i s o n b e t w e e n t h e o r y and e x p e r i m e n t a l r e s u l t s . A t t h e e n d , f u t u r e a p p l i c a t i o n s o f e l e c t r o n c o o l i n g a r e p r e s e n t e d .

I . INTRODUCTION

T h i s l e c t u r e i s i n t e n d e d t o p r o v i d e a n i n t r o d u c t i o n t o e l e c t r o n c o o l i n g a n d i t s m a j o r

a p p l i c a t i o n s i n a c c e l e r a t o r s a n d o t h e r f i e l d s o f p h y s i c s . I n t e n t i o n a l l y , i t i s k e p t a t a

f u n d a m e n t a l l e v e l and a r i g o r o u s t h e o r e t i c a l t r e a t m e n t o f t h e p r o c e s s i s s o m e t i m e s f o r f e i t e d

i n f a v o u r o f s i m p l e u n d e r s t a n d i n g and c l a r i t y .

A p a r t f r o a t h e e l e c t r o n c o o l i n g t h e o r y t h e p r a c t i c a l a s p e c t s a r e w o r k e d o u t a n d d i s ­

c u s s e d i n d e t a i l . F u r t h e r m o r e , a summary o f p a s t c o o l i n g e x p e r i m e n t s and a n o u t l o o k on

f u t u r e s y s t e m s i s g i v e n . P o s s i b l e e x p e r i m e n t s u s i n g t h e e l e c t r o n c o o l e r as a d e v i c e f o r

a t o m i c - p h y s i c s e x p e r i m e n t s w i l l a l s o be d i s c u s s e d .

E l e c t r o n c o o l i n g as a m e t h o d t o i m p r o v e t h e p r o p e r t i e s o f s t o r e d i o n beams was p r o ­

p o s e by C . Budker i n 1 9 6 6 . He a n d h i s g r o u p b u i l t t h e f i r s t e l e c t r o n c o o l i n g d e v i c e a n d

d i d t h e f i r s t c o o l i n g e x p e r i m e n t s a v N o v o s i b i r s k . T h e y a l s o l a i d down t h e t h e o r e t i c a l f r a m e ­

w o r k . A l i t t l e l a t e r , e l e c t r o n c o o l i n g e x p e r i m e n t s w e r e a l s o p e r f o r m e d a t CERN and a t F e r m i ­

l a b . T h e g e n e r a l a i m o f t h e s e e x p e r i m e n t s was t o t e s t a t e c h n i q u e w h i c h e v e n t u a l l y c o u l d

a l l o w t h e a c c u m u l a t i o n of. a n t i p r o t o n s . H o w e v e r , o w i n g t o t h e h i g h e n e r g y o f t h e p r o d u c t i o n

maximum f o r a n t i p r o t o n s , i t t u r n e d o u t t h a t t h e a p p l i c a t i o n o f s t o c h a s t i c c o o l i n g was more

a p p r o p r i a t e . I n s p i t e o f t h a t , t h e s p e c t a c u l a r r e s u l t s w h i c h came f r o m t h e f i r s t p i o n e e r i n g

e x p e r i m e n t s e n c o u r a g e d many p h y s i c i s t s t o use e l e c t r o n c o o l i n g i n l o w - e n e r g y i o n r i n g s i n

o r d e r t o i m p r o v e l u m i n o s i t y and r e s o l u t i o n i n e x p e r i m e n t s w i t h s t o r e d beams and i n t e r n a l

t a r g e t s .

2 . WHAT ELECTRON COOLING I S

E l e c t r o n c o o l i n g i s a f a s t p r o c e s s t o s h r i n k t h e s i z e , d i v e r g e n c e , and e n e r g y s p r e a d o f

s t o r e d c h a r g e d - p a r t i c l e beams w i t h o u t r e m o v i n g p a r t i c l e s f r o a t h e b e a n . S i n c e t h e number o f

p a r t i c l e s r e m a i n s u n c h a n g e d and t h e s p a c e c o o r d i n a t e s and t h e i r d e r i v a t i v e s a r e r e d u c e d ,

t h i s means t h a t t h e phase s p a c e o c c u p i e d by t h e s t o r e d p a r t i c l e s i s c o m p r e s s e d . I t a l s o

e n t a i l s t h e t e m p e r a t u r e o f t h e beam - - i f l o o k e d upon as a gas - - b e i n g r e d u c e d . We know o f

o t h e r c o o l i n c p r o c e s s e s w h i c h a c h i e v e s i m i l a r r e s u l t s . These a r e s t o c h a s t i c c o o l i n g , s y n -

c h r o t o n r a d i a t i o n c o o l i n g , and l a s e r c o o l i n g .

* ) V i s i t o r a t CERN, G e n e v a , S w i t z e r l a n d

3 WHY ELECTRON COOLING?

The most frequent a p p l i c a t i o n of e l e c t r o n c o o l i n g be:ng cons idered at present i s the

l o s s - f r e e compress ion of ion b e a n s . The r e d u c t i o n of beam s u e permits the l u m i n o s i t y t o

be c o n s i d e r a b l y i n c r e a s e d for c o l l i d i n g - b e a m experiments For f i x e d - t a r g e t exper iments the

small beam s i z e prov ides an e x c e l l e n t d e f i n i t i o n of the i n t e r a c t i o n v e r t e x . The r e d u c t i o n

of momentua spread paves t h e way for h i g h - r e s o l u t i o n e x p e n o e n t s with i n t e r n a l t a r g e t s .

L o s s - f r e e compress ion of phase space a l l o w s the accumulat ion of s p e c . e s of charged

p a r t i c l e s not abundantly a v a i l a b l e , such as l i g h t t o heavy ions or p o l a r i z e d protons and

d e u t e r o n s . E l e c t r o n c o o l i n g very r a p i d l y reduces the e m i t t a n c e of such beam p u l s e s i n j e c t e d

from a c y c l o t r o n or a Linac i n t o a s t o r a g e r i n g and s o c r e a t e s new space for subsequent

p u l s e s . Even with low primary-beam i n t e n s i t y h igh s tored-beam c u r r e n t s could be ach ieved

through s t a c k i n g and accumula t ion .

The p o s s i b i l i t y of working with small beams has another important a s p e c t , namely i t

a l l o w s one t o Jceep the d imens ions of the vacuum chambers and the magnet gap of s t o r a g e

r i n g s s m a l l , which c o n s i d e r a b l y reduces f i n a n c i a l expense .

In a d d i t i o n t o p h a s e - s p a c e compress ion , e l e c t r o n c o o l i n g may be a p p l i e d t o compensate

beam-heating e f f e c t ? . These are predominant ly intrabeam, r e s i d u a l - g a s , and i n t e r n a l tarnet

s c a t t e r i n g . The c o u n t e r a c t i o n a g a i n s t i n t r a b e a n s c a t t e r i n g p e r m i t s , for i n s t a n c e , to keep

even i n t e n s e s t o r e d beams small and t o have, a t the same t ime, a small momentum spread .

With e l e c t r o n c o o l i n g , r e s i d u a l - g a s s c a t t e r i n g l o s s e s can be reduced fron m u l t i p l e - s c a t t e r

ing (gradual growth of e m i t t a n c e u n t i l r ing a c c e p t a n c e i s reached) t o s i n g l e - s c a t t e r i n g

l o s s e s (on ly t h o s e p a r t i c l e s are l o s t t h a t s u r p a s s in a s i n g l e s c a t t e r the r ing a c c e p t a n c e

a n g l e ) , which c o n s i d e r a b l y i n c r e a s e s the s t o r e d bean l i f e t i m e . I t r e l a x e s s t r i n g e n t vacuum

requirements in q u i t e a nunber of c a s e s . On the o tner hand, i t a l s o g i v e s a c c e s s to the

o p e r a t i o n of s t o r e d beams at low e n e r g i e s , even of heavy i o n s , with reasonably long l i f o

t i m e s .

Compensation of energy l o s s and bean blow-up, coming from bean; i n t e r a c t i o n with an

i n t e r n a l t a r g e t , permits the o p e r a t i o n of t a r g e t s a t the maximum t h i c k n e s s and - - s i n c e the

beam p a r t i c l e s are c o n t i n u o u s l y r e c y c l e d -- the achievement of high - l u m m o s i t y , wel l de f ined

i n t e r a c t i o n v e r t i c e s , hiqh-momentum r e s o l u t i o n , and c l e a n background c o n d i t i o n s , p a r t i c u ­

l a r l y a t low energy .

The e l e c t r o n c o o l i n g arrangement p r o v i d e s more than an e x c l u s i v e c o o l i n g d e v i c e . It i s

a l s o , a t the same t ime, an e l e c t r o n t a r g e t wi th which the s t o r e d beam i n t e r a c t s . In par

t i c u l a x , e l e c t r o n - i o n i n t e r a c t i o n s a t low r e l a t i v e e n e r g i e s can be un ique ly s t u d i e d . Atonuc-

p h y s i c s exper iments i n v e s t i g a t i n g , for i n s t a n c e , r a d i a t i v e c a p t u r e , d i e l e c t r o n i c recombina­

t i o n , and Rydberg atoms f ind here near ly i d e a l c o n d i t i o n s . Such a t e c h n i q u e a l s o f i n d s

a p p l i c a t i o n s i n the eventua l product ion of a n t i h y d r o g e n . Moreover, r a d i a t i v e recombinat ion

may be s u i t e d t o the product ion ot monoenerget ic photons in the vacuum u l t r a - v i o l e t (UUVÎ

and the X-ray r e g i o n s The e l e c t r o n c o o l i n g arrangement may hence become a very iraporrant

t o o l for modern a t o n i c p h y s i c s .

4. TON-BEAM AMD STORAGE-RING PROPERTIES

We c o n s i d e r an ion beam of nominal momentum p c i r c u l a t i n g with the f r a c t i o n ß of the

speed of l i g h t c . The beam p r o p e r t i e s are c h a r a c t e r i z e d by i n v a r i a n t q u a n t i t i e s , which

d e s c r i b e the phase space occupied by the i o n s . The s i x - d i m e n s i o n a l phase space i s generated

by the t r a n s v e r s e and l o n g i t u d i n a l c o o r d i n a t e s and t h e i r d e r i v a t i v e s In the h o r i z o n t a l

plane we have, for i n s t a n c e , a s the c o o r d i n a t e s the d i s t a n c e of an ion from the nominal

o r b i t x. and i t s a n g l e xj = with r e s p e c t t o the nominal beam t r a j e c t o r y . In g e n e r a l , i t

i s assumed that the p a r t i c l e s of a beam are normally d i s t r i b u t e d m t h e s e c o o r d i n a t e s a c ­

cording t o

\ 1

The product of thp width of the s p a t i a l d i s t r i b u t i o n (beam s i z e ) and the angular d i s t r i ­

but ion (beam d i v e r g e n c e ) CJx , i s the beam emit tance e. This q u a n t i t y d i v i d e d by Pi i s the

normalized e m i t t a n c e . I t remains cons tant i n the absence of c o o l i n g and h e a t i n g a t any

energy

The t r a n s p o r t of ions in a r ing i s d e s c r i b e d by the f o c u s i n g f u n c t i o n s $ I s ) , where s

i s the c o o r d i n a t e a long the nonina l o r b i t . At any p o i n t i n the r ing i t r e l a t e s bean s i z e ,

d i v e r g e n c e , and e m i t t a n c e .

<*.> = yf~f7isï, < o = M— • i2ï 1 71 1 Jnß ís)

The q u a n t i t i e s of c, <x^>, < x j > > and (3 are u s u a l l y g i v e n in I'mmmrad, mm, mrad, and m,

r e s p e c t i v e l y . A f i n i t e t r a n s v e r s e beam si2e and d i v e r g e n c e are due t o betatron ( t r a n s v e r s e )

o s c i l l a t i o n s o£ the i o n s around the nominal o r b i t , which i s in s i n u s o i d a l approximation

g i v e n by

V . l - / T c « ß i . 6 ) . , 3 ,

where the period of the b e t a t r o n o s c i l l a t i o n Q i s c a l l e d the tune . The q u a n t i t y 6 i s the

i n i t i a l phase oE the ion and R i s the r ing c ircumference d i v i d e d by 2n.

In the l o n g i t u d i n a l plane the beam i s c h a r a c t e r i z e d by a d i s t r i b u t i o n of the ion

momentum around the nominal beam momentum. Only c o a s t i n g beams a i e c o n s i d e r e d . Again we

assume here a normal d i s t r i b u t i o n . Ions w^ch nomenta d i f f e r e n t from the nominal bean

•omentum c i r c u l a t e on d i f f e r e n t o r b i t s . Their r a d i a l d i f f e r e n c e r fron the n o i i n a l o r b i t a t

any p o i n t i s r e l a t e d t o t h e i r off-nonentum ûp^ by the d i s p e r s i o n func t ion D ( s ) ,

A p i

r . = D(a) - ~ , {A)

with D(s) u s u a l l y g i v e n in m.

At a g iven s p a t i a l p o s i t i o n i n the r i n g ( s , x , z ) we can d e f i n e the l o c a l beam tempera­

t u r e in the r e s t frame in a l l t h r e e p lanes as

T h = n c'ßV < x : > 2 , T v = « c 2 ß V <Z ; > J . T ( = ac2 p' ( A p ^ p ) * , T x = T h + T y . ( 5 )

In t h i s d e f i n i t i o n the t r a n s v e r s e beam temperature changes wi th the p o s i t i o n i n the r i n g . We

may d e f i n e an average t r a n s v e r s e beam temperature by us ing the beam emit tance [Eq. ( 2 ) ] and

the average p-£unct ion

Tj_ = „ V A _ ! _ . _ ! _

Beam l i f e t i m e

Repeated s m a l l - a n g l e s c a t t e r i n g of s t o r e d i o n s from r e s i d u a l gas p o l e c u l e s l e a d s t o an

e m i t t a n c e growth r a t e g i v e n by

d_£ = 10 S <P* >P _ 106RP

where P i s the r e s i d u a l gas p r e s s u r e in n i t r o g e n e q u i v a l e n t . When g i v i n g <r3 > and R in m and

P in Torr the dimension of d e / d t i s i:• mn'mrad" 5" 1 . The e m i t t a n c e grows u n t i l the machine

a c c e p t a n c e Ä i s reached and the beam g e t s l o s t . The beam l i f e t i m e T q s r e s u l t i n g from t h i s

p r o c e s s ( m u l t i p l e s c a t t e r i n g ) , i s g i v e n f o r a round vacuum chamber by

Q.a5(fl/ir)6-Vo, PR ( 9 1

For a r e c t a n g u l a r chamber À has t o be replaced by 2.5Ä (1/A = 1 /A h + 1 /A V ) -

Whenever m u l t i p l e s c a t t e r i n g i s c o u n t e r a c t e d by c o o l i n g , then only t h o s e p a r t i c l e s ge t

l o s t which undergo s c a t t e r s l a r g e r than the machine acceptance an g l e BQ . The c r o s s - s e c t i o n

for tha t i s g i v e n by

( 1 0 )

Here z i s the charge of the i on and Z tha t of the gas n u c l e i , and i g i s the c l a s s i c a l

e l e c t r o n r a d i u s ; m i s the e l e c t r o n mass. The s i n g l e s c a t t e r i n g l i f e t i m e i s g i v e n by

im •

Here A i s the a tomic weight and 0 the d e n s i t y of the r e s i d u a l g a s , N^ i s Avogadro's number

and cQ i s the r e s i d u a l gas d e n s i t y a t normal p r e s s u r e PQ = 760 Torr.

For a r e s i d u a l gas c o n s i s t i n g of two components wi th r e l a t i v e abundances K and 1 - K, t h e f o l l o w i n g h o l d s :

_ L - * , i__!<

5. HOW ELECTRON COOLING WORKS IK PRINCIPLE

In order to coo l a s t o r e d ion beam with e l e c t r o n s , a near ly monochromatic and p a r a l l e l

e l e c t r o n bean i s caused t o o v e r l a p with the ion beam i n one of the s t r a i g h t s e c t i o n s of a

s t o r a g e r i n g . The v e l o c i t y of the e l e c t r o n s i s made equal t o the average v e l o c i t y of the

Fig- 1 Schematic of the e l e c t r o n c o o l i n g

arrangement in the c o o l e r r ing

[ the dashed arrows are e l e c t r o n s )

F i g . 2 Proton beam as seen from the c o o r d i n a t e

s y s t e n where the e l e c t r o n s ( d o t s ) are a t

r e s t .

1 2 3 Passage

F i g . 3 I l l u s t r a t i o n of c o a l i n g a s an energy l o s s ot ions in a f o i l

i ons ( F i g . 1). A c l o s e - u p view of the over lap r e g i o n shows us the i o n s t r a v e r s i n g , under

d i f f e r e n t a n g l e s and d i f f e r e n t v e l o c i t i e s , the s t r e a n of p a r a l l e l e l e c t r o n s alL moving with

the same v e l o c i t y . However, i f we observe the s i t u a t i o n from a frame moving wi th the v e l o ­

c i t y of the e l e c t r o n s , the l a t t e r w i l l a l l be a t r e s t , w h i l e the ions w i l l pass through

the e l e c t r o n gas from any d i r e c t i o n wi th a v a r i e t y of v e l o c i t i e s , resembling the movement of

p a r t i c l e s in a hot gas ( F i g . 2). The ions w i l l undergo Coulomb s c a t t e r s in tha t gas and w i l l

l o s e energy , which i s t r a n s f e r r e d from t h e i o n s t o the e l e c t r o n s through t h i s Coulomb i n t e r ­

a c t i o n reducing the n o t i o n of the i o n s as seen from the r e s t f rane . The e l e c t r o n s are con­

t i n u o u s l y renewed. In t h i s p i c t u r e the e l e c t r o n c o o l e r can be understood a s a heat exchanger .

We may a l s o c o n s i d e r the e l e c t r o n s as being represented by a f o i l moving with the

v e l o c i t y v Q . Ions moving f a s t e r than the f o i l ( e l e c t r o n s ) w i l l p e n e t r a t e i t and w i l l l o s e

energy along the d i r e c t i o n of t h e i r momentum (dE/dx) during each passage u n t i l a l l t r a n s ­

v e r s e components a c e d iminished and t h e i r l o n g i t u d i n a l v e l o c i t y i s equal t o the f o i l v e l o ­

c i t y ( F i g . 3 ) . For s lower i o n s i t i s the same e f f e c t wi th t h e e x c e p t i o n tha t they t r a v e r s e

the f o i l from the o p p o s i t e s i d e . I d e a l l y , a t the end, a l l i o n s w i l l have the sane l o n g i ­

t u d i n a l v e l o c i t y a s the f o i l and no t r a n s v e r s e v e l o c i t y component.

6 . INTRODUCTION TO ELECTRON COOLING THEORY CFOR PEDESTRIANS I

We now want to d e r i v e the f o r e ; which i s r e s p o n s i b l e for the s lowing down of the i o n s

and the c h a r a c t e r i s t i c time which i t t a k e s . Our r e f e r e n c e frame i s s t i l l the system where

the e l e c t r o n s are a t r e s t . A)i q u a n t i t i e s measured with r e s p e c t to t h i s system are marked

VT wi th an a s t e r i s k . Let us cons ider f i r s t a

<on »— — - 1

' s i n g l e e l e c t r o n - i o n c o l l i s i o n . The ion moves

w i th the v e l o c i t y v. and s c a t t e r s from the

e l e c t r o n at an impart- parameter b (Fig 4 )

The momentum t r a n s f e r up from the ion t o

FLg. 4 Kinematics of e l e c t r o n - i o n c o l l i s i o n the e l e c t r o n i s :

1P" = Í «c dt - Í -T77 d t •

with <Í c be ing the Coulonb f o r c e . S in ce we c o n s i d e r t imes from n e g a t i v e t o p o s i t i v e i n f i n i t y ,

we can n e g l e c t t h e l o n g i t u d i n a l part of the f o r c e and can r e p l a c e v r by i t s t r a n s v e r s e com-

From t h i s we can c a l c u l a t e t h e energy l o s s of the i o n , which i s the energy taken by the

e l e c t r o n :

2 m e m v ' V e i

So far we have cons idered a s i n g l e c o l l i s i o n . Now we extend the p i c t u r e t o the s i t u a

t i o n where the i on p a s s e s through a l a r g e number of e l e c t r o n s ( F i g . 5 ) . We have t o i n t e g r a t e

over a l l p o s s i b l e impact parameters t o o b t a i n the energy l o s t by the ion a s i t t r a v e l s a

l ength dx through the e l e c t r o n c loud of d e n s i t y n :

P- = 2i [ b db n* AE* (b) = ^ " Z , f n* In : dx I e m « 2 e 1

This i s the f r i c t i o n a l (or c o o l i n g ) force F exper i enced by the ion: F - dK /dx

F i g . 5 I l l u s t r a t i o n of e l e c t r o n - i o n i n t e r a c t i o n in an e l e c t r o n gas

The l o g a r i t h m i c r a t i o of maximal t o minimal impact parameter i s c a l l e d the Coulomb

logar i thm:

We have t o f ind reasonable c u t - o f f s for the impact parameters The minmurn impact parameter

i s determined by the naxinuta »Omentum t r a n s f e r to the e l e c t r o n ( c l a s s i c a l head-on c o l l i s i o n )

- 540 -

U s i n g t h e c l a s s i c a l e l e c t r o n radius r = e /m we c a n w r i t e

I n a s y s t e m o f c h a r g e d p a r t i c l e s , we know t h a t t h e Coulomb f i e l d i s s h i e l d e d and f a l l s o f f

e x p o n e n t i a l l y w i t h i n a c h a r a c t e r i s t i c r a d i u s r Q , w h i c h i s tt- Debye r a d i u s

I t i s u s u a l l y s m a l l e r t h a n t h e e l e c t r o n beam r a d i u s . O t h e r w i s e , the l a t t e r has t o be t a k e n

a s t h e m a x i m a l i m p a c t p a r a m e t e r . So we c a n f i n a l l y w r i t e for t h e c o o l i n g f o r c e :

y' = _ * 1 L ± . n*Lc(v') = W e V r ^ L ^ / w * - * . ( 2 1

I t s i n v e r s e d e p e n d e n c e on t h e i o n v e l o c i t y l e a d s t o a d i v e r g e n c e as t h e L i t t e r a p p r o a c h e s

z e r o I F i g . 6 ) .

F i g . 6 Shape o f t h e c o o l i n g f o r c e f o c a f r o z e n e l e c t r o n bnas

Now we want t o c a l c u l a t e t h e r a t e o f v e l o c i t y change X a t w h i c h t h e i o n i s s lowed down

i n t h e e l e c t r o n g a s :

U s i n g t h e f o l l o w i n g r e l a t i o n s

2 i ' d t

t h e f r i c t i o n r a t e A c a n be e x p r e s s e d as

( 2 2 b )

U s u a l l y t h e i n v e r s e o f t h e f r i c t i o n r a t e l o r d a m p i n g d e c r e m e n t ) i s d e f i n t d as t n e c o o l i n g

t i m e i :

\ F

Hence we f i n a i o r t h e c o o l i n g t i n e , o b s e r v i n g t h a t r = e ? / m

T h i s i s t h e c o o l i n g t i m e i n t h e e l e c t r o n r e s t f r a m e . I n t h e l a b o r a t o r y f r a m e we o b s e r v e a

c o o l i n g t i m e w h i c h i s ( n o t i n g t h ^ t n = •yn )

w h e r e n i s t h e r a t i o o f t h e l e n g t h o f t h " c o o l i n g s e c t i o n t o t h e r i n g c i r c u m f e r e n c e .

So f a r we h a v e c o n s i d e r e d t h e e l e c t r o n s a s b e i n g s t a t i o n a r y . H o u e v u r , t h e y have a

f i n i t e t e m p é r a t u r e and h e n c e a v e l o c i t y d i s t r i b u t i o n f f v ^ ) , w h i c h we c o n s i d e r t o be

M a x w e l l î a n , c h a r a c t e r i z e d by i t s v e l o c i t y s p r e a d û e .

•y, i i f t v " ) = —, . I f ( v ' ) d V = ;

To a c c o u n t f o r t h i s , we h a v e t o r e p l a c e t h e i o n v e l o c i t y by t ' i e i o n e l e c t r ó n v e l o c i t y d i f

f e r e n c e u = V ( v^ and t o a v e r a g e o v e r t h e e l e c t r o n v e l o c i t y d i s t r i b u t i o n F ('.*) * CF , u ) .

w h i c h g i v e s

? ' l v ) --- • 4-tZ 2 e v1 r n' I . . { v * } f{v ) c? v' ¡ 2 7 ) i e e \ C i e J u | 0

Sine« 1 ihr; v a r i a t i o n o f t h e Coulomb l o g a r i t h m w i t h t h e i o n v e l o c i t y i s saa 1 1 we c a n p j ; i l m

f r o n t o f t h e i n t e g r a l

T h i s e x p r e s s i o n h a s an e l e c t r o s t a t i c a n a l o g y . T h e i n t e g r a l i n r e a l s p a c c r e s e m b l e s j u s t t h e

Coulomb f o r c e o f a c h a r g e d i s t r i b u t i o n a c t i n g on a t e s t c h a r g e . I t can be r e w r i t t e n as

Foc a d i s t r i b u t i o n

0 f o r | v ç | > û f i

i t c a n be 3olved a n a l y t i c a l l y and t h e 3 h a p e o f t h e c o o l i n g f o r c e i s shown i n F i g . 7 .

F o r a H a x w e l l i a n v e l o c i t y d i s t r i b u t i o n i t h a s t o be e v a l u a t e d n u m e r i c a l l y , a l t h o u g h

t h e r e e x i s t s a c o m p l e t e m a t h e m a t i c a l e x p r e s s i o n . The f r a c t i o n a l f o r c e c a n , S o w e v e r , t h e n be

a p p r o x i m a t e d ( F i g . 8 ) by

12iZ e c r n L , -

F i g . 7 Shape o f c o o l i n g f o r c e f o r a n e l e c t r o n beam w i t h r e c t a n g u l a r v e l o c i t y d i s t r i b u t i o n .

Using t h i s approximat ion the c o o l i n g time can be w r i t t e n as

2 v + 2o

Genera l ly the e l e c t r o n temperature i s independent of the beam energy and hence û g i s a

c o n s t a n t of the a p p a r a t u s . There fore we can d i s t i n g u i s h two domains of c o o l i n q

Í) COOLING OF HOT BE¿MS )> û g

Here the c o o l i n g t ime i s p r o p o r t i o n a l t o v ^ 3 . I t corresponds to the r e g i o n where

F -

i l ) Cool ing of 'warm' BEAMS v. << &e

In t h i s domain the c o o l i n g t ime i s p r a c t i c a l l y independent of s i n c e Afi i s c o n s t a n t

and one has an e x p o n e n t i a l damping This s imple nodel g i v e s the f o l l o w i n g s c a l i n g

behaviour of the cooling time:

T i s independent of p for ng - c o n s t

T i s independent of i on beam i n t e n s i t y

The s c a l i n g behaviour w i l l be modif ied l a t e r when r e f j n i n g the nodel For t y p i c a l

numbers (T = 0 . 2 eV, ù le = 1 0 ° , v* = û , n = 10B c m 0 , L_ = 10, n = 0 . 0 1 , i = 1 , ï e e i e e C we f ind a c o o l i n g t ime of about 5 s .

Before we go i n t o f u r t h e r d e t a i l s we w i l l f i r s t d i s c u s s how e l e c t i o n c o o l i n g i s

exper inentaVly r e a l i s e d .

?. EXPERIMENTAL REALIZATION OF ELECTRON COOLING

7.1 E l e c t r o n gun and a c c e l e r a t i o n

The e f f i c i e n t a p p l i c a t i o n of e l e c t r o n c o o l i n g on s t o r e d ion beams depends very much on

the q u a l i t y of the e l e c t r o n beam and the e x a c t matching of both beams. We w i l l now d i s c u s s

the g e n e r a t i o n of a c o l d e l e c t r o n beam.

E l e c t r o n s for c o o l i n g are produced in an M e e t ton gun, where they art3 a c c e l e r a t e d

e l e c t r o s t a t i c a l l y to the d e s i r e d energy . We w i l l d i s c u s s t h i s u s ing the Low Energy

Ant iproton Ring [LEAR) gun ( F i g . 9) as an example. A thormocathode which i s heated

r e s i s t i v e l y t o a température above 1000'C s e r v e s a s an electron .snurce. E l e c t r o n s l e a v e the

cathode in any d i r e c t i o n , forming a cloud in f ront of i t . The energy d i s t r i b u t i o n uf the

e l e c t r o n s f o l l o w s a Maxwel l ian d i s t r i b u t i o n wi th du average v e l o c i t y g i v e n hy the cathode

temperature T g = k T

c a t h (k = Boltzmann c o n s t a n t ) . E l e c t r o n s w i l l be e x t r a c t e d from t h i s rloud

with the h e l p of r i n g - s h a p e d anodes and a c c e l e r a t e d to the d e s i r e d energy with which they

e n t e r i n t o a d r i f t r e g i o n . U s u a l l y the cathode i s at high neqativ«? p o t e n t i a l and the anode

p o t e n t i a l s i n c r e a s e s t r a d i l y t o z e r o . In order to minimize t r a n s v e r s o r l c r l i i c f i e l d

1 Cathode W

2 P.erne shield Ta

3 Hear snik Mo

4 Gas cooled base Cu

5 Caihode feedtlirough AI.Oi

6 Anode (eedlhrgugh AljOi

? Bellows s.s

fl Afodes Ti

9 Anode Cu

10 Anode suDPOtl Al;0>

1 1 Solenoid

i .

F i g . 9 The LEAR e l e c t r o n gun

components t h e c a t h o d e i s s u r r o u n d e d by t h e P i e r c e s h i e l d , a n e l e c t r o d e on c a t h o d e p o t e n t i a l

w h i c h i s m a t c h e d t o such a shape as t o n u l l i f y , t o g e t h e r w i t h t h e p o t e n t i a l g i v e n by t h e

e l e c t r o n c l o u d , t h e e l e c t r i c f i e l d on t h e c a t h o d e s u r f a c e and t o p r o d u c e e q u i p o t e n _ i a J l i n e s

w h i c h a r e p e r p e n d i c u l a r t o t h e t e a m a x i s . W i t h t h e s u b s e q u e n t anodes one t r i e s t o m a i n t a i n

t h i s s i t u a t i o n a s much as p o s s i l e . H o w e v e r , a t t h e end o f t h e a c c e l e r a t i o n co lumn

t r a n s v e r s e f i e l d c o m p o n e n t s c u n a v o i d a b l e .

E l e c t r o n s a r e e m i t t e d f r o m t h e c a t h o d e b e c a u s e o f t h e i r t h e r m a l e n e r g y , i n any

d i r e c t i o n . T h e r e f o r e t h e e l e c t r o n gun i s embedded i n a l o n g i t u d i n a l a a g n e t i c f i e l d

( s o l e n o i d ) , w h i c h has t h e e f f e c t t h a t t h e t r a n s v e r s e m o t i o n s o f t h e e l e c t r o n s a r e

t r a n s f o r m e d i n t o s p i r a l s a b o u t - h e a a g n e t i c f i e l d l i n e s w i t h t h e c y c l o t r o n f r e q u e n c y g i v e n

by

The s p i r a l radius i s

v

r = , (32b) c - c

where v i s the e l e c t r o n v e l o c ty t r a n s v e r s e t o the magnet ic f i e l d . Moreover, the magnetic

f i e l d va lue i s chosen such tha t one r e v o l u t i o n (or a m u l t i p l e of i t ) i s completed when the

e l e c t r o n s enter i n t o the d r i f t r e g i o n . This has the advantage tha t the e f f e c t of r a d i a l

e l e c t r i c f i e l d components are minimized. Otherwise , i t would lead t o a cont inuous s c a l l o p i n g

of the beam. This t echn ique i s c a l l e e resonant a c c e l e r a t i o n ( o p t i c s ) . The c o n d i t i o n for

resonant o p t i c s i s t h a t t h e tim- the e l e c t r o n s need to pass through the a c c e l e r a t i o n column

i s equal t o a m u l t i p l e v of the nverse of the c y c l o t r o n frequency or t h a t , i n o ther words,

the l e n g t h of the a c c e l e r a t i o n r e g i o n XR i s approximate ly

v v

\> = v -r = v -I m yc . (33a)

This means the magnet ic f i e l d s a id have the va lue

B [kG] = 3.4 U m ]

So e l e c t r o n s a r e n o t e x t r a c t e d c r e c t l y f r o m t h e c a t h o d e , b u t f rom t h e s p a c e - c h a r g e c l o u d

i n f r o n t o f i t . U n d e r t h e s e c o n c . t i o n s t h e f i n a l e l e c t r o n c u r r e n t f o l l o w s C h i l d ' s l a w :

wheie the c h a r a c t e r i s t i c propor : o n a l i t y f a c t o r P i s c a l l e d the perveance . The perveance i s

e s s e n t i a l l y d e t e n i n e d by the r 10 of beam r a d i u s r and cathode-anode d i s t a n c e d:

where yP = 10 AV . In our • xanple here the gun perveance i s F = O.b uP. This geomet­

r i c a l perveance can, however, bi reduced by apply ing a smal l er cathode-anode pot*»~*.ial. The

- S I O -

Express ing n through P:

irr epc itr'eflc

one can r e w r i t e the above formula

u

One r e a l i z e s t h a t , for a c o n s t a n t perveance , the r e l a t i v e p o t e n t i a l i n c r e a s e a c r o s s the

e l e c t r o n beam remains c o n s t a n t .

The change of the e l e c t r i c p o t e n t i a l a c r o s s the e l e c t r o n beam has two major

consequences . F i r s t l y , i t l e a d s to a r a d i a l e l e c t r o n v e l o c i t y p r o f i l e of the same p a r a b o l i c

form; and second ly , the E x B s i t u a t i o n l e a d s to an azimuthal d r i f t of the e l e c t r o n beam

with a d r i f t v e l o c i t y g i v e n by

c 2f

As mentioned b e f o r e , the e l e c t r o n s emi t ted from the cathode have a t h r e e - d i m e n s i o n a l

Haxwel l ian v e l o c i t y d i s t r i b u t i o n g iven by Eq. 125) . Applying a v o l t a g e U a c c e l e r a t e s them t o

an energy E:

E = eU + T e f f 110)

where the r i p p l e of the h i g h - v o l t a g e system ÛU and the cathode temperature T r a t n adds up to

T . c : kT + e ûU a s the e f f e c t i v e cathorje temperature . Their f i n a l l o n g i t u d i n a l energy et t catn

spread i s

and the corresponding l o n g i t u d i n a l temperature i s , hence,

e l e c t r o n current can a l s o be reduced by h e a t i n g the cathode l e s s and thus reducing i t s

e l e c t r o n e m i s s i o n . For such a t e m p e r a t u r e - l i m i t e d gun Eg. 134) no longer h o l d s .

In F i g . 9b t h e c a l c u l a t e d e l e c t r o n t r a j e c t o r i e s in an e l e c t r o n gun are shown t o g e t h e r

with the e l e c t r i c p o t e n t i a l l i n e s In ti ie d r i f t reg ion the p o t e n t i a l l i n e s are p a r a l l e l t o

the e l e c t r o n t r a j e c t o r i e s w i th i n c r e a s i n g space between adjacent p o t e n t i a l l i n e s when going

from the c e n t r e t o the edge o f the bean. The r a d i a l behaviour o f the e l e c t r i c f i e l d i s des

c r i b e d by the p o t e n t i a l of a homogeneous charge d i s t r i b u t i o n with sharp boundaries .

c a t h 2 • m

we have, hence ,

In the t r a n s v e r s e d i r e c t i o n no change t a k e s p l a c e wi th r e s p e c t t o the s i t u a t i o n be fore

a c c e l e r a t i o n . Hen^e A = A and A << A . T h e r e f o r e , i n the a c c e l e r a t e d e l e c t r o n beam the

i o n s (observer ) f ind a l o n g i t u d i n a l l y compressed ( f l a t t e n e d ) e l e c t r o n v e l o c i t y d i s t r i b u t i o n

v.ith û << i

7 .2 E l e c t r o n beam t r a n s p o r t and i n t e r a c t i o n r e g i o n

From the gun e x i t onwards the m a g n e t i c a l l y conf ined e l e c t r o n s d r i f t t o the s e c t i o n

where they are bent i n t o the ion beam. This i s a c h i e v e d by a curved s o l e n o i d ( t o r o i d ) - - s e e

F ig . 10 — and an a d d i t i o n a l magnet ic d i p o l e f i e l d . The l a t t e r i s needed t o compensate the

c e n t r i f u g a l f o r c e e x p e r i e n c e d by the e l e c t r o n s and o b l i g e them t o f o l l o w the magnet ic f i e l d

l i n e s . Af ter the t o r o i d the e l e c t r o n s e n t e r the c o o l i n g r e g i o n , where they o v e r l a p with i o n s

a l s o e n t e r i n g t h e t o r o i d ( F i g . 10 ) .

The s i t u a t i o n i n the c o o l i n g s e c t i o n i s i l l u s t r a t e d i n F ig . 11. I t shows the p a r a b o l i c

e l e c t r o n - v e l o c i t y p r o f i l e and the s t r a i g h t l i n e of the i on d i s p e r s i o n . S ince the c o o l i n g

f o r c e F i s p r o p o r t i o n a l t o _ ( v ^ - v e ) / ( v ^ - v e ) 3 , the i o n s are dragged a long the s t r a i g h t l i n e

t o p o i n t A, e x c e p t when they are a t the r i g h t of po in t B. In the l a t t e r c a s e they are con­

t i n u o u s l y a c c e l e r a t e d and l o s t . Using the form of the c o o l i n g force g i v e n i n Eq. (30) (we

F i g . 12 The shape of the l o n g i t u d i n a l c o o l i n g force taking i n t o account r-lertron v e l o c i t y

p r o f i l e and ion-moreen tum d i s p e r s i o n for a zero u n i t t a n c o bi: m

c o n s i d e r here the l o n g i t u d i n a l force component on ly and assume vanish ing t r a n s v e r s e v e l o ­

c i t i e s ) , one f i n d s t h e shape of the l o n g i t u d i n a l f o r c e as g i v e n i n Fig. 12 when the p a r a b o l i c

e l e c t r o n v e l o c i t y p r o f i l e i s taken inco account . This means tha t the c o o l i n g force i s en­

hanced between A and B and a t t e n u a t e d l e f t of A. In order t o get s t a b l e c o n d i t i o n s and

e f f i c i e n t c o o l i n g the beams have t o be very we l l a l i g n e d and the v e l o c i t y of the e l e c t r o n s

c o r r e c t l y chosen . We w i l l come back t o t h i s po in t l a t e r .

At the end of the c o o l i n g s e c t i o n the e l e c t r o n s are separated from the ions aga in by a

t o r o i d a l magnetic f i e l d and d r i f t in a s o l e n o i d f i e l d to the c o l l e c t o r .

7 • 1 Electron, c o l l e c t o r

The c o l l e c t o r i s a very important component. I t has t o reduce the power I u " c a ( . n s t o r e d

in th¿ e l e c t r o n beam t o the lowest p o s s i b l e v a l u e s . This i s done by d e c e l e r a t i n g the e l e c ­

trons before they h i t the c o l l e c t o r . The remaining power 1 U C 0 ^ i s d i s s i p a t e d in the water-

c o o l e d c o l l e c t o r . The c o l l e c t o r i s u s u a l l y a few thousand voJts l e s s n e g a t i v e than the cathode

Another important task of the c o l l e c t o r i s t o gather the e l e c t r o n s wi th very high

e f f i c i e n c y . Secondary e l e c t r o n s c r e a t e d in the c o l l e c t o r or e l e c t r o n s r e f l e c t e d at i t s

e n t r a n c e may bounce back and f o r t h brtween gun and c o l l e c t o r be fore they are l o s t somewhere

on the grounded vacuum chamber w a l l s a t p r a c t i c a l l y f u l l energy. Apart from the gas load

which these l o s t e l e c t r o n s produce, the corresponding l o s s current has t o be provided by the

h i g h - v o l t a g e power supply . S in ce t h i s has to be a h igh ly s t a b i l i z e d power supply , the load

should be as smal l a s p o s s i b l e .

- bhj -

The d e c e l e r a t i o n o f t h e e l e c t r o n s b e f o r e tUe c o l l e c t o r c a n be done w i t h r e s o n a n t o p t i c s

s i m i l a r l y t o t h e gun a c c e l e r a t i o n c o l u a n , o r m e r e l y by p a s s i n g t n e e l e c t r o n s t h r o u g h a r i n g

anodu on t h e d e s i r e d p o t e n t i a l . When t h e e l e c t r o n s a r e d e c e l e r a t e d , a space c h a r g e c l o u d

b u i l d s up f o r m i n g a s i m i l a r s i t u a t i o n t o t h a t i n f r o n t o f t h e c a t h o d e . T h i s c h a r g e has

e i t h e r t o be c c a p e n s a t e d by o p p o s i t e c h a r g e s ( p o s i t i v e i o n s ) or t h e e l e c t r o n s h a v e t o be

s u c k e d away an - ' d i s t r i b u t e d o v e r a l a r g e v o l u m e ; o t h e r w i s e t h e e l e c t r o n b e a n i s r e f l e c t e d

f r o m t h i s v i r t u a l c a t h o d e . T h e l a t t e r i s a c h i e v e d by r e - a c c e l e r a t i n g t h e e l e c t r o n s i n t o t h e

c o l l e c t o r and r e d u c i n g t h e m a g n e t i c f i e l d t o z e r o . The v a n i s h i n g m a g n e t i c f i e l d i n t h e

c o l l e c t o r h e l p s a l s o t o p r e v e n t t h e s e c o n d a r y e l e c t r o n s f r o m l e a v i n g t h e c o l l e c t o r and

e n t e r i n g t h e c o o l e r . E x p e r i e n c e a t F e r m i l a b showed t h a t i t was q u i t e u s e f u l t o f o r m an i o n

t r a p b e t w e e n t h e d e c e l e r a t i o n c o l u m n a n d t h e c o l l e c t o r . T h i s was a c h i e v e d by r u n n i n g a

c y l i n d r i c a l c o l l e c t o r a n o d e a f e w h u n d r e d v o l t s a b o v e t h e c a t h o d e p o t e n t i a l , so t h a t i o n s

w e r e t r a p p e d and c o m p e n s a t e d t h e e l e c t r o n s p a c e c h a r g e

I n p r e v i o u s c o l l e c t o r s a t N o v o s i b i r s k and a t F e r m i l a b , e l e c t r o n l o s s e s a t t h e l e v e l o f

10 and b e l o w w e r e a c h i e v e d . I n F i g . 13 t h e N o v o s i b i r s k c o l l e c t o r i s shown.

F i n a l l y , a n e x a m p l e o f a w h o l e e l e c t r o n c o o l e r a s s e m b l y i s shown i n F i g . 14 , w h i c h

r e p r e s e n t s t h e LEAR e l e c t r o n c o o l e r .

1 solenoid 2 anode 3 electrostatic screen 4: the ihm iron screen 5.6: cooled copper cylinder 7: the thick ¡ion screen 8: insulator 9: the feed through for the electrostatic

screen potential 10: adjustment mechan. 11 : waterguide

F i g . 13 T h e e l e c t r o n c o l l e c t o r ol t h e HAP H e l e c t n - u c o o l e r

10 3 9

- V f - f , , " Y A: idi.»- p.

Fl'J. 11 An I : 1 I Î . - 1 [OI I roo l i t l'J j¿3i 'mUly (I.FAR e l e c t r o n r > . o ! . - )

- SSO -

7.4 H i g h - v o l t a g e s y s t e m

A t y p i c a l s c h e m a t i c d r a w i n g f o r t h e h i g h - v o l t a g e s y s t e m o f an e l e c t r o n c o o l e r i s shown

i n F i g . 1 5 . An e x t r e m e l y w e l l s t a b i l i z e d h i g h - v o l t a g e power s u p p l y ( r i p p l e < 1 0 " * ) p r o v i d e s

a n e g a t i v e p o t e n t i a l t o a h i g h - v o l t a g e p l a t f o r m . T h e c a t h o d e i s d i r e c t l y c o n n e c t e d t o t h e

P l a t f o r m . On t h e p l a t f o r m a d d i t i o n a l power s u p p l i e s p r o v i d e t h e b i a s f o r t h e c o l l e c t o r and

i t s e l e c t r o d e s . T h e p o t e n t i a l f o r t h e a c c e l e r a t i o n ( d e c e l e r a t i o n ) a n o d e t a r e e i t h e r d e r i v e d

f r o m a v o l t a g e , d i v i d e r o r f r o m a u x i l i a r y power s u p p l i e s . T h i s a r r a n g e m e n t h a s t h e a d v a n t a g e

t h a t a l l t h e e l e c t r o n c u r r e n t e s s e n t i a l l y f l o w s t h r o u g h t h e c o l l e c t o r s u p p l y , a n d t h e h i g h -

v o l t a g e power s u p p l y h a s o n l y t o d e l i v e r t h e l o s s c u r r e n t ( a n d t h e c u r r e n t t h r o u g h t h e

v o l t a g e d i v i d e r ) .

F i g . 15 Schema of h i g h - v o l t a g e s y s t e m s f o r c o o l e r s . The dashed

o u t l i n e shows t h e HT p l a t f o r m .

7 S Vacuum s y s t e m

E l e c t r o n c o o l e r s w i l l be used i n s t o r a g e r i n g s o p e r a t i n g under u l t r a h i g h - v a c u u m c o n ­

d i t i o n s (10" ' 0 - 1 0 1 2 T o r r ) . The vacuum s y s t e m o f t h e e l e c t r o n c o o l e r has t o match t h a t and

hence i t s h o u l d be b a x a b l e . T h e m a i n o u t g a s s i n g i n a n e l e c t r o n c o o l i n g d e v i c e comes f rom

a ) t h e h o t c a t h o d e ,

b ) t h e c o l l e c t o r ,

c ) t h e vacuum w a l l s h i t b y l o s t e l e c t r o n s w i t h f u l l e n e r g y .

M a i n l y h y d r o g e n and c a r b o n monox ide a r e p r o d u c e d

I n o r d e r t o k e e p t h e vacuum i n t h e r e g i o n w h i c h i s t r a v e r s e d by t h e i o n bean as low as

p o s s i b l e , a d i f f e r e n t i a l pumping s y s t e m has t o be b u i l t b e t w e e n t h e gun and t h e c o o l i n g

r e g i o n , and t h e c o l l e c t o r and t h e c o o l i n g r e g i o n . T h i s i s d i f f i c u l t s i n c e t h e w h o l j s y s t e «

i s c o n t a i n e d i n t h e s o l e n o i d and a c c e s s i s d i f f i c u l t . S u i t a b l e pumping sys tems a i e c ryopumps

and n o n - e v a p o r a b l e g e t t e r INEG) pumps.

The p r o p e r t i e s o f t h e m a g n e t i c f i e l d a r e v e r y i m p o r t a n t , f i r s t l y , t o p r e v e n t t h e

e l e c t r o n beam f r o m b e i n g h e a t e d up a n d , s e c o n d l y , t o g u a r a n t e e a good c o o l i n g e f f i c i e n c y

T h a t means v a r i a t i o n s o f t h e m a g n e t i c f i e l d s h o u l d n o t t a k e p l a c e o v e r d i s t a n c e s s m a l l e r

t h a n t h e s p i r a l l e n g t h o f t h e e l e c t r o n s o u t s i d e t h e c o o l i n g r e g i o n ( a d i a b a t i c ) . I n t h e

c o o l i n g r e g i o n , h o w e v e r , t h e a n g l e a Q b e t w e e n t h e m a g n e t i c f i e l d l i n e s [ e l e c t r o n t r a } e c -

t o r i e 5 ) and t h e i o n beam s h o u l d e v e r y w h e r e b e s m a l l compared t o t h e a v e r a g e t r a n s v e r s e a n g l e

o f t h e e l e c t r o n s g i v e n by t h e i r t r a n s v e r s e t e m p e r a t u r e

2 Etc 0 . 5 x 10 18

I n F i g 16 t h e m a g n e t i c f i e l d o f t h e LEAR e l e c t r o n c o o l e r b e f o r e f i n a l c o r r e c t i o n i s shown.

E v e n t u a l m a g n e t i c f i e l d e r r o r s a r e u s u a l l y c o m p e n s a t e d by s u i t a b l e c o r r e c t i o n c o i l s i n s i d e

t h e m a i n m a g n e t . T h e r e a r e a l s o s t e e r i n g c o i l s t o a l l o w f o r a d i s p l a c e m e n t o f t h e e l e c t r o n

b e a n .

F i g . 1 6 P l o t o f m a g n e t i c - f i e l d c o m p o n e n t s f o r LEAR e l e c t r o n c o o l e r ( b e f o r e f i e l d

c o r r e c t i o n s ) .

7 . 7 E f f e c t s o f t h e e l e c t r o n c o o l e r on t h e i o n beam

T h e m a j o r e f f e c t s o f t h e e l e c t r o n c o o l e r on t h e i c n beam a r e t h e d e f l e c t i o n a f t h e i o n

b e a i i n t h e t o r o i d s , t h e f o c u s i n g e f f e c t o f t h e e l e c t r o n beam w h i c h p r o d u c e s a t u n p s h i f t o f

t h e i o n b e a o , and t h e c o u p l i n g o f t h e v e r t i c a l and h o r i z o n t a l e m i t t a n c e r . i t : t h e s o l e n o i d .

T h e l a t t e r a l s o p r e c e s s e s t h e s p i n o f t h e i o n a n d t h e s o l e n o i d f i e l d has t o be c o m p e n s a t e d

i f p o l a r i z e d i o n s a r e t o be c o o l e d .

The d e f l e c t i o n o f t h e i o n beam i n t h e v e r t i c a l l y b e n t t o r o i d i s due t o t h e r i s i n g and

f a l l i n g m a g n e t i c f i e l d t h ^ r e , c a u s i n g a v e r t i c a l d i p o l e f i e l d . Thu d e f l e c t i o n a n g l e i s

JB d l BR

B [ r a d ] = -f- , — I n cos . [ 4 4 M

and t h e d i s p l a c e m e n t o f t h e bear* i s

( 4 4 b )

H e r e 6 i s t h e f i e l d o f t h e s o l e n o i d , BQ t 0 i s t h e i o n beam z i g i d i t y ( B n e o ¡ T a ] = 3 . 3 p ( w i t h

p i n GeV" c ' ] ) . R t i s t h e r a d i u s o f t h e t o r o i d and * q i t s a n g u l a r l e n g t h ( F i g . 1 0 ) .

The t u n e s h i f t p r o d u c e d by t h e e l e c t r o n s i s g i v e n by

¿v = 0 . 5 <fi. > r n @ Z i l l , ( 4 4 c ) h , v p e

w h e r e L i s t h e l e n g t h o í t h e c o o l i n g section. For a n e l e c t r o n gun o p e r a t i n g w i t h c o n s t a n t

p e r v e a n c e t h e t u n e s h i f t r e m a i n s unchanged when t h e beam e n e r g i e s a r e v a r i e d , s i n c e n i & 7

[ E q . ¡ 3 7 ) ] .

T h e s o l e n o i d a l n a g n e t i c f i e l d t w i s t s t h e i o n beam by an a n g l e w h i c h i s g i v e n by

T h i s e f f e c t i s m i n o r u n l e s s one i s w o r k i n g c l o s ç t o a mach ine r e s o n a n c e .

T h e s o l e n o i d r o t a t e s t h e s p i n by

v t r a d ] = i G ~ - , ( 4 4 e ) Vu

w h e r e G i s t h e G - f a c t o r o f t h e i o n (C = 1 . 7 9 3 f o r t h e p r o t o n ) T h i s has t o be c o m p e n s a t e d by

a n a d d i t i o n a l s o l e n o i d , o t h e r w i s e t h e beam w o u l d d e p o l a r i z e .

8 . MORE OH THE THEORY OF ELECTRON COOLING

8 . 1 Coo l i n f o r c e

T h e f o r c e g o v e r n i n g t h e e l e c t r o n c o o l i n g p r o c e s s was d e r i v e d by v a r i o u s a u t h o r s . One

d i s t i n g u i s h e s two b a s i c d e s c r i p t i o n s : t h e b i n a r y c o l l i s i o n model by D e r b e n e v and S k r i n s k y ,

a n d t h e p l a s m a p h y s i c a l a p p r o a c h by S t e n s e n and B o n d e r u p . A l t h o u g h t h e l a t t e r has some

a d v a n t a g e s compared t o t h e b i n a r y m o d e l , we w i l l f o l l o w t h e d e s c r i p t i o n o f Dathenev and

S k r i n s k y as i t i s m o i e p r a c t i c a l t o do so h e r e .

I n S e c t i o n 6 we d e r i v e d a c o o l i n g f o r c e mak ing t h e a s s u m p t i o n o f h a v i n g f r e e e l e c t r o n s

w i t h a s p h e r i c a l v e l o c i t y d : s t r i b u t i on. I n S e c t i o n 7 . 1 we l e a r n t t h a t t h e e l e c t r o n s h a v e ,

h o w e v e r , a f l a t t e n e d v e l o c i t y d i s t r i b u t i o n [ E q ( 4 2 c l ] a n d t h a t t h e e l e c t r o n s • • c o n f i n e d by

u l o n g i t u d i n a l f i e l d a r e p e r f o r o i n g r o t a t i o n s a b o u t t h e m a g n e t i c f i e l d l i n e s .

t i e r b e n e v a n d S k r i n s k y h a v e e v a l u a t e d t h e c o o l i n g f o r c e s f o r a f l a t t e n e d d i s t r i b u t i o n :

?*, = - « . Z ! e V r n* e e

( h e r e L , i s d e f i n e d by Eq . | 1 7 ) w i t h b n a x = r c 1 . and

. J ! 2 » 4nZ e c r n e e Lc [v> - r 1 J! W A < v < A (45b)

E | 1 E 4 -

Hence the f l a t t e n e d d i s t r i b u t i o n has the e s s e n t i a l consequence tha t the l o n g i t u d i n a l c o o l i n g

f o r c e f a l l s o f f ] e s s r a p i d l y f o r v. < û a s i n the c a s e of a s p h e r i c a l d i s t r i b u t i o n ; that 1 ej. neans the l o n g i t u d i n a l c o o l i n g i s f a s t e r than the t r a n s v e r s e c o o l i n g for cool ion beaas .

This i s e a s i l y unders tood . The i n f l u e n c e of the magnet ic f i e l d i s more d i f f i c u l t .

«her. an ion s c a t t e r s f r o * a s p i r a l l i n g e l e c t r o n a t impact parameters much l a r g e r than

the c y c l o t r o n rad ius r^ and the c o l l i s i o n t ime t * b/u i s long conpared t o the c y c l o t r o n

r e v o l u t i o n frequency , the e l e c t r o n makes many r o t a t i o n s and on ly the l o n g i t u d i n a l e l e c t r o n

v e l o c i t y has t o be taken i n t o account . S ince the t r a n s v e r s e e l e c t r o n motion i s f r o z e n , no

t r a n s v e r s e momentum i s t r a n s f e r r e d i n t h e s e s low c o l l i s i o n s ( l a r g e impact parameters s r ^ l l

i on v e l o c i t y ) . This type of c o l l i s i o n i s t h e r e f o r e o f t e n c a l l e d a d i a b a t i c c o l l i s i o n . This

f a c t has so far not been taken i n t o account and t h e prev ious c o o l i n g force has t o be

complemented with a magnet ic p a r t .

Usua l ly the c o l l i s i o n s are d i v i d e d i n t o two t y p e s depend in ; on the impact parameter b:

f a s t c o l l i s i o n s . - b . i b < r nun c

a d i a b a t i c c o l l i s i o n s : r < b . c

c o r r e s p o n d i n g l y , the t o t a l c o o l i n g f o r c e i s composed of a non-magnetic f o r c e ?° [Eq

and a magnet ic f o r c e F m . The l a t t e r i s a l s o d e r i v e d by Derbenev and Skr insky:

e e J u C äv » e ,

( 4 5 ) ]

H6a)

2in Z e e i r V jj- 1

: r e — LC -J u

Here L B i s the a d i a b a t i c Coulomb logar i thm with b m = r and b B = nin (r . u L/pc, , c n in If — 7 max o u / u p ) , mp be ing t h e e l e c t r o n plasma frequency up = m n ^ c .

For an i n f i n i t e l y nariow l o n g i t u d i n a l e l e c t r o n v e l o c i t y spread the i n t e g r a l s can be

e v a l u a t e d :

c e c i v-s i x ( 4 1 a )

„ 2 2 2

The c a s e _ i g c a n be e v a l u a t e d a l r e a d y f o r -x f i n i t e l o n g i t u d i n a l v e l o c i t y s p r e a d A p

w h i c h i s s i a j . a r t o t h e n o n - m a g n e t i c f a r c e w i t h a s p h e r i c a l v e l o c i t y d i s t r i b u t i o n o f t h e

w i d t h û V

T h e e f f r o f t h e m a g n e t i c c o o l i n g f o r c e i s t o e n h a n c e t h e c o o l i n g a t low i o n v e l o c i t y

c o n s i d e r a b l y T h i s r e s u l t s i n v e r y s h o r t c o o l i n g t i m e s f o r t h e damping o f s m a l l b e t a t r o n

o s c i l l a t i o n s ind t h e r e d u c t i o n o f s m a l l momentum s p r r a d s of t h e i o n beam. H o w e v e r , ( 4 7 a )

shows t h a t a so t r a n s v e r s e h e a t i n g may o c c u r i f v . < /2 v. . F u r t h e r m o r e , i t i n d i c a t e s t h a t

t h e t r a n s v e r c o o l i n g i s much s l o w e r t h a n t h e l o n g i t u d i n a l .

8 2 C o o l i n q ; . m e s

C o o l i n g t i m e s a r e u s u a l l y d e f i n e d as t h e t i m e i t t a k e s t o d a a p b e t a t r o n o s c i l l a t i o n

a m p l i t u d e s o a momentum s p r e a d by a f a c t o r 1 / e . T h i s assumes e x p o n e n t i a l damping ( c o n s t a n t

c o o l i n g t i m e w h i c h i s u s u a l l y n o t t h e c a s e . I t i s more a p p r o p r i a t e t o u s e a damping

( c o o l i n g ) r a i e . We w i l l t a k e t h e c o o l i n g t i m e i o be t h e i n v e / s e o f t h e d a m p i n g r a t e .

W i t h t h e c o o i n g f o r c e s g i v e n a b o v e , we o b s e r v e t h e f o l l o w i n g b e h a v i o u r o f t h e c o o l i n g

t i n e s :

c o n s t non m a g n e t i c f o r c e

- v . 3 m a g n e t i c f o r c e

i " c o n s t

T h i j i s du e s s e n t i e l . . u r r e c t i u u i J i t h . i m u L . y [Lqt, ( 2 4 ) .jtid I J l j J .

The d i v i s i o n of the c o l l i s i o n s i n t o two regimes i s rather crude . Recent ly the theory

deve loped by S t e n s e n and Bonderup overcame t h i s problem i n a natura l way by d e r i v i n g the

i r i c t i o n a l force from t h e p o l a r i z a t i o n the ion induces i n the m a g n e t i c a l l y conf ined e l e c t r o n

g a s .

S .3 Equi l ibrium

In the end phase of c o o l i n g , e q u i l i b r i u m i s reached between the e l e c t r o n and the ion

bean (under i d e a l c o n d i t i o n s ) , which means T = T . . Provided t h e r e i s no coupl ing between e i

t r a n s v e r s e and l o n g i t u d i n a l phase space and t h e r e a t e no other h e a t i n g e f f e c t s , the f o l l o w i n g

h o l d s :

T = T ar.d T = T. e x i x e A i (

The beam temperature i s r e l a t e d t o the beam d i v e r g e n c e by

Hence equal beam temperatures means

The i on beam d i v e r g e n c e and the momentum spread can then become /m^nT t imes s m a l l e r than

the corresponding v a l u e s for the e l e c t r o n beam. One n o t e s fur ther tha t - - s i n c e A& = c o n s t ,

A P^A 2 , and 8 = A/0c - - i d e a l l y , small i on-bean d i v e r g e n c e s and momentum spreads are e n e j _ .5

reached a t high e n e r g i e s . For 0 = 1 one could u l t i m a t e l y g e t B * 0 .5 mrad and 6 << 10 j_ e i |

Under t h e s e c o n d i t i o n s one could contemplate reaching a f l a t t e n e d i c n v e l o c i t y d i s t r i b u t i o n

with a l o n g i t u d i n a l temperature T- < 1*K. Gases a t t h e s e low temperatures undergo phase

t r a n s i t i o n s , and a t a c e r t a i n s t a g e c r y s t a l l i z a t i o n w i l l take p l a c e . Such e f f e c t s are

p r e d i c t e d to occur for ion beams c o o l e d down t o u l t r a l o w temperatures .

In p r a c t i c e , however, h e a t i n g e f f e c t s w i l l prevent us , in many c a s e s , from reach ing

t h i s r e g i o n . Dominant h e a t i n g p r o c e s s e s a i e t h e s c a t t e r i n g o í an ion from another ion in the

bean (intrabeant s c a t t e r i n g ) , r e s i d u a l gas s c a t t e r i n g , machine i m p e r f e c t i o n s , and m i s a l i g n

o e n t of the beams.

9. RECOMBINATION

When p o s i t i v e i o n s a r e c o o l e d by e l e c t r o n s , o c c a s i o n a l l y c o o l i n g e l e c t r o n s a r e r a d i a -

t i v e l y c a p t u r e d by beam i o n s i n t o a t o m i c s t a t e s w i t h m a i n q u a n t u m number n . The p r o c e s s

- r>M> -

i s i l l u s t r a t e d i n F i g . 7 . i t s c r o s s - s e c t i o n i s

q u a n t u m number . One n o t e s t h a t i t d i v e r g e s f o r d e c r e a s i n g e l e c t r o n e n e r g y E g •* 0 . I n

e l e c t r o n c o o l i n g E f i ( i n t h e r e s t f r a m e ) i s v e r y s m a l l . T h e v e l o c i t y d i s t r i b u t i o n o f t h e

e l e c t r o n s r e q u i r e s a v e r a g i n g a s i n t h e c a s e o f t h e c o o l i n g f o r c e . T h i s a v e r a g e i s c a l l e d

t h e r e c o m b i n a t i o n c o e f f i c i e n t a :

The e v a l u a t i o n of t h e i n t e g r a l f o r a f l a t t e n e d d i s t r i b u t i o n and v . << v r e s u l t s i n

where n Q a j ( i n d i c a t e s t h e s t a t e a b o v e w h i c h t h e i o n s a r e s t r i p p e d i n t h e m o t i o n a l e l e c t r i c

f i e l d o f t h e b e n d i n g a a g n e t ¿nd a r e r e c i r c u l a t e d . L e t us assume h e r e t h a t n ^ ^ = 4: t h e n

E 1 /n = 2 .

T h e r e c o m b i n a t i o n r a t e f-er s t o r e d i o n o b s e r v e d i n t h e l a b o r a t o r y i s

*r«c " " e V " ! ' ( 5 1 i

and u s i n g E q . ( 5 6 )

- 2 . 2 / 1 2 2„2 ,,,, ' r e c = V 1 U F o r e c 2 1 5 , 1

T h i s c a n be compared w i t h t h e c o o l i n g r a t e . L e t us t a k e , f o r i n s t a n c e , t h e i n v e r s e o f

E q . ( 3 1 ) f o r V* << L^-, t h e r . we Y. :ve

T h i s shows t h a t r e c o m b i n a t i o n i s much s l o w e r t h a n c o a l î r . ' i even f o r h<>dvy i o n s

1 0 , ELECTRON COOLING EXPERIMENTS

P i o n e e r i n g e l e c t r o n c o o l i n g e x p e r i m e n t s w e r e done i n t h e NAP M r i n g a t N o v i s i b i r s k , t h e

I n i t i a l C o o l i n g E x p e r i m e n t ( I C E ) r i n g a t CERN, and t h e F e r m i l a b c o o l e r r i n g The e x p e n o e n t s

were pertozaed uith s t o r e d p r o t o n s a t 1 .5, 35 , 4 6 , 8 5 , ' M , and 2 0 0 MeV The c o o l i n g of

c o a s t i n g and b u n c h e d beams was s t u d i e d , and t h e s t a c k i n g and a c - u n u l a t i o n o f p r o t o n p u l s e s

was t e s t e d .

T h e p a r a m e t e r s o f t h e s t o r a g e r i n g s a r e l i s t e d i n T a b l e 1 and t h o s e o f t h e e l e c t r o n

c o o l e r s a r e g i v e n i n T a b l e 2.

T a b l e 1

P a r a m e t e r s o f e l e c t r o n c o o l i n g s t o r a g e r i n g s

NAP-M I C E F e r m i l a b

C i r c u m f e r e n c e [m] 47 74 135

O p e r a t i o n e n e r g y [ M e V ] 1 . 5 - 8 5 46 114, 2 0 0

S t o r e d b e a n i n t e n s i t y 10S -1o" 1 0 6 1 0 s 5 . , C 6

A v e r a g e r i n g vacuum [ T o r r ] 5 * 10''° 2 • 1 0 " i . ,o-'° H o r i z o n t a l and v e r t i c a l

a c c e p t a n c e [ u m m m r a d ] 4 0 0 , 2 0 0 8 0 , 40 4 0 , 20

L o n g i t u d i n a l a c c e p t a n c e [ * ] i ' « 0 . 2 5 + 1

F r a c t i o n o f c o o l i n g s e c t i o n o f r i n g c i r c u m f e r e n c e

0.02 0 . 0 4 0 0 3 7

W o r k i n g p o i n t Q^, 1 . 7 4 , 1 . 3 4 1 71 , 1 16 3 5 7 , 5 57

A v e r a g e h o r i z o n t a l ß f u n c t i o n [m] 6 18 i n c o o l i n g s e c t i o n [ m ] 5 . 2 3 25

ß v i n c o o l i n g s e c t i o n [ m ] r, T 11 40

D i s p e r s i o n i n c o o l i n g s e c t i o n [m] 6 5 7 0 1

earn l i f e t i m e w i t h o u t c o o l i n g [ s ] 1 5 0 0 a 1 2 0 0 6 0 100

a n s i t i o n i t 1 .2 1 3 3 6

a ) A t 65 MeV

Table 2

Parameters oí electron coolers

NAP-H ICE Ferailab

Cathode diaaeter [cm] 1, 2 5 10

Beaa diaaeter [c»] 1, 2 5 5

Electron energy [fcev] 0. 7-46 26 62, 111

Electron current [A] 0.1-C . 8 0 6, 1.3, 2.2 0.5-2

Electron density [106 ciT 3] 0.09-3.7 0.2, 0.4, O B 0.1-0.6

Magnetic field [kG] 1 0.5 0.7, 0.93

Toroidal angle ['] 45 30 90

Length of cooling section [m] 1 3 5

Gun-collector voltage [kV] - 1 - 1.2 - 1

Electron current losses < 10'' - 2.5 x 10"1 < lo"'

1 0 . 1 C o o l i n g f o r c e s and c o o l i n g t i n e s

In these experiments transverse and longitudinal cooling times were measured under various conditions and equilibrium beaa properties were determined. The principal results are shown in Figs. 18 and 19, where the measured cooling times and longitudinal frictional force are plotted against the proton velocity and betatron amplitude, respectively. One distinguishes in Fig. 18b the clear scaling of the transverse cooling time with the trans­verse proton velocity. The data scale approximately as i ± « . From the previous section we would expect [Eg (49)]:

const from non-magnetic force

v*3 from magnetic force f o r v ( A_ - v

The longitudinal cooling force rises rapidly with decreasing velocity (Fig. 19). There the ICE results show foi snail velocities a bend over and a rapid decrease in contrast to the Novosibirsk results, A possible explanation could be that in ICE a beam misalignment or a magnetic-field ripple prevented a further rise of the force to the point where it should then decay linearly [Eqs. (15b) and (18b)], It also could indicate a considerable ripple on the high voltage, reducing the effect of the flattened distribution.

T,IH«V) l|TO*l • IK -

• ii

• Í Í

J NkP-W • D U 7 4 a <.t 4 3 <CE ; - 4 6 27 1 T

-• « TS mp-n • 1 ï 09

•

1 2 3 U 0 T Imrad)

Tnw« T

(sec) 1 14 Q n FW*j ¡

46 0 4.3 ICE 46 0 ??

• 1 5 D 01 MP-*

b)

I I i* 6 8 107 2 3 t. 6 VT k m / s )

C o m p i l a t i o n o f t r a n s v e r s e c o o l i n g t i m e m e a s u r e m e n t s

VL/f!c VL/[(-F i g . 11 C o m p i l a t i o n o f l o n g i t u d i n a l c o o l i n g t i m e / f o r c t - measi i remonl s

The ;:^"osibirsk r e s u l t s show a cont inuous r i s e of the c o o l i n g force with decreas ing

v* beyond v* /ßc = (ü /Sc} (* 2 .5 x 1 0 " 1 ) , whirh i s a c l e a r i n d i c a t i o n of the presence P | e i

of t h e magnetic f o r c e . In none ol t h e c o o l i n g exper iments , however, was the reg ion where

v p í ¿ e checked for a l i n e a r dependence o f the magnetic force on the proton v e l o c i t y r' The"Novosibirsk group has der ived semi -empir ica l formulae, which d e s c r i b e the data

rather we l l :

U - ! - ¡ — (-„Vc2 * »i . 1 ,v | ) It\ * v i . ,J (5

, ! , I A " . r

Here aQ accounts for a p o s s i b l e magnet ic f i e l d r i p p l e or a misal ignment angle between the

e l e c t r o n and the proton beam.

The l o n g i t u d i n a l f r i c t i o n a l force i n Eg. {59b) s c a l e s l i k e :

FM " - ¡ — for v_ ) Û II *2 II e,

and

F „ _ L _ f 0 I < t .

û V ±

10.2 Equil ibrium

The equ i l ibr ium proton beam p r o p e r t i e s were determined in the prev ious c o o l i n g measure­

ments for var ious c o n d i t i o n s . F ina l beam emi t tances wel l below 1 n m a u r a d were a c h i e v e d .

Depending on the va lue of the ^ - f u n c t i o n s i n the c o o l i n g s e c t i o n t h i s y i e l d e d d i v e r g e n c e s

of l e s s than 0 .1 arad or beam s i z e s of a f r a c t i o n of a m i l l i m e t r e . The equ i l ibr ium beam

momentum spread was i n most j a s e s l i m i t e d by intrabeam s c a t t e r i n g blow-up and ranged between

10" 6 and 1 0 * depending on the beam i n t e n s i t y . However, a t Novos ib irsk , wi th low beam i n t e n ­

s i t i e s , i n d i c a t i o n s for a l o n g i t u d i n a l order ing w i t h i n the proton bean were found t o po int

t o a c r y s t a l l i z a t i o n .

10.3 Recombination

The recombination of c o o l i n g e l e c t r o n s with c i r c u l a t i n g protons was observed and used

as beam d i a g n o s t i c s . In p a r t i c u l a r , i t served t o measure the beam s i z e and the o v e r a l l

e l e c t r o n beam temperature . The e x p e r i m e n t a l l y determined recombination c o e f f i c i e n t a ranged

between 0 . 8 x 10" ' ' and 2 .3 x 1 0 " , J cm 3 •s" ' , g i v i n g neutra l hydrogen r a t e s between a few

hundred and a few thousand per second.

10.4 Beam l i f e t i m e

In the absence of c o o l i n g and machine resonances the l i f e t i m e of the s tored beam i s

governed by the e m i t t a n c e i n c r e a s e due t o m u l t i p l e s m a l l - a n g l e s c a t t e r i n g of ions on r e s i ­

dual gas m o l e c u l e s . The bean emit tance c o n t i n u o u s l y i n c r e a s e s u n t i l the machine acceptance

i s reached and the beam g e t s l o s t . Cool ing c o u r t e r i c t s t i . i s beam blow-up and on ly t h o s e

p a r t i c l e s are l o s t which undergo a s i n g l e s c a t t e r 1 arger than the machine a c c e p t a n c e a n g l e ,

a n g l e .

In a l l c o o l i n g exper iments a c o n s i d e r a b l e i n c r e a s e o í beam l i f e t i m e uas observed

(approximate ly by a f a c t o r of 4 0 ) . The c a l c u l a t e d l i f e t i m e s , assuming s i n g l e s c a t t e r i n g

l o s s e s [Eq. ( 1 1 ) ] , were i n ra ther good agreement wi th the exper imenta l o b s e r v a t i o n s . For

i n s t a n c e , the l i f e t i m e o f a 50 MeV proton beam s t o r e d in ICE was about 1 h for a vacuum of

¿ x 10" 9 Torr and a r e s i d u a l gas compos i t ion of 5 0 \ R and 5 0 \ N ? . At lower e n e r g i e s

( 1 . 5 MeV) a beam l i f e t i m e of about 1 s was determined i n the N o v o s i b i r s k experiment

( p r e s s u r e - 1 0 " 1 0 T o r r ) .

11. SIMULATION OF ELECTRON COOLING IN STORAGE RINGS

In many c a s e s , t h e i n f l u e n c e of the ccabined a c t i o n of the c o o l i n g and h e a t i n g p r o ­

c e s s e s , the i n i t i a l i on beam p r o p e r t i e s , and the c h a r a c t e r i s t i c s of the e l e c t r o n beam on

t h e e v o l u t i o n of t h e beam e m i t t a n c e and momentum spread i n t i n e and on the e q u i l i b r i u m

/ a l u e s cannot be p r e d i c t e d by a s imple mathematical e x p r e s s i o n . Rather i t i s n e c e s s a r y t o

f o l l o w the f a t e of an e n s e n b l e of i o n s based on a r e a l i s t i c model for the s t o r a g e r i n g , the

e l e c t r o n c o o l e r , the matching of both s y s t e m s , and t o implement a maximum of beam dynamics .

This a l l o w s then t o c a l c u l a t e e m i t t a n c e d e c r e a s e r a t e s and t h e r e d u c t i o n of momentum spread

as a f u n c t i o n of the v a r i o u s machine parameters , and hence permi t s the o p t i m i z a t i o n of the

p r o c e s s for v a r i o u s o p e r a t i n g c o n d i t i o n s .

A computer code was deve loped for t h i s purpose i n the p a s t few years by the KfK group

at CERN, borne examples a r e d i s c u s s e d h e r e . F igure 20 shows t h e d i s t r i b u t i o n of a sample of

beam p a r t i c l e s i n the c o o l i n g r e g i o n , be fore c o o l i n g i s s t a r t e d . Also shown i s the d i s p e r ­

s i o n curve and the v e l o c i t y p r o f i l e of t h e e l e c t r o n s . The h o r i z o n t a l d i s t a n c e o£ the i o n s

from the d i s p e r s i o n s t r a i g h t l i n e i s a measure of t h e i r b e t a t r o n ampl i tude (a zero emi t ­

t a n c e beam would coincide wi th the d i s p e r s i o n . l i n e ) .

F i g . 20 S imulat ion of the e l e c t r o n c o o l i n g p r o c e s s ; d i s t r i b u t i o n of an ensemble of i o n s for

v a r i o u s t imes a f t e r onse t of c o o l i n g .

HORIZONTAL EMITTÀHCC MOMENTUM WIOTH

F i g . 21 Time dependence of averaged proton beaa p r o p e r t i e s dur ing s i m u l a t i o n . The emi t tances

c o n t a i n 5 3 . 2 \ of the beam p a r t i c l e s for a Haxwel l ian beam p r o f i l e

En F i g . 21 the e v o l u t i o n of the h o r i z o n t a l emi t tance and the momentum spread in time i s

shown. One observes two r e g i o n s w i th d i s t i n c t damping r a t e * and a reg ion where the e q u i ­

l i b r i u m i s reached. In the reg ion with the s m a l l e r damping r a t e e s s e n t i a l l y o n l y the non­

magnet ic c o o l i n g f o r c e p l a y s a r o l e , w h i l e , a f t e r the i n i t i a l compression of t h e phase

space , the o n s e t of the m a g n e t i c - f o r c e c o n t r i b u t i o n l eads t o f a s t e r c o o l i n g .

12.

12.1 Electron-beam d i a i i n o s t i c a

In order t o o p t i m i z e e l e c t r o n c o o l i n g , e f f i c i e n t d i a g n o s t i c a are needed. The important

p r o p e r t i e s of the e l e c t r o n beam have t o be measured. These are d e n s i t y d i s t r i b u t i o n and the

v e l o c i t y p r o f i l e acrosr. the e l e c t r o n bean a s we l l as the l o n g i t u d i n a l and t i n s v e r s e

e l e c t r o n beam tempera tures . For p r a c t i c a l reasons i t should be p o s s i b l e t o determine the

e l ec tron-beam p o s i t i o n a t var ious p l a c e s in the c o o l e r and the e l e c t r o n c o l l e c t i o n e f f i ­

c i e n c y . Of c o u r s e , the cathode temperature and the e l e c t r o n current should be e a s i l y d e t e r ­

minable .

The cathode temperature can be measured p y r o m e t r i c a l l y once and then determined from

the h e a t i n g power. The e l e c t r o n current i s e s s e n t i a l l y deduced from the c o l e c t o r c u r r e n t .

The l o s s current has tù be provided by the h i g h - v o l t a g e supply and i s hence known.

The beam p o s i t i o n can be measured from e l e c t r o s t a t i c p ick-up e l e c t r o d e s wi th an ac ­

curacy of a f r a c t i o n of a m i l l i m e t r e . F ? r * h i s the beam current has to be nodula ted .

The d e n s i t y d i s t r i b u t i o n can be determined by scanning across th? e l e c t r o n beam with a

small Faraday cup or c r o s s e d w i r e s .

Temperatures and v e l o c i t y p r o f i l e s are d i f f i c u l t to measure. The o v e r a l l t r a n s v e r s e

temperature of the e l e c t r o n beam can be determined from the microwave r a d i a t i o n spectrum

- 563 -

e m i t t e d by the e l e c t r o n s s p i r a l l i n g in t h e magnet ic f i e l d of the s o l e n o i d . The l e v e l o í t h i s

r a d i a t i o n can be measured and from t h a t the t r a n s v e r s e e l e c t r o n temperature be ieduced i f

the c o u p l i n g of the r a d i a t i o n f i e l d t o the antenna and t h e c h a r a c t e r i s t i c s of the d e t e c t i o n

system are known. This method was used i n ICE t o minimize the t r a n s v e r s e e l e c t r o n tempera­

t u r e . I t i s a l s o a p p l i e d f o r the LEAR e l e c t r o n c o o l e r .

Longi tudinal e l e c t r o n temperature and v e l o c i t y p r o f i l e have never been determined from

t h e e l e c t r o n beam a l o n e . I t was on ly from a c t u a l c o o l i n g exper iments thar t h i s in format ion

was d e r i v e d . However, i t can be measured by s c a t t e r i n g Laser l i g h t f r e t the e l e c t r o n beam.

The b a c k - s c a t t e r e d l a s e r l i g h t i s s h i f t e d in frequency owing t o t h e r e l a t i v i s t i c a l l y moving

e l e c t r o n s . Analysing the l i g h t i n frequency a l l o w s for the d e t e r m i n a t i o n of v e l o c i t y p r o f i l e

and l o n g i t u d i n a l temperature n o n - d e s t r u c t i v e l y . This method i s be ing at tempted a t t h e LEAR

e l e c t r o n c o o l e r .

12 .2 E l e c t r o n c o o l i n g d i a g n o s t i c s

During the e l e c t r o n c o o l i n g p r o c e s s t h e p o s i t i o n and a l ignment of the i on ind tbe e l e c ­

tron beam have t o be known p r e c i s e l y . Equal ly important i s the informat ion on the e v o l u t i o n

of t h e ion-beam e m i t t a n c e and romentum spread . For the measurement of tha s tored-beam pro­

p e r t i e s u s u a l l y the standard beam d i a g n o s t i c s , such a s Schottky and e l e c t r o s t a t i c p i c k - u p s ,

are a p p l i e d . A f a s t beam p r o f i l e monitor , which a l l o w s moni tor ing of t h e beam si-re d i c i n g

the c o o l i n g , i s a l s o very u s e f u l .

Much informat ion can be e x t r a c t e d from the measurement of the recombinat ion channe l . I f

protons are t o be c o o l e d , recombinat ion produces a n e u t r a l hydrogen beam which l e a v e s the

s t o r a g e r i n g t a n g e n t i a l l y . The bean s i z e can be measured with a u l t i w i r e p r o p o r t i o n a l cham­

bers or s o l i d - s t a t e d e t e c t o r s . I t a l l o w s the shr inkage of the beam t o be observed during the

c o o l i n g p r o c e s s ( F i g . 22) and p r o v i d e s in format ion on the e q u i l i b r i u m beam s i z e . The r a t e

1 G .3

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13 fci..i. kl.G l<i. 15.*.

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J LIJITX. LA Li .LL 1

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I« fil.J l.B.2 5.S 5.6

L. 5.9

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L. 5.9

0 13 7 0 '33 'MD '53 BO ' C I : -'. H U : ?S6 H i ! Ï C . T Ï : - ' Fl-.Z 5 .315. i

D 'ID 2 • '30 'uo 'SO fi D ID 8 0 5C

a) b)

F i g . 22 Neutral hydrogen beam p r o f i l e a s measured in ICE.

- 564 -

c a n be m e a s u r e d w i t h s c i n t i l l a t i o n c o u n t e r s . I t c o n t a i n s i n f o r m a t i o n on t h e a v e r a g e t r a n s ­

v e r s e e l e c t r o n t e m p e r a t u r e {see Section 9) F o r b u n c h e d b e a n s t h e t i m e s t r u c t u r e o f t h e

d o w n - c h a r g e d i o n beam c a n be m e a s u r e d , e n a b l i n g t h e d e t e r m i n a t i o n o f bunch l e n g t h and i n t e n ­

s i t y d i s t r i b u t i o n w i t h i n t h e b u n c h .

A n o t h e r i n t e r e s t i n g method o f d i a g n o s t i c s i s t h e l a s e r - i n d u c e d r e c o m b i n a t i o n ( s e e

S e c t i o n 1 3 . 2 . 1 1 . T h e r e t h e c a p t u r e o f a n e l e c t r o n by an i o n i s s t i m u l a t e d by i r r a d i a t i n g t h e

w h o l e s y s t e m w i t h l a s e r l i g h t o f s u i t a b l e f r e q u e n c y . M e a s u r i n g t h e r e c o m b i n a t i o n r a t e as a

f u n c t i o n o f t h e l a s e r f r e q u e n c y a l l o w s one t o scan a c r o s s t h e e l e c t r o n v e l o c i t y d i s t r i b u t i o n

a n d t o d e d u c e t h e L o c a l t r a n s v e r s e and l o n g i t u d i n a l e l e c t r o n beaci t e m p e r a t u r e s . T h e t h r e s h ­

o l d o f t h e r e c o m b i n a t i o n p r o v i d e s i n f o r m a t i o n on t h e e n e r g y of t h e beams

I f p a r t i a l l y s t r i p p e d i o n s a r e t o be c o o l e d , d i e l e c t r o n i c r e c o m b i n a t i o n ( S e c t i o n 1 3 . 2 . 3 ) c a n be u s e d t o m e a s u r e t h e e l e c t r o n v e l o c i t y d i s t r i b u t i o n and t h r o u g h t h a t t h e

t e m p e r a t u r e s . I t a l s o makes i t p o s s i b l e t o d e t e r m i n e t h e e n e r g i e s o f t h e beams p r e c i s e l y .

1 3 . APPLICATIONS OF ELECTROH COOLIHG

So f a r e l e c t r o n c o o l i n g e x p e r i m e n t s h a v e been done b e t w e e n 0 - 0 . 0 5 and 0 . 5 7 . A t

p r e s e n t e l e c t r o n c o o l e r s a r e b e i n g b u i l t w h i c h w i l l qo up t o p = 0 . 7 6 and e v e n t u a l l y t o B =

0 . 8 6 f f = 2 ) . M o r e o v e r , s t u d i e s f o r a n e l e c t r o n c o o l i n g s y s t e m w h i c h c o u l d go up t o -f = 6 . 9

a r e under w a y . The m a j o r d o m a i n f o r e l e c t r o n c o o l i n g w i l l , h o w e v e r , r e m a i n in t h e c o o l i n g o f

i o n Learns o f v e l o c i t i e s b e l o w 0 . 8 c .

One t a s k o f e l e c t r o n c o o l i n g w i l l t h e n be t h e c o m p r e s s i o n o f t h e p h a s e s p a c e o f c i r ­

c u l a t i n g beams a t ¡ - j e c t i o n e n e r g y t o a l l o w t h e a c c u m u l a t i o n o f p u l s e s a n ' 1 t h e b u i l d - u p

o f h i g h s t o r e d b e a » i n t e n s i t i e s e v e n w i t h l o w - c u r r e n t i n j e c t o r s ( r a r e i o n s , p o l a r i z e d

p a r t i c l e s ! .

1 3 . 1 I n t e r n a l t a r g e t s

The a c h i e v e m e n t o f h i g h - i n t e n s i t y s t o r e d i o n beams makes t h e p e r f o r m a n c e o f i n t e r n a l

e x p e r i m e n t s i n t h e r i n g v e r y a t t r a c t i v e . A p a i t f r o m c o l l i d i n g - b e a m e x p e r i m e n t s , t h e use o f

t h i n i n t e r n a l t a r g e t s p r o v i d e s an e f f i c i e n t way t o u t i l i z e t h e i o n s r e p e a t e d l y . I f t h e

t a r g e t t h i c k n e s s i s k e p t s m a l l e n o u g h , m u l t i p l e s c a t t e r i n g beam b l o w - u p and e n e r g y l o s s can

be c o m p e n s a t e d b y e l e c t r o n c o o l i n g ; a n d beam l o s s e s a r e t h e n o n l y d u e t o s i n g l e s c a t t e r s

w i t h a n g l e s l c r g e r t h a n t h e m a c h i n e a c c e p t a n c e a n g l e a t t h e t a r g e t p o s i t i o n , or t o n u c l e a r

r e a c t i o n s . The a d m i s s i b l e t a r g e t t h i c k n e s s D d can be e s t i m a t e d f r o m t h e e m i t t a n c e g r o w t h

r a t e

^ 1 = e = 1 9 . 2 8 , C n - m r a d - s ' 1 ] , (gd t a k e n i n g-cm 2 ) , ( 6 0 ) d t ms h , v 02 2

P 1

w h i c h i s c o u n t e r a c t e d by e l e c t r o n c o o l i n g . F o r s i m p l i c i t y l e t us t a k e a c o n s t a n t c o o l i n g

t i m e . The e q u i l i b r i u m e m i t t a n c e i s t h e n g i v e n by t h e s o l u t i o n o f t h e d i f f e c e n t t a L e q u a t i o n

w h i c h i s

( 6 1 )

( 6 2 )

- 565 -

Assuming a ring can be filled to its space-charge liait at injection energy

luminosities of

^ax NL L = -^jp jj-=— pd {NL = Avogadro's number) , (64)

targ can be achieved. The important point in this operation is that the ions are not lost after their passage through the target, but are recycled

13.2 Atomic D h v s i c s

A large fraction of future experiments with stored ions cooled by electrons will most likely be devoted to atomic-physics investigations, making direct use of the electron cooler as an electron target. This comes from the fact that electron-ion collisions car be made at very well defined energies which have at the same time a high resolution. The basic processes to be studied will be the recombination of an electron from the cooler with circulating ions.

13.2.1 Radiative recombination

We have already discussed the formation of hydrogen atoms during electron cooling of protons, which was an important diagnostic in previous experiments. In general the capture process is

e" + Av* + (A*"'1 U ) + hv (65}

This process is the gateway to an exciting new field. The reaction could be enhanced by irradiating the system with laser light of suitable frequency

hv + e + A V f * (A(v" ' ]> ) + 2hv . (66)

Steering the frequency allows one to populate, i n a well-defined Banner, specific atonic levels. This, in principle, allows Rydberg atoms to be formed i n a very clean way and their properties to be studied.

13.2.2 Antihydrogen production The (stimulated) radiative recombination is at present the only promising way to form

the never before observed anti-hydrogen atom, by replacing the electron with a positron and the proton with an antiproton in an arrangement similar to electron cooling This is a very tantalizing application of electron cooling.

13.2.3 Dielectronic recombination

If partially stripped ions are caused to overlap with electrons the latter can be cap­tured without the '.'mission of a photon. The energy of the free electron is then dissipated through siiu-j? taneous excitation of the residual electron core. It happens only at electron

- 566 -

T a b l e 3

E l e c t r o n c o o l i n g p r o j e c t s

, CEÍN, SmtierUnd 1968 1583 1990 H Si 193;

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1961

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CELSIUS, Uppidl ESR, C51, DaiBilidt, FFC TLB. KPÍ. Heideltwr], ffî *STPIU. Aarhuï, itCMrk CRT*]NJ, StockboU. Sueden

Frascati. Italy FerBilab. USA

Î Franltiuit. IRC

Ber.i.nj I .eld

• 1 LI Light lonr HI : Heavy ions UC Under construction F Funded P Planne«

e n e r g i e s w h i c h B a t c h w i t h i o n e x c i t a t i o n e n e r g i e s a n d i t h a s r e s o n a n c e c h a r a c t e r . T h i s

r e a c t i o n i s c a l l e d d i e l e c t r o n i c r e c o m b i n a t i o n and c o u l d o n l y be s t u d i e d p o o r l y so f a r The

e l e c t r o n c o o l i n g a r r a n g e m e n t c o u l d p r o v i d e h e r e a l s o a v e r y p o w e r f u l e x p e r i m e n t a l t o o l .

T h e r e a r e many o t h e r a s p e c t s w h e r e t h e e l e c t r o n c o o l i n g a r r a n g e m e n t p r o v i d e s d i r e c t l y

or i n d i r e c t l y a c l e a n e r and « o r e p r e c i s e a p p r o a c h t o i n t e r e s t i n g q u e s t i o n s i n a t o m i c ,

n u c l e a r , a n d p a r t i c l e p h y s i c s , w h i c h w e r e a l r e a d y d i s c u s s e d i n t h e l i t e r a t u r e o r w h i c h w i l l

come up w i t h t h e new g e n e r a t i o n o f c o o l e r r i n g s a t p r e s e n t under c o n s t r u c t i o n .

14. ELECTRON COOLING PROJECTS

G i v e n t h e enormous e x p e r i m e n t a l p o t e n t i a l o f s t o r a g e r i n g s e q u i p p e d w i t h e l e c t r o n

c o o l i n g , t h e i n t e r e s t i n t h i s f i e l d h a s i n c r e a s e d v e r y much i n t h e p a s t few y e a r s . I n

T a b l e 3 a l i s t i s g i v e n o f t h e c o o l e r s w h i c h a r e a t p r e s e n t o p e r a t i n g , o r u n d e r c o n s t r u c ­

t i o n , o r p l a n n e d .

- 567 -

BIBLIOGRAPHY

First ideas on electron coolinq G.L. Budker, Proc. Int. Symposium on Electron and Positron Storage Rings, Saclay, 1166 (PUF, Pari., '967) , p. II-1 -1 .

Reviews on electron coolinq in general G.I Budker and A.N. Skrinsky, Sov. Phys.-Usp. 21, 277 (1978). A.N. Skrinsky and V.V. Parkhomchuk, Sov. J. Part. Nucl. 12., 223 ( 1981 ) F.T. Cole and F.E. Mills, Annu. Rev. Nuct. Sei. 33, 295 (1981). Ya. Derbenev and A.N. Skrinsky, Physics Reviews, Vol. 3, ed. I.M. Khalati, .-.uv (Harwood

Academic Press, 1981), p. 165.

Topical conferences on electron cooling Proc. Workshop on Electron Cooling, Bad Honnef, 1982 [ed5. G. Berg, W Hurlimann and J Römer), KfA Spez 159, Jülich, 1982).

Proc. Workshop on Electron Cooling and Related Applications ECOOL 84, Karlsruhe, 1984 [t>d. H Poth) (KfK 3846, Karlsruhe, 19B5).

Electron coolinq at other international conferences Proc. Joint CERN-KfK Workshop on Physics with Cooled Low Energetic Antiprotons (1st LEAR

Workshop), Karlsruhe, 1979 (éd. H. Poth) (KfK 2647, Karlsruhe, 1979). Proc. 2nd LEAR Workshop on Physics with Low-Energy Cooled Antiprotons, Erice, 196? (eds.

U. Gastaldi and R. Klapisch) (Plenum Press, New York, 1984). Proc. 3rd LEAR Workshop on Physics in the ACOL Era with Low-Energy Cooled Antiproton,

Tignes, 1985 (eds. U. Gastaldi, R. Klapisch, J.H. Richard and Tran Thanh Van) (Editions Frontieres, Gif-su -Yvette, 1985).

Proc. Workshop on Nuclear Physics with Stored, Cooled Beams, Mccormick's Creek State Park, Indiana, 1984 (eds. P. Schwandt and H.O. Heyer) (AIP Conf. Proc. No. 128, New York, 19B5).

Proc. 11th Int. Conf. on High Energy Accelerators, Geneva, 1960 (Birkhäuser, Basle, 1980). Proc 12th Int. Conf. on High Energy Accelerators, Batavia, 1983 leas, F.T. Cole and

R Donaldson) (Fermilab, Batavia, 19B4). Beam Cooling Workshop, Madison, Wisconsin, 19>82. Workshop on the Physics with Heavy Ion Cooler Rings, Heidelberg, 1984 Pror Workshop on the Physics Program of CELSIUS, Uppsala, 1981 (eds. B R Karlsson and

G. Tibell), Vol. 1 (Tandem Accelerator Laboratory, Uppsala, 19B3! and Vol 2 (Tandem Accelerator Laboratory, Uppsala, 1984)

Electron cooling experiment;? G.I. Budker et al., Part. Accel. 7, 197 (1976) Ya. Derbenev and I Meshkov, CERN 77 08 11177) N.S. Dikansky et al , The study of fast electron cooling, INP Novosibirsk preprint 79 ">b

( 1979) . V.l. Kndeldinen et al . , Temperature lelaxation m Beignet i i'J el eel ron flux. INI' Novosibirsk preprint 82-78 (1982), to be published in :»ÏV Phys JETP

- Sb8 -

v v. parkhomchuk et al.. Measurement of momentum cooling rates with electron -ooling at NAP-M, INP Novosibirsk preprint 78-81 Í197S).

C.I. Buöker et al., New experimental results ol electron cooling, presented, at M l Union High Energy Accelerator Conference, Moscow, 1976, translated at CESN, CERN PS/DL/Note 76-25 [19761.

H. Bell et a) , phys. Lett. 275 1 1979). M Bell eta)., Nucl. Instruir,. Methods Jiû 237 ( 1981 ), R Forster et al., IEEE Trans. Nucl, Sei. NS-28. 2386 (1981). T. Ellison etal., IEEE Trans. Nucl, Sei- NS-30. 2636 (1983).

Electron cooling theory Ya. Derbenev and A.N. Skrinsky, Part. Accel. 8, 1 (1977). Ya Derbenev and A N . Skrinsky, Paît. Accel. 8, 235 ( 1977). Ya. Derbenev and A.N. Skiinsky, Or high energy electron cooling, INP Novosibirsk preprint

79-87 [19791. T. Ogino and A.G. Ruggiero, J. Phys, Soc. Japan 4 3 . 1654 (1980) and Part. Accel, jtp, 197

[1960). M. Bell, Part. Accel. JlQ, 101 (1980). J.5. Bell and H. Bell, Part. Accel. JJ, 233 (1981).

A H . S0rensen and E. BondcruD, Nucl. Instrum. Methods 211, 27 (1983).

Electron capture

L. Spitzer, Physics of fully ionized gases (Interscience, Neu York, 1956). H. Bell and J.S. Bell, Part. Accel. U, 49 (1982). R. Neumann et al., Z. Phys. A313. 253 (1983). Simulation of electron cooling M. Bell, Cooling in ICE, CERR-EP Internal Report 79-10 (1979). A. Wolf et al., simulating electron cooling of ion beams, to be submitted to Nucl, Instrum. Methods, 1986.

Applications and pnecial aspects of electron cooking H. Poth and A. Wolf, Phys. Lett. 9_i$, 135 ( 1983). H Poth, Nucl Instruí», Methods 2Û1, 5*7 ( 19821. J.P. Schiffer and P. Kienle, Z. Phys. AJL2J, 181 ( 1985). n . Wohl et al., Nucl. Instrum. Methods ¿ U 2 , 427 (1982). A. Wolf et al., Electron cooling of low-energy antiprotons and production of fast anti-

hydrogen atoms, Preprint CERN-EP/86-10 (1986).

Vacuum systems of electron coolers C. Habfast et al., Vakuun-Technik 2, 1)5 (1985). C. Habfast et al., Das Ultrahochvakuuo-íysteo des Elektronenkiihlers für LEAR, Karlsruhe report XfK Î816 (19651

- 5bi) -

Magnetic field A Wolf et al., Magnetic field measurements in the cooling device for LEAR, Karlsruhe

report KfK 3718 (1964)-

Electron beams J.R. Pierce, Bell Syst. Tech. j. Jfi, B25 (1951). J.R. Pierce, Theory and design of electron beans (Van Nostrand, New York, 195-)! P.T. Xirstein, G.S. Kino and W.E. Waters, Space-charge flow (McGraw-Hill, New yorx, 1967) C. Rubbia, On the form?'.ion of intense electron beams with small transverse velocities for

(anti-)proton cooling, CERN-EP Internal Report 77-2 (1977J.

Electron collectors V.l. Kudelainen. et al., Sov. j. Tech. Phys. 4£, 1678 ( 1976). V.l. Kokoulin et al., Sov. J. Tech. Phys. 50, 1475 (1980).

Diagnostics of electron cooling T. Hardek and W. Keils, T-EE Irans. Nucl. Sei. NS-2B, 2219 (1981). W. Kells, Laser diagnostics for electron cooling beam, Fermilab Technical Memo TM-771, 1978,

unpublished. W. Kells, Detector of microwave radiation from cooler electron beam, Fermilab Technical Memo TM-798, 1978, unpublished.

C. Rubbia, Microwave radiation from the transverse temperature of an intense electron beam confined by a longitudinal magnetic field, CERN-EP Internal Report 77-4 E1977).

B. Schnitzer and E. Farnleitner. Acta Phys. Austriaca £2, 225 (1980).

Internal Targets M. Giesen et al., Implications of an internal target for antineutron production at LEA«, PS/DL/LEAR Note 81-4, 1981.

H.O. Meyer, Nucl. Instrum. Methods filÛIU 342 (1965).

Theses about electron cooling P. M0ller Petersen, Studies of electron cooling in the ICE storage ring at CERN, ur ;versity of Aarhus (1982).

A. Wolf, Electron cooling for low-energy antiprotons, University of Karlsruhe Report KfK 4023 (1986).

D.J. Lùrson, Intermediate energy electron cooling for antiproton sources, University of Wisconsin, Hadison (1986).

- " 0 -

ELECTRON DYNAMICS WITH RADIATION AND NONLINEAR WIGGLERS J O H N M . . I O W E T T

(ERS. Ge.ntv.fi. Switzerland

A B S T R A C T : T h e phys ics of e l e c t r o n mi . ' . ion in s t o r a g e r ings is descr ibed hy s u p p l e m e n t i n g t h e U a m i l t o n i a n e q u a t i o n s r f m o t i o n w i t h f l u c t u a t i n g r a d i a t i o n r e a c t i o n forces t o d e s c r i b e t h e effects o f sy chrotTOn r a d i a t i o n . T h i s l eads !o a d e s c r i p t i o n o f r a d i a t i o n d a m p i n g a n d q u a n t u m d i f fus ion in s i n g l e - p a r t i c l e p h a s e - s p a c e by m e a n s o f F o k k e r - P l a n c k eq j a t i o n s . F o r p r a c t i c a l p u r p o s e s , m o s t s t o r a g e r i n g s r e m a i n in t h e r e g i m e o f l i : ea r d a m p i n g a n d d i f f u s i o n ; this is d iscussed in s o m e d e t a i l w i t h e x a m p l e s , c o n c e n t r a t i n g o n l o n g i t u d i n a l phase s p a c e . H o w e v e r s p e c i a l dev ices s u c h as n o n l i n e a r w i g g l e r s m a y p e r m i t the n e w g e n e r a t i o n o f v e r y l a r g e r ings t o go b e y o n c th is i n t o r e g i m e s o f n o n l i n e a r d a m p i n g . I t is s h o w n h o w a spec ia l c o m b i n e d - f u n c t i o n w i g g l e r c a n he used i o m o d i f y t h e e n e r g y d i s t r i b u t i o n a n d c u r r e n t p ro f i l e of e l e r t r o n b u n c h e s .

1 . I N T R O D U C T I O N

I n th is l e c t u r e w e s h a l l p r e s e n t s o m e m a t h e m a t i c a l tools w h i c h a r e p a r t i c u l a r l y usefu l in t h e s t u d y of

e lec t ron or p o s i t r o n d y n a m i c s in s t o r a g e r ings a n d a p p l y t h e m t o s o m e i m p o r t a n t p r o b l e m s . Jlo«ev<>r l l ie

c;., j . i a s i s is o n u n d e r s t a n d i n g t h e p h y s i c a l c o n t e n t r a t h e r t h a n t h e m a t h e m a t i c s i tsel f . A c c o r d i n g l y , some

m o r e t e c h n i c a l m a t e r i a l has b e e n p laced in a p p e n d i c e s .

I t is a s s u m e d t h a t t h e r e a d e r has s o m e f a m i l i a r i t y w i t h g e n e r a l a c c e l e r a t o r t h e o r y a n d has , in par ­

t i c u l a r , been i n t r o d u c e d t o t h e p h e n o m e n o n of s y n c h r o t r o n r a d i a t i o n a n d h o w i t afloras C l o d r o n m o i i o n .

K x c e ü e n t i n t r o d u c t i o n s t o these top ics w e r e g i v e n in t h e f irst Schoo l of l l i is scries. 1 A m o n g o t h e r h i t rod uc-

t i o n s , t h e classic l ec tures of S a n d s 2 a r e espec ia l l y w o r t h r e a d i n g . T e x t s in H a m i l l o i i i a i i d y n a m i c s ? ' 4 a n d

classical a n d q u a n t u m e l e c t r o d y n a m i c s 5 , 6 p r o v i d e t h e p h y s i c a l b a c k g r o u n d w h i l e b o o k s on t h e t h e o r y of

s tochas t i c p r o c e s s e s w i l l g i v e m o r e de ta i l s of s o m e of t h e t e c h n i q u e s e m p l o y e d h e r e . S ince a f igurons

m a t h e m a t i c a l d iscuss ion of t h e l a t t e r w o u l d get us i r r e c o v e r a b l y s i d e t r a c k e d , w e s h a l l a d o p t a f o r m a l

a p p r o a c h . t r u s t i n g t o i n t u i t i o n Tor t h e m e a n i n g of w o r d s l ike " r a n d o m " a n d '"noise".

T h e last p a r t of th is l e c t u r e is d e v o t e d to t h e spec ia l top ic of n o n l i n e a r w i g g l e r s w h i c h g i v e rise in new

d y n a m i c a l p h e n o m e n a a m e n a b l e to d e s c r i p t i o n in t e r m s of a F o k k e r - I ' l a n c k e q u a t i o n a n d ideas f r o m the

s t a b i l i t y t h e o r y of d i s s i p a t i v e s y s t e m s . T h e i r use is l i m i t e d t o very la rge s t o r a g e r ings such as I.F,I* nn<\

no such w i g g l c r has y e t been o p e r a t e d . H o w e v e r t h e y o p e n u p a n i n t e r e s t i n g n e w r a n g e of possib i l i tés for

c o n t r o l l i n g i h e p a r a m e t e r s o f t h e b e a m s .

R a d i a t i o n effects o n b e t a t r o n m o t i o n a re n o t discussed in d e t a i l h e r n because m o s t of the m a t h e m a t i c a l

Ji t ' b c i i j u t i r a n be i l l u s t r a t e d in c o n n e x i o n w i t h s y n c h r o t r o n m o t i o n a n d t h e m o s t i n t e r e s t i n g effects of

n o n l i n e a r w igg le rs are l o n g i t u d i n a l . W c h a v e t a k e n t h e o p p o r t u n i t y t o p r o v i d e a H a m i l t o n i a n f o r m u l a t i o n

of syiit h r o t r o n m o t i o n , s p e c i a l l y geared for e l e c t r o n m a c h i n e s w i t h loca l ised R F c a v i t i e s . T h i s f r a m e w o r k

alli j ' , .s a n a t u r a l d e v e l o p m e n t o f l o n g i t u d i n a l c h r o m a t i c effects a n d t h e i m p o r t a n t n o t i o n of t h e d a m p i n g

: « ' r ! u r e . H e t a t r o n m o t i o n w a s t r e a t e d in R e f s . 2 . 10, 11 a n d . w i t h t h e a p p r o a c h used h e r e , in Rof. Kt.

2 . T H E D Y N A M I C S O F E L E C T R O N S IN A S T O R A G E R I N G

W e s h a l l f o r m u l a t e t h e e q u a t i o n s o f m o t i o n for e l e c t r o n s (o r p o s i t r o n s ) in a s t o r a g e r i n g . T h e H a m i l t o -

n i a n d e s c r i p t i o n o f p a r t i c l e m o t i o n i n a c i r c u l a r a c c e l e r a t o r , r e g a r d e d as a s p e c i a l c o n f i g u r a t i o n of e x t e r n a l

e lec t r ic a n d m a g n e t i c fields, is f a m i l i a r t o t h e r e a d e r f r o m o t h e r l ec tures in t h i s S c h u o l . 3 S ince i t p r o v i d e s

t h e s h o r t e s t r o u t e f r o m t h e g e n e r a l e q u a t i o n s o f m o t i o n o f a c h a r g e d p a r t i c l e in a n e l e c t r o m a g n e t i c f ield to

t h e speci f ic f o r m s w h i c h these e q u a t i o n s t a k e in a. s t o r a g e r i n g ( H i l l e q u a t i o n s for b e t a t r o n m o t i o n etc.),

w e sha l l e m p l o y i t f r ee ly t o a v o i d a l o n g r e c a p i t u l a t i o n o f basic a c c e l e r a t o r p h y s i c s .

O n t h e o t h e r h a n d , o u r m a i n i n t e r e s t h e r e is t h e ef fect o f s y n c h r o t r o n r a d i a t i o n o n e l e c t r o n d y ­

n a m i c s . A n d t h i s c a n n o t b e d e s c r i b e d so le ly i n t h e c o n t e x t o f H a m i l t o n ' s e q u a t i o n s . W e m u s t a d d

d i s s i p a t i v e t e r m s t o d e s c r i b e t h e e n e r g y loss t h r o u g h r a d i a t i o n . M o r e o v e r , t h e 1 7 - t e r m s m u s t s o m e h o w

i n c l u d e t h e e s s e n t i a l l y r a n d o m n a t u r e o f t h e p h o t o n e m i s s i o n process . A p p r o p r i a t e m a t h e m a t i c a l tools

a re f o u n d in t h e t h e o r y o f s t o c h a s t i c processes , n o t a b l y i n s t o c h a s t i c differential e q u a t i o n s a n d t h e associ­

a t e d Fokker-Pianck equations}* H e r e w e s h a l l f o l l o w t h e t r e a t m e n t o u t l i n e d i n R e f s . 15 a n d 13 a l t h o u g h

t h e a p p l i c a t i o n s w i l l b e s o m e w h a t d i f f e r e n t ; s o m e a l t e r n a t i v e a p p r o a c h e s t o t h e m a t h e m a t i c a l s ide of t h e

p r o b l e m m a y b e f o u n d i n R e f s . 1 6 , 17 a n d 18 .

2 . 1 Coordinate system and Hamîltonian

W e s h a l l use t h e c u r v i l i n e a r c o o r d i n a t e s y s t e m o f C o u r a n t a n d S n y d e r 1 9 a n d f o l l o w t h e c o n v e n t i o n s

of m o s t o f t h e s t a n d a r d o p t i c s p r o g r a m s ( M A D , T R A N S P O R T , etc.).

F i g . 1 T h e re fe rence o r b i t a n d C o u r a n t - S n y d e r c o o r d i n a t e s y s t e m

F o r s i m p l i c i t y , w e a s s u m e t h a t t h e m a g n e t s a r e p e r f e c t l y a l i g n e d in t h e sense t h a t t h e r e exists a

closed p l a n a r r e f e r e n c e c u r v e r o ( s ) , p a s s i n g t h r o u g h a l l t h e i r c e n t r e s w h i c h is a lso t h e closed orbit for a

h y p o t h e t i c a l r e f e r e n c e p a r t i c l e o f c o n s t a n t m o m e n t u m po w h i c h n e i t h e r r a d i a t e s n o r coup les t o t h e R F

• Not, that ÜJ, provided we only contemplate including the degrees of freedom of the particle (leaving out tbote of the electromagnetic 6eld) and forego a fully quantum-mechanical treatment.

accelerating fields. The position of a rea] particle of kinematic momentum p is then described by giving the azimuthal position a of the closest point on this curve and its radial and vertical deviations r and y from that point, as shown in Fig. 1.

If we neglect edge effects in the magnets and use the Coulomb gauge then, in many important cases, the fields can be described in the electromagnetic fields can be derived from a single scalar function,19 thc-

A , ( z , y , l , s ) = A • e, (1 + G(s)i)

C ( s ) - \ + I / r . i a W x 1 - i / 2 l + -K,tt.tx* - 3 i u ! l T (2.1)

- t f { * G [ , ) (i+c(»)|) + i/r,W(x! - y 7 ) + \ k a ^ ' - 3 * ! / ' ) T •

This includes only the fundamental accelerating mode of a set of RF cavities with peak voltages Vk located at positions s¿; Sc i S

a 5-function, periodic on the circumference C — 2irR. Our assumption about the closed orbit of the reference particle can only hold if the normalised dipole field strength is equal to the curvature of the reference orbit:

we take s as

independent variable so that ( i ,y , t) may be tajeen as canonical coordinates and ( p x - P y , — E ) as canonical

momenta. The Hamiltonian of a particle with kinematic momentum p is

H a { x , y . t , p x ¡ p y , - E \ s ) - (p + (e /c)A) • e 8 [1 + G{s)x)

= - { e / c ) A , ( z , y , t , s ) (2.3)

- (1 + G(s)x) yjE*/c* - m 2 c ! - pi - p ¡ .

Before proceeding further, it is convenient to replace the energy, E, by the magnitude of the total mo­

mentum p; this requires a canonical transformation of one pair of variables,

( ( , - £ ) " ( 2 , , p ) , (2,1)

effected by means of the generating function

F2(p,() = - c l f y T m 2 c 2 , (2.5)

and resulting in the new Hamiltonian

f/(x,y,z, ,p„p ! „p; S ) = - ^ , ( i , y , t ( 2 , , p ) , s ) - (1 + C(a)i) ^ p 2 - p\ - pj. (2.G)

1 This recently in t roduced 2 0 term laveB a good deal oí círtuinlocution.

The new canonical variables are related to the old by

p = ^Etjc* - m 2 c 2 ,

Z l = -etyfl - mlc*/E* (2 ~>

- - ( i n s t a n t a n e o u s velocity) x (time particle passes s).

i.t, z( is not immediately related to the path length, except while the energy is constant . The explicit

form of Hamilton's equations in these variables is

-' - (1 -r G i l , P x = ~ (1 + Gx)^ r - Ti- P¡

: (1 + G i ) f _ P'_ = (1 + C i ) ^ P¿ - P Í - Py

z[ = -(1 + Gx) P = * -(1 + Gx)

yJpí-PÍ-pl (2.8)

p¿ = -G(p - PO) - P O ( G 2 + Ki)x - IpùKtix2 - y2) + • • -

p'y = poKiy + ^p0K2xy -\

p' = - ~ M a ~ ^k)5in{urízt/c + 4>k). t

With these variables, the reference m o m e n t u m po factors out of all terms in H except those describing tin-

cavities. T h u s , particle motion in magnets will be "geometric", depending only on £ — (p - p. . ) /pj am!

not on the mass or absolute value of p . The price paid comes through the more complicated expression of

the t ime-dependence of A3 in (2.6). There, t(zt,p) denotes the solution of (2.7) for f in terms of z; and p.

This does not matter much since the motion of high energy electrons is cxtrcme-relativistir and the third

argument of AB in (2.6) may be set equal to - zt/c. From this also follows the excellent approximation

t' « (1 f Gx)/c ( 2 . 9 )

Vor later convenience, let us define the normalised magnetic field strength b through

B{x,y,s) •= V x A P 0 i b ( i , y , j t ) ( 2 . 1 0 )

and also write b(x,y,s) for | b ( r , y , s ) | .

2.2 S t a t i s t i c a l p r o p e r t i e s o f i n c o h e r e n t s y n c h r o t r o n r a d i a t i o n

Wo review the essential facts about incoherent synchrotron radiation 1 1 2 "' , C and recast lhe;n in a

notation suited for our present purposes.

As a particle is accelerated transversely in a magnetic field, it rrnits photons. Because this is ;i quantum-mechanical phenomenon, the emission t imes and the quanta of energy carried away by tin-

- S74 -

Provided the energies ana magnetic fie Ida ire nal too híg

photons are random quantities. However certain average quantities such as the mean emission rate and the mean radiated power may be calculated with good accuracy " within classical electrodynamics- ln the classical picLure, the accelerated particle emits a continuous beam of radiation in a narrow cone around its momentum vector. Quantum mechanically, the momentum vector of each photon is almost coMinear with the particle's momentum.

Since the orbital quantum numbers of electrons in typical storage rïniçs are very large, we may u=c classical arguments to construct the equations of motion provided we do not attempt to describe the emission process itself. In fact it will be represented simply as an instantaneous jump in energy. This is acceptable since, for an electron of energy E = ^mc1 m a magnetic field B = E/tp, photon emission occurs within a time

where fl is a frequency characteristic of betatron or synchrotron oscillations. In addition, the fact that

— « w e, (2.12)

where w( is the frequency corresponding to the critical energy (defined below), means that the frequency

(or energy) spectrum of the photons is locally well-defined.2

Let us now develop these ideas formally.

An individual photon emission event, in which a photon of energy u ; is emitted at s ~ s}, is specified by the ordered pair of random variables (uy.Sj). The distribution function of (UJ,S:) depends only on

the local magnetic fíeld and the particle's momentum. Since these conditions vary as the particle moves, there is no very meaningful way of relating time averages (along the trajectory of a given particle) and ensemble averages (over many hypothetical particles experiencing the same conditions).

For definiteness, let us define the expectation vaJue of a dynamical variable A(x, y, zt, px, py, p; s).

associated with the instantaneous state of the particle, to be the average of A over all possible realisations of [vj, Sj), that is to say, all the ways in which the particle's photon emission history might occur, weighted appropriately. W e denote such ensemble averages by

{A{x,y,zt,px,pv,p;s)) or, more briefly and generally, {A)x* (2.13)

as it suits us; here X is a shorthand notation for the set of (usually canonical) variables describing the instantaneous state of the particle in whatever representation we happen to be using. The averaging is understood to be taken while the azimuthal position s and the phase-space coordinates X—and thereby the magnetic field felt by Ihe particle—are supposed fixed. Parameters characterising the synchrotron radiation may be regarded as dynamical variables of the particle since they too are determined by A' and s. For example, the critical energy,

def 3 hcpo j,, . . . 2 (^53" 4 ( W ) ' ( 2 H )

may be thought of as a parameter determining the overall scale of the distribution in energy of the photons which the particle has a t.' -pensity to emit.

The exact density of a given realisation (holding X fixed) in (u,a) space is

n x(u , a ) = £ R 5 ( 5 - Sj)6{u - * , ) , ( 2 . 1 5 )

J

where the sum is taken over all events which actually take place. Its expectation value is the distribution function of [uj,s}) {more correctly termed the probability density function)

(nX(u,s)) = Hx{s)fx(v;3)/c, (2.16)

which factorises, reflecting the statistical independence of s} and U j . Here,

„ 5\/3 e 3 S\/3crtPo

N x { 8 ) = - 6 - ^ | B t l - y ' j ) l = - 6 " — b ( l ' S , ' s ) " ' is the distribution function of s}-, or the average photon emission rate, and is independent of the par'icle'a momentum; rt = « 2/mc 2 is the classical electron radius.

| ' V ' ' | ' ' ' ' J ' • ' ' | ' ' ' ' |

0 u - 1 , • I , I ! , ..... .1 . . . . I . . . . 1

0 0.6 I IB 2 26 3

f = u/u e r i t

Fig. 2 Distribution of photons in energy

The distribution of photon energies, fx(u\ s ) . ¡ s closely related to the classical frequency spectrum of synchrotron radiation

w"'s> - ~T hep, pW^y.sr 1 1

where we follow the standard definition2

F{() S ( £ ) / t . S ( f ) d ° , 9

1 £ 3 í / t f S / 3 ( í ) < ' í ( 2 - 1 9 )

and / i 5 / 3 is a modified Bessel function (see e.g. Reís. 2 , 5, 1 0 for derivations of ( 2 . 1 7 ) and (2. 1 R ) ) . The

universal functions and S(£) are plotted in Fig. 2; note that a non-algebraic dependence of /y [»;.•;} on the momentum and magnetic FIT Id arises through the factor u- in the argument of F.

t - / T 0 = s / 2 7 r R

Fig. 3 A realisation of Px[s)

The instantaneous radiated power is

PxW =c j«nx(«,í|¿u = í £ u , - í ( a - s , ) . (2.20)

A typical realisation cf Pxi-t), obtained by simulation, is shown in Fig. 3; the parameters are such

thai the expectation value of the number of photons emitted in one revolution period is

i V x ( s ) 7 b = 1000 (2.21)

Counting the peaks, we find 1049, a 1-55*7 deviation. In preparing the figure, r 7 was been taken to be

fixed and equal to the width of a line on the printer so that the distribution of the heights of the peaks,

displayed in units of the critical energy v.c, is given by the function fx{u\s) defined in (2.18). It is worth

remarking that, although half the energy is carried away by photons with energies greater than tic, there

are few such photons. In fact 9 1 % of the photons have u < r e and 50% have u < 0-1 uc.

The expectation value of Px [s) is the classical power, given by the relativistic Larmor formula? Using

the Lorcrttz force equation in 4-veclor form (p*1 - (E/c,p) and proper time - ) , for a particle in a purely

magnetic field with p • B ~ 0

dp" - ( 0 , p x B ) , (2.22)

we express this in terms of the canonical variables

2 e 2 dp»dp„ I n V dr d-

2e ! r ,p !

;|D(w)l'

3 rrrc*1

P 2 6 ( i , ï , i ) 2 (2.23)

' Nx(s)Wx

The last form uses the mean photon energy

4 hcpo

Similarly, the mean-square quantum energy is the second moment of /x(u;.s):

(u2)x. = = H í j ^ y ^ , , , , ) ' . ( 2. 2 5 )

0

The exact power (2.20) may be split into its mean and fluctuating parts:

PxM = ftM) +Px[*), (2 26)

where Px[s) is just the difference between the classical power (2.23) and the instantaneous power in a given realisation.

The two-time correlation function of such a quantity is given by (a generalised version of) Campbell's Theorem, 2 1 , 9 closely related to the well-known Schottky formula,

(Px[s)Px{s')) = c//xWu')x.f{, - s')

55 uhSpl , 3 (2.27)

The -function expresses the fact that Px{¿) and Px(s') a-re uncorrected when s j£ s'. 1 his really means \s - s'\ » cr7 with r 7 as defined in (2.11).

Let us introduce a unit noise source, £(s), known technically8'7 as a centred, Gaussian Markov-process. It is defined to have the formal properties

( f W > = 0 , <íWÍ(s')) = í ( í - A (2.28)

with respect to our ensemble-averaging operation (...}. With this, we can confect a formal representation of the stochastic power which reproduces the essentia) properties derived above, namely (2.23) and (2.27),

PX(>) = W i l ^ ï . ' ) ' + v ^ M * . » . " ) ! ' " ^ ) 1 I 2 20)

where the constants ci and c¡ are

, del 2r.pg d d 55r.ftpjj

Statistically, there is no way of distinguishing our conceptual model of discrete random photon emission and this formal representation.

Noting that ti or ft, we see that, in this formalism, the classical radiation power has been corrected by a stochastic term of order yfh . W e also observe that, in general, the average radiation power and its quantum fluctuations depend nonlinearly on the particle's coordinates through the spatial dependences of the magnetic field.

• These are not "fundamentar constants because they tf.il! depend on the absolute v*lue of the bending field in the ring through the reference momentum po-

2.3 Radiation reaction forcea

Now that we know the distribution of photons, we can include their effect on the motion of the electrons by adding radiation reaction forces to Hamilton's equations (2.8).

A single photon emission of energy u¿ at azimuth A = 3 } (when T = i ; , say) will produce an abrupt (since r 7 is short) change in the momentum but leave the spatial position of the particle unchanged. At high energy the opening angle of the beam of radiation is 5

im« - ^~ - 0-26mrad at E0 = 1 GeV. (2.31)

It is therefore an excellent approximation to take the photon's 3-momentum vector Uj.'c = « ;p/pc to be collinear with the momentum p and apply momentum conservation to evaluate the changes in the canonical momenta

P P - "j/c,

Pv - P y - ZÏJ7-

If we consider a time-interval surrounding the moment of photon emission, which is so short that the probability of more than one photon being emitted can be neglected, then we know that the energy of the emitted photon is equal to the time-integral of the fluctuating radiation power (2.20),

«y = j Px{s) ds/c (with probability ~» 1 as e —> 0*), (2.33)

and we may reinterpret (2.32) as stochastic differentia! equations

dp= -Px(s)dt/c = ~Px[3)t'ds/c = -Px(s)zt'ds/c2 + 0{-y-2)ds,

dp* = ~Px{s){x'/t')dt/c2 = -Px(s)x'ds/c\ (2.34)

dpv = -PxWv'ß') dt/c* = -PX[s)y' ds/c2.

To complete the equations of motion, we must restore the forces due to the direct action of the external fields given by Hamilton's equations, x' = dH/ÔPx, tic. In this way we find

, _dH_ , = dH_ _ Px[f)dJl_ x ~ dPx

1 P* dx c2 dpx ' . dfí dH PX(s)dH

dpv

v dy c¿ opy

, _ d]l , _ _ dH_ PX[s) dH *l ' dp' P ~ d z t

+ c2 dp'

The Hamiltonian part of these equations has already been written out explicitly in (2.8), but it is instruc-

> We are neglecting a very, very small increase in Zi due to the small reduction in the velocity of the electron; see [2.7). I l is easy to check thai Una U utterly negligible.

live to write out the radiation terms in detail:

PÍ = ~ - (1 + Gllpp^cHi,!/,.)' + /í¡Mx,v,»)3'!£(»)] "¿' +

?; = • — - ( ! -Gi)pi>,[ci*(i.y.í),+v^»(i.».')S/JíW]'1= c ?n., (2.36)

P' = - | f - (i + a x ) p \ l b { ^ ? + K . ' ) 5 / î e w ; "= - Ç n „

This also serves to define the radiation coupling functions Ilx, JTr and îlt.

Notice the dependence on the canonical momenta—this is at the root of Robinron's Theorem on the damping partition numbers. 2 2 , 2

3. N O R M A L M O D E S A N D O P T I C A L F U N C T I O N S

In principle, the equations (2.35) completely describe electron motion under the combined influences of the applied electromagnetic fields and synchrotron radiation. At a fundamental '.evel their physical content is manifest but they are not in a form suitable for many practical calculations. Other lectures in this School have shown how useful it is to describe particle trajectories first of all in terms of the three normal modes of linearised motion around the closed orbit in the 6-dimensional phase space and then in terms of the optical functions which characterise the storage ring lattice and determine the frequencies of these modes. In a planar ring with x-y coupling terms, such a we have assumed, these are the familiar modes of linearised betatron and synchrotron motion.

In the coordinates (x.y.if), the Hamiltonian contains linear coupling terme between i and p due to the spectrometer effect of the (horizontal) bending magnets, but these may be eliminated by introducing the dispersion functions.

The remainder of this section may be skimmed by the reader who does not wish to be convinced of each step in the introduction of the dispersion, functions and the concepts of synchrotron and betatron motion as they emerge in the Hamiltonian formulation used here. He will be familiar with these notions from other lectures in the School. The following sections are includ'd principally to cover certain aspects peculiar to electron machines.

Behind the formalism, however, there He a few key physical ideas which are essential to the under­standing of what follows. In particular the reader should be aware of the distinction between the two components of the momentum deviation 6t and £ (to be introduced) and the way in which the energy loss by synchrotron radiation is coupled into the transverse oscillations through the dispersion functions.

3.1 Synchrotron motion

Let us simplify the Hamiltonian (2.6) by neglecting higher order kinematic terms in the transverse momenta and in the momentum deviation S. Accordingly, we approximate the square root term by

^ + Gx)Jp> - p>- p> * (1 + Cx)p-?L^i + — . (3.1) Before writing down the Hamiltonian, we perform a simile reseating of variables which makes all the

momenta di mens ion less:

H ~ Hi = B/pa, P x px = P l / p o , P , « - Py = r>„/po, P = p/Po- (3.2)

The canonical coordinates and the independent variable all have dimensions of length and the Hamiltonian is

pi + p2 n-.ix.Sf,^,^,^,?;«) =: -Gx(P-l) +• x

2 p

y

•+• + ^ i ( x 2 - V2) + g / f S (* 3 - W ) -r - - - (3.3)

_ V " _íZL¿c(s _ sk)cos[wriz,/c + tj>t). ^ PO^rf

The dispersion functions r/ and c are designed to eliminate the linear coupling appearing in the first term. Some higher-order couplings can be eliminated at the same time by allowing these functions to depend on momentum 2 0 and this approach is often used in nonlinear optics studies when synchrotron motion is neglected. On the other hand, for electron rings, where the value of p oscillates relatively rapidly and certainly must be included as a dynamical variable, it appears al first sight that the simplest approach would consist in defining n and f with respect to the reference momentum p0. This avoids having a Hamiltonian which depends on a canonical momentum through functions which have to be calculated (and, eventually, differentiated) numerically.^

However ve can do a little better than this if we recognize that, depending on the precise value of the RF frequency, the equilibrium momentum of the beam may not be equal to p 0; synchrotron oscillations will then take place around a slightly different value of the momentum which we shall denote as po(l + ¿a), with 6e <$L 1. With this in mind, the dispersions may be introduced by means of a canonical transformation

whose generating function is

Í2 (p^,p v , e ,x ,y,2 () = Pß\x- tï(5i,s)(5, + + ic(¿«,a)(¿« + E) -

+ (1 + St + E) [Z( 4 Z0{s)\ (3.5)

In this expression, ZQ{S), *){6tls) and f(ó,,s) are as yet unspecified functions of s; natural choices for them wil! emerge in the following. In order to take proper account of chromatic effects in cases where the equilibrium value of p is other than pq, they have also been allowed to depend para.metrica.Hy on tiie constant 63. Later we shall show how the value of 8t is determined naturally by the RF frequency. When the equations of motion are constructed from the new Hamiltonian there is no need to differentiate rj or c with respect to 6t.

• E.g. for liadron colliders where the synchrotron oscillation frequency is very low. » Despite what is said in the following paragraphs, it may yet prove convenient to define t h * dispersion in this way because

it provides a measure of by how much the equilibrium orbit differs from the reference orbit, presumed lo pass through the reference points of the beam position monitors al the centres of the magnet apertures. In practice a value quoted for the dispersion will almost always be this one, denoted below as TJ{0, »).

- 581 -

The new coordinates and momenta are given by

P „ _ a ¡ ( _ p „ (3.0)

-' = ^ = zt + Z0{s) - „ p „ + fzß + 7 f ( í , + e), P = = 1 - i. +

H, = n¡ + ^- = H- p,(6, + <r)V + [X0 + + e)] (¿> + *k' (3.7)

The spl itt ing of x into its betatron and "energy" components should be familiar. It is perhaps less

well-known that , in order to preserve the canonical structure (symplect ic i ty) , one must also use a new

longitudinal coordinate z which takes account oT local changes in the length of the particle's orbit due to

its betatron oscil lations. S ince there is no vertical bending in our perfect machine, there is no vertical

dispersion and the y transformations are trivial.

Expressing the new Hamiltonian H% in terms of the new coordinates , we can elinr ite coupling terms

linear in x$ by imposing the condit ions

dH2 âH2

oxß apß

Writing these out explicit ly, we find that rj and ç must satisfy first-order difT. iitial equations and a

periodicity condit ion,

l ' = r - V , (' -•= G - ( i f , + G 2 ) i ) - \ K2t,H„ n(6„s I - 2 II) • n(i„5), (:l.9)

which are nothing but the familiar equations defining the dispersion functior ' we emphasise again thai

f>, appears as a paraincier and that primes denote differentiation with respor' to <:. A common pracl cal

moans of determining these functions for a range of values of 6¡ is Lo cxp I them ^

V{S;*) = m(s) +• 171 (a)fif + .... ffA.a) = O J ( Ä ) .(.<0¿, - - (3.10)

and equate coefficients of 6a in the equations (3.9). Then each function may be evaluated once and for

all, indeper,Jcntly of fla.

F iom (2.36) and (3.6) it is straightforward to work out* the new equat ions of motion

/ dlh „ , . 0 1 ! .„ ,

X ß = d~p- + U t 1 ] , c ' P ß - o ' [ i " ' ••'<>:r°

V ' - | - - , ' !- -Ily/c, 13-11)

dpy

v iy

z' = -— nXT¡/c ntcT¡;c, e d^ \\,¡c. Defining an effective quadrupole gradient^ for a particle - ih the reference moinenlmif which happens

»• Soiiidimes r> is denoted D, v'' or <*v{>) hut there are no otlir iiutatimi.i íot the faiijiiR.iie fumti"U •;(•'. introdu.-^d ii i exhibit the fact that (3.9) can themselves he d m v e d from a Haiiiiltcuiian

t Alternatively we c i u l d use ihe transformation theory ef A p p . dix f!.

Î Had we included out upóle fields, it would also have bren n:i «ral to doli no an cITerlivc pixtiiji.ilc Rradii-n

fashion.

Lo be at the position of the off-momentum o r b i t a

ki{ó„s) --= Kx{s) + ±K7(i)v(6g>3)Sa ( 3 .12 )

we find, after a good deal of algebra, exploit ing the cancellations implied by (3.8) and dropping several

constant terms, that the Hamiltonian is

A ~ p\ G*x' ^ + ^ - + î*.(xî-»,) + >.(*î - 3xßy)

Y l —— M J - at) cos I —— [2 - Za[s) + nPe - (X0 - C>l(í, 1- s) • i t \ \

2 ( 1 + 6 , ) 2 2 K ß íi

' PO^rf

(3.13)

Although the terms describing betatron motion are simplified, the local fon. i lation of synchrotron motion

appears fairly complicated. However we recall that, in order to avoid dangerous synchro-betatron coupling

effects, storage rinrjs are almost always designed so that the dispersion functions vanish at the locutions

of the Ii 1'* cavit ies:

VM --- í (*k) = 0. for each k. (3.1-1)

'In practice, of course, imperfections will usually create some horizontal and vertical dispersion in the

cavities.] Then , thanks to the ^-functions, the phase of the cosine describing the RY waveform simplilirs

(3.15)

and all coupling effects between the longitudinal and transverse motions have been eliminated. We remain

free to choose the function Zo(s) to our best advantage. A formal analogy with (3.8) prompts us to demand

that

^ ~ - 0 for Xp = pp = y ~ pv - z - c - 0. (:> 1G)

The physical interpretation of this condition is clear if we notice that (in the absence of radiation effects)

the change in z around an orbit will be

2*R 2wR

I (2TTÄ ) - z ( Û ) = j z'ds^ J d~^ds, (3.17)

0 0

and the condit ion (3.16) determines a shift in the origin of phase space to a fixed point of the mapping

which describes the evolution of the phase space coordinates over one turn. Moreover we are insisting

that this hold true at every point on the circumference. Alternatively, we can describe this as a canonical

transformation to a reference frame moving with the synchronous particle.

§ A further step might be to divide the Hamilionian and all magnetic Beld terms by (1 t St] so that the rôle of the original reference momentum j v would be Laken over by p,,(l -t 6, ). This u «omet¡mea convenient. However since it is often useful ty take advantage of the properties of the separated-Tunc lion lattice which is need tor most electron machines, w« shall refrain from taking this step. The interested reader may consult Reí. 20.

Working out (3.16) explicitly, we find that Lhe effects of the sextupole terms in the longitudinal pan of the Hamiltonian cancel, leaving us with

| r ^ - ( G S + '-l)'7 2}<.-l-Zo,W 1 = 0 . (.1-19)

which is straightforwardly integrated to yield

ZoU) = -** + S.J ] * [ * , . j)rfs. [;, V

where zt is a constant, related to the stable phase angle and we defined the loca! path lengt!". slippage funct ion by

In Appendix C we give the details of the Fourier analysis of this function, sfiov. -:>¿,

is just the negative of the momentum compaction factor, ac, and how utw <"•• -OUSIR., i :U> J.

approximation to synchrotron motion from this local description.

Neglecting unimportant sextupole terms, the Ilainiltonian for local synchrturon motion i-i now

l - V J l )

Finally the requirement that this Hamillonian be periodic in s,

/ / , { * . € • , * - 2»/f) - flf(z.e,s),

means th?.t the argument of the cosine must advance by an integer mulliple (>f 2TT |>er revohiiion. In :hr limit Ss - * 0, the RF frequency has to be an integer multiple of the rovoii. t i.iti frequency on the r e f e r e n t

orbit:

-Vf "= 2 7 T / R F -- '2*h fa r h c : o ) , %U)

where h is called the harmonic number. From tiiis it follows that ( 3 . 2 2 ) is equiva1- t io

c c J 0

The equilibrium momentum of Ihe beam may be dcfcr/r.ined by imnJI shifts of the .'."F i-cqurnry. Wriiin.j

h fu ' A /rf, we may write the familiar linearised version of this relationship.

" . . . ( Q ) Sri ' 1

and exhibit the dependence of the average radius of the equilibrium orbit on the momentum compaction

factor:

R(S.) ¥ h i = R(l , a.,(i,)i.), \ d j i - - . , ( < . ) . ( 1 . 2 6 ,

In Appendix 0, the details of the phasing of the RF cavities are worked out and it is shown that, i:i smooth approximation, one may replace (3.13) with the simplified Hamillonian

(3.2i ' C 2 ;

2(1 + Í )

Including the radiation reaction effects, the equations of motion are given by (3.11 ). To make the radiation

tenn<¡ explicit, we have to work forward through the chain of variable subst i tut ions from the original forms

of \ \ z and fT( as functions of (xtp,s). These operations are deferred to the next section.

In this formulation of synchrotron motion it might appear thai we are a lways above transition energy

and have somehow neglected the possibility of the revolution frequency's increasing with momentum as

it does below transition. This appearance is only a consequence of our having used s, and not time /,

as independent variable. Transition energy does indeed occur when the increase in time taken to cover

a greater orbit length due to a momentum deviation is exactly compensated by the greater velocity of

the particle on that orbit. If the transformation of independent variable is made, and if higher order

terms are included in (2.9), the velocity (and hence the familiar -j 2 factor) enters explicitly. A canonical

transformation " 4 then restores variable sign in the coefficient of fr2.

4. RADIATION D A M F I N G

The deterministic parts of the equations (3.11) show how the transverse momrnta are damprd directly

by photon emission and how, moreover, the disperrion function couples the longitudinal damping ii lo the

radial phase space. As mentioned in the Introduction, we shall not discuss b n a l r o n motion any further

than this.

4.1 Damping ir; longitudinal phase space

I,et us set Xß ~ pß - y ---- pv = 0 and study the effects of radiation on the dynamics of longitudinal

[tlüi.se space in smooth a p p r o x i m a t e Carrying through the change.* of var iab le , wo find fr iii (2..in),

(3 6) , (3.11) and (3.27) that the deterministic parts of the equations of synchrotron motion are

„,,, sin (A(z + 2,) <fl) L- / ds(n,,•<:;, 0

~ J ds. ( n ^ / c - n , f n / c )

**ro on average

The only diss ipative term which does not average out is that in the equation for c ' and is s imply related

to the average energy loss. Integrating the definition of Iii contained in (2.35) and (2.36) we find that

2*R i*R

po J Utds^ J Px[s)t'ds = U(6S + E ) (4.2) 0 0

is the energy loss per turn of a particle with total momentum po(l + ¿>s + e).

The normalised magnetic field strength at a displacement x in the median plane is

6 ( i , 0 , 5 ) = G{s) + Ki{s)x + ^Kj{s)z7 + (4.3)

Taking i.ito account the energy lost in the lattice dipolcs and quadrupoles, we can write out the first few

terms in the expansion of the expectat ion value in powers of 8¡ and e:

;{63'E)~PIC J ds {1+ (2 + G ( s ) r i ) ( 6 i + £ r ) } c l 6 ( r 7 ( 6 i + £ ) , 0 , s ) 3

(4,1)

= c l P l c {h r b,{2h + U) - E{2I2 + U) 4 (¿ a

2 ^ 2e6t)(h - 2/< I /„) ( 0{s2)} .

The arguments of n have been suppressed and the definitions of the synchrotron radiation integrals J 5 , it

and /g will be found in Appendix D.

A fixed point of the equations (4.1) is a point where E' ~ z' - 0 and it is easy to see that one exists

on the line £ = 0 in the phase plane. The still undetermined constant z, can now be chosen according to

eVs\n{hze/R) - V[6t) = c l P l c { ] 7 + ( 2 / 2 •+ lA)bt 4 6;{12 i 2JA ^ / B ) } (-1.5'

50 as to move the origin to this natural position.

In the case of a stable fixed point , ¡j>t = hzsjK is called the stable phase angle. A particle which

maintains this phase relationship with the HF wave will find that its acceleration just balances its average

energy loss by synchrotron radiation.

negligible

we can linearise ( 4 . 1 ) (still neglecting the fluctuation terms) to find

2' = -Q e f f , £ '= rj2, z - Jt ¿ « M - r / o e - 4-7 2i¡R2hpoc PQC¿

These, of course, are the equations of a damped linear oscillator with natural frequency f l , given by

c » n ; = - ^ — c o s ç i , , (4 .S)

and are equivalent to a single second-order differential equation

i + 2 6 , ¿ + n¡2 = 0, (4 .9 )

where the damping rate Q e and the damping t ime f e are defined by

^ J_ y A dgf *r - W ^ f t ) / o = / » V ( . , j ( r f e 2 r , 2 2 P û c ; u 3 Kmc) 1 V '

The damping rate ö e coincides with the quantity ct e when J e — 2. This case is a useful reference point.

as will be explained below.

•1.2 D a m p i n g p a r t i t i o n n u m b e r s a n d d a m p i n g a p e r t u r e

Although we have not given the derivations here, damping partit ion numbers analogous to (1.0) alan

exist for the radial and vertical betatron osci l lat ions. 1 0 While the damping is linear, they satisfy the sum

rule known as Robinson 's Theorem:

M£i) " - M Ä « ) + T J 4 - ( 4 - n )

for all values of &t such that an off-momentum closed orbit given by r¡(6g,s) exists. This holds even if one

of the partition numbers is negative. In most lattice designs, J¡, — 1.

I- " varying the RF frequency, and thereby 6,, it is therefore possible lo redistribute the damping

between the longitudinal and radial modes. In a storage ring, one must ensure that each damping

partition number remains positive. The range of values of 6S in which this is true is called the tinnijiing

aperture, and is determined by the values of the synchrotron integrals ¡2, ¡A <wd

<*-<'•&• Together with (3.25), this translates directly into an allowable range of va.iation of RK frequencies or

an allowable displacement of the equilibrium orbit. For further details , including the use of lioltinson

wigglers to shift the damping aperture, the reader may consult Ref. 10.

In small storage rings, the physical aperture is usually smaller than the damping aperture.

The damping aperture is easily measured by varying the RF frequency and watching for beam blow-up

on a synchrotron light monitor.

5. Q U A N T U M F L U C T U A T I O N S A N D F O K K E R - P L A N C K E Q U A T I O N S

Wlien certain conditions are satisfied, sets of stochastic ordinary differential equations can be replaced by a partial differential equation for a distribution function on phase space. In the limit of vanishing cor­relation time of the random terms, this partial differential equation takes the form of a Fokker-Planck equation. Its physical meaning and precise relation to the stochastic equations are discussed in Ap­pendix D. Fokker-Planck equations have been applied to several problems in accelerator physics; for some examples, see Refs. 12 (several articles), 14, 17, 23 and the reference lists which they contain.

5 . 1 Q u a n t u m fluctuations in longitudinal phase space

W e now reintroduce the fluctuating part of the radiation power into the longitudinal equations of motion. To prepare the ground for writing down the Fokker-Planck equation, we write them in the form

i' = /r,(z,<0+ « « ( * , £ ) { • ( < ) , E ' = jf,(*,e)+ e,(*.oew (5.1)

where the K- and Q-functions are

KAz,c) = -ace, K,(z,n) = (fl./i)' z - — c (5.2)

Strictly speaking, these are in something of a hybrid form since the smooth approximation has not yet been applied to the fluctuation terms. These, by their nature, must be approximated in a root-mcan-

sguare fashion, rather than directly. This is easier to understand in terms of the Fokker-Planck equation. Since

the recipe for writing it down (t;ee Appendix A) simplifies by virtue of the lack of "spurious drift" terms

and we find

oF[z,£,s) dF[z,E,s) (n,\* dF{z,e,s) Jeat 3 . „,

ds dz \ c } de c àe ^ ^

Weh d^Fjz.e.s)

2(2irfi) ¿ V where F(z,e,s) is tne distribution function in longitudinal phase space and we have made the smooth approximation of the diffusion term (i.e. the one with second derivatives) in terms of the synchrotron integral / 3 defined in Appendix D.

This equation can be solved completely in terms of its Gleen function23,9 or by eigenfunction expansions8 but we can simplify it further by making a phase-mixing assumption

OO CO

(z> = j dz J dEzF{z,E,.s) = 0 (5.5) -00 -oo

which will be true in many situations, including that of equilibrium. Then we can integrate (5.4) over z

to get an equation for the reduced distribution function

F(e,s) --= / f ( 2 , e , s ) < Í 2 ,

namely

ds 2[2nR) de* (5.7)

To find the equilibrium solution Fo(^). we simply set dtF = 0 and integrate once to find

-eFo[e) = Wohld

2Jcae

When we integrate this again, choosing the constant of integration to normalise the distribution to unity,

we find the familiar gaussian distribution of momentum (or energy deviations)

where ac is the r.m.s. energy spread in the beam for a linear damping rate determined by the value of Je:

The quantity oe, which can be regarded as a measure of the strength of quantum excitation, is defined to

be the energy spread for the reference case Je = 2.

Since hfh « p, the bending radius, in an ¡somagnelic ring there is very little which can be done,

beyond varying J e , to reduce the energy spread of an electron storage ring. Moreover, the energy spread is

directly proportional to E Q . A very small decrease of at can in principle be achieved with wiggler magnets

but their usual effect is to increase it.

This is an important l imitation since it determines the energy resolution of particle physics experiment which may be trying, for example, to detect, or measure the widths of, narrow resonances in ; :.e in. ->

spectrum. 2 6 In fact, since design considerations Tor colliding beam rings usually imply p ex A",', it aliiio^:

always turns out that ae 0-1 % at the top energy of a given ring. Nevertheless e ' e rings still provide

a much finer energy resolution than any foreseeable linear collider or liadron collider and there remains

the possibility of enhancing it still further with the so-called "monochromator" insertions.' 7

The gaussian distribution of energy deviation is by no means inevitable—nonlinear terms (dissipative

or conservative) may well change it, especially in the tails (as we shall see shortly). Arguments based on

Lhe "Central Limit Theorem" should only bt applied in linear approximation and the analogy with the

Maxwel l -UolUmann velocity distribution in a gas is not a complete one.

• Often no notational distinction is made between a, and it,, so one should always be careful to imd rsr ;itnl whirl, is iiionin

(5.9)

(5.IUI

- S89 -

5.2 Fokker-Planck equation in action-angle variables

W e transform to action-angle variables of linearised synchrotron motion aj.d make A rcsraiing to variables (x, I) with

z = - — \ / 2 Î C O S { K . X ) , £ = v^siníícv). (511}

Kg CtcC

The constant can be thought of as a conversion factor between energy deviation and length -Jnits:

and Qt — f í B / 2 ^ / 0 is the synchrotron tune. With these variables, the longitudinal Ilamilinnian reii.nf-

and, by applying the results of Appendix B (or otherwise), we can derive slochasiu v<v

equivalent to (5.4)

x'=Kx[x,l) + Qx(x, !)({'), l' = K,(x,l) • Q,(x./)£ls)

where

h\lx,l) = - a t - ^ p s i n f Z . c . x ) , K,(XJ) = -à^l {l - co S(2«,x)! .

Now Qi and Qx depend on / and \ and their derivatives will contribute spurious drift terms and S O M E

algebraic complications to the Fokker-Planck equation

Here, the complete drift terms are

(S IT)

and Ihe diffusion terms are

Ç! - - 1 r , < ° £ 2 / [ i - C O 3 ( 2 « R X ) ; . = - ^ s i n ( 2 K , x ) , Q\ [1 C O S ( 2 « , 0 . ¡ L | . S ]

lu actioi -angle variables, it is of course easy to apply the averaging method ' a::d write D O W R , NIL nvcnig •<! Fokker-I'lanck equation for the action variable which will be valid on liine-SC;ILES Ion TIER (Ii,-.!' LLI;it <<l .i

• Tliis is <\ ?l>'[> lipyond tlic smooth approximation.

-\ in ii nitron oscil lation, notably the clamping t ime scale:

damping of amplitude phase advance amplitude diffusion p n a „ diffusion

The absence of a damping term for the phase x guarantees that the phase diffusion term superpose; o::

i In rapid oscillatory phase advance will lead to a uniform distribution in x o n 0 , 2 * Moreover, we have

not in- . jded the dependence of the synchrotron frequency on amplitude which produces a filamentatiori

. ifeci, further accelerating the phase-mixing.

We can solve (5.19) lor the stationary distribution

*•<>('> = ¿i « p ( - i j )

from winch wc ran evaluate the longitudinal emittance (in these units!)

2 '

7 (/)= / IF0(I)dl = c , ¡ ^ ( E - ^ í Z - ) . (5.21)

The distribution (5.20) is equivalent to a joint gaussian in z and e:

, ) = + c»/2) = ^ «P ( - ¿ - ¿ ) . {*•»)

where the natural (or zero-current) bunch length is

a z — iíC¡KT. (5.2:i)

'Chi'- iJisîrtbution is shown in Fig. 4 for some typical values of Jt r-nd ot\ the meaning of the parameters b

¿ii.ii it will be explained in a later sect'iin. In this, and similar plots to be shown later, we also show the

projection of the action distribution along one axis , given by the integral

-co 0 *

My virtue of the special properties of the g aussi an, this is just the same as (5.9). In general il nerd not be.

as we shall see later. Such a projection corresponds to the energy distribution or the longitudinal current

density profile of the bunch. It is not the shadow or the phase space distribution. The second equality in

(5.24) holds only for rotationally symmetric distributions.

0. N O N L I N E A R W I G G L E R S

In small and medium-sized e + e " storage rings, it is an excellent approximation to assume that the

radiation damping and quantum excitat ion effects remain linear to large ampl i tudes . On the other hand,

the new generation of large rings (such as T R I S T A N , LEP or the HCRA, electron ring) begin to enter

a regime where these effects can develop ampli tude-dependences which may have to be included in ¡i realistic calculation. Such nonlinear effects will tend to produce equil ibrium distributions whose cores are

fatter and whose tails decay more s l o w l y 1 3 than predicted by the linear theory of the previous sections.

Such effects may generally be expected to be detrimental lo beam stabil i ty and lifetime.

More optimistically, we might regard the existence of such effects i s an opportunity to favourabK

influence the distribution function by means of intentional dissipative noiilincarities. In this section, wi­sh all introduce the idea of a nonlinear w / g g l e r 2 5 which allows one some freedom' to shape the energy

distribution in an e + e~ ring. Such wigglers have been studied in the context of the I .KP design as a means

of reducing the severity of certain collective instabilities and, possibly, depolarizing effects.

Nonlinear wigglers are special combined-function magnets which modifify the low-intensity particle

d i s t r i b u t i o n 2 8 , 1 * ' 2 9 , 3 0 in longitudinal phase space. The original i d e a 2 5 for a nonlinear wiggler in LFI' wa>

a combined function dipole-octupolc magnet . Although this works, it produces a large additional energy

loss through its itribution to the integral / 2 . One, and only one, other multipole combinai ion e\i-i>

which produces the same nonlinear damping effect with negligible additional energy loss.

G.l Q u a d r i i p o l p - s c x t u p o l o wiggirrr

We shall consider the quadrupo/e-sextupoJe wigglcr in which, as the name suggests , there are su]>cr-

posed quadrupole and sextupole fields. The vertical component of B in the y - 0 plane is

-Í Pf{KtvX r Kimx*f2)t ((i.lj

alternating in sign between adjacent blocks of the- device, so tiiat the integrated quadrupo'.e and sexlupole

field components vanish. For simplicity, let us assurr. ? that K\w and K^w have constant nmgnit'ide in

the blocks, whose tota! length is L w . To correct this approximation for a real wiggler, one must first find

« Heyond the variation of damping partition numbrra.

- 5 9 : -

the correct gradient profiles, either numerically or by measurements on real magnets . Each term in the

ene'gy loss given below is of the form of a product of L w and powers of field-gradients and dispersion

functions and simply has to be replaced by the corresponding synchrotron radiation integral.

0.2 N o n l i n e a r d a m p i n g

Particles with a momentum deviation pn£ from the synchronous value po( 1 + St) will pass through the

wiggler v.ith a horizontal displacement rjw{6t + e) off the axis. The additional contribution to the toi al

energy loss of such particles due to the wiggler is equal to

Vw[6. ± E ) = c l P l c L w [ K^njè] + {2Klvnl + K 1 v , K M I ) $

(Ï)

{ 1 K ] w n l 6 t + 3 ( 2 X 7 n* + KlulK3wr¡Í)^}e

(iv)

Each term in this expression has a different physical significance and must be examined in its turn:

(i) These terms are independent of £ and simply add to the total value of V(6à); they include a contri­

bution to the integral Igt reducing the damping aperture somewhat .

(ii) The coefficient of e 1 is related to (¡) and adds to the linear damping rate.

(iii) lîeing proportional to e2 these average out over the phase of the oscillation.

(iv) With these terms we find a (juaJîtativeJy new effect; being proportional to f 3, they provide a damping

proportional to I 2 . The most significant contribution comes from the quadrupole-sextupole cross

term which only exists by virtue of the fact that the quadrupole and sextupole fields are spatiaiiy

superposed. Building a wiggler with separate quadrupole and sextupole blocks will not produce the

same effects, no matter how closely the blocks are spaced.

On*, can verify numerically that the nonlinear quantum excitation due to the wiggler is much less

impo'cant than the nonlinear damping.

Taking (6.2) into account, the equation of motion for I is given by (5.14) where Qi is given by (5.15)

b_- now

Kdx, 1) = - ±&>*\1 - ««(S*,*)! + (¿/2) I \,m(*.x)\< . (6.3)

where the nonlinear damping coefficient for a quadrupole-sextupole wiggler is defined by

fcd=*r I / { 2 K l u n i + KlvXTVT,l)da « Z - ^ [ 2 K \ ^ l - K l v K l v t n l ) . ( 6 . 4 ) h J h

Since the average of (sinff)H is 3 / 8 the nonlinear term makes an important contribution after phase-

averaging.

- 595 -

0 . 3 N e g a t i v e Je a n d b i r t h o f a l i m i t c y c l e

From ( 6 - 3 ) we c a n see that , even when Jt is negative, so that small amplitude synchrotron oscillations

are anti -damped, the nonlinear terms generated by the quadrupole-sextupoW wiggler can restore positive

damping at larger amplitudes. It is possible to choose 6t so that the central momentum of the beam lies

outside the damping aperture on the side of negative Je.

Returning to cartesian coordinates in phase space, ¡t is easy to show that the deterministic equation

(4.9) should be replaced by a van der Pol equation™

i + at[Je + b*?zz2\z + f ljz = 0. (6.5)

- 0 . 0 0 4 - 0 . 0 0 2 0 0 . 0 0 2 0 . 0 0 4 - 0 . 0 0 4 - 0 . 0 0 2 0 Ü.002 0 .004

/ C 22 Fig. 5 Approximate solutions of the Van der Pol equation

In Fig. 5, we show solutions of this equation which are obtained analytically (for a e / f l , <£ 1} by an

application of the averaging method. They are equivalent to integrating (6.3) for twc qualitatively distinct

cases. For clarity the damping time has been artificially shortened to a few times the synchrotron period.

T w o cases are plotted:

( a ) Here Je = 2., b = 0 and the origin of phase space is a simple attracting point. All orbits within the

separatrix of the RF bucket (not shown here) are attracted to it. If b is given a positive .aluc no

qualitative change occurs but particles with large amplitudes are damped more rapidly.

(b) Now Je = — 1 and 4 - 5 x 10 s ; small amplitudes are anti-damped but positive damping is restore^

at larger amplitudes. The linear anti-damping and nonlinear damping balance an the value

/ = / = - - - . (6.6)

corresponding to a limit cycle of (6.5) , clearly visible i s a periodic orbit which attracts particles

from both largeT and smaller amplitudes. The fixed point at the origin has become unstable via the

so-called fiopf bifurcation.

- 594 -

0 . 4 F o k k e r - P l a n c k e q u a t i o n w i t h n o n l i n e a r w i g g l e r

To sec how this new phase space structure affects the distribution function, we construrt the averaged

Fokker-Planck equation in the action variable by generalizing ( 5 . 1 9 ) and integrating over phase

" ™ = - h {" ['•' + I'' - * * ] ™ } + • <c 7 > G.5 E q u i l i b r i u m s o l u t i o n

Integrating (6.7), we find a non-gaussian equilibrium distribution,

n ( I ) ~ Z i J . , < , , e , ) - > w ( - ± l - ± I > ) . «..„

where the normalisation constant Z[Jtlb,at) will be discussed in detail bciow.

For b > 0 , the tails of this distribution decay very much faster than gaussian ones [ as exp( f 4) rallier

líian c.xp( c¿) ' and this can considerably improve the lifetime and stability of the particle beam. F.wu

wlu-u Jt is made negative, the balance between linear ant i -damping at small amplitudes and nonlinear

damping at larger amplitudes results in a stable distribution, i .e. , one which can be normalised.

By adjusting the two free parameters Jc and b we find that we have an additional degree of freedom

in moulding the longil - ' 'inal profile of :he bunch. In addition, our freedom to vary Jt is extended by the

possibility of moving ÍL on' ,1 : negative real axis.

It can he s h e w n " that distriLJtions with the same value of the ditnensionless parameter

R -- - - ¡ ¿ - («.«)) s/Tbc;

* geometrically similar.

- 595 -

J,=0 b=5.E5 tr (=0.7E-3 R = C V<e2)=1.26E-3

Fig. 8 The critical value of J e on the bifurcation

Each of the Figs. 6 -11 is analogous to Fig. 4 but includes the influence of a nonlinear wiggler of n

certain strength. To ease comparison, ail three scales of Figs. 6-11 are the same as in Fig. 4.

In this sequence of figures, we can follow what happens as a single ¡irameter, Jt. is varied from

positive to negative values. All other relevant parameters , namely the wiggler strength b and the quant um

excitation ot are held constant.

In Figs. 6 and 7 the distribution is similar in form to the gaussian Fig. 4 when b 0 except thai the

tails decay faster. Few particles lie in the region of phase spa^e where the wiggler has much influence and

therefore the r.m.s. energy spread is only very sl ightly reduced. However if off-momentum particles are

responsible for unwanted effects (e.g. depolarization) , such a distribution may be very beneficial.

- 596 -

J,=-0.15 b=5.E5 <T g=0.7E-3 R=0.21<"' V ( 6 2 ) = l

Fig. 9 r negative, a crater appears

J (=-0.3785 b = 5.E5 o t=0.7E-3 R=0-5409 V(e*)=1.51E-3

Fig. 10 Special value of Jt gives flat current profile

Win 1: .ecreased to zero, Fig. 8 (.here is no quadratic term in the exponent of (6.S); the distributir.ü

remain -» but has spr* 1 out consii b r v sine? small amplitude particles are hardly damped at all.

When Je goes negative (Figs 9-11) .he : at the centre cl" the phase space distribution becomes

a crater. The maximum density then occurs approximately above the attracting limit cycle of the deter­

ministic equations of motion (4.7). For sutficiently small negative values of J t , (Fig. 9) the profile uf I he

bunch in -cal space still contains only one hump because the craier is so shallow that it is wiped out in

the integration across phase space.

This s tate of affairs persists until Jt has passed through a special v a l u e 1 4

Jt < - 0 5409v / 2op t . (6.10)

- 597 -

Jt = -0 . 6

b = 5 . E 5

0 \ = O . 7 E - 3

R = D.8r» 7 l

V ( e 2 ) = 1 . 6 3 E - 3

Fig. 11 T w o peaks appear ¡n current profile

Around this transition value (Fig. 10), the profile is very flat. Beyond it two humps appear in the curreni

prolile (Fig. 11).

Such a flattening-out of the current distribution can be of considerable utility in tho attempts to increase the amount of stored current in a storage ring. The peak current in the bunch is lowered and the energy spread and bunch length are increased, tending to reduce the wake-fields.

At first sight, it may seem that a much larger RF voltage would be necessary to accomodate the larger

energy spread with reasonable quantum lifetime. Generally however this is not a serious problem for tw-i

reasons:

(i) In the case oT a storage ring which is also an accelerator, the bunch-lengthening effect would be needed mainly in the lower part of the energy range of the storage ring, notably at inji-rtion em-r^j and, there, there ought to bp RF voltage to spare.

(it) For a given energy spread and it F voltage, the quantum lifetime for a distribution such as (ii.H) is much longer than that of (5,20) because of the much faster decay of its tails. In other words one can fill up a much larger proportion of the RF bucket with particles without increasing the loss rale across the separat rix -

0.0 Partition function and momenta

The distribution (6.8) is normalised to unity by means of the partition Function

where u' is the error function for complex arguments (sometimes known as the plasma dispersion function 1

Considered as a function of the parameters Jt, b and ot, the partition function contains a lot of informal io:i

o

- S9S -

( J e - » 0 , b > 0) , (6.13)

which describes the neighbourhood of the bifurcation as Jt «*h\nges sign. For comparison with (;. ' 1 ) , the

longitudinal emit tance is given by

0

Evaluating this in the limit of B m a l l 6, with the help of (6.12), shows that it does indeed reduce lo ( )

when the nonlinear wiggler is turned off

More generally, all the moments of the equilibrium distribution can be found from

< 0 = j rFt[J)dI=i~2ai)nZ'l^ß. (6.16)

0 . 7 Q u a n t u m l i f e t i m e

The longitudinal quantum lifetime of the beam is the inverse of the loss rate of particles across the

separatrix due to quantum fluctuations. Other loss m e c h a n i s m s 1 1 may also contribute to determine the

net lifetime of the beam.

To calculate the quantum lifetime, one interprets the Fokker-Planck equation as a continuity equation

in phase space and identifies the diffusive component of the particle flux across the separatrix. This

component const i tutes the loss rate. At the separatrix, it is not balanced by a flux cf particles damping

down from larger ampli tudes . The details of such a calculation for the gaussian distribution were given

about the global properties of the distribution and is a f.onvcnient tool for calculation. In this respect, il

is analogous to the partition functions of equilibrium statistical mechanics.

As an aid to physical understanding, it is particularly useful lo make two distinct asymptotic expan­

sions of the reciprocal:

which is useful as we consider the transition from positive linear damping to nonlinear damping (small

values of 6), and

- 59'.) -

in Ref. 11 and can be generalized for (6.8); the details of this part may be found in Ref. M. The result is

r"= "i"U(°¡b + ïj<) "UT|

where the parameter £ is half the squared bucket half-height in units of ot\

Except for the fact that we have taken Jc =- 2 as a reference case, this is precisely the same definition

as introduced in Ref. 2 . The energy aperture £ - m a x is usually thought of as arising from the RF voltage

limitation but it may also arise from a limitation of the vacuum chamber aperture in a dispersive region

or e\ en a reduced dynamic aperture at large m o m e n t u m deviation.

The formula (6.17) includes the result for a gaussian distribution^'" as a special case.

0.8 Practical aspects

A study of the available gradients and apertures of combined-function quadrupole-sextupole magnets

would be out of place here. We only mention that , to make the nonlinear damping effert noticeable, we

need 2bo2 ~ 1. From (6.4) , we see that , for given gradients, b oc p ¿ 2 while from (5.10), we have o ' ce p¿, so

I hat ooj is independent of energy for a given set of wiggtcrs and storage ring. In addition the dependence

on the bending radius cancels out from the product. Because the dispersion function is of the order of

1 m in a lmost i l l lattice designs, it follows that the length of wiggler required is roughly independent of

the size of the storage ring and the energy at which it is operated. Since several tens of metres of wiggler

are required, we can contemplate installing them only in the largest rings.

Taking the example of LEP, we find that if a set of quadrupole-sextupole wigglers were installed with

effective parameters

= lOteslam"" 1 , = 4 0 0 t e s l a m " 2 , Lw - 4 0 m , ij«, =7 1-8 in. (Ü.Ití) ox ax1

then, at 1 beam energy of 20 GeV, where ot — 0-3 x 1 0 " 3 we would have b ~ 5 x 10^, the value we used in.

Figs. 6 - 1 1 . The energy spread used in the figures would correspond to a beam energy of around 45 GeV

in LEP.

The only serious adverse effect of a quadrupole-sextupole wi^gler seems to be the reduction in damping

aperture due to its contribution to the integral /g. This requires a slightly more elaborate control of the

RF frequency to achieve the desired value of Jt s ince the variation of Jc will be coupled with the excitation

of the wiggler.

~ 6 0 0 -

ACKNOWLEDGEMENTS I acknowledge the influence of colleagues too numerous io mention in CERN, other laboratories p.nd

several universities; some, by no means all, of their names are among the references.

The task of writing un a paper containing so many formulae and graphics has been wonderfully eased by Donald E. Knuth's program TgX; I am grateful to SLAC for access to it during a visit.

REFERENCES 1. CERN Accelerator School, General Accelerator Physics, (Orsay, 3-14 September 1984), published

as CERN 85-19 (1985).

2. M. Sands, The Physics of electron storage rings, SLAC-121 (1970).

3 . J.S. Dell, Hamiltonian. mechanics, these proceedings.

4. V. Arnold, Mathematical methods of classical mechanics. Springer-Verlag, New York, 1978.

b. J.I). Jackson, Classical electrodynamics, Wiley, New York, 1975.

6. A.A. Sokolov and l.M. Ternov, Synchrotron radiation, Pergamon Press, Berlin, 1968.

7. C.W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences,

Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.

8. R.L. Stratonovich, Topics in the Theory of Random Noise, Vols I and II, Gordon and Breach, New

York, 1967.

9. N. Wax (ed.), Selected papers on noise and stochastic processes, Dover, New York, 1954.

10. K. Hübner, Synchrotron radiation and Radiation damping in Ref. 1.

1 [. A. Piwinaki, Beam losses and lifetime in Reference 1.

12. J.M. Jowett, M. Month and S. Turner (eds.), Nonlinear Dynamics Aspects of Particle Accc'erators,

Proceedings of the Joint US/CERN School on Particle Accelerators, Sardinia 1985, Springer-Verlag, 3erlin, Heidelberg, New York, Tokyo, 1986.

13. J.M. Jowett, Non-linear dissipatwe phenomena in electron storage rings, m Ref. 12.

14. A. Hofmann and J.M. Jowett, Theory of the Dipoie-Octupole Wiggler, Part I: Phase Oscillations,

CERN/ISR-TH/81-23 (1981).

15. J.M. Jowett, Dynamics of electrons in storage rings including non-linear damping and quantum

czcitation effects, Proc. 12th International Conference on High-Energy accelerators. Batavia, 1983 and CERN LEP-TH/83-43, 1983.

16. C. Bernardini and C. Pellegrini, Linear theory of motion in electron storage rings, Ann. Phys. 40,

174 (1968).

- 601 -

17. II. Bruck, Accélérateurs circulaires des particules. Presses Universitaires de France, Paris, i960.

18. M. Dell and J.S. Bell, Radiation damping and Lagrange invariants, Particle Accelerators, 13, 13 (1983).

19. E.I). Courant and U.S. Snyder, Theory of the alternating gradient synchrotron Ann. Phys. 3, I (1958).

20. R.D. Ruth, Single particle dynamics and nonlinear resonances in circular accelerators, in Ref. 12.

21. M. Sands, Phys. Rev. 97, 470, (1955), and SLAC-SPEAR Note 9 (1969).

22. K.VV. Robinson, Radialion effects in circular accelerators, Phys. Rev. I l l , 373 (1958).

2Í1 S. Kheifets, Blowup of a weak beam due to interaction with a strong beam in an electron storage

ring, Proc. 12th International Conference on High-Energy accelerators, Batavia, 1983 and C E R N LEP-TH/83-43, 1983.

24. P.L- Morton, private communication.

25. A. flofmann, Attempt lo change the longitudinal particle distribution by a dipole-octupole wiggler,

CERN-LEP Note 192 (1979).

26. J . M . Jowett, Luminosity and energy spread in LEP, CERN-LEP-TH/85-04 (1985).

27. A. Rcnicri, Possibility of achieving uery high energy resoiution. in efectron-positron storage rings, Frascati Preprint /INF-75/6(R), (1975).

28. A. Hofmann, J.M. Jowett and S. MyerB, Change of the energy distribution in an electron storage

ring by a dipole-octupole wiggler, IEEE Trans. Nud. Sei. NS-28, 2392 (1981).

29. J.M. Jowett, Theory of the Dipole-Octupole Wiggler, Purl .7. Coupling of Phase and Betatron Os­

cillations, CERN-ISR-TH/81-24 (1981).

30. J . M . Jowett, Non-linear wigglers for large e+e~ storage rings, Proc. I2th International Conference on High-Energy accelerators, Batavia, 1983, and CERN-LEP-TH/83-40 (1983).

31. C.N. Lashmore-Davies, Kinetic theory and the Vlasov equation, thvse proceedings.

32. R.H. Helm, M.J. Lee, P.L. Morton and M. Sands, Evaluation of synchrotron radiation integrals,

IEEE Trans. Nucl. Sei. NS-20, 900 (1973).

- Ml -

A P P E N D I X A : P h y s i c a l m e a n i n g o f t h e F o k k e r - P l a n c k : e q u a t i o n

A f t e r s o m e d iscuss ion o f i ts r e l a t i o n s h i p t o L i o u v i l l e ' s T h e o r e m , w e s t a t e a r e c i p e fo r w r i t i n g d o w n

t h e F o k k e r - P l a n c k e q u a t i o n c o r r e s p o n d i n g t u a g i v e n set o f s t o c h a s t i c d i f f e r e n t i a l o u a t i o n s . A v a r i e t y of

d e r i v a t i o n s of t h i s r e l a t i o n s h i p m a y h e f o u n d i n t h e l i t e r a t u r e o n s t o c h a s t i c p r o c e s s e s 7 , 8 a n d o n e t a i l o r e d

to the p r e s e n t p r o b l e m s a n d n o t a t i o n s has b e e n g i v e n p r e v i o u s l y . ' 3

C o n s i d e r s o m e v e c t o r o f c o o r d i n a t e s X ( i ) = ( A f i , . . . , X j v ) [e.g. in a n e a r - H a m i l l o n i a n s y s t e m , t h e

c a n o n i c a l c o o r d i n a t e s a n d m o m e n t a ( < j i , . . . ,qn,pi, •.. , p n ) , w h e r e /V — 2 n ) . I t evolves in t i m e a c c o r d i n g

t o a set o f first-order s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s w i t h a s ing le no ise s o u r c e £ ( t ) s a t i s f y i n g

( { ( < ) > = 0 , ( f(i ) f ( ! ' ) > = * ( < - 1 ' ) • ( A l )

T h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s a r e t a k e n t o b e

X(l)=K(X,i)+Q(X,i)£(<)- (A2)

N o t e t h a t p u t t i n g ( 2 . 3 5 ) in th is f o r m requ i res K t o i n c l u d e t h e t e r m s in

W h e n w e s p e a k o f a r e a / i s a t i o n o f f ( ( ) w e m e a n j u s t o n e o f the e n s e m b l e of poss ib le h is tor ies o f t h e

s tochas t i c process . O f course th is e n s e m b l e w o u l d c o n t a i n m u c h m o r e i n f o r m a t i o n t h a n w e c o u l d possibly

cope w i t h so w e b e a r in m i n d t h a t w e m u s t a f t e r w a r d s a v e r a g e o v e r a l l r e a l i s a t i o n s w i t h a n a p p r o p r i a t e

w e i g h t i n g .

T o each r e a l i s a t i o n o f £(») t h e r e c o r r e s p o n d s a r e a l i s a t i o n o f the s o l u t i o n X ( f ) . L e i us i n t r o d u c e t h e

e x a c t p h a s e - s p a c e d e n s i t y for th is s o l u t i o n

F(X,() = {(X - X ( 0 ) = J!* (*." - *.<<)) ('«)

w h e r e X is a f ree v a r i a b l e a n d X ( / ) is t h e r e a l i s a t i o n o f t h e s o l u t i o n of ( A S ) s t a r t i n g f r o m i n i t i a l c o n d i t i o n s

X ( 0 ) . T h e c o n t i n u i t y e q u a t i o n fo r F is

a,F(X,¡) +• V x • |X(!)F(X,!)] = 0 . (A - l )

or , less a m b i g u o u s l y ,

a ( F ( X , i ) + V x • Í K ( X , I ) F ( X , i ) ] t V x • [Q(X,í)eCOm.í)l = 0 . ( A 5 )

a n d is c o m p l e t e l y e q u i v a l e n t t o ( A 2 ) . S ince X ( i ) d e p e n d s o n t h e values of t h e fluctuations £ { / ) , i " ( X , r )

is also a fluctuating q u a n t i t y :

F ( X , i ) = ( f ( X , ( ) ) + F ( X , t ) . ( A G )

a n d i t w o u l d b e j u s t as d i f f i cu l t t o d e a l w i t h — l e * a l o n e find—the e n s e m b l e o f its s o l u t i o n s as i t w o u l d be

t o d e a l w i t h t h a t o f ( A 2 ) .

• The generalisation to several noise sources is easy; we only need one in our application since the changes in the three canonical momenta of the electron due to a particular photon emission are correlated.

- 605 -

In the particular case of a Hamiltonian system, where

K ( ' " " ' dp* ' ' dqn ) ' ^ °' ^A7'

the gradient operator Vx = [â/dq,d/dp) in (AS) may be commuted to the right, past the X(i), giving Liouville's theorem.

To derive the Fokker-Planck equation describing the time evolution of the average phase-space density of a system, subject to deterministic drift K and diffusion Q, one must average (A5) but tak account of the way in which the fluctuations make the sharply-peaked phase space density (A3) spreí I out to become, upon averaging, a smooth function.13 In this process, information about the fine details of the distribution function is lost and we make the transition to an irreversible description of the time-evolution Reinterpreting ,F(X, i) to mean this average or amoothed-out function, the final result is

2 ¿ - a x j '

The second term inside the curly brackets is known as the "spurious drift" and is absent when the diffusion Q(X,() = Q(i) independently of amplitude. Because it often vanishes, this term is sometimes overlooked although it is essential for a complete description—see e.g. its rôle in section 5.

It is possible to avoid this apparent complication by writing stochastic differential equations which incorporate the spurious drift terms into the definition of K. One must then work with the so-called Ho calculus, in which the ordinary rules of differentiation are replaced by rules which bring in extra terms designed precisely to maintain the simple relationship between the differential equations and the Fokker-Planck equation. A change of variables is then rather more work than it is when one works in the Sí ra ton o vich interpretation as we have done in this lecture. What you gain on the swings of Fokker-Planck equations, you lose on the roundabouts of variable transformations. We spent more time on the roundabouts so the Stratonovich interpretation was the better choice. In this scheme, Markovian stochastic difFerential equations are written down as limits of equations governing real processes, as was implicit in the formulation of the electron equations of motion (2.35). Ito equations of motion can be written down to describe the same physical system but, since the Fokker-Planck equation has to be the same, they look different from (2.35).

For a system, like the electron in a storage ring, which is close to being Hamiltonian, (A2) would take the form

p = - - - ^ - P — + <Kp(q.P.O + v^Qp(q,P.')e(0-

Here the ordering of terms with respect to the small parameter Í has been chosen so that the dissipative terms have an effect proportional to £ At in a short time interval Ai. That this is so is clearer when,

- 604 -

following (AS), we write down the Fokker-Planck equation corresponding to (A9)

(AW)

+ i w i + i + \kk

where summation over repeated indices is to be understood. The first two terms represent the incom­pressible flow in phase space described by Liouville's Theorem; the second pair represents an at leasi partially irreversible drift [e.g. damping); finally the last three terms describe diffusion.

Since this School has included a course on plasma kinetic theory,31 it may be helpful to briefly discuss the relationship of the Fokker-Planck equation described there to that given here.

Mathematically speaking, these equations are not of the same form, but they are nevertheless related. Physically, of course, they describe quite different phenomena—this is one resemblance between high energy e 1er iron bunches and plasmas which turns out to be rather superficial.

itere, the most notable difference is that (A8) is aiways /inear, while in plasma physics, the name is commonly applied to an equation (sometimes also called the Landau equation) in which the distribution function appears quadratica/Jy in order to desrribe collisions between particles of the same, or different, species. Thus, while we have been exclusively concerned with single particles subject, in effect, to random externa] forces, the collision terms of the "plasma" Fokker-Planck equation describe forces acting between pairs of particles. This richer physical content makes it considerably more difficult to solve. Yet, when the distribution is linearised about the equilibrium Maxwell-Boltzmann form, an equation of '.he form (A8) is recovered. This, of course, is not surprising since, in that case, we may think of the equation as describing the distribution function of a single test particle subjected to random forces arising from the thermal background plasma. In this limit, the Rosenbluth potentials31 become independent of the perturbation of the distribution function and are related to the functions K and Q used here.

- b05 -

A P P E N D I X B : C a n o n i c a l t r a n s f o r m a t i o n s for d i s s i p a t i v e s y s t e m s

The title of this appendix verges on the self-contradictory. Of course, we do not suggest that dissi­

pative sys tems can be made canonical , only that, when their equations of motion contain a Hamiltonian

part, upon which it is convenient to make canonical t ransformat ions 3 we may extend the formalism of

generating functions to transform the dissipative parts too. This is of obvious util ity in electron storage

ring theory because the radiation terms in the equations of motion are small on average compared wiiii

those describing the applied electromagnetic fields.

bet us consider the case of a free (i.e. such that old and new coordinates are independent) transfor­

mation from (<i,p) = ( ç i , . . • ,<7n,Pi,. -- ,pn) to new variables ( Q , P ) = [Q\,. . . , Q„, P\,... , Pn).

Call the old Hamiltonian H(q,p,s) and the new K ( Q , P , s ) . Both in the old and new variables.

Hamilton's equations have to be supplemented by terms ( a , b ) and ( A , B ) which describe the dissipation:

OLD N E W

dH dp

_3H

r a ( q , p , i )

+ b ( q , p , a )

Q ' = | £ + A ( Q , F , * )

( H I )

Since Ihe transformation is canonical, there exists a generating function S(Q,q,.s), depending on the

iind new coordinates and no m o m e n t u m / such that the relationship between old and new coordin

and momenta is obtained by solving

dS

3 q ' as

K - H \ a s

ds 1 1 1 2 ]

In a sys tem with 3 degrees of freedom, 6 different types of generating function arr necessary to generate

all possible canonical transformations, and in practice each of them conies up sooner or later 3''' The other

cases can be worked by analogy to this one; for a different example , sec Rcf. 13.

The equivalence of the two descriptions of the Hamilton*.? n sys tem is guaran teed by these relationship. .

Given that the two se ts of equations in ( B l ) are supposed to describe the same dissipative system, » v

need to know how to calculate the new dissipative terms ( A , B ) .

[As in Appendix A, we must now acknowledge that any stochastic terms arc to be interpreted in

the Stratonovich, not the l lo , sense. The following results would be much more complicated in the l(<i

calculus.)

To a u cumbersome notat ions , let us denote partial deri\ atives by subscripts where convenient; ihu«,

for example, we may write a vector or a matrix

/ .

a s

dq2

dqn

a 2 s a ' s a 7 s ,

SQ¡dq¡ ' 3Q¡dq„ d'S d'S iPS

dQïdqi d Q 2 d q i ' OQ,dq„

6Q„d<ii ' 3Q„aq„

the transformation lo aclic n-angl. variMik

(113)

- 606 -

By hypothesis, the arguments of the generating function provide a unique labelling of points in extended phase space? Hence, their differentials form a basis for the vector space of differentials of dynamical variables and any such differential can be expressed as a linear combination of thera. In particular, from (B2), we may use the ordinary chain rule to calculate

dp = Sqqifq + £ q qdQ + Svds. (B4)

When the postulated equations of motion (Bl), are used to project this identity along the local direction

of time-evolution of the system, we find

dp = Sc^Hp + a)ds + 5 q q{ / f P + A)ds + Sqtds

= -H^ds + bds J [ S q q a + S"qqA = b. ( B 5 )

where we separately identified the Hamiltonian and dissipative parts of the equality.

In an analogous way, we can calculate dp and conclude

Sqq/Zp t- SqqKp + Sqg ~ Kp,

- 5q qa — SqqA = B. (06)

Since the canonical transformation is free, the Jacobîan determinant of either act of canonical variables with respect to the independent coordinate (Q,q) cannot vanish, i.e.

3p 9p Op a P

9Q a¡¡ act aq

dq aq o 1

a q

det S„<j jí 0. (BT)

(BS)

Hence the matrix 5 q q is invertible and it follows that the last members of (B5) and (B6) may be solved simultaneously to yield a líneas relationship between the diasipative terms in the old and new represen­tations:

A = {5 q Q r , ( b-5 q q a ) B = - S Q q a - 5qq[5 qqr'(b - S q q a ) .

By means of these formulae, the dissipative terms can be transformed in parallel with the canonical transformation; (B8) may be regarded as a convenient recipe for carrying out complicated transformations. It should be remembered that, ¡n the course of this work, all derivatives of the generating funrtion must be expressed in terms of the variables (Q,q) and the expressions for q(Q,P) and p(Q,P) should only be substituted afterwards. Of course, exactly the same constraint applies while transforming the ilamiltonian.

' When thete term* have fluctuating part* this property u not at all t: the Stratenovich interpretation.

inly aa a coniequence of our uie of

- 607 -

(c:i)

( C I )

h d , , ^ = i { - JL L - ( C ! + * ) » ' } e< = { ( , „ ' ) ' C „ } e \ ( C I )

w h e r e w e used ( 3 . 9 ) . W e m a k e a F o u r i e r a n a l y s i s o f i h e f u n c t i o n

r ( í „ 5 ) = ¿ r„(í„K"/", ( C 2 )

w h e r e t h e coef f ic ient ! , w i l i be y i v o n b y

r-<4'> = S5 / d a n 6 " s l ' 0

I n t e g r a t i n g b y p a r t s t w i c e a n d u s i n g p e r i o d i c i t y a r g u m e n t s , i t is n o t d i f f i c u l t t o s h o w t h a t

r.(i.) = r:.[ i .) = /",(«...) l^-'l - c W ]

0

In p a r t i c u l a r , t h e c o n s t a n t t e r m is j u s t

2*R 2wli

r ° « < > -Û^hïTJ,- < c * + K^)ds - Í G " d s " " ' < * ' > • ( C 5 )

0 0

w h e r e ae[ós) is calle--' •> momenwm compaction factor w h i c h , l i ke t h e d i s p e r s i o n f u n c t i o n s m a y be

e x p a n d e d in p o v * '-, ""eneral ly , a c <£ I .

N e g l e c t o f t h t •• ••: m o n i e s o f T{6a, s) ( a n d t h e R F v o l t a g e t e r m s ) g ives t h e s m o o t h a.pproxin)nt ion.

I n a d d i t i o n , s i n . . -.¿ a r g u m e n t s s h o w t h a t t h e first t e r m in t h e i n t e g r a l ( 0 4 ) is m u c h s m a l l e r t h a n

t h e s e c o n d .

T o s i m p l i f y t h e a n a l y s i s o f t h e R K v o l t a g e d i s t r i b u t i o n w e d e f i n e a f u n r l i o n

-- • j r{&4ta) do, (C(i) o

w h i c h g ives t h e i n c r e a s e i n p a t h l e n g t h p e r u n i t m o m e n t u m d e v i a t i o n 6- in t h e sector o f t h e r i n g b e t w e e n

a z i m u t h s 0 a n d 3.

A P P E N D I X C : L o c a l s y n c h r o t r o n m o t i o n a n d s m o o t h a p p r o x i m a t i o n

A s i n g l e s y n c h r o t r o n o s c i l l a t i o n t a k e s m a n y t u r n s of t h e m a c h i n e ; in Tact t h e l a r g e s t s y n c h r o t r o n tunes

a r e r e a l i s e d in l a r g e m a c h i n e s l i ke L E P a n d a r e o f t h e o r d e r o f 0 . 1 . I t is t h e r e f o r e n a t u r a l to s i m p l i f y t h e

d e s c r i p t i o n o f t h i s m o t i o n b y m a k i n g a F o u r i e r d e c o m p o s i t i o n of t h e H a m i l t o n i a n o n t h e c i r c u m f e r e n c e

a n d l o o k i n g o n l y a t t h e m o s t s l o w l y - v a r y i n g t e r m s .

L e t us c o n s i d e r t h e t e r m s in ( 3 . 2 1 ) i n d i v i d u a l l y .

C I M o m e n t u m c o m p a c t i o n

F r o m ( 3 . 2 1 ) t h e " k i n t t i c e n e r g y " o f s y n c h r o t r o n o s c i l l a t i o n s is g i v e n b y t h e t e r m

- « I S -

C2 Effective RF voltage

A similar Fourier analysis of the RF voltage term in (3.21 ), may he effected by subst i tut ing the icjciuii}

and expanding the cosine into complex exponentials . The result is

n ( s - sk) h{z I zs) h

[ í n ( s - s f c ) h{z -\- : . h , (t:s)

Hearing in mind the role of the RF harmonic number, we recognize that nearly ¡di llie terms in -.lie

expansion are rapidly oscillating functions of the independent variable .s and will not produre significan',

average effects on the beam. The terms which do count are those in which the .s-depen<leuce can be nnule

to cancel from the arguments of the exponentinls. From (3.23), it follows thai there nre precisely two of

these, namely the term with ri - \ h in the first group and that with n h in the second, Combining

l l iem, we can reassemble a slowly-varying cosine function

(cit)

We separate the function Ï1 into a contribution from the n — 0 term in (C2) anil a remainder:

= ac{S,)jt ' £{6t, s) (CIO)

where, by virtue of (3 .26) , (C l ) and (C2),

(CIl)

Then (treating the two terms in (CIO) differently with respect to the original i-func*îons) the .^-dependence

cancels and the argument of the cosine becomes

hst k(z + zt) h.6, - ,, .

-Tr'-^r-mf11--*^*1- (c,2)

It is clear that for the most efficient use of the fiF s y s t e m ' one should choose ihe relative phases of (lie

* We are ignoring all collccliv« effects here.

- ooy -APPENDIX D: Common synchrotron radiation integrals

In the following table we collect the definitions of some of the more important synchrotron radiation

integral; together with indications of the contexts in which they occur.

Definition Uses

h ^ fiends momentum compaction factor

I2-f0

2'RG*ds energy loss, energy spread, damping t imes,

etnittances, damping partition numbers

h = f**\G*\ds energy spread, pola r izat ion t ime, polarization

level

polarization level

h = !¿*RtG2 + 2Kl)Gr}ds energy spread, crnil lanres, damping partition

numbers

emittance

/Cl = £ ' R K\ß* ds energy loss in quadrupoles, nonlinear radiation

damping

damping partition number variation

The function M is d e f i n e d 2 ' 1 1 by

H{s) = M s ) 8 + (fl(i)f(j) + o(s]lW)')/ÍW- ( D l )

Integrals 1-5 <vere defined in Ref. 32 which also describes useful algorithms for evaluating them. These

are implemented in several computer programs [e.g. BEAMPARAM, COMFORT, PATRICIA . . . ) . Further

information on the use of these integrals will be found in Refs. 2, 10, 11, 13, 17, 25, 26 and, especially, 32.

- 6 1 0 -

BEAU BREAK UP

J . Le Duff

Laborato ire de l ' A c c é l é r a t e u r L i n é a i r e , Orsay, France

ABSTRACT A f t e r a b r i e f review of the experimental ev idence £or the beam break up i n s t a b i l i t y i n e l e c t r o n l i n a c s , emphasis i s g i v e n to the t r a n s v e r s e d e f l e c t i o n which a r i s e s from radio frequency f i e l d s . Typ ica l t r a n s ­v e r s e RF mode, in c i r c u l a r i r i s loaded waveguides are then d e s c r i b e d . F i n a l l y two types of beam break up are d i s c u s s e d : the r e g e n e r a t i v e ÖBU which o c c u r s in a s i n g l e a c c e l e r a t i n g s e c t i o n and the cumulat ive BBU which i s a m u l t i - s e c t i o n e f f e c t .

I . EXPERIMENTAL EVIDENCE

The beam break up (BBU), a l s o known a s beam blow up, i s a t r a n s v e r s e i n s t a b i l i t y

observed i n RF e l e c t r o n l i n a c s and in induc t ion l i n a c s . As e a r l y as 1957 the phenomenon

was observed on shor t l i n a c s operat ing with long p u l s e s (> 1 us) in the range of 500 mA

peak. Above the current thresho ld the beam p u l s e l e n g t h , as observed a t the output of the

a c c e l e r a t o r , i s s h o r t e n e d " . This s u g g e s t s induced f i e l d s by the head of the p u l s e which

a c t back on the t a i l to make i t u n s t a b l e . This mechanism which can occur i n a s i n g l e

a c c e l e r a t i n g s t r u c t u r e was c a l l e d r e g e n e r a t i v e beam break up.

Later on , wi th longer l i n a c s made of many s u c c e s s i v e a c c e l e r a t i n g s t r u c t u r e s , the

same p u l s e s h o r t e n i n g e f f e c t could be observed but a t a much lower t h r e s h o l d , of the order

of 10 mA peak, s u g g e s t i n g a cumulat ive e f f e c t from a l l the s t r u c t u r e s . This second mani­

f e s t a t i o n ef t i .° beam break up has been i n t e n s i v e l y s t u d i e d on the Stanford Linear A c c e l e ­

r a t o r , t w o - m i l e s l o n g , s i n c e i t appeared a s the main l i m i t i n g e f f e c t ' 1 . The e x p e r i m e n t a l

o b s e r v a t i o n s can be summarized as f o l l o w s :

a ) At any l o c a t i o n a long the a c c e l e r a t o r the t y p i c a l p i c t u r e s of the beam p u l s e s below

and above thresho ld are i l l u s t r a t e d in ' i g . 1 . The s h o r t e n i n g i s more pronounced a s the

current from the i n j e c t o r i s i n c r e a s e d

below threshold

above threshold

0 , 5 / i s / d i v i s i o n

F i g . 1 Osc i l lograms of beam p u l s e s be lov and above bean break up thresho ld

- 611 -

b) Above the t h r e s h o l d the amount of t ransmi t ted charge a long the a c c e l e r a t o r d e c r e a s e s

e r r a t i c a l l y , and i f the current i s fur ther increased the l o s s e s v i l l appear e a r l i e r a long

the a c c e l e r a t o r path ( F i g . 2 ) .

Û 5 10 15 20 25 30 35 40 — sector number F i g . 2 Transmitted charge a long the a c c e l e r a t o r

for d i f f e r e n t i n j e c t e d c u r r e n t s

c ) Above the t h r e s h o l d the beam c r o s s s e c t i o n as observed at the end of the a c c e l e r a t o r

i s randomly i n c r e a s e d in both t r a n s v e r s e d i r e c t i o n s , s u g g e s t i n g a t r a n s v e r s e i n s t a b i l i t y .

d ) A s u i t a b l e e x t e r n a l magnet ic f o c u s i n g sys t em, a s provided for i n s t a n c e by quadrupole

magnets , can improve the BBU t h r e s h o l d .

e ) The BBU e f f e c t s t r o n g l y depends on misal ignment of the a c c e l e r a t i n g s t r u c t u r e r ( o f f -

a x i s beam) and on the n o i s e l e v e l from the HF power s o u r c e s -

The beam break up, as d e s c r i b e d above , e i t h e r r e g e n e r a t i v e or c u m u l a t i v e , has been

i d e n t i f i e d a s a beam i n t e r a c t i o n v i t h p a r a s i t i c d e f l e c t i n g modes which can propagate i n

the a c c e l e r a t i n g s t r u c t u r e s . Such s i n g l e modes can have r e l a t i v e l y h igh shunt impedances

and high Q v a l u e s ; hence the beam-induced wake f i e l d has a long memory which can a f f e c t

the t a i l of l ong beam p u l s e s . In t h e s e c a s e s the e q u i v a l e n t impedance of the s u r r o u n d i n g s ,

which c a u s e s d e f l e c t i o n , i s a narrow-band type r e s o n a t o r .

I n more recent t i m e s , l i n a c s are be ing operated with very s h o r t , h igh peak c- irrent ,

p u l s e s . Th i s i s for i n s t a n c e the c a s e at SLAC, where the SLC p r o j e c t (SLAC Linear

C o l l i d e r ) r e q u i r e s a c c e l e r a t i o n of s i n g l e , h igh d e n s i t y , RF bunches, In that c a s e a new

type of beam break up has been i d e n t i f i e d i n which the head of the bunch a l s o i n f l u e n c e s

the t a i l 3 ' - However s i n c e the p u l s e i s very short the mechanism needs a much f a s t e r

induced wake f i e l d . Such a f a s t wake f i e l d can be generated by the e q u i v a l e n t low-Q, wide­

band t r a n s v e r s e impedance of the a c c e l e r a t i n g s t r u c t u r e s , corresponding to an average

e f f e c t of a l l the narrow - band p a r a s i t i c modes up to q u i t e high f r e q u e n c i e s . S ince the

frequency spectrum of very short bunches i s wide, the e x c i t a t i o n of t h i s impedance i s made

p o s s i b l e . The e f f e c t i s very s i m i l a r to the h e a d - t a i l e f f e c t in s t o r a g e t i n g s , a l t h o u g h

h e r e there i s no synchrotron o s c i l l a t i o n to enhance the a m p l i f i c a t i o n mechanism. In a

l i n e a r a c c e l e r a t o r the head of the bunch does not a l l o w d e f l e c t i o n , on ly the t a i l i s

a f f e c t e d , l e a d i n g to a banana shape as shown on F i g . 3 a. However t h i s corresponds to

the most s imple o s c i l l a t i n g mode ( d i p o l e mode) for the t a i l motion under d e f l e c t i n g

- 6 1 2 -

Tail

Fig. 3 BBU induced by short beam pulses a) dipole mode b) quadrupole mode

According to the similarity of the banana effect with the head tail, it will not De treated further in this lecture,

2. TRANSVERSE DEFLECTION OF CHARGED PARTICLES IN RADIO-FREQUENCY FIELDS

Consider an electron travelling parallel to the axis of an accelerating structure vith a velocity v < c. If the structure develops an electromagnetic field having trans­verse components, the transverse force applied to the electron is s

? x = e [E ± + v x JBJ e i eo where v = v u, u being the unit vector along the longitudinal axis oz, 3í± = u JCj_ and 0^ is the initial phase difference between the particle and the longitudinal component of the vave.

First considering a travelling wave propagating a?ong oz one has : Cj_ = EjL^y) exp i(»t - ßz)

JC± = B±(x,y) exp i(wt - ßz) where the phase velocity is defined as v^ = w/ß Hence :

P ± = e [E ± • vu Cu x BjJJ exp i(wt - ßz + 6 q)

Analysis of rtaxwel1 's equations leads to the folloving identity relating the transverse components of the field to the longitudinal electric component :

l>(u x H l ) . - i- KL . Í Vj. E z

and the force now becomes

F l - el<i - ~ ) Ei + i I TL Ea l e J tP ^ M - & + e

0>

Assuming the particle travels in synchronism vith the vave, v = v^, and since t = z/v one gets wt = tuz/v = ßz. Hence*1

"i = I 'i E* having choosen 9 o = -n/2 which means that the particle is in phase vith the transverse components of the travelling wave.

forces. Figure 3 b for instance shows a quadrupole oscillating mode for vhich the center of gravity of the tail does not move. In both cases the effective transverse emittance is increased.

- l>15 -

H t 0 " - í j ¡ IsrJ A r e l a t i v i s t i c e l e c t r o n t r a v e r s i n g t h e c a v i t y o f l e n g t h L n e a r t h e a x i s w i l l r e c e i v e

a t r a n s v e r s e momentum i m p u l s e f r o m t h e H component :

up » - e <™0v ¥ • s h o w i n g t h a t a s t a n d i n g wave TH mode c a n d e f l e c t r e l a t i v i s t i c p a r t i c l e s . I n f a c t t h i s

f i n i t e d e f l e c t i o n comes f r o m t h e i n t e r a c t i o n o f t h e p a r t i c l e w i t h t h e b a c k w a r d w a v e , w h i c h

o f c o u r s e i s n o t s y n c h r o n o u s w i t h t h e p a r t i c l e , and c a n o n l y h a v e a l i m i t e d i n t e r a c t i o n

l e n g t h .

T h e p r e v i o u s e x p r e s s i o n i s g e n e r a l and can be a p p l i e d t o a l l "ypes o f t r a v e l l i n g

w a v e s . I t shews t h a t f o r a s y n c h r o n o u s wave t h e combined e f f e c t f lo ra t h e t r a n s v e r s e

e l e c t r i c and m a g n e t i c f i e l d s i s p r o p o r t i o n n a i t o t h e t r a n s v e r s e g r a d i e n t o f t h e l o n g i t u ­

d i n a l e l e c t r i c f i e l d c o m p o n e n t . A p p l i e d t o c l a s s i c a l waves one c a n c o n c l u d e t h a t 5 1 :

- f o r TE w a v e s , s i n c e E z = 0 , t h e r e i s no t r a n s v e r s e d e f l e c t i o n w h a t e v e r t h e p a r t i c l e

v e l o c i t y i s , p r o v i d e d t h e s y n c h r o n i s m c o n d i t i o n i s s a t i s f i e d . I n e t h e r w o r d s t h e t r a n s ­

v e r s e m a g n e t i c f i e l d e x a c t l y c o m p e n s a t e s t h e t r a n s v e r s e e l e c t r i c H e l d f o r t h i s c a s e .

- f o r TM w a v e s , s y n c h r o n o u s w i t h t h e p a r t i c l e , t h e d e f l e c t i n g f o r c e i s f i n i t e b u t

d e c r e a s e s a s v = v ^ i n c r e a s e s . For v ^ a p p r o a c h i n g c , i t c a n be s e e n f r o m Mnxwel 1 ' s e q u a t i o n s

t h a t 9j_ E^ t e n d s t o z e r o ¡ h e n c e t h e t r a n s v e r s e d e f l e c t i o n f r o m s y n c h r o n o u s TM w a v e s g o e s

t o z e r o f o r u l t r a - r e l a t i v i s t i c p a r t i c l e s .

As a f i r s t c o n c l u s i o n , one d o e s n o t e x p e c t any t r a n s v e r s e d e f l e c t i o n f r o m c l a s s i c a l

t r a v e l l i n g w a v e s , s y n c h r o n o u s w i t h u l t r a - r e l a t i v i s t i c p a r t i c l e s . N o t i c e t h a t a b o v e a few

HeV e l e c t r o n s c a n b e c o n s i d e r e d a s u l t r a - r e l a t i v i s t i c ( v = c ) .

L e t us c o n s i d e r now t h e c a s e o f c l a s s i c a l s t a n d i n g - w a v e modes , k n o w i n g t h a t a l t h o u g h

a n a c c e l e r a t i n g s t r u c t u r e has been d e s i g n e d t o p r o p a g a t e a p e c u l i a r a c c e l e r a t i n g mode

w i t h o u t r e f l e c t i o n , h i g h e r - o r d e r s t a n d i n g - wave modes can d e v e l o p l o c a l l y f o r i n s t a n c e

w h e r e m e c h a n i c a l t r a n s i t i o n s o c c u r ( c h a n g e i n i r i s d i a m e t e r f o r t a p e r e d i r i s - l o a d e d

s t r u c t u r e s t o k e e p t h e a c c e l e r a t i n g g r a d i e n t c o n s t a n t ) . D e f l e c t i n g p r o p e r t i e s o f s t a n d i n g -

wave modes c a n be e m p h a z i s e d w i t h a v e r y s i m p l e e x a m p l e . C o n s i d e r f o r i n s t a n c e a c y l i n ­

d r i c a l c a v i t y i n w h i c h a T H j ^ ^ mode i s e x c i t e d ; t h e f i e l d components a r e ' :

E z = E o J l ( k r ) C 0 S *

H r * _ l T U^kO/ktUin* E „

JJ ( k r ) c o s *

2 n / X Q s w / c , w h e r e XQ i s t h e f r e e s p a c e w a v e l e n g t h .

E x p a n d i n g t h e B e s s e l f u n c t i o n s and a s s u m i n g t h e wave i s p o l a r i z e d i n t h e h o r i z o n t a l

p l a n e , o n e g e t s n e a r t h e a x i s :

- 614 -

A t t h i s p o i n t one s h o u l d m e n t i o n t h a t t h e o f f - a x i s p a r t i c l e w i l l a l s o i n t e r a c t w i t h

t h e l o n g i t u d i n a l e l e c t r i c component o f t h e d e f l e c t i n g mode, l e a d i n g to a n e n e r g y i n c r e -

men t :

D e p e n d i n g on t h e s i g n o f t h i s q u a n t i t y t h e p a r t i c l e can e i t h e r g e t m o r t e n e r g y o r

l o s e a f r a c t i o n o f i t s i n i t i a l e n e r g y . I n t h e l a t t e r c a s e t h e p a r t i c l e g i v e s e n e r g y t o t h e

d e f l e c t i n g mode.

A s i m i l a r t r e a t m e n t i n t h e c a s e o f s t a n d i n g wave TE mode w o u l d shov t h a t no d e f l e c ­

t i o n i s e x p e c t e d i n t h a t c a s e , w h i c h i n f a c t i s q u i t e o b v i o u s s i n c e b o t h t h e b a c k w a r d and

f o r w a r d waves h a v e no & z c o n p o n e n t .

3 DEFLECTING HOPES I N CIRCULAR I R I S LOADED WAVEGUIDES

Up t o now o n l y TM and TE modes h a v e beer, c o n s i d e r e d , and f o r w h i c h l i t t l e t r o u b l e

c a n be e x p e c t e d , s i n c e f o r r e l a t i v i s t i c p a r t i c l e s no s y n c h r o n o u s d e f l e c t i o n can o c c u r .

H o w e v e r , t h e s e modes h a p p e n t o be i n d e p e n d e n t s o l u t i o n s o f M a x w e l l 1 , e q u a t i o n s o n l y i n t h e

c a s e o f s i m p l i f i e d s t r u c t u r e s s u c h as s m o o t h w a v e g u i d e s and c l o s e d b o x e s . I n p r a c t i c e

t h e r e must be h o l e s , f o r i n s t a n c e i n a r e s o n a n t c a v i t y , f o r t h e t ^ a m p a s s a g e and i f one

c o n s i d e r s a TH mode i n such a r e a l c a v i t y , t h e m a g n e t i c component o f t h e f i e l d , w h i c h : . o r -

m a l l y l i e s i n a p l a n e p e r p e n d i c u l a r t o the a x i r , w i l l be d i s t o r t e d i n the n e i g h b o u r h o o d o f

t h e h o l e s ( F i g . 4 ) r e s u l t i n g i n a n a d d i t i o n a l a x i a l m a g n e t i c f i . I d c o m p o n e n t . T h u s t h e

p r e s e n c e o f t h e end h o l e s r e s u l t s i n a mode w h i c h i s no l o n g e r a p u r e TH mode , b u t a TH

l i k e mode w i t h a n a s s o c i a t e d l o n g i t u d i n a l m a g n e t i c f i e l d . Such a mode i s c a l l e d a h y b r i d

mode .

I n o r d e r t o s a t i s f y M a x w e l l ' s e q u a t i o n s i t can be shown t h a t i n t h e g e n e r a l c a s e two

i n d é p e n d a n t h y b r i d modes a r e f o u n d 1 ' ; t h e y a r e c a l l e d HE ( h y b r i d e l e c t r i c ) and HM ( h y b r i d

m a g n e t i c ) a n d t h e y become TE a n d TH modes i n t h e s p e c i a l c a s e o f s i m p l e b o u n d a r y c o n d i ­

t i o n s . H y b r i d n o d e s a r e a l s o v e r y o f t e n c a l l e d HEM ( h y b r i d e l e c t r o m a g n e t i c ) modes I n t h e

1 i t e r a t u r e .

To a c c e l e r a t e r e l a t i v i s t i c e l e c t r o n s one m a i n l y uses t r a v e l l i n g i r i s l o a d e d

w a v e g u i d e s * ' 9 1 ( F i g . 5 ) w h i c n change t h e p h a s e v e l o c i t y t o t h a t o f l i g h t .

F i g . 4 F r i n g i n g f i e l d i n t h e c u t - o f f o f a r e s o n a n t c a v i t y

F i g . 5 I r i s l o a d e d s t r u c t u r e

- 615 -

Analysis of TE and TH nodes shows that they tend to become plane waves in the limiting case where v, = c i thus they are no longer independent solutions, and here again hybrid

9

inodes are necessary Co aarisfy completely Maxwell's equations. More generally the irises of a loaded waveguide will have an effect similar to the end holes of a cavity since they will distort the field lines and introduce additional field components.

The first hybrid deflecting mode is the HEH^j since it can be snovn tnat the ȣi1gj lends to split into two independent TH^j and TE^j, the former being used for acceleration.

The general expressions for the components of this deflecting hybrid mode are compli­cated 7 1. However in the limiting case where the phase velocity v^is equal to the light velocity c, one gets simple algebraic terms for an iris loaded structure 1 :

E f = - iE o [(jlta)2 + (\kr)2] exp(it) exp(ikz-iwt)

S = Eo l<Z k a> 2 - (jfcO'l exp(H) exp(ikz-iut) Ez = 2 E o ^ 2 k r ) ] e x P i d * > expiikz-iut)

Z o H r = Eo 1 1 " " Í2 k r) 2)J exp(i*) exp(ikz-iwt)

Z Q H ç = iE o [1 - {(jka)2 -^kr)*]] exp(i*) exp(ikz-iwt)

Z QH z = iZEQ |(jkr)I exp<i*) exp(ikz-iwt)

where Z = p c is the free space wave impedance, k = 2n/>.o is the free space propaga­tion constant, and a is the iris radius. Since the transverse force vector is given by

F l = v l ez ( k = w / c = 0>

it can be seen that in the present case the nagnetic and electric forces no longer compen­sate each other at light velocity. The components of the transverse force are :

F = eE sin^ exp(ikz-iwt)

Note that space harmonics also exist in order to satisfy the periodic boundary conditions due to the irises, but they do not contribute to the deflection since in general they are not in synchronism with the particle.

Hybrid modes can exhibit some curious properties ; depending on the choice of the waveguide parameters, such as 2a and 2b (see Fig. 5), the group velocity can be either positive or negative. The group velocity is given by :

v = P/w g s where w g is the stored energy per unit length and P the tine average ycvec transmitted across the waveguide, for instance the closed surface S defined by the iris hole :

l i s

Since dS - rdr d*

one gets :

- M u

From the f i e l d components s e t p r e v i o u s l y for the H E M J J mode one g e t s f i n a l l y ;

showing that the group v e l o c i t y i s n e g a t i v e i f ka <\f6.

For a s tandard i r i s loaded waveguide, such as the SLAC one at 2 GHz, the f i r s t hybrid

mode e x h i b i t s a n e g a t i v e group v e l o c i t y ( F i g . 6 a ) , but not the next one ( F i g . 6 b ) .

As w i l l be seen l a t e r the f i r s t hybrid mode, which frequency roughly 1 .5 t imes the

a c c e l e r a t i n g mode frequency, i s the d e f l e c t i n g mode of i n t e r e s t for BBU because of i t s

n e g a t i v e group v e l o c i t y .

However, s i n c e most of the i r i s - l o a d e d s t r u c t u r e s used for e l e c t r o n a c c e l e r a t i o n are

tapered to keep the a c c e l e r a t i n g g r a d i e n t c o n s t a n t , or quas i c o n s t a n t , the B r i l l o u i n

diagram w i l l show a s many d i s p e r s i o n c u r v e s as there are d i f f e r e n t i r i s d iameters a l o n g

the s t r u c t u r e . This i s for i n s t a n c e i l l u s t r a t e d in F ig . 7 for the SLAC c a s e 6 1 .

F i g u r e s of merit are a l s o de f ined for d e f l e c t i n g modes, such as r, Q and r/Q. The

e x p r e s s i o n for r_j_ r e l e v a n t in c a l c u l a t i n g the t r a n s v e r s e d e f l e c t i o n i s g i v e n by :

which takes account of the f a c t that = 0 on the a x i s but not apart from the a x i s

( V J _ E z it 0 ) .

4 REGENERATIVE BEAH BREAK UP

R e g e n e r a t i v e BCU occurs in one a c c e l e r a t i n g s e c t i o n and i s due to the d e f l e c t i n g

H E H J J wave t r a v e l l i n g in the d i r e c t i o n of the e l e c t r o n motion with a phase v e l o c i t y

s l i g h t l y l o v e r than the l i g h t v e l o c i t y s o that approximate s y n c h r o n i s a i s p o s s i b l e . The

smal l d i f f e r e n c e in v e l o c i t y w i l l make the e l e c t r o n s l i p ahead. Depending on i t s

i n i t i a l phase the e l e c t r o n can be d e f l e c t e d in one or the o ther d i r e c t i o n ( F i g . 8 ) .

INPUT COUPLER OUTPUT COUPLER

AXIS

ELECTRON BUNCHES ENERGY FLOW

F i g . 8 D e f l e c t i n g HEH.. node in an i r i s - l o a d e d s t r u c t u r e

- 618 -

If, for instance, the electron enters the structure vUh an initial phase such that the transverse deflecting force has its maximum value, then the longitudinal electric component of the mode is zero, but since the electee,., slips ahead it will enter in a longitudinally decelerating field, off-axis, and give energy to the mode. As a consequence a noise - generated H E H J J wave can be aaplified by the beam itself as soon as it has been brought off-axis.

This can be better understood by using a schematic representation of the Lorentz force near the axis, in a system co-moving with the H E M . . wave (Fig. 9).

Electrons entering the structure at phase points 1 and 3, corresponding to maximum deflecting forces, will move off axis and since they travel faster than the wave they will enter in a retarding field and thus will transfer energy to the field. If these electrons leave the structure at points 1' and 3' corresponding to a it-slippage they will have reached the maximum deflection. Electrons which enter the structure at intermediate phases, such as point 2 will in general also transfer a positive (or zero) amount of energy to the field. The optimum ph^se slip, giving a maximum deflection, depends on the initial electron phase relative to the wave, and it can be as much as 180° as seen before-

In a dispersive structure there will be in general some frequency at which the phase slip is optimized and it is near this frequency that beam break up is most likely to occur

In addition to the amplification mechanism which has been described, regeneration (or enhancement of the amplification) can occur due to the backward wave characteristic of the HEM jj mode (v g < 0). As a matter of fact the energy deposited flowing back upstream will reinforce the original deflecting field.

Finally if the corresponding generated power exceeds the power losses into the walls, both the field and the deflection will grow exponentially leading to a transverse instabi­lity.

»- LORENTZ FORCE ELECTRON TRAJECTORIES

/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

Fig. 9 Lorentz force near the axis in the HEM^ aode, in a system co-moving with the wave

(Fig. 7).

- 619 -

The starting current, or instability threshold, for the backward wave oscillation has been estimated by several authors 1' 6 1 by equating the power generated by the beam to the power propagating into the vave and lost into the walls.

The energy given by the beam to the deflecting mode can be written as :

q E t dz < fair x J bd z

where q is the accelerated charge, the longitudinal electric component of the deflec­ting mode, x the beam transverse displacement relative to the axis. The index "b" means that the integral must be performed along the beam path.

The generated power, time averaged over an RF cycle, in complex notation, becomes :

i I A

rsE • x dz rb " - 2

where I is the beam current, L the length of the accelerating structure.

The first ter» in the bracket is obtained from the expressions of the field compo-

5 T i (w t -ßz )

where t = z/c assuming the particle velocity renitins constant and equal to c and where ß is assumed to be slightly different from k = u/c. expanding ß near the point 0 q - W Q / C where the straight line v^ = c crosses the dispersion curve (see Fig. 6 a) one gets :

S , (¡o - <„ _ «>o)/|vg| .

Introducing the following quantity :

3 E z , _ i iß z r— = k E e 3x o The second term in the bracket is obtained as follows ; since :

„ d x

p» • p! ai

Using the results of Section 2 for the transverse momentum

ie

- OJO -

o n e f i n a l l y g e t s

L z ' z '

pb = - < W 1 k E o * e dz d z ' e i 6 6 ( 2 - - z )

0 o O

h a v i n g assumed p z = p = c t e .

I n t e g r a t i o n o f t h e p r e v i o u s e x p r e s s i o n l e a d s to :

P b = 2 k ( L / n ) 3 ( e / p c ) I g ( 0 ) v i t h

a = L &ß

g(a) = j ( 1 - c o s a - % a sLn<x)/(a/K)%

I t i s f o u n d t h a t t h e r e i s a n opt imum v a l u e f o r t h e p h a s e - s l i p p a r a m e t e r a w h i c h g i v e s t h e

maximum e f f e c t :

* = 2 . 6 5 > g ( a ) = 1 . 0 4 .

T h e power l o s s c a n be deduced f r o n t h e d e f i n i t i o n o f the t r a n s v e r s e s h u n t i m p e d a n c e

p e r u n i t l e n g t h :

( 1 / k ) 2 ( 3 E z / 3 x ) 3

r i = dT7dl S i n c e :

0 = dT7dT V = P / v

s g

w h e r e v s i s t h e s t " ^d e n e r g y p e r u n i t l e n g t h , t h e n :

v Q 1 Í 3 E 1 2 v Q

E q u a t i n g t h e power l o s s t o t h e g e n e r a t e d p c v e r l e a d s t o t h e c u r r e n t t h r e s h o l d f o r

r e g e n e r a t i v e BBU :

t v g Q oc i X o l 3

1th = g " c - r w w h e r e kQ i s t h e E r e e - s p a c e w a v e l e n g t h .

N u m e r i c a l a p p l i c a t i o n

C o n s i d e r a 1 m e t e r l o n g S -band s t r u c t u r e o p e r a t i n g a t 2 , 8 CHz i n the f t /2 mode, w i t h

a n a c c e l e r a t i n g g r a d i e n t o f 15 H e V / m . The e s t i m a t e d c h a r a c t e r i s t i c s o f t h e f i r s t d e f l e c ­

t i n g mode a r e :

f r e q u e n c y = 4 . 2 5 GHz

v / c = - . 0 2 e r l K

= 2 0 0 B

H e n c e , o n e g e t s f o r t h e t h r e s h o l d , 1 ^ = 6 6 mA. I t must be n o t i c e d t h a t i n s i m i l a r

s t r u c t u r e s BBU has been o b s e r v e d a t t h e l e v e l o f 1 0 0 mA, a f t e r s e v e r a l m i c r o s e c o n d s .

5 CUHULATIVE BEAM BREAK UP

The cumulative beam break up, or multisection beam break up, differs considerably from the previous one. Here each section acts like an amplifier which provides a small increase in the amplitude nt the transverse deflecting wave (Fig. 10). Even though the gain per stage is very small the total gain in a long accelerator can be ^ery large. At each amplifying cavity there is a transverse displacement modulation and a transverse momentum modulation on the beam.

LOCAL H E M , , O S C I L L A T I O N S

UMI I l\( I I I I I / / I I I I IUI I I II E L E C T R O N B U N C H E S

Fig. 10 Multisection HEMj^ transverse deflection

The transverse displacement modulation exci tes the cavi ty through the interact ion vith the off- axis E z field component oí the H E M j j mode, and the resulting field component provides an additional momentum kick to the bean.

In the drift space between cavities the transverse momentum is converted into addi­tional displacement.

This modulation further excites the resonant field In the downstream cavities which in turn deflect the next bunches even more until finally they scrape th» accelerator vails.

The effect manifests itself at beam currents veil belov the threshold for regenera­tive BBU. It was first observed and extensively studied at SLAC^'.

The model for cumulative BBU assumes that the effect of an entire accelerator section is equivallent to an impulse at a single point (Fig. 1 1 ) , This description applies parti­cularly to machines which use tapered sections for which the synchronous length at any frequency is very short.

L

n-1 n n+l

Fig. 11 Schematic representation for cumulative BBU

H e n c e t h e w h o l e s y s t e m c a n be c o n s i d e r e d a s a t r a n s p o r t s y s t e m

pci _ r»n "12] r 1 0

t p j n ["21 ™22j WPri-l 1 " V l

O b v i o u s l y c u m u l a t i v e BBU w i l l depend on the m a g n e t i c f o c u s s i n g c h a n n e l a l o n g t h e

a c c e l e r a t o r w h i c h g i v e s a d d i t i o n a l t r a n s v e r s e momentum k i c k s t h a t c a n c o m p e n s a t e p a r t i a l l y

t h e c a v i t y d e f l e c t i o n s .

S i n c e t h e i n f o r m a t i o n i s t r a n s f e r e d f r o m one u n i t t o t h e n e x t b t h e beam i t s e l f , a

s l i g h t c h a n g e i n t h e g e o m e t r y o f e a c h s t r u c t u r e , w h i c h a l s o c h a n g e s t h e d e f l e c t i n g mode

F r e q u e n c y , w i l l p r o v i d e a d e t u n i n g e f i e c t t h a t c a n l o w e r t h e e f f i c i e n c y o f c u m u l a t i v e beam

b r e a k u p . T h i s t r i c k h a s been used on r e c e n t l i n a c s and i s v e r y s i m i l a r t o t h e o n t w h i c h

c o n s i s t s o f t a p e r i n g a s i n g l e s t r u c t u r e to l o v e r the r e g e n e r a t i v e e f f e c t .

T h e s t a n d a r d n o d e l f o r c u m u l a t i v e BBU does n o t use t h e f i e l d c o n f i g u r a t i o n f o r t h e

mode , b u t r a t h e r assumes t h e mode i s c h a i ac leL i zed by a ' . T H J L r j i e u , • nictKes

use o f t h e f a c t t h a t t h e t r a n s v e r s e k i c k i s d i r e c t l y r e l a t e d t o t h a t v e c t o r p o t e n t i a l 1 :

k~c

3 e™ Î 5 T '

T h e d e f l e c t i n g mode i s c o n s i d e r e d as a s i n g l e c a v i t y e i g e n mode and i s c h a r a c t e r i z e d

by a v e c t o r p o t e n t i a l A^ t h a t o b e y s t h e H e l m h o l t z e q u a t i o n ¡

However , s i n c e a beam t r a v e l s a l o n g t h e c a v i t y t h e a c t u a l t i m e d e p e n d a n t v e c t o r p o t e n ­

t i a l A ( r , t ) must obey t h e wave e q u a t i o n :

c' St' ° A s s u m i n g t h e f o l l o w i n g e x p a n s i o n 1 2 ' 1 :

A ( r , 0 . o ( t ) A x < r )

one g e t s :

" ' "X * " X • F J • O

I f the c u r r e n t i s f l o w i n g i n t h e z d i r e c t i o n , •> J v = 0 and t h e component A , ^

r e m a i n s .

T a k i n g t h e s c a l a r p r o d u c t o f t h e p r e v i o u s e q u a t i o n w i t h A^ and i n t e g r a t i n g o v e r t h e

v o l u m e o c c u p i e d by t h e f i e l d l e a d r t o :

o + to, u

Assuming x i s indépendant of z in the a c t i v e reg ion and i f in a d d i t i o n hy^ v a r i e s

l i n e a r l y v i t h x , one can w r i t e :

J A. dV = I x 3A

Xz dz z Xz

The i n t e g r a l in the denominator i s r e l a t e d to the s t o r e d energy :

Mixing p r e v i o u s formulae l e a d s to U 2 „ e I x 6

The t o t a l t r a n s v e r s e shunt impedance o f the c a v i t y i s such that

OTP s

Hence one f i n a l l y g e t s

e I x or

2 c ' «1 '

Losses i n the c a v i t y can be taken i n t o account in the bracket by adding a term

(<d/Q)d/dt, and d e f i n i n g the damping f a c t o r a s a = u/2Q.

I n t e g r a t i n g the prev ious e x p r e s s i o n by apply ing the G r e e n - f u n c t i o n method

g i v e s the s o l u t i o n r e l a t i n g Up at the n 1 * 1 un i t to x at the same unit :

Up = *xn ~ 2 2c I ( t ' ) x ( t ' ) e"' s i n w ( t - t ' ) d t '

having in troduced the i n i t i a l c o n d i t i o n s such that up = 0 and d ( û p ) / d t - 0 at t = 0 . The

i n t e g r a l shows that at time t . i n c a v i t y number n , t h e e f f e c t depends on the sum of the

d i s p l a c e m e n t s of each part of the beam which has a lready passed the c a v i t y . The d i s p l a c e ­

ment of each part of the beam depends on the momentum kick which was g i v e n to that part of

the beam in the prev ious c a v i t y ana can be computed from the t r a n s f e r matr ix . A computer

code would d i v i d e the beam in e lementary p o r t i o n s corresponding to t r a n s i t t imes At' and

r e p l a c e the i n t e g r a l by a sum. In fac t s i n c e the beam in a l i n a c i s hunched t h e s e p o r t i o n s

could be taken corresponding to the RF micro bunches. A s o p h i s t i c a t e d t h e o r y 1 4 1 has

inc luded the bunching in the s t a r t i n g assumptions g i v i n g the f i n a l r e s u l t in c a v i t y number

N in fXTxsof a summation over the bunch number H.

AfUîxpis to f ind a n a l y t i c s o l u t i o n s of the equat ions of cumulat ive BBU have been m;¡j.-by s e v e r a l a u t h o r s 6 ' 1 1 * L 1 ' 1 * 1 .

It is found that the beam break up can be characterized by three regimes. The first corresponds to an exponential increase of bunch displacement w i t h time (or Hf" b u n c h number). Tlie second corresponds to the maximum displacement, while tlie thirc is tlie steady s : a t e

regime.

In the exponential growth regime, for a coasting beam and no f o c u s inj; t h e e-fuldinji ííictoi i s given by :

.1/3 33/2

F e ( t > = 3 7 3 z2 I c t ft2

L V X* 0

where L is the distance between the input of two successive cavities, e V Q = ym c , X the wave length of the active mode, z the position along the accelerator.

The maximum steady state displacement arising from an initially modulated beam has an e-folding factor which can be written, with no focusing :

3'4 í 1 * 1o R J - Ï 1 / 2

valid for an accelerated beam with energy much greater than the initial energy. V'=dV7dz represents a uniform accelerating gradient.

In the case where a smooth focusing is introduced and if the accelerating gradient and the focusiny strength are constant along the accelerator the e-folding factor beco­me? :

F ' = F il - C kl z'/ F 1 ] . e e L 0 ej Here k^ is the betatron wave number of the focussing system and C is equal to 1/2 for the steady state case and 3/4 for the transient case.

Compensation of cumulative beam break up can be partly attained by good design of the focusing system. Improvement can also be made by minimizing the positioning errors (beam off-axis) and the noise fron RF sources.

- bZS -

REFERENCES

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s h o r t e n i n g i n a l i n e a r a c c e l e r a t o r w a v e g u i d e , P r o c . I E E E , 112 9 ( 1 9 6 5 ) .

2 ) R. N e a l , e d i c o r , The S c a n f o r d Two M i l e A c c e l e r a t o r , W .A . B e n j a m i n , I n c . 1 9 6 8 .

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f i e l d s i n a l i n e a r a c c e l e r a t o r , N u c l . I n s t r u í a . M e t h o d s , 178 1 ( 1 9 8 0 ) .

ù) Y . G a r a u l t , CERN 6 4 - 4 3 ( 1 9 6 4 ) .

5 ) U . K .H. P a n o f s k y , U . A . W e n z e l , R e v . S e i . I n s t r . 27 9 6 7 ( 1 9 5 6 ) .

6 ) R . H . H e l m , G . A . Loew, L i n e a r A c c e l e r a t o r s , e d i t e d by P . M . L a p o s t o l l e and A . L . S e p t i e r , J o h n W i l e y and S o n s , 1 9 7 0 , page 1 7 3 .

7 ) H . H a h n , D e f l e c t i n g mode i n c i r c u l a r i r i s l o a d e d w a v e g u i d e , R e v . S e i . I n s t r . 10 ( 1 9 6 3 ) .

8 ) G . A . L o e w , R. T a i m a n , L e c t u r e s o n e l e m e n t a r y p r i n c i p l e s o f L i n e a r A c c e l e r a t o r s , A I P C o n f e r e n c e P r o c e e d i n g s n o . 1 0 5 , SLAC Summer S c h o o l on P h y s i c s o f H i g h E n e r g y P a r t i c l e A c c e l e r a t o r s ( A m e r i c a n I n s t i t u t e o f P h y s i c s , New Y o r k , ) 9 8 3 ) p. 1 .

9 ) J . Le D u f f , D y n a m i c s and A c c e l e r a t i o n i n L i n e a r S t r u c t u r e s , L A L / R T / 8 5 - 0 1 , LAL/ORSAY ( 1 9 8 5 ) , a l s o a v a i l a b l e P r o c . CERN A c c e l e r a t o r S c h o o l , G e n e r a l A c c e l e r a t o r P h y s i c s , C i f - s u r - Y v e t t e , 1 9 8 4 (CERN 8 5 - 1 9 , CERN, G e n e v a , 1 9 8 5 ) p . 1 4 4 .

1 0 ) B.W. M o n t a g u e , P a r t i c l e S e p a r a t i o n a t H i g h E n e r g i e s : I I R a d i o f r e q u e n c y s e p a r a c i ó n , P r o g r e s s i n N u c l e a r T e c h n i q u e s and I n s t r u m e n t a t i o n , ^.i ( 1 9 6 8 ) , N o r t h H o l l a n d P u b l . C o .

1 1 ) W. ,K. H. P a n o f s k y , , T r a n s i e n t b e h a v i o u r o f beam b r e a k u p , SLAC-TN 6 6 - 2 7 ( 1 9 6 6 ) .

12 ) E. . U . C o n d o n , J . A p p l . Phys . 1 2 , 129 ( 1 9 4 1 ) .

1 3 ) V. ,K. N i e l , L . S . H a l l , R . K . C o o p e r , F u r t h e r t h e o r e t i c a l s t u d i e s o f t h e beam b r e a k up i n s t a b i l i t y , P a r t i c l e A c c e l e r a t o r s , 9_, 213 ( 1 9 7 9 ) .

14 ) R . L . G l u c k s t e r n , R . K . C o o p e r , P . J . C h a n n e l 1 , C u m u l a t i v e Beam B r e a k up i n RF l i n a c s , P a r t i c l e A c c e l e r a t o r s , ]6_> 125 ( 1 9 8 5 ) .

BEAM LOAPIMC

- 626 -

D. B o u s s a r d

CERJJ, G e n e v a . S w i t z e r l a n d

ABSTRACT Beam l o a d i n g on RF c a v i t i e s may s e r i o u s l y l i m i t t h e p e r f o r m a n c e o f h i g h - i n t e n s i t y c i r c u l a r a c c e l e r a t o r s o r s t o r a g e r i n g s . The RF p o w e r r e q u i r e m e n t s t o c o r r e c t f o r beam l o a d i n g w i l l b e f i r s t e x a m i n e d i n s e v e r a l t y p i c a l c a s e s ( l e p t o n and h a d r o n m a c h L n e s ) . T h e n , t h e m e t h o d s t o c o n t r o l t h e RF s y s t e m ( f e e d b a c k and f e e d f o r w a r d ) and a c h i e v e s t a b i l i t y u n d e r h e a v y beam l o a d i n g c o n d i t i o n s w i l l be r e v i e w e d .

1 . IKTROD'JCTIOM

I n a c c e l e r a t o r l a n g u a g e , beam l o a d i n g u s u a l l y r e f e r s t o t h e e f f e c t s i n d u c e d by t h e

p a s s a g e o f t h e beam i n t h e r a d i o f r e q u e n c y c a v i t i e s . As s u c h , i t c o u l d be c o n s i d e r e d t o

be one p a r t i c u l a r e x a m p l e o f t h e more g e n e r a l p r o b l e m o f t h e beam i n t e r a c t i o n w i t h i t s

s u r r o u n d i n g s , i n t h i s c a s e t h e c a v i t y i m p e d a n c e -

H o w e v e r t h e beam l o a d i n g p r o b l e m d e s e r v e s a s p e c i a l t r e a t m e n t , f o r s e v e r a l r e a s o n s .

F i r s t l y , t h e RF c a v i t i e s a r e v e r y o f t e n t h e l a r g e s t c o n t r i b u t o r t o t h e t o t a l r i n g

impedance ( i n t h e f o l l o w i n g We s h a l l c o n c e n t r a t e on c i r c u l a r m a c h i n e s ) a n d , c o n s e q u e n t l y ,

p o w e r c o n s i d e r a t i o n s p l a y a V e r y i m p o r t a n t r ô l e i n b e a m - l o a d i n g p r o b l e m s . S e c o n d l y ,

c o n t r a r y t o many o t h e r m a c h i n e e l e m e n t s , t h e RF c a v i t i e s a r e w e l l known i t e m s b e i n g

c a r e f u l l y d e s i g n e d and m e a s u r e d , f r o m t h e RF p o i n t o f v i e w , a n d a r e e a s i l y a c c e s s i b l e f r o m

t h e o u t s i d e w o r l d v i a t h e RF p o w e r a r o p l i f i f D e d i c a t e d c o r r e c t i o n t e c h n i q u e s c a n

t h e r e f o r e b e u s e d w h e r e n o t o n l y t h e c a v i t i b u t a l s o i t s a s s o c i a t e d RF a m p l i f i e r a r e

i n c l u d e d .

I n t h e f o l l o w i n g we s h a l l f i r s t c o n s i d e r t h e s t a t i o n a r y s i t u a t i o n e s t a b l i s h e d i n t h e

b e a m - c a v i t y s y s t e m , t h e two e x t r e m e c a s e s b e i n g when t h e b u n c h e s a r e w i d e a p a r t and when

e v e r y bucfcet i s f i l l e d . T r a v e lI i n g -wave cavities with their inherent a d v a n t a g e s e s f a r a s

beam l o a d i n g i s c o n c e r n e d w i l l be e x a m i n e d i n t h i s c o n t e x t .

B e f o r e s e t t l i n g t o t h e s t a t i o n a r y s i t u a t i o n , t h e b e a m - c a v i t y s y s t e m u n d e r g o e s a

t r a n s i e n t p h a s e w h i c h may be v e r y h a r m f u l t o t h e t h e b e a m , e s p e c i a l l y f o r h a d r o n m a c h i n e s

w i t h o u t n a t u r a l d a m p i n g . To c i r c u m v e n t t h i r p r o b l e m , i t w i l l be shown t h a t RF p o w e r must

b e a v a i l a b l e . F i n a l l y t h e v a r i o u s m e t h o d s sed t o c o n t r o l t h e RF a m p t i f i e r - c a v i t y

c o m b i n a t i o n i n o r d e r t o s u p p r e s s b e a r o - l o a d i r g e f f e c t s w i l l be r e v i e w e d -

2 . SIMGLE-BUMCH PASSAGE I H A CAVITY

/hen t h e d i s t a n c e b e t w e e n b u n c h e s i s v e r y l a r g e compared t o t h e f i l l i n g , t i m e o f t h e

c a v i t y , t h e f i e l d s i n d u c e d b y t h e p r e v i o u s b u n c h e s , o r t h e p r e v i o u s bunch p a s s a g e s o f t h e

same b u n c h , h a v e d e c a y e d s u f f i c i e n t l y a n d c a n b e n e g l e c t e d . C o n s e q u e n t l y , b e f o r e t h e

b u n c h p a s s a g e t h e RF w a v e f o r m i s a p u r e s i n e w a v e p r o d u c e d by t h e RF g e n e r a t o r ( F i g . l a ) .

The e f f e c t o f t h e b u n c h p a s s a g e i s t o e x c i t e an a d d i t i o n a l f i e l d i n t h e c a v i t y ( F i g .

l b ) . F o r a s h o r t bunch ( s h o r t compared t o t h e RF p e r i o d ) a n d c o n s i d e r i n g o n l y t h e

f u n d a m e n t a l r e s o n a n c e o f t h e c a v i t y , t h e e x c i t e d w a v e f o r m i s an e x p o n e n t i a l l y d e c a y i n g

s i n e w a v e o s c i l l a t i n g a t t h e r e s o n a n t f r e q u e n c y o f t h e c a v i t y u ^ .

C o m b i n i n g t h e g e n e r a t o r d r i v e n and beam d r i v e n w a v e f o r m s , one o b t a i n s t h e t o t a l

v o l t a g e V ( t ) a t t h e c a v i t y g a p ( F i g . l c ) .

F i g . 1 S i n g l e - b u n c h p a s s a g e i n a c a v i t y

I n v e c t o r r e p r e s e n t a t i o n t h e p o w e r d e l i v e r e d t o t h e beam by t h e RF g e n e r a t o r i s

s i m p l y :

( 1 )

w h e r e V i s t h e g e n e r a t o r d r i v e n v o l t a g e b e f o r e t h e b u n c h p a s s a g e and 1 i s t h e

- b2S -

w h e r e i s t h e r e s o n a n t f r e q u e n c y o f c a v i t y , Qq t h e u n l o a d e d c a v i t y q u a l i t y f a c t o r

a n d R t h e shLnt impedance o f t h e c a v i t y ( c i r c u i t c o n v e n t i o n ) . O b v i o u s l y V ^ o = q / C , and

t h e e n e r g y l o s t b y t h e b u n c h and s t o r e d i n t h e c a v i t y j u s t a f t e r t h e b u n c h passage amounts

t o :

« = r C V. = 7 q V L . ( 3 ) 2 bo 2 1 bo

T h e n e t p o w e r r e c e i v e d b y t h e beam P i s s i m p l y , r e m e m b e r i n g t h a t i ^ and a r e i n p h a s e :

H e r e V i s t h e e f f e c t i v e RF v o l t a g e , d e l i v e r i n g t h e n e t p o w e r P ' t o t h e beam.

I n o t h e r w o r d s , t h e beam " s e e s " o n l y o n e - h a l f o f i t s own i n d u c e d v o l t a j e :

V ^ s l / Z V. . T h i s r e s u l t i s s o m e t i m e s q u o t e d as " t h e f u n d a m e n t a l t h e o r e m o f beam l o a d i n g " , b bo 2 j

and can be d e m o n s t r a t e d more g e n e r a l l y ( p . W i l s o n ) u s i n g l i n e a r i t y and s u p e r p o s i t i o n .

S i m i l a r l y , i t i s e a s y t o show t h a t , i n f a c t . r e p r e s e n t s t h e sum o f a l l beam i n d u c e d

v o l t a g e s f o r a l l c a v i t y m o d e s . E q u a t i o n ( 5 ) l e a d s t o t h e v e c t o r d i a g r a m o f F i g . 2 , w h i c h

shows t h e v o l t a g e s b e f o r e and a f t e r t h e b u n c h p a s s a g e and t h e i r r e l a t i o n s w i t h t h e bunch

c u r r e n t . O b v i o u s l y t h e v o l t a g e t o be d e l i v e r e d b y t h e g e n e r a t o r i s h i g h e r f o r t h e

same e f f e c t i v e v o l t a g e V , t h a n i n t h e c a s e o f no beam l o a d i n g . T h e e x c e s s power c a n b e

e a s i l y computed f r o m t h e c a v i t y s h u n t r e s i s t a n c e and beam c u r r e n t .

"V| ; generator d r i v e n vintage

V _ ; vultagii aftiir buncli passage

V : net volta^i* exper ¡ ';nceú by

F i g . 2 V e c t o r d i a g r a m - S i n g l e - b u n c h p a s s a g e i n a c a v i t y

f u n d a m e n t a l component o f t h e beam c u r r e n t . When c r o s s i n g t h e gap t h e c h a r g e q ,

( i . = q / T . ; T b e i n g t h e b u n c h d i s t a n c e ) i n d u c e s t h e v o l t a g e V. , and l o s e s a f r a c t i o n b b b bo

o f i t s e n e r g y w h i c h i s f i n a l l y t r a n s f o r m e d i n t o h e a t i n t h e c a v i t y w a l l s b e f o r e t h e n e x t

bunch p a s s a g e .

I n t h e t r a n s i e n t p h a s e ( s h o r t t i m e s c a l e compared t o t h e RF p e r i o d ) , t h e c a v i t y gap

impedance c a n be r e p r e s e n t e d b y a s i n g l e c a p a c i t o r C r e l a t e d t o t h e c a v i t y p a r a m e t e r s b y :

3 . MULTIPLE-BUNCH PASSAGES

We l o o k f o r a s t a t i o n a r y s o l u t i o n , when an I n f i n i t e t r a i n o f b u n c h e s , s p a c e d b y h ^

RF p e r i o d s , c r o s s e s t h e c a v i t y g a p . F o l l o w i n g P . W i l s o n ' s a n a l y s i s 1 * we s h o u l d r e p l a c e

V i n E q . ( 5 > , w h i c h r e p r e s e n t s t h e v o l t a g e j u s t b e f o r e t h e b u n c h p a s s a g e , by t h e

c o m b i n a t i o n o f t h e g e n e r a t o r - d r i v e n v o l t a g e and t h e v o l t a g e r e s u l t i n g frr -¡11 pr . iv iou"

b u n c h p a s s a g e s . T h e d e c a y o f t h e v o l t a g e b e t w e e n two s u c c e s s i v e b u n c h p a s s a g e s i s s i m p l y

Ä = T . / T _ w h e r e T , i s t h e c a v i t y t i m e c o n s t a n t (T..=2C) / « , Q : l o a d e d c a v i t y q u a l i t y o f f f L c L

f a c t o r ) , and t h e p h a s e s h i f t w i t h r e s p e c t t o t h e RF g e n e r a t o r amounts t o i ^ u ^ T ^ - 2 » * ) ^ .

T h e r e l a t i o n V=V + 1 / 2 V . w i l l t h e r e f o r e t r a n s f o r m i n t o : g bo

-* -* -* -Ä i * -2<5 2 i * ~ 1 -* •* -• V = V + V L ( e e J T + e e J T * - . . . ) + ir V L = V + V L It g bo 2 bo g b

i n w h i c h t h e t e r m i n b r a c k e t s r e p r e s e n t s t h e c o n t r i b u t i o n s o f a l l p r e v i o u s b u n c h p a s s a g e s ,

w h e r e a s t h e l a s t one r e f l e c t s t h e e f f e c t o f t h e b u n c h on i t s e l f ( F i g . 3 ) .

F i g . 3 V e c t o r d i a g r a m - M u l t i p l e - b u n c h p a s s a g e s

U s i n g t h e sum o f t h e g e o m e t r i c s e r i e s :

• b o

one o b t a i n s :

w h i c h , " h e n s e p a r a t i n g r e a l and i m a g i n a r y p a r t s l e a d s t o :

2 ( l - 2 e cos ii *• e )

( 1 0 )

- 630 -

I f we i n t r o d u c e now t h e m o r e u s u a l c a v i t y p a r a m e t e r s :

t a n 4 ( d e t u n i n g a n g l e ) = 2Q

ß ( c o u p l i n g c o e f f i c i e n t ) ; QL = Qq ( 1 2 )

a n 0 *o = Tb/T£o * r f o b e i n ß t h e f l y i n g t i m e o f t h e u n l o a d e d c a v i t y ) , E q . ( 9 ) becomes:

O o I C

w h e r e 1 i s t h e DC beam c u r r e n t .

D - 1 - 2 e - V 1 + P > cost: ( 1 + P > t a n » ] + e " 2 V 1 + [ i i

From t h e s e e x p r e s s i o n s , i t i s p o s s i b l e t o c a l c u l a t e t h e g e n e r a t o r p o w e r needed t o

p r o d u c e a g i v e n a c c e l e r a t i n g v o l t a g e V . F o r a g e n e r a t o r w h i c h i s assumed t o be m a t c h e d ,

b y u s i n g , f o r i n s t a n c e , a c i r c u l a t o r b e t w e e n g e n e r a t o r and c a v i t y , one o b t a i n s 1 ^ :

üäfii- -* J j 4 * + B ' | * ß 2R

w h e r e A and B a r e c o m p l i c a t e d f u n c t i o n s o f c a v i t y and beam p a r a m e t e r s N u m e r i c a l

c o m p u t a t i o n s a r e r e q u i r e d t o o p t i m i z e t h e v a r i o u s p a r a m e t e r s i n o r d e r t o m i n i m i z e P

4 . L I H I T 1 H G CASE 5 Q = 0

When t h e b u n c h d i s t a n c e T, i s s h o r t compared t o t h e u n l o a d e d c a v i t y f i l l i n g t i m e b

( F i g . 4 ) , Eqs . [10) s i m p l i f y t o :

i ( 1 + ß ) ( 1 + t a n * )

- 631 -

T ,

J I I I I Bunch trun

RF envelope

F i g . 4 C e s e è0 = O. The RF w a v e f o r m i s a q u a s i s i n u s o i d

5 < l + ß > ( 1 + t a n * , )

Combined w i t h ( 1 3 ) , o n e o b t a i n s :

1+ß 1 - j t a n $

I n t h i s c a s e t h e c a v i t y g a p w a v e f o r m i s a p p r o x i m a t e l y s i n u s o i d a l ( F i g . 4 ) , a n d t h e

e q u i v a l e n t c i r c u i t o f F i g . 5 , w h e r e t h e beam c u r r e n t i s r e p r e s e n t e d b y i t s c o m p o n e n t a t

t h e RF f r e q u e n c y ( i ^ = 2 i Q f o r s h o r t b u n c h e s ) , c a n be u s e d . T h e r e t h e c o u p l i n g

c o e f f i c i e n t ß i s s i m p l y r e l a t e d t o t h e c a v i t y and g e n e r a t o r s h u n t r e s i s t a n c e s b y : B =

R / R ^ . O b v i o u s l y , V*b g i v e n b y B q . ( 2 0 ) i s t h e c a v i t y v o l t a g e ( s i n u s o i d a l i n t h e

a p p r o x i m a t i o n 4 = 0 ) d e v e l o p e d when i = 0 .

F i g . S E q u i v a l e n t c i r c u i t f o r t h e c a s e <S0 = 0

I n t h e v e c t o r d i a g r a m o f F i g . 6 a , t h e t o t a l c u r r e n t i . - i + 1 . d r i v e s t h e PLC t g b

c i r c u i t and p r o d u c e s t h e g a p v o l t a g e V . F o r a g i v e n V , t h e v e c t o r i ^ f o l l o w s t h e d o t t e d

l i n e l n F i g . 6 a , when t h e d e t u n i n g a n g l e <J» i s v a r i e d . T h i s i s b e c a u s e t h e a d m i t t a n c e

o f t h e e q u i v a l e n t RLC c i r c u i t h a s a c o n s t a n t r e a l p a r t .

I f we a g a i n assume a g e n e r a t o r c o n n e c t e d t o t h e c a v i t y v i a a c i r c u l a t o r , t h e r e q j i r e d

RF p o w e r :

P = ~ R i » E 2 S S

( 2 1 )

i s a minimum f o r g i v e n V , i and $ , i f t h e two c o n d i t i o n s :

R LOS 4» t a n * cm L b ( 1 + p ) v ( 2 2 )

R s i n $ _S

C23) V

a r e f u l f i l l e d . T h e minimum RF p o w e r f o r * =<> and fl=P i s g i v e n b y :

Í 2 ¿ )

t h e f i r s t t e r m c o r r e s p o n d i n g t o t h e c a v i t y l o s s e s and t h e s e c o n d t o t h e p o w e r d e l i v e r e d t o

t h e beam. F o r t h e o p t i m u m c o n d i t i o n s w h e r e no p o w e r i s r e f l e c t e d t o w a r d s t h e c i r c u l a t o r ,

i t i s e a s y t o see f r o m E q . ( 2 2 ) t h a t $ = c o r r e s p o n d s t o i and V b e i n g i n p h a s e

c cm g

( F i g . 6 b ) . U s u a l l y t h e r e i s a s e r v o - t u n e r w h i c h m e a s u r e s t h e p h a s e d i f f e r e n c e b e t w e e n RF

d r i v e and gap v o l t a g e , and c o n t r o l s t h e c a v i t y t u n e v i a a m e c h a n i c a l t u n e r o r f e r r i t e

b i a s , f o r i n s t a n c e . A t e q u i l i b r i u m o f t h e s e r v o - t u n e r , E q . ( 2 2 ) i s a u t o m a t i c a l l y

s a t i s f i e d .

On t h e c o n t r a r y , t h e c a v i t y c o u p l i n g i s u s u a l l y f i x e d by c o n s t r u c t i o n , and c a n o n l y

be o p t i m i z e d f o r a g i v e n v a l u e o f and 4>s- H o w e v e r f o r a h a d r o n s t o r a g e r i n g , w h e r e

4 ^ = 0 , t h e c r i t i c a l c o u p l i n g ( f l = 1 ) c o r r e s p o n d s t o t h e o p t i m u m s i t u a t i o n .

a

b

F i g - 6 V e c t o r d i a g r a m s f o r t h e c a s e á Q = 0 . Opt imum t u n i n g i n ( b ) .

- 6 5 5 -

5- THE CASE OF A TRAVELLING-WAVE STRUCTURE

I t i s Known t h a t i n a l o n g c h a i n o f c o u p l e d r e s o n a t o r s t r a v e l l i n g w a v e s c a n p r o p a g a t e

w i t h i n some f r e q u e n c y l i m i t s i . e . p a s s b a n d s o f t h e s t r u c t u r e . I n t h e t r a v e l l i n g mode o f

o p e r a t i o n , t h e s t r u c t u r e i s t e r m i n a t e d by i t s c h a r a c t e r i s t i c i m p e d a n c e and b e h a v e s l i k e a

t r a n s m i s s i o n l i n e ( F i g . 7 ) . A t s y n c h r o n i s m , t h e p h a s e v e l o c i t y i> o f t h e wave e q u a l s t h e

p a r t i c l e v e l o c i t y v^, g i v i n g maximum v o l t a g e s e e n by t h e b e a m , l i k e a n RLC c i r c u i t a t

r e s o n a n c e .

I I I I I I I M I ! I I I j

I I I I I I I 1 I I I I I I i

F i g . 7 S c h e m a t i c s o f a t r a v e l l i n g - w a v e s t r u c t u r e

F o r a s i n g l e - b u n c h p a s s a g e , i t i s u s u a l l y p o s s i b l e t o n e g l e c t t h e c a v i t y c o u p l i n g as

t h e e n e r g y t r a n s f e r f r o m c e l l t o c e l l i s much s l o w e r t h a n t h e b u n c h v e l o c i t y

( u << v : v : g r o u p v e l o c i t y ) . T h e p r e v i o u s a n a l y s i s c a n t h e r e f o r e be a p p l i e d t o t h e B P S

q u a s i u n c o u p l e d r e s o n a t o r s . I t i s g e n e r a l l y a p p l i e d a l s o f o r t h e s t a n d i n g - w a v e mode o f

o p e r a t i o n o f m u l t i c e l l c a v i t i e s , w h i c h a r e n o n - t e r m i n a t e d s t r c t u r e s 3 ^ . H o w e v e r , f o r a

r e p e t i t i v e t r a i n o f many b u n c h e s , t h e RLC e q u i v a l e n t c i r c u i t m o d e l w o u l d f a i l i n t h e

t r a v e l l i n g - w a v e mode b e c a u s e t h e waves e x c i t e d b y p r e v i o u s bunch p a s s a g e s a l s o p r o p a g a t e

a l o n g t h e s t r u c t u r e .

F o r i n s t a n c e , a t e x a c t s y n c h r o n i s m ( o = v ) , t h e w a v e s e x c i t e d i n e a c h c e l l by t h e V P

beam p a s s a g e add l i n e a r l y i n t h e f o r w a r d d i r e c t i o n , a n d , on a v e r a g e , c a n c e l i n t h e r e v e r s e

d i r e c t i o n , i n a f o r w a r d t r a v e l l i n g wave s t r u c t u r e . I n o t h e r w o r d s , t h e d e c e l e r a t i n g

e l e c t r i c f i e l d E^ i s s i m p l y p r o p o r t i o n a l t o t h e d i s t a n c e a l o n g t h e s t r u c t u r e c o u n t e d

f r o m t h e f e e d p o i n t .

I f t h e s y n c h r o n i s m i s n o t p e r f e c t , we m u s t i n t r o d u c e a p h a s e f a c t o r

exp j ( w t - ß z ) f o r e a c h i n d i v i d u a l w a v e , w h e r e z = v t and ß i s t h e wave p r o p a g a t i o n <p P M>

c o n s t a n t , w i t h t h e r e s u l t t h a t t h e i n d u c e d f i e l d E ( z ) i s p r o p o r t i o n a l t o ' .he i n t e g r a l :

exp j ( u t - ß z ) dz = / e x p j 6 d z

- t>34 -

I n t r o d u c i n g v = ÛWAB and t h e p h a s e s l i p a n g l e T d e f i n e d b y :

( 2 7 )

L b e i n g t h e s t r u c t u r e l e n g t h , one o b t a i n s :

e z ( 2 8 ) L

a n d :

z

e x p j ô dz = 1 - e x p ( - j j z )

( 2 9 )

I n p a r t i c u l a r , f o r z = 0 , t h e i n t e g r a l v a n i s h e s : t h e beam i n d u c e d f i e l d i s z e r o on

t h e u p s t r e a m e n d o f t h e s t r u c t u r e ( g e n e r a t o r s i d e ) . T h i s i s a v e r y i m p o r t a n t r e s u l t as i t

shows t h a t , f o r a t r a v e l l i n g - w a v e s t r u c t u r e , t h e r e i s no beam l o a d i n g e f f e c t s e e n b y t h e

HP g e n e r a t o r , w h i c h a l w a y s r e m a i n s m a t c h e d w i t h o u t t h e n e e d f o r a c i r c u l a t o r . I n t h e c a s e

o f a b a c k w a r d - w a v e s t r u c t u r e , w h e r e t h e g e n e r a t o r i s c o n n e c t e d t o t h e d o w n s t r e a m er J o f

t h e s t r u c t u r e , t h i s r e s u l t i s s t i l l v a l i d . Beam l o a d i n g o n l y c h a n g e s t h e f i e l d on t h e

l o a d s i d e : n o t a l l t h e g e n e r a t o r power g o e s i n t o t h e l o a d , some f r a c t i o n i s t r a n s f e r r e d t o

che beam.

The t o t a l v o l t a g e s e e n b y t h e beam i s o b t a i n e d b y i n t e g r a t i n g t h e e l e c t r i c

f i e l d , g i v e n b y ( 2 9 ) a l o n g t h e s t r u c t u r e :

w h e r e t h e p r o p o r t i o n a l i t y f a c t o r R , c a l l e d t h e s e r i e s impedance o f t h e s t r u c t u r e , I s 4) 7

c h a r a c t e r i s t i c o f i t s g e o m e t r y F i g u r e B shows a p l o t o f e q u a t i o n ( 3 1 ) i n t h e c o m p l e x

impedance p l a n e .

( 3 0 )

o

I t g i v e s f i n a l l y :

( 3 1 )

- 635 -

F i g . 8 I m p e d a n c e s e e n b y t h e beam o f a t r a v e l l i n g - w a v e e t r u c t u r e

* . TRAHSIEST CORRECTION

C o n s i d e r a g a i n t h e ca.ee o f a c a v i t y r e p r e s e n t e d b y i t s RLC e q u i v a l e n t c i r c u i t . E v e n

i n t h e c a s e á = 0 ( q u a s i s i n u s o i d s ) t h e s t a t i o n a r y s o l u t i o n o f S e c t i o n 4 , w h e r e o n l y t h e o

RF f r e q u e n c y component i s c o n s i d e r e d , c a n n o t d e s c r i b e t r a n s i e n t s i t u a t i o n s , when V o r l f c

c h a n g e r a p i d l y .

T h e w o r s t c a s e s i t u a t i o n c o r r e s p o n d s t o s s u d d e n c h a n g e o / V ( e . g . t r a n s i t i o n ) o r

* b ( * n - i e c t * o n ° ^ a P r e h u n c h e d beam, f a s t e j e c t i o n o f p a r t o f t h e b e a m ) . T h e r e s u l t i n g

u n w a n t e d t r a n s i e n t m u s t o f c o u r s e be damped f n r t h e s t a t i o n a r y s o l u t i o n d e s c r i b e d a b o v e t o

s e t t l e down p r o p e r l y , b u t I t must a l s o b e s h o r t c o m p a r e d w i t h t h e s y n c h r o t r o n p e r i o d

T . T h i s c o n d i t i o n w i l l e n s u r e t h a t t h e e f f e c t s on t h e beam s u c h a s m i s m a t c h and a

s u b s e q u e n t b l o w - u p , o r e v e n l o s s o f p a r t i c l e s , w i l l b e m i n i m u m , o r i n o t h e r w o r d s t h a t

beam l o a d i n g w i l l be p r o p e r l y c o r r e c t e d .

We s h a l l now c o n s i d e r t h e e x a m p l e o f a p r e b u n c h e d beam i n j e c t e d i n t o a n empty

m a c h i n e . B e f o r e i n j e c t i o n t h e s e r v o - t u n i n g k e e p s i , = i and V i n p h a s e . I m m e d i a t e l y t g

a f t e r i n j e c t i o n t h e new v e c t o r i b d e s t r o y s t h e e q u i l i b r i u m , and V c h a n g e s b y a l a r g e

amount u n t i l t h e t u n i n g l o o p r e t u n e s t h e c a v i t y t o a d i f f e r e n t v a l u e . U n l e s s one u s e s

v e r y f a s t t u n e r s 5 * , w h i c h stay lead to multiloop s t a b i l i t y p r o b l e m s 6 ' , i t w i l l t a k e more

t h a n a s m a l l f r a c t i o n o f a s y n c h r o t r o n p e r i o d f o r t h e t u n i n g l o o p t o s e t t l e a t I t s new

v a l u e , t h e r e s u l t b e i n g a s t r o n g d i s t o r t i o n o f t h e l o n g i t u d i n a l p h a s e p l a n e .

T h e o n l y way t o m a i n t a i n V c o n s t a n t d u r i n g t h e t r a n s i e n t p h a s e o f t h e t u n e r i s t o a c t

v i a t h e RF p o w e r ge . o r a t o r w h i c h p r o v i d e s a f a s t c o n t r o l o f V . The o b v i o u s s o l u t i o n ( P i g .

9 ) i s t o c h a n g e i i n t o i ' when t h e beam i s i u j e c t a d - I f we make :

F i g . 9 C o r r e c t i o n o f b e a m - l o a d i n g t r a n s i e n t w i t h t h e p o w e r g e n e r a t o r

( 3 2 )

t h e t o t a l c u r r e n t i n t h e c a v i t y d o e s n o t c h a n g e a n d , a t c o n s t a n t t u n i n g , V s t a y s c o n s t a n t .

I n t h e s i m p l e c a s e o f no a c c e l e r a t i o n , t h e a m p l i t u d e o f t h e p e a k c u r r e n t i 1 w h i c h

muet b e d e l i v e r e d b y t h e RF power t u b e d u r i n g t h e t r a n s i e n t p h a s e o f t h e t u n e r , i s g i v e n

b y :

T h i s e x t r a c u r r e n t must be d e l i v e r e d i n a n o n - m a t c h e d l o a d i n t h i s s i m p l i f i e d

e x a m p l e . W i t h a c i r c u l a t o r i n s e r t e d b e t w e e n t h e RF a m p l i f i e r and t h e c a v i t y ( F i g . 1 0 ) ,

t h e g e n e r a t o r i s a l w a y s m a t c h e d and t h e e x t r a c u r r e n t a l s o means e x t r a p o w e r . A g a i n f o r

* =0 a s i m i l a r a n a l y s i s c a n be m a d e ; i t g i v e s t h e p e a k p o w e r P n e e d e d d u r ' . n g t h e t r a n s i e n t

p h a s e o f t h e t u n e r ' ' :

w h e r e P q i s t h e p o w e r f o r no beam ( m a t c h e d c a v i t y ) ; t h e e x c e s s p o w e r P - P q i s s i m p l y

w a s t e d i n t o t h e l o a d t o k e e p V c o n s t a n t . One can o p t i m i z e P by s e l e c t i n g t h e b e s t c a v i t y

( 3 3 )

( 3 4 )

impedance (R opt = 2 V / Í . ) and o b t a i n t h e s i m p l e r e s u l t :

p o p t = 2 P 0 = ) V | | i „ ] / 2 ( 3 5 )

Circulator

Cavity I F i g . 10 A c i r c u l a t o r t o m a t c h t h e RF p o w e r g e n e r a t o r

Remember, n e v e r t h e l e s s , t h a t t h i s i s t h e w o r s t c a s e s i t u a t i o n and i n c e r t a i n c a s e s i t

i s p o s s i b l e t o m i n i m i z e t h e r e q u i r e d p e a k p o w e r o r p e a k c u r r e n t . I n p a r t i c u l a r , by

p r e t u n i n g t h e c a v i t y b e f o r e i n j e c t i o n , one c a n make t h e two p o w e r s , b e f o r e and a f t e r

i n j e c t i o n , e q u a l and o b t a i n i n t h i s c a s e P Q p t = f V | | ] ( f o r » s = 0 ) . W i t h s u p e r c o n d u c t i n g

c a v i t i e s , u s u a l l y w i t h o u t v a r i a b l e t u n e r s , t h e p e a k p o w e r c a n e v e n be r e d u c e d t o

9)

| V | i i b l / 8 One can a l s o r e d u c e t h e t r a n s i e n t on w i t h m u l t i p l e i n j e c t i o n s o f

s m a l l e r c u r r e n t s , o r by a d j u s t i n g t h e b u n c h i n g f a c t o r o f t h e i n j e c t e d beam.

I n t h e a b o v e a n a l y s i s , we assumed t h e f i l l i n g t i m e o f t h e c a v i t v t o b e l o n g compared

t o t h e r e v o l u t i o n p e r i o d T = 1 / f b u t s m a l l w i t h r e s p e c t t - . T , w h i c h means t h a t a a g a l l b u n c h e s a r e s u b m i t t e d t o t h e same RF v o l t a p r . i f t h i s i s n o t t h e c a s e ( Q ^ < h ; h :

h a r m o n i c number) , u n e q u a l f i l l i n g o f t h e : i n g w i l l g i v e a m o d u l a t i o n o f v a t f and i t s

m u l t i p l e s . T h e same a n a l y s i s a l l i e s h e r e : a t e a c h " b a t c h " p a s s a g e t r a n s i e n t beam

l o a d i n g must be c o r r e c t e d t e make a l l b u n c h e s s e e t h e same RF v o l t a g e . T h i s e f f e c t i s

p a r t i c u l a r l y i m p o r t a n t i n l a r g e m a c h i n e s n o t o n l y a t i n j e c t i o n b u t a l s o a t t r a n s i t i o n . As

b e f o r e , c o n d i t i o n ¡34) i s v a l i d i n t h e w o r s t c a s e s i t u a t i o n , i ^ b e i n g now t h e b a t c h

e u r r e n t .

7 . RF DRIVE GENERATION

D u r i n g t h e t r a n s i e n t p h a s e o f t h e t u n e r , we must s y n t h e s i z e i ' t o meet c o n d i t i o n

( 3 2 ) and c o r r e c t f o r t h e e f F e c t o f beam l o a d i n g . r t o b v i o u s l y i m p l i e s t h a t i ' (or t h e * g

c o r r e s p o n d i n g p o w e r P ) i s a v a i l a b l e f r o m t h e RF g e n e r a t o r , o t h e r w i s e t r a n s i e n t beam

l o a d i n g c a n n o t be c o r r e c t e d c o m p l e t e l y . V a r i o u s t e c h n i q u e s u s e d t o g e n e r a t e t h e p r o p e r

i ' w i l l now be e x a m i n e d .

7 . 1 A m p l i t u d e a n d p h a s e s e r v o l o o p s

T h e s y n t h e s i s o f i n o r d e r t o k e e p V c o n s t a n t i r r e s p e c t i v e o f t h e beam l o a d i n g

c a n be done w i t h two s e r v o l o o p s ( F i g . 1 1 ) : t h e f i r s t a c t i n g on t h e a m p l i t u d e o f i

( a m p l i t u d e l o o p ) c o n t r o l s | V | , and t h e s e c o n d m a i n t a i n s t h e r e l a t i v e p h a s e o f V and i f a

c o n s t a n t t h r o u g h t h e c o n t r o l o f t h e p h a s e o f i ^ ( p h a s e l o o p ) . The c u t - o f f f r e q u e n c y

f £ o f t h e l o o p s must be much l a r g e r t h a n t h e s y n c h r o t r o n F r e q u e n c y f g , w h i c h means

v e r y s t r o n g d a m p i n g &f beam o s c i l l a t i o n s . T h i s j u s t i f i e s t h e s i m p l i f i e d s t a b i l i t y

a n a l y s i s ^ i n w h i c h t h e beam t r a n s f e r f u n c t i o n i s n e g l e c t e d . The c u t - o f f f r e q u e n c y

i s o b v i o u s l y l i m i t e d b y t h e d e l a y s i n t h e s y s t e m , i n c l u d i n g t h e cavity b a n d w i d t h , b u t more

f u n d a m e n t a l l y b y t h e r e v o l u t i o n f r e q u e n c y f Q . T h e s i m p l e c o n f i g u r a t i o n o f F i g . 11 w i t h

h i g h loop g a i n s c a n n o t c o r r e c t t r a n s i e n t beam l o a d i n g a t and i t s m u l t i p l e s .

S t e a d y beam l o a d i n g w i t h i t s a s s o c i a t e d c a v i t y d e t u n i n g c o u l d e x c i t e mode n = 0

( R o b i n s o n i n s t a b i l i t y 1 0 ' ) i f i t w e r e n o t h e a v i l y damped by t h e p h a s e l o o p . H o w e v e r ,

mode n = 1 ( o n e w a v e l e n g t h p e r t u r n ) w h i c h i s n o t damped may show up a l s o d u e t o c a v i t y

- OAS -

F i g . 1 1 T u n i n g , a m p l i t u d e and p h a s e l o o p s . F e e d f o r w a r d c o r r e c t i o n ( d o t t e d l i n e ) .

d e t u n i n g and roust be s u p p r e s s e d b y d e d i c a t e d f e e d b a c k c i r c u i t r y a c t i n g t h r o u g h t h e RF

c a v i t y i t s e l f .

I n d e p e n d e n t a m p l i t u d e and p h a s e c o n t r o l o f V i s a w e l l known t e c h n i q u e f o r p r o t o n

m a c h i n e s . I t w o r k s s a t i s f a c t o r i l y f o r r e l a t i v e l y s m a l l beam c u r r e n t s , i . e . when t h e g a p

v o l t a g e i s p r e d o m i n a n t l y d e t e r m i n e d b y t h e g e n e r a t o r c u r r e n t ( t y p i c a l l y | i b l < U g l > -

F o r h i g h e r beam c u r r e n t s , a v a r i a t i o n o f t h e a m p l i t u d e o f i , f o r i n s t a n c e , n o t o n l y

r e s u l t s i n a v a r i a t i o n o f t h e a m p l i t u d e o f V b u t a l s o o f i t s p h a s e . I n o t h e r w o r d s , t h e

two l o o p s , w h i c h w e r e i n d e p e n d t n t a t l o w beam c u r r e n t s , become c o u p l e d t o g e t h e r and a n

u n s t a b l e b e h a v i o u r o f t h e s y s t e m r e s u l t s a b o v e a c e r t a i n beam c u r r e n t t h r e s h o l d .

P e d e r s e n ' s d e t a i l e d a n a l y s i s , 6 ^ c o n f i r m e d b y e x p e r i m e n t s on t h e CERN PS b o o s t e r , l e a d t o

t h e g e n e r a l i z e d R o b i n s o n s t a b i l i t y c r i t e r i o n , v a l i d f o r * = 0 :

( 3 6 )

w h e r e f , f and f T a r e t h e u n i t y g a i n f r e q u e n c i e s o f t h e l o o p s ( a m p l i t u d e , p h a s e

and t u n i n g r e s p e c t i v e l y ) . A l t h o u g h t h e t h r e s h o l d i s w e a k l y d e p e n d e n t on t h e l o o p c u t - o f f

f r e q u e n c i e s , L t m i g h t be d a n g e r o u s i n t h i s c o n f i g u r a t i o n t o i n c r e a s e t h e s e r v o - t u n e r

b a n d w i d t h .

A l t h o u g h i t i s i n p r i n c i p l e p o s s i b l e t o c o m p e n s a t e l o o p c o u p l i n g b y an a d d i t i o n a l

d e c o u p l i n g c i r c u i t r y so i n c r e a s i n g t h e i n s t a b i l i t y t h r e s h o l d , a much s i m p l e r s o l u t i o n i s

o f f e r e d by f e e d f o r w a r d c o r r e c t i o n .

7 . 2 F e e d f o r w a r d c o r r e c t i o n

W i t h a p i c k - u p e l e c t r o d e f o l l o w e d by a f i l t e r c e n t e r e d a t f R f l , one can o b t a i n a

s i g n a l p r o p o ' - M o n a l t o - i i n d e p e n d e n t l y f r o m t h e RF s y s t e m , and g e n e r a t e i ' (RF

- 639 -

d r i v e w i t h b e a n ) a c c o r d i n g t o < 3 2 ) w i t h a s i œ p l e a d d e r . A p p l i e d t o t h e a m p l i t u d e and

p h a s e s e r v o l o o p s d e s c r i b e d i n s e c t i o n 7 . 1 , t h e method c o n s i s t s o f i n j e c t i n g i n t o t h e

i n p u t o f t h e RF a m p l i f i e r t h e p i c k - u p s i g n a l , w i t h p r o p e r a m p l i t u d e and p h a s e ( g , ip^ t o

g e n e r a t e t h e - i ^ c u r r e n t a t t h e g a p ( F i g . 1 1 Ï . T h e a m p l i t u d e a n d p h a s e l o o p s now a c t o n

t h e q u a n t i t y i , c o r r e s p o n d i n g t o no beam l o a d i n g , i n s t e a d o f i • , and t h e c r o s s c o u p l i n g s S ^ 1 1 )

b e t w e e n l o o p s a r e r e m o v e d , as c a n be shown a n a l y t i c a l l y and e x p e r i m e n t a l l y ( F i g . 1 2 ) .

As a r e s u l t , t h e i n s t a b i l i t y t h r e s h o l d can b e c o n s i d e r a b l y i n c r e a s e d a n d , f o r i n s t a n c e ,

s t a b l e o p e r a t i n g c o n d i t i o n s h a v e b e e n o b s e r v e d i n t h e CERN PS f o r | i . | / | i | = 8 t o 1 0 .

( a ) ( b )

F i g . 12 T r a n s i e n t r e s p o n s e o f a m p l i t u d e l o o p w i t h ( b ) and w i t h o u t ( a ) f e e d f o r w a r d c o r r e c t i o n {CERN PS m a c h i n e ) . T h e l o o p r e s p o n s e become-: o s c i l l a t o r y a t h i g h i n t e n s i t y ( b o t t o m t r a c e ) w i t h o u t f e e d f o r w a r d c o r r e c t i o n

T h e s i g n a l c o r r e s p o n d i n g t o - i ^ d o e s n o t n e e d t o be s y n t h e s i z e d w i t h t h e u l t i m a t e

p r e c i s i o n a s i t o n l y r e m o v e s t h e l o o p c o u p l i n g s a n d r e s t o r e s s t a b i l i t y . F o r a v a r y i n g RF

f r e q u e n c y , t h e p i c k - u p t o c a v i t y d e l a y must b e c o n t i n o u s l y a d j u s t e d , and t h e v a r i a t i o n s in

g a i n and p h a s e o f t h e RF p o w e r a m p l i f i e r ( a s s u m e d l i n e a r ) c o r r e c t e d . I n t h e CERN P S , a

c o a r s e f e e d f o r w a r d c o r r e c t i o n ( c a v i t y c o m p e n s a t i o n ) c o v e r s t h e w h o l e RF f r e q u e n c y s w i n g

d u r i n g a c c e l e r a t i o n , b u t more p r e c i s e s e t t i n g s a r e p o s s i b l e a t a f e w c r i t i c a l ( f i x e d

f r e q u e n c y ) p o i n t s .

F e e d f o r w a r d , c o r r e c t i o n c a n a l s o b e c o n s i d e r e d as a means t o r e d u c e t h e e f f e c t i v e

i m p e d a n c e o f t h e c a v i t y s e e n b y t h e beam. A t t h e RF f r e q u e n c y , t h e beam i n d u c e d v o l t a g e

o n t h e c a v i t y a m p l i f i e r c o m b i n a t i o n i s z e r o f o r a p e r f e c t c o r r e c t i o n . From t h i s p o i n t o f

v i e w , h i g h a m p l i t u d e and p h a s e l o o p g a i n s a t f g a r e no l o n g e r r e q u i r e d t o c o r r e c t beam

l o a d i n g a s V i s a u t o m a t i c a l l y k e p t c o n s t a n t b y t h e f e e d f o r w a r d c o m p e n s a t i o n . A p p l i c a t i o n

o f t h i s t e c h n i q u e ( l o w l o o p g a i n s ) w a s , f o r i n s t a n c e , u s e d on t h e B r o o k h a v e n AGS d u r i n g

a d i a b a t i c c a p t u r e .

I t i s i n t e r e s t i n g t o m e n t i o n a v a r i a n t o f t h e f e e d f o r w a r d t e c h n i q u e d e r i v e d f r o m t h e

A l v a r e z l i n e a r a c c e l e r a t o r t e c h n o l o g y . I f t h e g e n e r a t o r Ls a g r l d d e d t u b e ( t e t r o d e o r

t r i o d e ) , i t s o u t p u t i m p e d a n c e i s h i g h i f maximum RF p o w e r l s t o be e x t r a c t e d f r o m t h e

t u b e . When c o n n e c t e d t o t h e c a v i t y b y a l o n g l i n e , i t f u l l y r e f l e c t s t h e beam l o a d i n g

wave t r a v e l l i n g f r o m t h e c a v i t y t o t h e g e n e r a t o r . One can c h o o s e t h e l e n g t h o f t h e l i n e

t o make t h e r e f l e c t e d wave c a n c e l t h e beam i n d u c e d v o l t a g e a t t h e g a p , t h e h i g h i m p e d a n c e

o f t h e g e n e r a t o r i s t h e n t r a n s f o r m e d i n t o a q u a s i - s h o r t c i r c u i t a t t h e c a v i t y . M o t e t h a t ,

e v e n w i t h no v o l t a g e i n d u c e d o n t h e g a p , t h e g e n e r a t o r s e e s a m i s m a t c h e d l o a d w i t h b e a a

and must b e a b l e t o d e l i v e r t h e c u r r e n t u n d e r t h i s c o n d i t i o n . T h i s t e c h n i q u e i s l n use o n

t h e CERN PS 2 0 0 MHz RF s y s t e m , w i t h t r o m b o n e s I n s e r t e d on t h e f e e d e r l i n e s o f t h e

f i x e d - t u n e c a v i t i e s .

I f t h e p i c k - u p t o c a v i t y d e l a y i s a d j u s t e d t o be e x a c t l y one t u r n Í T ) , beam

l o a d i n g c a n c e l l a t i o n c a n be a c h i e v e d , n o t o n l y a t f B t . . h u t a l s o a t f r e q u e n c i e s f 1 2 )

RF n f

T h i s i s r e l a t i v e l y e a s y a t f i x e d RF f r e q u e n c y , f o r e x a m p l e i n t h e CERN ISR

b u t w i t h m o d e r n s a m p l e d o r d i g i t a l f i l t e r s and v a r i a b l e d e l a y s i t i s a l s o p o s s i b l e t o

f o l l o w a v a r y i n g RF f r e q u e n c y . T h e o v e r a l l r e s u l t i s a r a p i d l y c h a n g i n g i m p e d a n c e ,

i d e a l l y 2 e r o a t f r e q u e n c i e s n f Q , b u t t w i c e a l a r g e a t i n t e r m e d i a t e f r e q u e n c i e s ,

C n + t 4 ) f Q , w h e r e t h e r e a r e no beam c u r r e n t comp n e n t s ( F i g . 1 3 ) . W i t h a one t u r n d e l a y

a n d p e r f e c t c a n c e l l a t i o n , t h e v o l t a g e p e r t u r t i t i o n o n l y l a s t s w h i c h i s s m a l l compared

w i t h T s i n c e Q = ( f T ) 1 i s u s u a l l y « 1 . n o t h e r w o r d s t h e r e d u c t i o n o f t h e m a g n i t u d e s s o s 1 °

o f t h e c a v i t y i m p e d a n c e a t t h e s y n c h r o t r o n SÍ e j l i t e s n f ± m f i s a l s o l a r g e ( f a c t o r - 1 O S

(2sinmirQ ) ) f o r a s m a l l Q .

/ \ F i g - 13

R

V

-rof,

.3 RF f e e d b a c k a ~ound t h e c o w e r ampl e r

R e s i d u a l impedance a t s y n c h r o t r o n s i d e b a n d s f o r a one t u r n d e l a y f e e d f o r w a r d c o r r e c t i o n

We c a n c o n s i d e r t h e c a v i t y i t s e l f a s a iara p i c k - u p t u n e d a t f R p a n d o b t a i n t h e

- i ^ s i g n a l f r o m t h e g a p i t s e l f . T h i s l e a d s :> t h e c o n f i g u r a t i o n o f F i g . là i n w h i c h one

o b v i o u s l y r e c o g n i z e s a f e e d b a c k l o o p b u i l t a Jund t h e RF p o w e r a m p l i f i e r . From t h e l o o p

e q u a t i o n s one o b t a i n s :

GZ L

w h i c h , f o r GZ » 1 <GZ: l o o p g a i n , Z c a v i t y in »edance) r e d u c e s t o e q u a t i o n ( 3 2 ) :

i b e i n g h e r e t h e g e n e r a t o r c u r r e n t w i t h no oeara.

- 641 -

A * —1

amplitude ond phase loops

R L Cí

-Tilling

F i f i - 1 * RF f e e d b a c k « r o u n d t h e p o w e r a m p l i f i e r

T h e f e e d b a c k l o o p a u t o m a t i c a l l y g e n e r a t e s t h e c o r r e c t c o m p e n s a t i n g s i g n a l , w h i c h i s

a n o t h e r way o f s a y i n g t h a t i t k e e p s t h e c o n t r o l l e d p a r a m e t e r V c o n s t a n t . One c a n c o n s i d e r

RF f e e d b a c k a s a means t o r e d u c e t h e o u t p u t i m p e d a n c e o f t"ne RF a m p l i f i e r , a w e l l known

d e s i g n b e i n g t h e c a t h o d e f o l l o w e r w i t h i t s l o w o u t p u t i m p e d a n c e w h i c h s h u n t s t h e c a v i t y .

H o w e v e r , s t a b i l i t y o f t h e c a t h o d e f o l l o w e r w i t h a r e a c t i v e l o a d n e e d s c a r e f u l s t u d y . 1 3 '

Even s i m p l e r , b u t o f l i m i t e d e f f i c i e n c y , i s t h e u s e o f a t r i o d e i n s t e a d o f a t e t r o d e

as t h e RF p o w e r t u b e , t h e i n t e r n a l p l a t e t o g r i d f e e d b a c k r e d u c i n g t h e o u t p u t i m p e d a n c e .

I n t h e same way p u l s i n g t h e DC c u r r e n t o f t h e RF t u b e o r p o w e r i n g second t u b e , i n

1 4 )

p a r a l l e l • h a s b e e n u s e d t o r e d u c e t h e o u t p u t i m p e d a n c e o f t h e RF a m p l i f i e r f o r s h o r t

p e r i o d s .

I n t h e c a s e o f F i g . 1 4 , t h e c a v i t y p a r a m e t e r s ( p o l e a t f R F

/ 2 0

L í a n d t n e t o t a l

d e l a y o f t h e f e e d b a c k p a t h d e t e r m i n e t h e l o o p s t a b i l i t y , p r e a m p l i f i e r s w h i c h a r e

s e l e c t e d f o r t h e s h o r t e s t p r o p a g a t i o n d e l a y m u s t b e l o c a t e d v e r y c l o s e t o t h e p o w e r

a m p l i f i e r - c a v i t y c o m b i n a t i o n . As an e x a m p l e T a b l e 1 g i v e s t h e p a r a m e t e r s f o r t h e CERN PS

b o o s t e r s e c o n d h a r m o n i c s y s t e m , o p e r a t i n g b e t w e e n 6 and 16 M H z : 1

T a b l e 1

F e e d b a c k p a r a m e t e r s o f t h e CERN PSB s e c o n d - h a r m o n i c s y s t e m

P r e a m p l i f i e r g a i n

B a n d w i d t h

P o w e r

P r o p a g a t i o n d e l a y

I m p e d a n c e r e d u c t i o n f a c t o r :

25 x

1 5 0 MHz

1 3 0 U ( 1 dB c o m p r e s s i o n )

5 ns

21 dB a t t> KHz

1 4 , 5 dB a t 16MHz

F o r a v a r y i n g RF f r e q u e n c y one c o u l d , i n p r i n c i p l e , a d j u s t t h e d e l a y o f t h e r e t u r n

p a t h t o Veep t h e 1 8 0 " p h a s e c o n d i t i o n a t f R J , . H o w e v e r , i n many d e s i g n s , f o r i n s t a n c e

t h e s e c o n d h a r m o n i c PS b o o s t e r and t h e f u t u r e PS RF s y s t e m , a w i d e b a n d w i d t h p r e a m p l i f i e r

i s u s e d t o k e e p t h e t o t a l d e l a y s h o r t enough t o e n s u r e s t a b i l i t y o v e r t h e e n t i r e RF

~ 642 -

( 3 9 )

E q u a t i o n ( 3 9 ) shows t h a t t h e u l t i r a a c e performance of wideband RF feedback only d e p e n d s

o n T a n d t h e c a v i t y g e o m e t r y ( R / Q ^ p a r a m e t e r ) .

I f a p p l i c a b l e , i . e . i f T c a n be made s m a l l e n o u g h , t h i s i s t h e b e s t s o l u t i o n t o t h e

p r o b l e m o f beam l o a d i n g s i n c e i t p r o v i d e s w i d e band c o v e r a g e and a v o i d s t h e need f o r

c r i t i c a l a d j u s t m e n t s . V e r y l a r g e i m p e d a n c e r e d u c t i o n f a c t o r s o f s e v e r a l o r d e r s o f

m a g n i t u d e h a v e been a c h i e v e d a t l a w RF v o l t a g e s i n t h e CERN AA f o r i n s t a n c e u s i n g f i x e d

c a v i t y t u n e w i t h o u t a s e r v o l o o p . I f , h o w e v e r , a s e r v o t u n e r i s u s e d , i t may be n e c e s s a r y

t o c o n t r o l i t b y t h e n o r m a l i z e d r e a c t i v e p o w e r o f t h e a m p l i f i e r 1 6 ' .

7 . 4 T h e RF f e e d b a c k w i t h l o n g d e l a y

I n l a r g e RF s y s t e m s , t h e CERN SPS f o r i n s t a n c e , l o n g d e l a y s may b e u n a v o i d a b l e and

t h e c o n v e n t i o n a l RF f e e d b a c k w o u l d h a v e a t o o r e s t r i c t e d b a n d w i d t h , much s m a l l e r t h a n t h e

c a v i t y b a n d w i d t h i t s e l f i n t h e SPS c a s e . T r a n s i e n t beam l o a d i n g a t m u l t i p l e s o f f Q

w o u l d n o t be c o r r e c t e d , l e a d i n g t o p h a s e o s c i l l a t i o n s o f f r a c t i o n s o f t h e beam and

p o s s i b l y c o u p l e d b u n c h i n s t a b i l i t i e s -

I n o r d e r t o s o l v e t h e p r o b l e m , we o b s e r v e t h a t a l a r g e g a i n G i s o n l y n e e d e d Ln t h e

v i c i n i t y o f t h e r e v o l u t i o n f r e q u e n c y h a r m o n i c s w h e r e beam c u r r e n t components e x i s t .

O u t s i d e t h e s e b a n d s , t h e p h a s e cotation due to the e x c e s s i v e d e l a y w i l l be u n i r a p o r t a n L i f

G can b e made s m a l l e n o u g h . W i t h a r e t u r n p a t h t r a n s f e r f u n c t i o n h a v i n g a c o m b - f i l t e r

f r e q u e n c y r a n g e , w i t h o u t p r o g r a m m i n g t h e p h a s e . I n t h i s c a s e i t i s e x t r e m e l y i m p o r t a n t t o

damp t h e h i g h e r r e s o n a n c e s o f t h e c a v i t y o r t o r e j e c t t h e c o r r e s p o n d i n g s i g n a l s i n o r d e r

t o a v o i d p a r a s i t i c o s c i l l a t i o n s o f t h e f e e d b a c k s y s t e m a t h i g h f r e q u e n c i e s

T h e RF f e e d b a c k t e c h n i q u e i s v e r y a t t r a c t i v e s i n c e i t r e d u c e s t h e e f f e c t i v e i m p e d a n c e

o f t h e c a v i t y n o t o n l y a t t h e RF f r e q u e n c y b u t a l s o o v e r a l a r g e b a n d w i d t h . T h i s f e a t u r e

i s p a r t i c u l a r l y h e l p f u l t o a v o i d s e l f - b u n c h i n g i n s t a b i l i t i e s i n s t o r a g e r i n g s f o r

d e b u n c h e d beams a n d i s u s e d a t t h e CERN ISR and AA f o r i n s t a n c e .

I n s u c h a c o n v e n t i o n a l f e e d b a c k s y s t e m t h e t o t a l phase s l i p s h o u l d be l e s s t h a n a b o u t

î w/A o v e r t h e u n i t y g a i n b a n d w i d t h 2 Au o f t h e s y s t e m , g i v i n g t h e c o n d i t i o n :

fiw = * / 4 T ( 3 8 )

w h e r e T i s t h e o v e r a l l d e l a y i n t h e f e e d b a c k p a t h . F o r a f i x e d t u n e d c a v i t y and a small

d e t u n i n g a n g l e , t h e c a v i t y i m p e d a n c e (RLC a p p r o x i m a t i o n ) f a r f r o m t h e w r e s o n a n c e i s

g i v e n b y 2 = R / 2 J Q ^ { Û U / L > c ) . The o v e r a l l l o o p g a i n , GZ, a t t h e ± Aw p o i n t s i s

o f t h e o r d e r o f u n i t y : t h i s g i v e s a n u p p e r l i m i t f o r G2 and a m in i inu iQ v a l u e o f t h e

i m p e d a n c e s e e n by t h e beara, R . , g i v e n by ;

- t>43 -

shape wi th maxima a t every harmonic, t h i s c o n d i t i o n can be s a t i s f i e d - In a d d i t i o n ,

the o v e r a l l d e l a y of the system must be extended t o e x a c t l y one machine tumCT^) to

ensure a zero phase a t the f ^ + n fQ f r e q u e n c i e s .

The comb f i l t e r t r a n s f e r f u n c t i o n ( F i g . 15) i s of the form:

K exp ( - j i w T )

where C and K are c o n s t a n t s (0<K<1).

R E F L E V E L

1 2 . O D O d B rv

5 . OQOdB

S T A R T 1 0 ODO. OOOHa S T O P 2 5 0 O D D . O O O H E

F i g . 15 C o m b - f i l t e r t r a n s f e r f u n c t i o n K = 7 / 8 H = 462

Combined w i t h the one turn d e l a y ( t r a n s f e r f u n c t i o n : e x p ( - j i u T ^ ) ) , the o v e r a l l

open loop t r a n s f e r f u n c t i o n becomes:

C (jto)Z(jüi)

r e p r e s e n t e d i n the complex p l a n e by a c i r c l e f o r a s l o w l y varying 2 ( j w J . The complex

p lane o r i g i n i s e n c i r c l e d and t h e r e f o r e the ga in of the sys tem i s l i m i t e d by the s t a b i l i t y

c o n d i t i o n . In the v i c i n i t y of t h e c a v i t y r e s o n a n c e , where Z ia maximum and r e a l , ( n o t e

t h a t f o r a t r a v e l l i n g wave s t r u c t u r e Z i s a lways r e a l * " 1 ) , the c i r c l e c r o s s e s the

n e g a t i v e r e a l a x i s a t a d i s t a n c e -C Z/( l+K) from the o r i g i n .

I 1-K

F i g . 16 Open-loop t r a n s f e r f u n c t i o n for RF feedback wi th long d e l a y

- 6 4 J -

S t a b i l i t y o b v i o u s l y r e q u i r e s t h a t | G q 2 | < l - t - X , and i t can b e shown t h a t t h i s

c o n d i t i o n i s a l s o s u f f i c i e n t e v e n o u t s i d e r e s o n a n c e f o r an RF c a v i t y a p p r o x i m a t e d by a

s i n g l e RLC e q u i v a l e n t c i r c u i t .

A g a i n f o r Z r e a l , t h e a p p a r e n t i m p e d a n c e o f t h e c a v i t y Z ' :

exp (jéwTQ) - K Z = Z exp <ji«To) - K - G Q Z

i s r e a l f o r f r e q u e n c i e s :

f R F * " f O ' Z ' ' 2 1 -\\\z " Z "

and :

V + ( " ^ > f o '• 2' - 2 rn^Vi " To s t a y a t a r e a s o n a b l e d i s t a n c e f r o m t h e s t a b i l i t y l i m i t , t a k e f o r i n s t a n c e G Q Z =

( l + K ) / 2 . T h i s g i v e s , a t f r e q u e n c i e s f R p + ( n + ' A ) ^ , Z ' = 2Z a s i n t h e c a s e o f f e e d f o r w a r d

c o r r e c t i o n , w h e r e a s f o r t h e r e v o l u t i o n f r e q u e n c y h a r m o n i c s o n e o b t a i n s :

Z ' = Z < 1 - K )

f o r ( 1 - K ) « 1 .

By m a k i n g K c l o s e t o u n i t y , RF f e e d b a c k a p p r o a c h e s t h e t h e o r e t i c a l p e r f o r m a n c e o f t h e

f e e d f o r w a r d c o r r e c t i o n b u t w i t h a l l t h e i n h e r e n t a d v a n t a g e s o f c l o s e d l o o p s y s t e m s i n

p a r t i c u l a r no c r i t i c a l a d j u s t m e n t s a r e n e e d e d . S i m i l a r l y , t h e t i m e r e s p o n s e o f t h e RF

f e e d b a c k i s e n t i r e l y d e t e r m i n e d by t h e one t u r n d e l a y as i n t h e f e e d f o r w a r d c a s e . N o t e

t h a t t h e u n i t y g a i n f r e q u e n c y o f t h e s e r v o i n t h i s c a s e i s o f t h e o r d e r o f f c ' 2 .

T h e r e s i d u a l i m p e d a n c e a t t h e s y n c h r o t r o n s i d e b a n d s i s a p p r o x i m a t e l y t h e same as f o r

a o n e t u r n d e l a y f e e d f o r w a r d c o r r e c t i o n ( f o r K'¿1 and G Q Z = 1 ) ; i t s p h a s e changes s i g n a t

e a c h n f h a r m o n i c r e s u l t i n g i n a r o t a t i o n o f t h e c o m p l e x s y n c h r o t r o n f r e q u e n c y s h i f t

c u r v e . T h e c o u p l e d - b u n c h , c a v i t y - d r i v e n , i n s t a b i l i t y t h r e s h o l d s must be o b t a i n e d

n u m e r i c a l l y

E x c e p t f o r r e l a t i v e l y s m a l l m a c h i n e s w i t h f i x e d RF f r e q u e n c y , l o n g d e l a y f e e d f o r w a r d

o r f e e d b a c k t e c h n i q u e s c o u l d o n l y be e n v i s a g e d w i t h t h e h e l p o f modern s i g n a l p r o c e s s i n g

t e c h n o l o g y , i . e . s a m p l e d o r d i g i t a l f i l t e r s . The d i g i t a l comb f i l t e r i s d e r i v e d f r o m t h e

w e l l known f i r s t - o r d e r l o w - p a s s r e c u r s i v e f i l t e r shown i n F i g . 1 7 . W i t h a s a m p l i n g

f r e q u e n c y N f Q l o c k e d t o a s u b h a r m o n i c o f t h e RF f r e q u e n c y , t h e t h e o r e t i c a l b a n d w i d t h o f

t h e f i l t e r i s N f 12, c o r r e s p o n d i n g t o N / 2 maxima i n t h e comb f i l t e r r e s p o n s e ( N = ¿62

i n t h e SPS d e s i g n ) . I m p l e m e n t a t i o n o f t h e o n e t u r n d e l a y i s s t r a i g h t f o r w a r d i n d i g i t a l

t e c h n o l o g y w i t h a memory ( R . A . M . o r f i r s t - i n f i r s t o u t - t y p e ) .

Memory tOO worts

12 bit

Auxiliary memory

—J Phasing J

F i g . 17 T h e d i g i t a l f i l t e r and d e l a y

The s p e e d o f t h e v a r i o u s e l e m e n t s , l i m i t e d by the c y c l e t i m e (T^/ti), may become

v e r y c r i t i c a l r e q u i r i n g t h e f a s t e s t A - D c o n v e r t e r s ( f l a s h c o n v e r t e r s ) , m e m o r i e s and

m u l t i p l i e r s ( p a r a l l e l m u l t i p l i e r s ) . F o r t h i s r e a s o n t h e number o f b i t s i s l i m i t e d , 8

b i t s i n t h e ADC a n d 12 b i t s i n t h e m u l t i p l i e r a r r a y i n t h e c a s e o f t h e S P S , b u t no a d v e r s e

e f f e c t s f r o m t h e q u a n t i z a t i o n e r r o r s c a n b e o b s e r v e d . H o w e v e r , K c a n n o t b e made v e r y

c l o s e t o u n i t y w i t h a s m a l l number o f b i t s a n d t h e r e s i d u a l i m p e d a n c e Z ' a t t h e r e v o l u t i o n

h a r m o n i c s i s e s s e n t i a l l y d e t e r m i n e d b y t h i s t e c h n o l o g i c a l l i m i t a t i o n C l - K = 1 / 8 f o r t h e

SPS c a s e ) .

The SF s i g n a l s may h a v e t o be t r a n s l a t e d i n f r e q u e n c y t o be c o n v e n i e n t l y p r o c e s s e d .

C o h e r e n t m i x i n g w i t h s e p a r a t e c h a n n e l s f o r i n - p h a s e and i n - q u a d r a t u r e c o m p o n e n t s i s

n e c e s s a r y t o i e j e c t t h e u n w a n t e d image f r e q u e n c i e s , ( m e a s u r e d r e j e c t i o n > 3 5 d B ) , and t o

make t h e o v e r a l l e l e c t r o n i c c h a i n l o o k a l i n e a r s y s t e m . F o r a v a r y i n g RF f r e q u e n c y t h e

c o r r e c t p h a s e c a n e v e n be m a i n t a i n e d w i t h a n a r t i f i c i a l d e l a y i n s e r t e d b e t w e e n t h e o u t p u t

a n d i n p u t l o c a l o s c i l l a t o r s as i n F i g . 1 8 .

F i g . 18 L a y o u t o f t h e RF f e e d b a c k s y s t e m

REFERENCES

- Mo -

1 ) P. W i l s o n , CERN I S R - T H / 7 8 - 2 3 ( 1 9 7 8 )

2 ) P. W i l s o n , I X t h i n t . C o n f . on H i g h E n e r g y A c c e l e r a t o r s , SLAC, S t a n f o r d ( 1 9 7 4 ) , p . 57

3 ) H. H e n k e , CERN I S F - R F / 7 8 - 2 2 , ( 1 9 7 8 ) .

4 ) G. Oöme, 1 9 7 6 P r o t o n L i n a c c o n f e r e n c e , C h a l k R i v e r , C a n a d a , p . 1 3 8 .

5 ) 1. M. E a r l e y , G. p . L a w r e n c e , J . H . P o t t e r , I E E E T r a n s , on N u c l . S e i . N S - 3 0 , ( 1 9 8 3 ) , P . 3 5 1 1 .

6 ) P. P e d e r s e n , IEEE T r a n s . N u c l . S e i . N S - 2 2 . C 1 9 7 5 ) , p . Ï 9 0 6 .

7 ) 0 . B o u s s a r d . CERN S P S / A R F / N o t e 8 4 - 9 ( 1 9 8 4 ) .

8 ) D- B o u s s a r d , IEEE T r a n s . N u c l . S e i . N S - 3 2 , ( 1 9 8 5 ) , p . 1 8 5 2 .

9 ) E. H a e b e l , C E R N / E F / R F 8 4 - 4 , ( 1 9 8 4 ) .

1 0 ) K- W. R o b i n s o n , CEA r e p o r t CEAL - 1 0 1 0 U 9 6 4 ) .

11) 0 . B o u s s a r d , CESS/SPS/ARF Hate 7 8 - 1 6 ( 1 9 7 8 ) .

12) H. F r i s c h h o l z , W. S c h n e l l , I E E E T r a n s , on N u c l . S e i . H S - 2 4 , ( 1 9 7 7 ) , p . 1 6 8 3 .

1 3 ) S. C i o r d a n o , M. P u g l i s i , IBEE T r a n s , on N u c l . S e i . N S - 3 0 . ( 1 9 B 3 ) . p . 3 4 0 8 .

I t ) G. G e i a t O e t a l . , I E E E T r a n s , o n N u c l . S e i . N S - 2 2 , ( 1 9 7 5 ) , p . 1 3 3 4 .

1 5 ) J . H . B a i l l o d e t a l . , I E E E T r a n s , o n N u c l . ¡ s c i . N S - 3 0 , ( 1 9 8 3 ) , p . 3 4 9 9 .

1 6 ) F . P e d e r s e n , I E E E T r a n s , on H ü c l . S e i . N S - 3 2 . ( 1 9 8 5 ) . p . 2 1 3 8 .

1 7 ) D. B o u s s a r d , C. L a m b e r t , I E E E T r a n s , on N u c l . S e i . N S - 3 0 . ( 1 9 8 3 ) , p . 2 2 3 9 .

- 6 4 7 -

POLARIZATION IN ELECTRON AND PROTON BEAKS J. Buon

Laboratoire de l'Accélérateur Linéaire, 9 1 * 0 5 O R S A Y , France

ABSTRACT One first introduces the concept of polarization for spin 1 / 2 particle beatas and discusses properties of spin kinetics in a stationary magne­tic field. Then the acceleration of polarized protons in synchrotrons is studied with emphasis on depolarization unen resonances are crossed and on the ^eihoda- of reducing ir. Finally, transverse polarization of electrons in storage rings is discussed as an equilibrium between po­larizing and depolarizing effects of synchrotron radiation. Means for obtaining longitudinal polarization are also treated.

INTRODUCTION

Spin is an important feature of nuclei and subnuclear particles, as well as their mass and electric charge. In general, interactions between them depend on their spin. The expe­rimental study of these interactions vith unpolarized beaics and targets cannot investigate this spin dependence and is incomplete. Polarization experiments are able to reveal impor­tant and new aspects of Nature. There have been in the past many examples of unexpected results obtained in such experiments, the most famous one being the discovery of parity violation in B-decay. One then could ask why so few polarization experiments are done in Nuclear and Subnuclear Physics. The reason is that these experiments are generally more difficult and more delicate. In particular, polarized beams are more elaborate to produce than unpolarized beams- Usually their intensity and their reliability are lower. If it was not so, all experimentalists would ask for polarized beams ! Surely progress in the deve­lopment of polarised beams would be valuable.

The physics of polarized beams is a wide topic, not often familiar to accelerator physicists. It is difficult to cover it completely in a limited time. Ve will restrict ourselves to the acceleration of polarized protons in synchrotrons and to the polarization of electrons in storage rings, i.e. to the most common high-energy polarized beams. Ue will not consider other polarized beams like secondary beams, electron beams in linear accelerators and synchrotrons, nuon beans, deuteron beams, ... We will concentrate on the spin kinetics of electron and proton polarized beams and we will not study other aspects like polarized-ion sources and polarization monitoring.

The aim is to explain the physics of spin kinetics in these polarized beams to acce­lerator physicists. No attempt will be made to use less familiar mathematical formalisms (like the S U ( 2 ) representation of spin rotations), to derive the basic formulae (such as the Thomas-BMT and Froissart-Stora equations or the formulae of Sokolov-Ternov and Derbenev-Kondratenko), or to treat particular details or more advanced topics, reserving these developments to specialists. Ue prefer to limit ourselves to an analysis of the physical contents of the basic equations and of their consequences, illustrated by expe­rimental results.

Ve v i l l not try to quote in r e f e r e n c e s a l l the authors and c o n t r i b u t o r s in the f i e l d

of p o l a r i z e d beams. Ue l i m i t the b ib l iography to a few genera l and recent r e p o r t s which

were models for prepar ing t h i s l e c t u r e and which can be recommended to tbe n o n - s p e c i a l i s t

r eader . The l a t t e r v i l l f ind in them a i l the re l evant r e f e r e n c e s .

Th i s l e c t u r e i s d iv ided i n t o three p a r t s . The f i r s t one i s devoted to g e n e r a l i t i e s on

the p h y s i c s of p o l a r i z e d beams which are u s e f u l for understanding the behaviour of p o l a ­

r i z e d protons and e l e c t r o n s in a c c e l e r a t o r s , e s p e c i a l l y c i r c u l a r a c c e l e r a t o r s . Ve f i r s r

reir-eober the concept of s p i n and we e x t e n s i v e l y d i s c u s s the meaning of p o l a r i z a t i o n for

s p i n 1/2 p a r t i c l e beams. Some knowledge of Quan tun Mechanics i s not r e a l l y needed a s ve

w i l l e s s e n t i a l l y take a s e m i c l a s s i c a l point of v iew, apart from two p a r t i c u l a r p o i n t s

vh i ch can p o s s i b l y be omit ted by the reader . Then the k i n e t i c s of s p i n n o t i o n i n a s t a ­

t i o n a r y magnetic f i e l d i s e x t e n s i v e l y s t u d i e d , s t a r t i n g from the Thomas-BHT e q u a t i o n of

s p i n mot ion, and with emphasis on the s p i n - o r b i t c o u p l i n g . A genera l d i s c u s s i o n of

d e p o l a r i z a t i o n re sonances i s based on the consequences of s p i n - o r b i t c o u p l i n g . F i n a l l y

the g r e a t s i m i l a r i t y with Nuclear Magnetic Resonance i s s t r e s s e d , r ecogn iz ing that the

b a s i c f e a t u r e s of s p i n k i n e t i c s are the same.

In the second part the a c c e l e r a t i o n of p o l a r i z e d protons in synchrotrons i s s t u d i e d

with emphasis on d e p o l a r i z a t i o n when resonances are c r o s s e d and on the cures for reducing

i t . In p a r t i c u l a r s p i n k i n e t i c s in a ring equipped v i t h "Siber ian Snakes" i s q u a l i t a t i v e l y

d i s c u s s e d a s "S iber ian Snakes" appear e s s e n t i a l for very high e n e r g i e s .

The t h i r d and l a s t pari i s devoted to the p o l a r i z a t i o n of e l e c t r o n s in s t o r a g e r i n g s ,

which has very d i f f e r e n t a s p e c t s . As in beam dynamics, the synchrotron r a d i a t i o n dominates

s p i n k i n e t i c s in e l e c t r o n s t o r a g e r i n g s . Synchrotron r a d i a t i o n prov ides a p o l a r i z i n g me­

chanism ( t h e Sokolov-Ternov e f f e c t ) and enhances a l s o beam d e p o l a r i z a t i o n - We q u a l i t a t i ­

v e l y d i s c u s s both p o l a r i z a t i o n and d e p o l a r i z a t i o n phenomena induced by synchroton r a d i a ­

t i o n and how to manage with them for o b t a i n i n g a high degree of p o l a r i z a t i o n . At the end

we b r i e f l y d i s c u s s s p i n r o t a t o r s for o b t a i n i n g l o n g i t u d i n a l p o l a r i z a t i o n and i n d i c a t e two

p a r t i c u l a r and important problems : d e p o l a r i z a t i o n enhancement by l a r g e energy spread of

beams a t h igh e n e r g i e s and d e p o l a r i z a t i o n by the beam-beam i n t e r a c t i o n .

- o4y -

I. GENERALITIES ON ÛLARIZATION AND SPIN MOTION

] . 1 5pin and magnetic raoaent of a p a r t i c l e

The s p i n S* of - p a r t i c l e ( e l e c t r o n , proton, . . . ) i s an i n t e r n a l degree of freedom

which behaves l i k e angular momentum. I t i s an a x i a l v e c t o r wi th quant i zed v a l u e s of i t s

modulus \ $ { 2 and of i t s component on any a x i s Oz :

|S*|* = s ( s + 1 h 2

Is = - s , - s+1 . . . , s - 1 , s ti z where h i s the Piar k cons tant d i v i d e d by 2n.

The s p i n va e s i s a h a l f - i n t e g e r for Fermions ( 1 / 2 for e l e c t r o n s , muons, p r o t o n s ,

n e u t r o n s , . . . ) an an i n t e g e r for Bosons (0 for H and K mesons, 1 for photons and d e u t e -

r o n s )

p a r t i c l e s .

Charged p a r t i c l e s have a magnetic noment p propor t iona l to t h e i r s p i n S* :

M = g j f - t CI -1 -1 ) o

where e and mQ are the e l e c t r i c charge and the r e s t i a s s of the p a r t i c l e , r e s p e c t i v e l y

(u i s p a r a l l e l to f íor a proton and a n t i p a r a l l e l for an e l e c t r o n accord ing to the s i g n of

t h e i r e l e c t r i c cha: e ) .

The g y r o m a g n e t - r a t i o g i s 2 for p o i n t - l i k e Fermions in the Dirac theory . There a r e

c o r r e c t i o n s and n e d e v i a t i o n from 2 i s measured by the gyromagnet ic anomaly a = (g-2)/2 ( v e r y o f t e n a l s o df i gnated by G i n the l i t e r a t u r e ) :

e l e c t r o n muon proton deuteron

a = 1 .1596x10"' 1 .1659x10"' 1.7928 - 0 . 1 4 3 0 ( 1 - 1 - 2 )

A charged par c l e p laced in a magnetic induc t ion B* has a magnetic energy U g i v e n by •

W r - P . í ( 1 . 1 . 3 )

Here we wi l? c o n s i d e r s p i n 1/2 p a r t i c l e s ( e l e c t r o n s and pro tons ) which have two s t a t e s of magnetic ?nergy o n l y .

1 ,2 P o l a r i z a t i o n s p i n 1/2 p a r t i c l e s

A bunch of sp i 1/2 p a r t i c l e s i s p o l a r i z e d i f t h e i r s p i n s have a pre ferred d i r e c t i o n .

Th i s s i t u a t i o n i s a r a c t e r i z e d by a p o l a r i z a t i o n v e c t o r P* p o i n t i n g in t h i s d i r e c t i o n . The l e n g t h |r"| i s the .gree of p o l a r i z a t i o n .

- Ü 5 0 -

H e r e we w i l l d e f i n e t h e p o l a r i z a t i o n v e c t o r P* i n t h e most g e n e r a l c a s e . E s s e n t i a l l y

t h e p o l a r i z a t i o n v e c t o r i s a c l a s s i c a l q u a n t i t y ( f o l l o w i n g a c l a s s i c a l e q u a t i o n o f n o t i o n )

w h i c h d e t e r m i n e s c o m p l e t e l y any s p i n s t a t e o f a s p i n 1 / 2 p a r t i c l e e n s e m b l e . These two p r o ­

p e r t i e s j u s t i f y t h e s e m i c l a s s i c a l d e s c r i p t i o n o f p o l a r i z a t i o n f o r s p i n 1 / 2 p a r t i c l e s ,

w h i c h i s b a s e d on t h e e v o l u t i o n o f t h e p o l a r i z a t i o n v e c t o r P*. T h i s s e m i c l a s i c a l d e s c r i p ­

t i o n i s t o t a l l y e q u i v a l e n t t o a p u r e l y q u a n t u m - m e c h a n i c a l d e s c r i p t i o n . T h e p r o o f o f t h e s e

two p r o p e r t i e s w h i c h n e e d s some k n o w l e d g e i n Quantum M e c h a n i c s , i s g i v e n i n s e c t i o n s 1 . 2 . 2

a n d 1 . 2 . 3 . T h e y h a v e been put i n a p p e n d i x a t t h e end o f t h i s p a p e r such t h a t t h e y can be

o m i t t e d i f one i s n o t f a m i l i a r w i t h Quan cum M e c h a n i c s .

1 . 2 . 1 D e f i n i t i o n o f _ t h e p o l a r i z a t i o n v e c t o r ?

I n a p u r e s p i n s t a t e o f a n i n d i v i d u a l p a r t i c l e the d i r e c t i o n o f s p i n ? i s t h e d i r e c ­

t i o n a l o n g w h i c h t h e s p i n component t a k e s t h e maximum v a l u e ( + h / 2 ) w i t h p r o b a b i l i t y 1 . T h e

p o l a r i z a t i o n v e c t o r P* i s d e f i n e d a s t h e u n i t v e c t o r i n t h i s s p i n d i r e c t i o n .

Now, f o r a bunch o f N p a r t i c l e s w i t h d i f f e r e n t p o l a r i z a t i o n v e c t o r s P*. ( i = 1 , N ) , t h e

p o l a r i z a t i o n v e c t o r ? i s d e f i n e d as t h e b a r y c e n t r e o f a l l t h e i n d i v i d u a l p\ :

i N

? = -jr- Z ? • ( 1 - 2 . 1 ) " i = l

T h e d e g r e e o f p o l a r i z a t i o n |P*| v a r i e s f rom 0 t o 1 d e p e n d i n g on the r e l a t i v e d i r e c t i o n s o f

t h e v e c t o r s P*j.

Uhen a l l t h e s p i n s a r e p a r a l l e l t o Oz , t h e component o f the p o l a r i z a t i o n v e c t o r p\

f o r one p a r t i c l e i s +1 i f i t s s p i n i s " u p " ( S ; = + t i / 2 ) and - 1 i f i t i s "down" ( S 2 = - h ' 2 ) .

I f N and N a r e t h e numbers o f p a r t i c l e s w i t h s p i n " u p " and "down" r e s p e c t i v e l y , t h e

p o l a r i z a t i o n v e c t o r P* i s p a r a l l e l t o Oz and i t s component measures t h e a s y m m e t r y i n

t h e p o p u l a t i o n s o f t h e s e two s p i n s t a t e s :

N + N (1-2.2)

A bunch o f N p a r t i c l e s i s u n p o l a r i z e d (P^ = 0 1 when N + = N _ , and c o m p l e t e l y p o l a r i z e d

( P z = + 1 ) when e i t h e r N + o r N v a n i s h e s , i . e . when a l l t h e s p i n s a r e e i t h e r p a r a l l e l o r

a n t i p a r a l l e l t o Oz .

F i n a l l y , a b r u p t t r a n s i t i o n s ( s p i n - f l i p ) b e t w e e n "up" and "down" s t a t e s may o c c u r , as

f o r e l e c t r o n s r a d i a t i n g i n a s t a t i c m a g n e t i c f i e l d . The s p i n s t a t e o f such an e l e c t r o n i s

m i x e d : a s t a t i s t i c a l m i x t u r e o f t h e two s p i n s t a t e s " u p " and "down" w i t h p r o b a b i l i t i e s q

a n d 1 -q r e s p e c t i v e l y . I n t h i s c a s e t h e p o l a r i z a t i o n v e c t o r P* i s d e f i n e d as t h e s t a t i s t i c a l

a v e r a g e o f t h e p o l a r i z a t i o n v e c t o r s ? o f t h e two p o s s i b l e s t a t e s !

? - q ? + • ( 1 - q ) ? _ •

T h i s s t a t i s t i c a l a v e r a g e i s e q u i v a l e n t t o a n e n s e m b l e a v e r a g e w i t h q • N / ( N + N )

and t h i s c a s e does n o t need t o be d i s t i n g u i s h e d i n t h e f o l l o w i n g .

- ()51 -

I . 3 Spin p r e c e s s i o n in s t a t i c e l e c tromaine t i c f i e l d s

F o l l o w i n g the t r a d i t i o n we w i l l , from now on, use the e x p r e s s i o n "spin v e c t o r S*™ for d e s i g n a t i n g the p o l a r i z a t i o n v e c t o r P* of an i n d i v i d u a l p a r t i c l e . The e x p r e s s i o n " p o l a r i z a ­t i o n v e c t o r w i l l be reserved to the c a s e of a p a r t i c l e ensemble .

In t h i s s e c t i o n we w i l l s tudy the c l a s s i c a l motion of s p i n v e c t o r ? in s t a t i c e l e c ­tromagnet ic f i e l d s . Ue w i l l not c o n s i d e r the e f f e c t of e m i s s i o n (or a b s o r p t i o n ) of e l e c ­t romagnet i c r a d i a t i o n , which happens when e l e c t r o n s r a d i a t e . S ince t h e s e r a d i a t i v e e f f e c t s occur in very short t i m e s , we w i l l d e s c r i b e s p i n motion only between two c o n s e c u t i v e r a d i a t i v e e f f e c t s - For protons t h e s e e f f e c t s are normally n e g l i g i b l e and can be i g n o r e d .

The c l a s s i c a l e q u a t i o n of s p i n motion w i l l be f i r s t w r i t t e n down in the n o n - r e l a t i -v i s t i c and r e l a t i v i s t i c c a s e s ( s e c t i o n s I . 3 - Î and 1 . 3 . 2 ) , Then the g e n e r a l p r o p e r t i e s i n c l u d e d in the r e l a t i v i s t i c ec uat ion w i l l be emphasized ( s e c t i o n 1 . 3 . 3 ) . F i n a l l y , i t w i l l be e x p l i c i t l y shown that the . . a s s i c a l e q u a t i o n i s s t r i c t l y e q u i v a l e n t to the Schrodinger e q u a t i o n for s p i n o r s in a quantum-mechanical formalism. Again t h i s l a s t part ( s e c t i o n 1 . 3 , 4 ) has been put in appendix and can be omit ted i f one i s not f a m i l i a r wi th Quantum Mechanics .

I . 1 . 1 N o n ^ f e l a t i v i s t i c _ p a r t i c 'S

The s p i n - v e c t o r motion c. : an i n d i v i d u a l p a r t i c l e i s g i v e n by the i n t e r a c t i o n of i t s magnet ic noment u with the magnetic i n d u c t i o n B* :

( 1 - 3 . 1 )

T h i s e q u a t i o n of motion can b* r e w r i t t e n , u s i n g formula ( 1 . 1 . 1 ) :

3f.^*í (1.3.2,

"ith K • - f i r 1 - - ( U a > r 1 • o o

The motion i s a p r e c e s s i o n around the f i e l d B* at the "Larmor" frequency jjjpB t imes the gyroraagnetic r a t i o g .

This p r e c e s s i o n i s s i m i l a r to the v e l o c i t y r o t a t i o n in a magnetic f i e l d :

w i t h the " c y c l o t r o n " frequency |Q c J = jj¡-B .

The r e l a t i v e frequency of s p i n and v e l o c i t y p r e c e s s i o n s i s p r o p o r t i o n a l to the gyromagnet i c anomaly a ;

K " K ~ K = aK ' ( 1 . 3 . 3 )

The measurement of 3 a i s the bas - of a l l the "g-2" exper iments which intend to measure t h i s gyromagnet ic anomaly.

1 . 3 . 2 R e l a t i v i s t i c p a r t i c l e s

The e q u a t i o n o í s p i n - v e c t o r n o t i o n in an e l e c t r i c and magnetic f i e l d becomes :

where B*j_ (B*. ) i s the t r a n s v e r s e ( l o n g i t u d i n a l ) component of the induc t ion f i e l d B* r e l a t i v e

to the p a r t i c l e v e l o c i t y ; Y I S the r e l a t a v i s t i c Lorentz Eactor and I the r a t i o of the ve ­

l o c i t y ? to the l i g h t v e l o c i t y c ( a l l q u a n t i t i e s in HKS u n i t s ) .

In t h i s "Thomas~Bargnian,Hichel,Telegdi" e q u a ' i o n , r e f erred to as Thomas-BHT e q u a t i o n ,

the f i e l d s Ê* and B\ and the time t , are c a l c u l a t e d in the laboratory frame, but the s p i n

v e c t o r S* i s c a l c u l a t e d i n the r e s t frame of the p a r t i c l e for a v o i d i n g compl icated Lorentz

t rans format ion of s p i n .

For comparison the v e l o c i t y r o t a t i o n in a t r a n s v e r s e magnetic f i e l d $± i s g i v e n by :

I T

with the r e l a t í v í s t í c "cyc lo tron" frequency | °

1 . 3 . 3 O e n e r a l p r o p e r t i e s o f s p i n p r e c e s s i o n

The s p i n - v e c t o r motion a s g i v e n by the Thomas-BHT equat ion ( 1 . 3 . 4 ) i s a r o t a t i o n about the r o t a t i o n v e c t o r 2g H T with an angular frequency

d T

ar :

and with the f o l l o w i n g g e n e r a l p r o p e r t i e s :

i ) The e f f e c t of an e l e c t r i c f i e l d E has near ly the same ampl i tude as the e f f e c t of a

magnet ic f i e l d B =• E / c . Therefore an e l e c t r i c f i e l d of 3 x 10 V/m i s comparable to a ma­

g n e t i c f i e l d of one T e s l a . The e l e c t r i c f i e l d s normally found in a c c e l e r a t o r s have then a

n e g l i g i b l e e f f e c t a s compared to Che magnetic f i e l d s and1 t h e s e e 2 e c t r ï c f i e l d s viil now be

i g n o r e d .

i i ) The s p i n r o t a t i n g power of a l o n g i t u d i n a l f i e l d B, i s i n v e r s e l y p o r p o r t i o n a l to the

p a r t i c l e momentum p\ e x a c t l y l i k e the v e l o c i t y r o t a t i n g power of a t r a n s v e r s e f i e l d Bj_.

Kore p r e c i s e l y the l o n g i t u d i n a l - f i e l d i n t e g r a l J*B.,ds, needed for r o t a t i n g the s p i n by one

r a d i a n , i s :

10 .479 . P(GeV/c) J B,.ds (Tm, rad) =

( 1 . 3 . 5 )

Th i s i n t e g r a l becomes very large at high e n e r g i e s .

- uS5 -

iii) The relative frequency P-a of spin and velocity precessions in a transverse magnetic field :

( 1 . 3 . 6 )

is exactly independent of the particle energy, and is ya larger than the cyclotron fre­quency 3 c :

3 a = ra 5c . ( 3 . 3 . 7 )

The vector 3g is the spin rotation vector vith respect to a frame following the par­ticle motion (usually named orbit frase) as this frame rotates at cyclotron frequency °-c.

The transverse-field integral, needed for rotating the spin by one radian in the orbit frame, JB ±ds is :

j B ±ds (Tm/rad) = 5'* 8* . |^ for a proton, ( 1 . 3 . 8 )

J B ±ds (Tm/rad) = ¿ ' ^ 1 8 . for an electron,

where E is Che total relativistic energy of the particle. In a given transverse field, a proton and an electron with the same velocity have nearly the same spin rotation, .u the larger mass of the proton is compensated by its larger gyromagnetic anomaly (formula 1.1 . 2 ) .

iv) At high energies, i.e. when ya » 1, the spin rotating power of a longitudinal field becomes much smaller than the power of a transverse field. Therefore transverse fields are usually preferred for spin manipulations at high energies. Moreover the abso­lute precession frequency Sg H T in a transverse field becomes nearly energy-independent and spin rotation appears to be easier to realize than trajectory bending.

v) In a circular accelerator vith distributed bending magnets, the spin motion is a succession of rotations. In one turn the napping of spin is a rotation : product of all the successive rotations in individual magnets. This mapping is characterized by a preces­sion axis n and angle f, which play an important role for the spin kinetics in circular accelerators•

I - 4 S pin-orb i t coup1 ins

According to the Thomas-BHT equation (1.3.4), the spin motion at a given energy is determined by the magnetic fields encountered by the particles. These fields depend on the individual trajectories followed by the particles. The spin motion is coupled to the orbi­tal motion.

For instance, in an ideally planar ring, the reference orbit lies in the horizontal plane. All along this orbit the magnetic field is vertical and spin precesses around the vertical line.

- dí4 -

On the o ther hand, a long a v e r t i c a l beU ron t r a j e c t o r y , r a d i a l f i e l d s p r o p o r t i o n a l

T O v e r t i c a l d i sp lacement are experienced in quadrupoles . A smal l l o n g i t u d i n a l f i e l d i s

a l s o e x p e r i e n c e d i n bending magnets where the t r a j e c t o r y has a v e r t i c a l s l o p e .

In g e n e r a l s p i n - o r b i t c o u p l i n g i s r e p o n s i b l e for d e p o l a r i z i n g e f f e c t s s i n c e p a r t i c l e s

in a beam have s l i g h t l y d i f f e r e n t t r a j e c t o r i e s and e n e r g i e s . Their spin v e c t o r s £> r o t a t e

about d i f f e r e n t f i e l d s v i t h d i f f e r e n t s p e e d s . They tend to spread out in a l l d i r e c t i o n s

and the p o l a r i z a t i o n v e c t o r d e c r e a s e s i n l e n g t h . These d e p o l a r i z i n g e f f e c t s are the main

concern for s p i n motion in a c c e l e r a t o r s . Their most gen era l a s p e c t s w i l l be s t u d i e d in

s e c t i o n 1 . 6 . The on ly e x c e p t i o n where s p i n - o r b i t c o u p l i n g does not cause d e p o l a r i z a t i o n i s

a h o r i z o n t a l l y f l a t beam, p o l a r i z e d in the d i r e c t i o n of the v e r t i c a l bending f i e l d , for

i n s t a n c e in an i d e a l l y planar r i n g . Whatever the p a r t i c l e energy and motion in the h o r i ­

z o n t a l p l a n e , s p i n p r e c e s s i o n i s about the v e r t i c a l l i n e and the v e r t i c a l l y a l i g n e d s p i n

v e c t o r s do not r o t a t e a t a l l . In t h i s i d e a l s i t u a t i o n the r ing i s s a i d to be " s p i n - t r a n s ­

parent" • In g e n e r a l for reducing d e p o l a r i z a t i o n one t r i e s to approach s p i n - t r a n s p a r e n c y a s

much a s p o s s i b l e .

On the o t h e r hand, the r e v e r s e c o u p l i n g , an o r b i t per turbat ion depending on s p i n

s t a t e , i s e x p e c t e d as in 5 tern-Ger lach exper iments . However t h i s coupl ing i s very weak at

a c c e l e r a t o r e n e r g i e s as the magnetic energy g i v e n by the Hamiltonian B (formula 1 . 3 . 1 1 i n

appendix) i s at most of the order of a(eh/2m)B = 1 0 ~ 1 4 MeV and i s very much s m a l l e r than

the k i n e t i c energy . An e f f e c t of the Stern-Gerlach type cannot be observed in p r a c t i c e -

Now, tak ing account of the s p i n - o r b i t c o u p l i n g , the q u e s t i o n may be r a i s e d whether

s p i n manipu la t ions are at a l l p o s s i b l e s i n c e the t r a j e c t o r y o p t i c s in a c i r c u l a r a c c e l e ­

r a t o r i s almost complete ly determined by many imposed c o n s t r a i n t s .

In p a r t i c u l a r , one could argue t h a t , in one turn of a r i n g , the o v e r a l l s p i n p r e c e s ­

s i o n would be s t r i c t l y propor t iona l to the v e l o c i t y r o t a t i o n s i n c e s p i n r o t a t i o n i n a

t r a n s v e r s e f i e l d i s near ly Ta t i n e s the v e l o c i t y r o t a t i o n (formula 1 . 3 . 7 ) . Spin p r e c e s s i o n

could not then be changed without grea t m o d i f i c a t i o n of beam o p t i c s . This argument i s

wrong a t h igh energy because v e l o c i t y r o t a t i o n s in magnets are r e l a t i v e l y small and n e a r l y

commute between themse lves , and a t the same time s p i n r o t a t i o n s are l a r g e and do not

commute. Then the r e s u l t of s u c c e s s i v e r o t a t i o n s for sp in and for v e l o c i t y can be very

d i f f e r e n t from a s imple p r o p o r t i o n a l i t y r u l e .

- ¡IJS -

In other words s p i n manipulat ions are p o s s i b l e at high energy due to the non-cossiuta-

t i v i t y of r o t a t i o n s . An example of t h i s p o s s i b i l i t y Is a sp in r o t a t o r nade of s e v e r a l

t r a n s v e r s e l y bending magnets , which bends the sp in by 90° but not the t r a j e c t o r y <see

s e c t i o n I I I . 6 ) :

s

V

In c o n c l u s i o n , at high energy , in s p i t e of the s p i n - o r b i t c o u p l i n g , s p i n motion can be

c o n s i d e r e d a s a new degree o ' freedom to some e x t e n t which a l l o w s s p i n m a n i p u l a t i o n s .

I • 5 Spin c l o s e d solution and s p i n cune

H e r e a f t e r we w i l l r e s t r i c t our c o n s i d e r a t i o n s to s p i n motion in c i r c u l a r a c c e l e r a t o r s .

In t h i s s e c t i o n we only c o n s i d e r on-momentum p a r t i c l e s c i r c u l a t i n g on a r e f e r e n c e c l o s e d

o r b i t in a r ing with non-uniform magnetic f i e l d such that the r e f e r e n c e o r b i t i s not

n e c e s s a r i l y p l a n a r .

The o n e - t u r n [tapping, s t a r t i n g a t azimuth s ,

i s a r o t a t i o n T ( s ) v i t h a p r e c e s s i o n a x i s n ( s )

and a p r e c e s s i o n a n g l e -| ( i n genera l d i f f e r e n t

from 2ft for a v o i d i n g d e p o l a r i z a t i o n r e s o n a n c e ) .

Ue w i l l prove the f o l l o w i n g theorem :

Theorem : The o n e - t u r n p r e c e s s i o n a x i s n ( s )

i s the p e r i o d i c s o l u t i o n of s p i n mot ion, nased

the s p i n c l o s e d s o l u t i o n , and any s p i n - v e c t o r

d i r e c t i o n r o t a t e s by 2n\i about ñ ( s ) in one turn,

where v I s the s p i n tune and i s independent of '(sj ^lS0) ¿Isl r(S)

the i n i t i a l azimuth s .

Proof : the one - turn r o t a t i o n s T ( s ) and T ( S q ) , s t a r t i n g at azimuth s and S q r e s p e c t i ­

v e l y , can be r e l a t e d by :

T ( s ) = R ( s o , s ) T ( s o ) R " i ( s o , s )

where R(s , s ) i s the s p i n r o t a t i o n between t h e s e two az imuths . The s p i n d i r e c t i o n n{s) i s

the e i g e n v e c t o r of the r o t a t i o n T ( s ) , corresponding to the e i g e n v a l u e 1 :

n ( s ) = T ( s ) í ¡ (s ) or

f î (s) = R ( s Q , s ) T ( s o ) R " 1 ( s o , s ) n ( s )

then :

R _ 1 ( s o , s ) n ( s ) = T ( * o ) R " ' f s o , s ) ñ ( s ) .

- ö56 -

Th i s r e l a t i o n shows that R~ ( s , s ) n ( s ) i s a l s o an e i g e n v e c t o r of the one - turn r o t a ­

t i o n T ( s ) , for the same e i g e n v a l u e 1. The u n i c i t y of t h i s e i g e n v e c t o r for a r o t a t i o n ,

d i f f e r e n t from the i d e n t i t y , l e a d s to :

R ~ J ( s o , s ) £ ( s ) = n ( s o ) or

n<s) = R ( s o , s ) r i ( s o )

showing that i î ( s ) i s e f f e c t i v e l y a s o l u t i o n of sp in motion. This s o l u t i o n i s p e r i o d i c a s ,

i n a second turn , the s p i n mapping i s the same, the r e f e r e n c e o r b i t be ing p e r i o d i c t o o .

For proving the second part of the theorem, l e t one c o n s i d e r another s p i n - v e c t o r

d i r e c t i o n ? < s ) a t az inuth s , orthogonal to the s p i n c l o s e d s o l u t i o n n ( s ) . After one turn

t h i s d i r e c t i o n i ( s ) i s mapped i n t o f ' ( s ) and the one- turn p r e c e s s i o n ang l e i s :

* = ( ? ( s ) , ? ' ( s ) ) .

Again the mapping between f ( s ) and Í ' ( S ) : T'(s> = T ( s ) i ( s )

can be vr i t ten :

? ' ( s ) = K ( s o , s ) T { s o ) R ~ l ( s o , s ) tf(s)

R " l ( s o , s ) r{s) = T ( s o ) R " 1 ( s o , s } t{s)

showing that f ( s ) = R _ 1 ( s o > s ) ¿"(s) i s mapped i n t o ^ ' ( S q ) = R _ 1 ( s o , s ) f'(s) in one turn

s t a r t i n g at azimuth s o . I t f o l l o w s that the s p i n p r e c e s s i o n ang l e (t(so), t'is^)) i s a l s o

* a s the d i r e c t i o n s Î < S q ) , t ' ( s o > and £ ( s ) , £ ' ( s ) are r e l a t e d r e s p e c t i v e l y by the same r o ­

t a t i o n R" ( S Q , S ) which c o n s e r v e s the ang l e between them. The s p i n tune v •= +/2n i s then

independent of the azimuth s .

E x e r c i s e : Prove t h i s theorem by us ing the Floquet theorem,

( n o t e : a r o t a t i o n by an ang l e * has three e i g e n v a l u e s : l . e 1 * and e - 1 * )

Consequent ly , i t i s o f t e n convenient to look at s p i n motion as a r o t a t i o n about the

s p i n c l o s e d s o l u t i o n n ( s ) , s i n c e the ang l e between the s p i n - v e c t o r d i r e c t i o n and n ( s ) i s

conserved for p a r t i c l e s c i r c u l a t i n g on the r e f e r e n c e o r b i t . This r o t a t i o n has a 2nv phase

advance per turn.

In an i d e a l r ing w i th a uniform v e r t i c a l f i e l d , the s p i n c l o s e d s o l u t i o n n ( s ) i s v e r ­

t i c a l everywhere and the s p i n tune v, in the o r b i t frame, i s g i v e n by :

E l e c t r o n s Protons Deuterons

y = ya =

E < G e V > EfCeV) EfGeV) ( 1 . 5 . I J

.44065 .52335 13.13

a s f u n c t i o n of the t o t a l r e l a t i v i s t i c energy E.

According to formula ( 1 . 3 . 6 ) , the s p i n p r e c e s s i o n frequency r e l a t i v e to the o r b i t

frame, i s then :

3 . V Í .

a c

In a r i n g with a non-uniform bending f i e l d , the s p i n tune i s in ge nera l d i f f e r e n t from

Ya. The most famous example i s a r ing equipped v i t h a "Siber ian Snake" ( s e e s e c t i o n I I . 5 ) ,

where the s p i n tune i s 1/2 whatever the energy .

1,6 Resonant perturbations of spin notion

Noraally polarized particles circulate in a circular accelerator with their spins pointing in the direction of the spin closed solution ñ(s>, vhich is the only stable di­rection of polarization as will be seen in part II for protons and part III for electrons. This direction n(s) corresponds to on-nomentum particles circulating on the reference orbi t.

However, particle motion and energy slightly differ from these references, due to closed-orbit distortions, betatron and synchrotron oscil­lations- These perturbations of orbital motion lead to a perturbed spin motion, via the spin-orbit coupling. They produce a perturbing magnetic field b* which bends Che spin vector t away from the spin closed solution n(s). Only the perturbing field component orthogonal to n(s) needs to be considered here.

These perturbations are small and rapidly varying in time. Usually they tend to cancel out on average. Therefore a large spin deviation from n can only occur if there is some piling-up of small perturbations. Such a coherent effect is observed when the perturbing field b(s) has a component precessing about n at the same frequency as r ve spin vector S*. This is illustrated in the following figure which shows their orientations at different times as seen in the plane transverse to n (the dotted arrows show the direction in which 3 will be tilted).

For analyzing in frequency the perturbing field fj(6) let us define the complex quantity b(8) = b

1 + < e = s/R , R = average radius)

where b^ and b 2 are the two components of b* on two axes ê and e 3, orthogonal to n (e( lying in the transverse plane xOz). The frequency spectrum of b(0) :

iv.e b(6) = E b. e •*

J J

= K

integer) o,x,z,s since the perturbing field b* results from closed-orbit distortions with integer harmonics, betatron oscillations with Q and Q tunes, and synchrotron oscillations with Q tune.

- (i5S -

the frequency Vy Th i s component and the sp in vec tor S* preces s at the same frequency vhen

the resonant condi ton :

* • ko * kx°x * V z * ks Qs "- 6- 1» i s f u l f i l l e d . Then l a r g e d e v i a t i o n of ? from ñ w i l l occur, vhich depends on p a r t i c l e e n e r ­

gy and o s c i l l a t i o n a n p l i t u d e s , and vhich in genera l l eads to some d e p o l a r i z a t i o n j u s t i ­

f y i n g the name : " d e p o l a r i z a t i o n resonance".

1 . 6 . 1 General c l a s s i f i c a t i o n of d e p o l a r i z a t i o n r e s o n a n c e s

Linear resonances are mostly produced by t ransverse quadrupole f i e l d s :

b - * * r 2 • s * r x ( I - 6 - 2 ' where x and z are the t r a j e c t o r y d i sp lacements in the d i r e c t i o n of the rad ia l £ and

3 B x 3 B z

ver i c a l 2 un i t v e c t o r s r e s p e c t i v e l y , and the corresponding f i e l d g r a d i e n t s .

These l i n e a r resonances are c l a s s i f i e d i n t o the fo l l owing f a m i l i e s :

i ) V e r t i c a l b e t a t r o n . e s o n a n c e s ( a l s o named i n t r i n s i c resonances)

» - k ± 0 , These are produced by a v e r t i c a l betatron o s c i l l a t i o n :

z = a z ß z cos ( Q z 6 + * z )

whenever the s p i n c l o s e d s o l u t i o n ñ i s not p o i n t i n g in the r a d i a l d i r e c t i o n £ . TMs i s in

g e n e r a l the c a s e as n i s v e r t i c a l for an a c c e l e r a t o r r ing l y i n g in an h o r i z o n t a l p l a n e .

Normally the i n t e g e r kQ i s a m u l t i p l e of the r ing s u p e r p e r i o d i c i t y P (V.Q = kP) , as i t

r e s u l t s from harnonics of the p e r i o d i c funct ions 0 z , # 2 and 3B / 3 z when ana lyz ing the p e r ­

turbing f i e l d b in frequency. However, in a r ing with grad ient e r r o r s , k Q may be not s u -

p e r p e r i o d i c ( k Q * kP) .

i i ) H o r Í 2 o n t a l _ b e i a t r o n resonances

These are produced by a h o r i z o n t a l betatron o s c i l l a t i o n :

x = a x Jsx cos (Qjie . y every time the s p i n c l o s e d s o l u t i o n n i s not v e r t i c a l , as happens in a non-planar r ing

(very o f t e n due to small i m p e r f e c t i o n s ) . The1' can a l s o be produced by an x-z coupl ing :

z = £ (Qxe . • x;,

when ñ i s not r a d i a l .

i i i ) I n t e g e r resonances ( a l s o named imper fec t ion r e s o n a n c e s )

These are produced e i t h e r by a v e r t i c a l c l o s e d - o r b i t d i s t o r t i o n ( 2 C 0 * o ) when ñ i s

not r a d i a l , or by a h o r i z o n t a l c l o s e d - o r b i t d i s t o r t i o n ( * c o * ° ) "hen ñ i s not v e r t i c a l .

These c l o s e d - o r b i t d i s t o r t i o n s are due to magnet i m p e r f e c t i o n s . For random i m p e r f e c t i o n s ,

a l l i n t e g e r harmonics are p r e s e n t and i n t e g e r resonances are separated by one u n i t i n s p i n

tune , i . e . by 440 HeV for e l e c t r o n s , 523 MeV for protons and 13 .1 GeV for deuterons

( formula 1 - 5 . 1 ) . For s y s t e m a t i c i m p e r f e c t i o n s , supe rp er i o d i c resonances (kQ-= kp) are

produced a l s o .

i v ) Synchrotron resonances

These are produced by synchrotron o s c i l l a t i o n s :

y = D y jß- c o s ( Q s 9 + * s ) , y = x , z

in the R or 2 d i r e c t i o n , p r o p o r t i o n a l l y to the corresponding d i s p e r s i o n D y and t o the am­p l i t u d e 5P/P of energy d e v i a t i o n .

I t i s worth n o t i n g the absence of parametric resonances v = k/2 (k i n t e g e r ) . Horeo-

v e r , f o r p o l a r i z e d beams in s t o r a g e r i n g s , a h a l f - i n t e g e r s p i n tune i s g e n e r a l l y the bes t

o p e r a t i n g p o i n t , which i s midway between d e p o l a r i z a t i o n re sonances .

Now, rpjujjiear re sonances are produced by h i g h e r - o r d e r m u l t i p o l e f i e l d s :

b ( 6 ) a. x p z q (p + q > 1)

The frequency a n a l y s i s of b (9 ) l e a d s to a resonant c o n d i t i o n :

v - k o + k x Q x + k z Q z

w i th | k x | •= p and | k z | t q.

For i n s t a n c e a s e x t u p o l e f i e l d w i l l d r i v e n o n l i n e a r resonances wi th | k x | + jk 2 | •= 2 .

The beam-beam i n t e r a c t i o n in s t o r a g e r i n g s w i l l a l s o d r i v e s e r i e s of n o n l i n e a r r e s o ­

n a n c e s .

Moreover, l a r g e - a m p l i t u d e synchrotron o s c i l l a t i o n s cause a l a r g e frequency modulat ion

of s p i n tune which i s normally propor t iona l to p a r t i c l e energy . S i m i l a r l y to frequency

modulat ion In RF-waves, s e v e r a l synchrotron s a t e l l i t e - l i n e s appear in the frequency

spectrum of s p i n motion :

" „ - k s Q s < i > g > i >

where \»o= ya i s the usua l s p i n tune for v a n i s h i n g synchrotron ampl i tude . Then s e v e r a l s y n ­

c h r o t r o n s a t e l l i t e s :

(i"sl >o appear on each s i d e of any d e p o l a r i z a t i o n resonance ( v » v^) of the prev ions types .

1 . 6 .2 S i m i l a r i t y wi th NuclearHagrie t i c . ïesonance (NWR) phenooena

Summarizing the mechanism of an i s o l a t e d d e p o l a r i z a t i o n resonance , a perturbing f i e l d

b* r o t a t e s about the s p i n c l o s e d s o l u t i o n ñ at frequency u^. The s p i n v e c t o r S* i s p r e c e s -

s i n g about ñ a t frequency \>. Resonant s p i n motion occurs vhen both f r e q u e n c i e s are

equa l ( v = v ^ ) .

3 k M L .

7 U R F

D e p o l a r i z a t i o n resonance Nuclear magnetic resonance

In a standard NMR exper iment , the magnet i za t ion vec tor M* of a nuc l ear magnetic s u b s ­

tance i s p r e c e s s i n g about a s t a t i o n a r y magnetic f i e l d B* a t the Larmor frequency u. . A -» 0 -+

t r a n s v e r s e R F - f i e l d i s superimposed with a frequency K ^ . This b R F f i e l d can be decom­posed i n t o tvo f i e l d s b and b*' r o t a t i n g in o p p o s i t e d i r e c t i o n s . Nuclear magnetic resonance o c c u r s when both f r e q u e n c i e s are equal ( ( ^ = U p ) , 6* and H* s t a y i n g in phase . This resonance i s e x p e r i m e n t a l l y observed as a s i g n a l in RF-energy absorpt ion by the s u b s t a n c e , c o r r e s ­ponding to the p o p u l a t i o n of h igher -energy s t a t e s .

The n o t i o n s of Í and S* look, s i m i l a r and appear s impler w i t h i n

a new frame :

Let us c o n s i d e r a frame r o t a t i n g about the c l o s e d s o l u t i o n ( \

n, a t the resonance frequency v D . Vith r e s p e c t to t h i s r o t a t i n g I 1

•* * ' 1

frame, the per turb ing f i e l d b i s a t r e s t and the s p i n vec tor S _^ i

p r e c e s s e s about n at the frequency v - v^. C] j

On re son an ce , the s p i n p r e c e s s i o n about n i s v a n i s h i n g and

one i s l e f t wi th on ly the s p i n p r e c e s s i o n about the s t a t i o n a r y

p e r t u r b i n g f i e l d B*. The s p i n v e c t o r becomes "up" and "down" pe­

r i o d i c a l l y , e x p l a i n i n g the popu la t ion of the tvo s p i n s t a t e s .

In f a c t , an experiment of NMR type i s used in e l e c t r o n s t o r a g e r ings for a very a c c u ­

r a t e (=• 10~5) energy calibration. Vhen the applied R F - f i e l d e n t e r s i n t o resonance vith

s p i n p r e c e s s i o n a sharp d e p o l a r i z a t i o n o c c u r s . Then the frequency of the R F - f i e l d i s equal

to the s p i n tune ( v = ya) and i s proport iona l to the beam energy.

- oui -

II. ACCELERATION OF POLARIZED PROTONS IN SYNCHROTRONS

P o l a r i z e d proton beams have been a c c e l e r a t e d s u c c e s s f u l l y in s e v e r a l synchro trons :

the ZGS at Argonne (up to 12 GeV), the AGS at Brookhaven ( a t present up to 16 .5 GeV),

Saturne a t Sac lay (up to 3 GeV) and the KEK PS at Kyoto ( a t present in the 0 . 5 GeV

b o o s t e r ) .

The scheme for a c c e l e r a t i n g p o l a r i z e d protons

in a s y n c h r o t r o n i s near ly the same for a l l

machines . One can take the example of the ACS

scheme, shown in F i g . 1. I t i n v o l v e s a p o l a r i z e d

ion s o u r c e d e l i v e r i n g a 25 yA H beam which i s

a c c e l e r a t e d up to 200 MeV in a l i n a c . Af ter

i n j e c t i o n w i t h e l e c t r o n s t r i p p i n g , about 1 0 1 0

protons per p u l s e are a c c e l e r a t e d in the s y n ­

chrotron r i n g . Af ter reach ing the top energy

( 1 6 . 5 GeV i n 1984, Z6 GeV planned for 1 9 0 5 ) ,

protons are e x t r a c t e d and transported to expe ­

r imenta l a r e a s for bombarding f ixed t a r g e t s

( p o s s i b l y p o l a r i z e d t a r g e t s ) . The degree of

p o l a r i z a t i o n i s measured by po lar i mete .'S at

s e v e r a l s t a g e s of the a c c e l e r a t i o n proces s : a

200 MeV P o l a r i m e t e r a t the end of the l i n a c , an

i n t e r n a l p o l a r i n e t e r i n s i d e the main r ing and an

e x t e r n a l h i g h - e n e r g y Po lar imeter in front of

e x p e r i m e n t s . The measurement of p o l a r i z a t i o n i s

based on the asymmetry in the s c a t t e r i n g of

p o l a r i z e d protons through a t h i n t a r g e t .

RFQ _ Polarized-ion source

ZOO MeV Polanmelei

F i g . 1

Internal Polarimeter High-energy Polarimeter

AGS layout for a c c c e i e r a t i o n o f p o l a r i z e d p r o t o n s .

During a c c e l e r a t i o n in the main r i n g , the sp in tune i n c r e a s e s l i n e a r l y wi th t ime and

e n e r g y , and s e v e r a l d e p o l a r i z a t i o n re sonances ace c r o s s e d . D e p o l a r i z a t i o n i s reduced by

c o r r e c t i o n d e v i c e s o f two types : d i p o l e s c o r r e c t i n g harmonics of the v e r t i c a l c l o s e d

o r b i t d i s t o r t i o n when imperfec t ion resonances are c r o s s e d , and pulsed quadrupoles dr iven

by s p e c i a l power s u p p l i e s e n a b l i n g rapid jupping of i n t r i n s i c r e sonances . In the AGS f i v e

i n t r i n s i c and 31 imper fec t ion resonances are c r o s s e d when a c c e l e r a t i n g up to 16 .5 GeV.

F ina l p o l a r i z a t i o n i s about 40 X and r e p r e s e n t s near ly 60 X of the p o l a r i z a t i o n a t i n ­

j e c t i o n i n t o the main r i n g . In Saturne, on ly four i n t r i n s i c and s i x i m p e r f e c t i o n r e s o ­

nances are c r o s s e d when a c c e l e r a t i n g up to 3 GeV. A remarkably high degree of p o l a r i z a ­

t i o n , about 80 X, i s c u r r e n t l y o b t s i n e d .

The low i n t e n s i t y (10 ppp) i s the p r i c e to be paid for o b t a i n i n g p o l a r i z e d p r o t o n s .

However, p r e s e n t developments in p o l a r i z e d - i o n s o u r c e s and i n j e c t i o n t echn iques support

the hope that i n the near future polar ized-beam i n t e n s i t y v i l l reach present v a l u e s

(* 1 0 1 3 pps) of unpolarized-beam i n t e n s i t y .

The c r o s s i n g o f d e p o l a r i z a t i o n resonances i s the main problem to s t u d y . In the f o l ­lowing s e c t i o n s we i n v e s t i g a t e the mechanise of d e p o l a r i z a t i o n when c r c s s i n g an i s o l a t e d resonance and the cures for low-energy synchrotrons a s w e l l as the proposed "Siber ian Snakes" at h igher e n e r g i e s .

I I . 1 D e p o l a r i z a t i o n resonances in proton synchrotrons

Ue c o n s i d e r on ly planar r i n g s l y i n g in a h o r i z o n t a l p lane , as usual up to now. The

spjn c l o s e d s o l u t i o n i s then v e r t i c a l , i . e . p a r a l l e l to the magnetic f i e l d in bending

magnets .

For unders tanding d e p o l a r i z a t i o n phenomena in proton r i n g s , the most important f e a t u r e to c o n s i d e r is the e f f e c t of beam energy spread. As the s p i n tune i s propor t iona l to e n e r ­gy ( v = Y 3 In a planar ring v i t h v e r t i c a l bending f i e l d ) , energy spread l e a d s to s p i n tune spread- Spin v e c t o r s of p a r t i c l e s with d i f f e r e n t e n e r g i e s p r e c e s s a t d i f f e r e n t r a t e s and w i l l r a p i d l y ge t out of phase . For i n s t a n c e two 100 MeV p r o t o n s , d i f f e r i n g in t o t a l energy by 10"*, reach a s p i n phase s h i f t of 2n a f t e r only 50ÛÛ t u r n s .

Due to t h i s s p i n phase mixing, any h o r i z o n t a l component of the p o l a r i z a t i o n v e c t o r v a n i s h e s r a p i d l y . Only i t s v e r t i c a l component P g can s u r v i v e . Consequently the beam must be i n j e c t e d i n the r i n g with the p o l a r i z a t i o n vec tor p a i n t i n g iti the v e r t i c a l d i r e c t i o n .

Now, any p e r t u r b a t i o n of sp in motion, due to a per turbat ion of o r b i t a l mot ion, w i l l , on re sonance , l ead to l a r g e d e v i a t i o n s of the sp in vec tor S* away from the v e r t i c a l Oz. The v e r t i c a l component ? z d e c r e a s e s and d e p o l a r i z a t i o n i s observed . I t i s worth n o t i n g that ampl i tudes of o r b i t a l - m o t i o n per turbat ion may d i f f e r from one proton to another; the s p i n v e c t o r S* w i l l d e v i a t e more for l a r g e ampl i tudes than for small ones . An averaged ampl i tude must be taken for c a l c u l a t i n g the amount of d e p o l a r i z a t i o n .

The most important p e r t u r b a t i o n s , and the resonances they d t i v e , are of two types :

i ) V e r t i c a l be ta tron o s c i l l a t i o n s are r e s p o n s i b l e for r a d i a l f i e l d s a long the t r a j e c t o r i e s , which bend the s p i n vector a«ay from Oz. They d r i v e v e r t i c a l b e t a t r o n r e s o n a n c e s , named i n t r i n s i c resonances . They are caused by the f i n i t e v e r t i c a l e m i t t a n c e of the i n j e c t e d beam.

i i ) V e r t i c a l c l o s e d - o r b i t d i s t o r t i o n s are r e s p o n s i b l e for r a d i a l f i e l d s a l s o . They dri" e i n t e g e r r e s o n a n c e s , named imperfec t ion resonances , and they are caused by f i e l d e r r o r s and magnet misa l ignments .

I I . 2 Resonance s t r e n g t h and width

As s t r e s s e d in s e c t i o n 1.6.2, the resonance phenomenon looks s impler when seen in a r o t a t i n g frame. Th i s frame r o t a t e s at the resonance frequency u^, r e l a t i v e l y to the o r b i t frame, about the v e r t i c a l l i n e .

- ob.ï -

In t h i s frame, according to formula 1 . 3 . 6 , sp in v e c t o r S* p r e c e s s e s about the v e r t i c a l

at the frequency 5 = {v - v^) Q^, whsre i s the cyc lo t ron frequency in the f i e l d B of

the r ing magnets. On the o ther hand i t a l s o p r e c e s s e s about the s t a t i o n a r y component b R

of the perturbing r a d i a l f i e l d a t the frequency zQ , vhere

(1 * Ta)

i s the resonance s t r e n g t h . G loba l ly , spin v e c t o : o p r e c e s s e s

about the r e s u l t i n g r o t a t i o n v e c t o r 3 :

3= (l î + E î ) S

vhere I i s a uni t v e c t o r in the d i r e c t i o n of b^. The d e v i a ­

t i o n of 3 from Oz i s proport iona l to the resonance s t r e n g t h e .

The r o t a t i o n v e c t o r ÎÎ i s a l s o the s p i n c l o s e d s o l u t i o n in the

r o t a t i n g frame.

ü/a

On top of resonance the p r e c e s s i o n about the v e r t i c a l van i shes (5=o) and the r o t a t i o n

v e c t o r Ö i s t r a n s v e r s e , p a r a l l e l to 8* . The spin r o t a t i o n frequency i s 5 ^ cQ^, shoving

that the s t r e n g t h E i s the r a t i o of the sp in r o t a t i o n angle * t-> the v e l o c i t y r o t a t i o n

ang l e a : d*

e = & •

In o ther words, the resonance s t rength e i s tue sp in r o t a t i o n angle per radian of

v e l o c i t y r o t a t i o n , and i s t i i m e n s i o n l e s s .

The resonance s t r e n g t h e can a l s o be cons idered as the resonance v idth s i n c e the ang l e

of the r o t a t i o n v e c t o r 3 with Oz i s l a r g e r than n/4 in the (v^- z, £) s p i n tune i n ­

t e r v a l .

A rough e s t i m a t e of the resonance s t rength e can e a s i l y be obta ined by c o n s i d e ­

r i n g on ly the r a d i a l quadrupole f i e l d s as g iven by formula ( 1 . 6 . 3 ) :

b ( 9 )

where z i s the v e r t i c a l d isplacement due to e i t h e r a v e r t i c a l betatron o s c i l l a t i o n for an

i n t r i n s i c resonance or a v e r t i c a l c l o s e . i - o r b i t d i s t o r t i o n for an imperfec t ion resonance .

The s t a t i o n a r y component b^ i s the one-turn average of 6*(9) in the r o t a t i n g frame (noted

f ,m,n frame with ñ p a r a l l e l to Oz), g iven by :

(oi - U ) . b(6) de

in complex n o t a t i o n . The r e s u l t i n g cooplex express ion of the resonance s t r e n g t h e i s

(1 - ya) 7nK (S (m + i r ) . K z ds

1 where K i s the q u a d r u p l e s t rength :

For i m p e r f e i ^ o n resonances , the resonance s t rength c t i c a l c l o s e d - o r b i t d i s t o r t i o n and the t o t a l proton energy. I t the AGS at Brookhaven, and would reach 10 1 at most

:cales l i n e a r l y with the ver­

i s in ihe 10 ' - 1 0 3 range in

in the Tevatron at Fern i lab ,

- b M -

For i n t r i n s i c r e s o n a n c e s , c s c a l e s a s the square root of the v e r t i c a l i n v a r i a n t e m i t ­

tance and of e n e r g y . I t i s in the 1 0 ~ 3 - 1 0 ~ ! range in the AGS and would reach 10" 1 a t most

in the Tevatron . The s t r e n g t h c, cons idered h e r e , i s an average over a l l the b e t a t r o n

a m p l i t u d e s in the proton beam. However, vhen c o n s i d e r i n g a s i n g l e p a r t i c l e , the s t r e n g t h

depends on i t s b e t a t r o n ampl i tude and v i l l vary from one p a r t i c l e to another .

I I . 3 Linear c r o s s i n g of an i s o l a t e d resonance

For smal l resonance width as in the AGS, the d i s t a n c e in energy between resonances i s

very l a r g e compared to t h e i r w i d t h s . Each resonance , crossed during a c c e l e r a t i o n , can be

c o n s i d e r e d a s i s o l a t e d .

Then a s i m p l e p i c t u r e of s p i n - v e c t o r S* morion, vhen c r o s s i n g an i s o l a t e d resonance, can be o b t a i n e d in the r o t a t i n g frame a g a i n .

Far below the resonance energy , the r o t a t i o n v e c t o r 5 i s

v e r t i c a l and downward ( 6 « - e ) . When approaching the r e s o ­

nance , 3 s t a r t s to d e v i a t e from O 2 , jtid écornes e x a c t l y h o r i ­

z o n t a l ( i = o) on top of the resonance . Above the resonance ,

5 moves s y m m e t r i c a l l y and becomes v e r t í c a l , in the upvard

d i r e c t i o n , at the end ( 5 » e ) . G l o b a l l y , the r o t a t i o n v e c t o r 2 undergoes a complete r e v e r s a l of d i r e c t i o n when the resonance

i s c r o s s e d .

Does the s p i n v e c t o r S* of an i n d i v i d u a l p a r t i c l e , which

p r e c e s s e s about 3, f o l l o w i t during i t s r e v e r s a l ? I f y e s , the

s p i n v e c t o r S*, assumed v e r t i c a l i n i t i a l l y , w i l l a l s o be r e v e r ­

sed as 5 i s and there i s an a d i a b a t i c s p i n f l i p . I f a l l the

p a r t i c l e s do the same, the p o l a r i z a t i o n v e c t o r P* i s only r e v e r ­

s e d . I n i t i a l l y v e r t i c a l , i t becomes v e r t i c a l again a f t e r c r o s ­

s i n g , but p o i n t i n g in the o p p o s i t e d i r e c t i o n . There i s no depo­

l a r i z a t i o n .

above resonance

on top of resonance

below resonance

The a d i a b a t i c i t y c o n d i t i o n for s p i n f l i p is a s p i n p r e c e s s i o n about 3 much f a s t e r man

the motion of 3 i t s e l f . More p r e c i s e l y , assuming a l i n e a r v a r i a t i o n of energy with t ime,

i . e . a l i n e a r v a r i a t i o n of s p i n tune v v i t h azimuth 9 :

v = \ *

the crossing "time" Û8 is about :

During t h i s time the s p i n p r e c e s s i o n an g l e n> i s :

The a d i a b a t i c i t y c o n d i t i o n >• 1) can be w r i t t e n

2

s- » 1 .

On the c o n t r a r y , for very f a s t c r o s s i n g :

^ « i , a

the s p i n v e c t o r S* has not enough time for s t a r t i n g to rove during the resonance c r o s s i n g .

In t h i s c a s e the v e r t i c a l d i r e c t i o n of S* i s n o . changed. There i s no change in p o l a r i z a ­

t i o n e i t h e r , assuming f a s t c r o s s i n g for a l l p a r t i c l e s .

What happens between these two extreme c a s e s ? One e x p e c t s an incomplete s p i n f l i p ,

w i th S* f i n a l l y p o i n t i n g in a n o n - v e r t i c a l d i r e c t i o n . Const î e n t l y the v e r t i c a l component

| S z I of S*, which i n i t i a l l y was u n i t y , has decreased at the ^nd. The v e r t i c a l component ? z

of the p o l a r i z a t i o n v e c t o t has a l s o decreased and some d e p o . a r j z a t i o n has r e s u l t e d .

A q u a n t i t a t i v e e s t i m a t e of the f i n a l v e r t i c a l component , compared to i t s i n i t i a l

v a l u e , i s g i v e n by the F r o i s s a r t - S t o r a formula :

S f i n a l - il— „ _ 2 e ^* - 2 S2 i n i t i a l

which i n c l u d e s the two extreme c a s e s of a d i a b a t i c sp in f l i p (5^ f i n a l = - S 2 i n i t i a l ) and

of f a s t c r o s s i n g ( S z f i n a l = + S 2 i n i t i a l ) , as w e l l as the i n t e r m e d i a t e c a s e s .

The e f f e c t of a p a r t i c u l a r resonance depends on i t s s t r e n g t h E as compared to the

a c c é l é r â t ' - i r a t e a. Moreover, for i n t r i n s i c r e s o n a n c e s , p a r t i c l e s with very smal l

b e t a t r o n ampl i tude w i l l e x p e r i e n c e a weak resonance and t h e i r s p i n v e c t o r w i l l not be

r e v e r s e d - On the contrary par t i d e s with l a r g e amp11tude w i l l e x p e r i e n c e a s t r o n g

re sonance and t h e i r s p i n v i l l be r e v e r s e d . For t h i s type of resonance the amount of

d e p o l a r i z a t i o n i s g i v e n by an average over the be ta tron ampl i tudes among the p a r t i c l e s .

As an example, F ig . 2 shows the v a r i a t i o n of the p o l a r i z a t i o n a f t e r c r o s s i n g the im­

p e r f e c t i o n resonance M = 3 in Saturne as a func t ion of a d i p o i e c o r r e c t i o n which changes

the s t r e n g t h c of t h i s resonance . The observed maximum corresponds to a t o t a l compensat ion

1

F i g . 2 P o l a r i z a t i o n P, a f t e r c r o s s i n g the imper fec t ion resonance v = 3 in Saturne , versus c o r r e c t i o n ampl i tude of v e r t i c a l c l o s e d - o r b i t harmonics

- í»frí> -

o f i t s n a t u r a l s t r e n g t h by t h e d i p o l e c o r r e c t i o n . U i t h a c o i r e e t i o n , e i t h e r n u l l or oppo­

s i t e i n s i g n , t h e p o l a r i z a t i o n has t h e o p p o s i t e v a l u e , i n d i c a t i n g a s u c c e s s f u l l a d i a b a t i c

s p i n f l i p . Such a n a l m o s t p e r f e c t s p i n f l i p i s o b s e r v e d when c r o s s i n g f i v e i m p e r f e c t i o n

r e s o n a n c e s and t v o i n t r i n s i c r e s o n a n c e s i n S a t u r n e , e x p l a i n i n g t h e h i g h d e g r e e of p o l a r i ­

z a t i o n ( a b o u t 8 0 X) m a i n t a i n e d d u r i n g t h e a c c e l e r a t i o n c y c l e up t o the t o p e n e r g y ( 3 G P V ) .

I I . i* C u r e s f o r l o v - e n e r g y s y n c h r o t r o n s

V e r y o f t e n , c r o s s e d r e s o n a n c e s h a v e a s t r e n g t h w h i c h i s h a r m f u l t o p o l a r i z a t i o n as t h e

s p i n v e c t o r i s ben t away f rom t h e v e r t i c a l upon c r o s s i n g - The u n d e r s t a n d i n g of the d e p o ­

l a r i z i n g mechanism i n d i c a t e s Eour methods f o r r e d u c i n g d e p o l a r i z a t i o n . Tvo o f t h e s e me­

t h o d s h a v e been s u c c e s s f u l l y a p p l i e d , f o r i n s t a n c e i n t h e AGS s y n c h r o t r o n .

i ) D e c r e a s e t h e r e s o n a n c e s t r e n g t h E. T h i s i s a c o r r e c t i o n method ( a l s o named

h a r m o n i c s p i n m a t c h i n g ) w h i c h a ims t o c a n c e l out t h e r e s o n a n c e s t r e n g t h . For i m p e r f e c t i o n

r e s o n a n c e s t h i s method uses s e v e r a l d i p o l e c o r r e c t o r s w h i c h c o n t r o l h a r m o n i c s o f t h e v e r ­

t i c a l c l o s e d - o r b i t d i s t o r t i o n . By v a r y i n g t h e c o s i n e and s i n e components o f t h e most i m ­

p o r t a n t h a r m o n i c s one can compensate t h e d r i v i n g f i e l d o f a p a r t i c u l a r r e s o n a n c e . I n

g e n e r a l t h i s i s done a f t e r o r b i t c o r r e c t i o n and t h e needed h a r m o n i c c o r r e c t i o n i s s u f f i ­

c i e n t l y s m a l l f o r n o t c a u s i n g any t r o u b l e to the c l o s e d o r b i t .

T h e s i g n a l f o r m o n i t o r i n g t h i s c o r r e c t i o n

i s t h o p o l a r i z a t i o n P i t s e l f . T o t a l c o r r e c t i o n

i s a c h i e v e d when p o l a r i z a t i o n a f t e r r e s o n a n c e

c r o s s i n g i s maximum as shown i n F i g . 3 .

T h i s method has been s u c c e s s f u l l y used f o r

c o r r e c t i n g a b o u t t h i r t y i m p e r f e c t i o n r e s o n a n ­

c e s i n t h e AGS, t h e c o r r e s p o n d i n g c o r r e c t i o n s ID 0 sine 10 0 casme b t i n g t u r n e d on s u c c e s s i v e l y d u r i n g r e s o n a n c e F i g . 3 S i n e and c o s i n e h a r m o n i c s

c o r r e c t i o n f o r the v = 9 c r o s s i n g . i m p e r f e c t i o n r e s o n a n c e a t

1 3 . S G e V / c i n AGS.

I n p r i n c i p l e a s i m i l a r c o r r e c t i o n m e t h o d ,

u s i n g q u a d r u p o l e c o r r e c t o r s , can be used f o r i n t r i n s i c r e s o n a n c e s , but i s l i m i t e d t o r a ­

t h e r weak r e s o n a n c e s as i n the case o£ t v o n o n - s u p e r p e r i o d i c i n t r i n s i c r e s o n a n c e s i n

S a t u r n e .

i l ) I n c r e a s e t h e c r o s s i n g r a t e a . T h i s method (named r e s o n a n c e ' u m p i n g ) a ims to

r e a l i z e a f a s t c r o s s i n g d u r i n g v h i c h t h e s p i n v e c t o r has no t i m e f o r moving a v a y f r o m t h e

v e r t i c a l . T h i s method i s e s s e n t i a l l y d e s i g n e d f o r i n t r i n s i c r e s o n a n c e s .

D u r i n g a c c e l e r a t i o n t h e s p i n t u n e i n c r e a s e s

l i n e a r l y w i t h t i m e . Vhen a p p r o a c h i n g an i n t r i n s i c

r e s o n a n c e , a t t i m e t Q , t h e v e r t i c a l b e t a t r o n t u n e

Q i s a b r u p t l y d e c r e a s e d such t h a t t h e r e s o n a n c e

i s c r o s s e d i n a v e r y s h o r t t i m e . T h e r e a f t e r , t h e

i n i t i a l b e t a t r o n t u n e i s r e s t o r e d more s l o w l y .

- lih -

This method i s app l i ed in the AGS for c r o s s i n g four s trong i n t r i n s i c resonances . A s e t

oi pulsed quadrupoles , povered by s p e c i a l power s u p p l i e s , i s used for d e c r e a s i n g ihe ver -

l i c a l s p i n tune by ûQz= 0 .25 with a r i s e t i m e At of 1 . 6 - J S . The c r o s s i n g r a t e a, which i s

normally 3 . 1 0 " 4 per turn, i s increased by two orders of magnitude such that the resonance

crossed in l e s s than one turn. Figure i shows the p o l a r i z a t i o n a f t e r c r o s s i n g as a func­

t i o n sf the l i c e t at which pulsed quadrupoles are f i r e d . One observes a p o l a r i z a t i o n

maximum when the time tQ i s properly s e t for c r o s s i n g the resonance during the r i s e time

of the pulsed quadrupoles ( the observed secondary maximum could be an a r t e f a c t ) .

F i n a l l y , the o ther tvo methods aim to ach ieve com­

p l e t e a d i a b a t i c s p i n f l i p by e i t h e r i n c r e a s i n g the

resonance s t r e n g t h E or decreas ing the c r o s s i n g rate a.

They are not commonly used .

A l l these four methods seem to be l imi t ed to low-

energy synchrotrons; the l i m i t in energy may we l l be of

the order of the AGS top energy. There are two r e a ­

s o n s : i ) the s t r e n g t h and width of resonances i n c r e a s e

w i th energy ( s e e s e c t i o n I I . 2 ) making them more d i f f i ­

c u l t to compensate or to jump ; i i ) the number of r e ­

sonances to be crossed i n c r e a s e s l i n e a r l y v i t h energy,

r e q u i r i n g higher e f f i c i e n c y for curing each resonance

in order to o b t a i n a u s e f u l degree of p o l a r i z a t i o n at

the top energy . One could imagine a r r i v i n g at a e o s -

p í e t e sp in f l i p for most resonances . However, they

become wide and can o v e r l a p , and new harmful e f f e c t s

are expected when over lapping o c c u r s .

II.5. "Siber ian Snakes"

A very d i f f e r e n t method, which would work at h igher e n e r g i e s , has been proposed for

a v o i d i n g d e p o l a r i z a t i o n on Lesonance c r o s s i n g . The idea i s to equip the synchrotron r ing

with one or s e v e r a l magnetic d e v i c e s , named "Siberian Snakes".

In p r i n c i p l e a S i b e r i a n Snake r o t a t e s the spin v e c t o r £> by a n angle about an a x i s G l y i n g in the h o r i z o n t a l plane of the r i n g .

The s p i n motion in t h i s h o r i z o n t a l plane i s i l l u s t r a t e d in Fig . 5 for a ring equipped

wit. i a s i n g l e S iber ian Snake. S t a r t i n g at the point 0 o p p o s i t e to the Snake, a f t e r one

turn a h o r i z o n t a l sp in d i r e c t i o n (1 ) i s transformed i n t o the d i r e c t i o n ( 4 ) , which i s

symmetric to d i r e c t i o n (1 ) with respect to the a x i s û, as seen in the o r b i t frame. In par­

t i c u l a r the d i r e c t i o n G at point 0 is transformed i n t o i t s e l f 2nd then c o i n c i d e s with the

s p i n c l o s e d s o l u t i o n n at t h i s p o i n t . This sp in c losed s o l u t i o n l i e s in the h o r i z o n t a l

plane at any point in the r i n g . Horeover, the above symmetf property oE d i r e c t i o n s (1)

and ( 4 ) in the h o r i z o n t a l plane shows that they are connected by a i t -rotat ion about the

d i r e c t i o n u. The one-turn sp in napping i s a n -ro ta t ion about the sp in c l o s e d s o l u t i o n n

and the sp in tune i s 1 /2 .

y Fig . 4 P o l a r i z a t i o n P,

a f t e r c r o s s i n g the i n ­t r i n s i c resonance v=Q at ¿ .486 GeV/c in AGS? v e r s u s f i r i n g time t of pulsed quadrupoles'.

- <>ti8 -

F i g . 5 a) Spin motion in the h o r i z o n t a l p lane of a r ing equipped v i t h a S i b e r i a n ^nake ( S S ) .

b) S u c c e s s i v e h o r i z o n t a l s p i n d i r e c t i o n s s e e n i n the o r b i t frame. ( 1 ) I n i c i a l d i r e c t i o n a t point 0 of a test spin. (2) Spin d i r e c t i o n a t the Snake e n t r a n c e . ( 3 ) Spin d i r e c t i o n a t the Snake e x i t . ( 4 ) F ina l s p i n d i r e c t i o n a t po int 0 a g a i n .

The e s s e n t i a l f e a t u r e of a r ing equipped v i t h S i b e r i a n Snakes i s that the 1/2 s p i n

tune i s independent o f energy contrary to the usual l i n e a r dependence. Then bean energy

spread does not l ead to any sp in tune spread , a id there i s no s p i n phase n i x i n g , a t l e a s t

for an i d e a l S i b e r i a n Snake producing an exact i t - r o t a t i o n for a l l p a r t i c l e s . One can e x ­

pect a l a r g e r e d u c t i o n of d e p o l a r i z i n g e f f e c t s .

Spin motion in a r i n g equipped v i t h a snake i s analogous to a ve i l -known NMR pheno­

menon, named Spin Echo. In a Spin Echo "thought" experiment ( F i g . 6), a nuc lear magnet ic

s u b s t a n c e i s z a g n e t i z e d such that the magnet i za t ion v e c t o r ÍÍ p r e c e s s e s about a s t a t i o n a r y

f i e l d # q in a t r a n s v e r s e p l a n e . Due to l o c a l f i e l d i n h o m o g e n e i t i e s , n a g n e t i c moments ¡Íj 2 of d i f f e r e n t n u c l e i 1,2,.. preces s a t s l i g h t l y d i f f e r e n t f r e q u e n c i e s . I f they were a l i g n e d

in the same d i r e c t i o n o r i g i n a l l y , they spread out a f t e r and magnet i za t ion d e c r e a s e s . At

time T a t r a n s i e n t f i e l d i s app l i ed vhich r o t a t e s a l l the magnetic moments by n about the

a x i s Û. The f a s t e s t morr°nt ( U j ) , which was the former, becomes the l a t t e r a f t e r t h i s n-

rotation. Then a t time 2T the magnetic moments are a l i g n e d toge ther aga in and m a g n e t i z a ­

t i o n i s r e s t o r e d .

F i g . 6 Scheme of an NMR Spin Echo experiment . a) p r e c e s s i o n of three magnetic moments u. „ - about magnetic f i e l d B

with Jt -rotat ion a t time T. ' ' b) v a r i a t i o n of magnet i za t ion M with time t .

In a t i n g equipped with a Snake, the s p i n v e c t o r s of a p a r t i c l e bunch wi th some energy

s p i e a d , have e x a c t l y the sane behav iour . The Snake p lays the r o l e of the Transient f i e l d

in the Spin Echo exper iment . S t a r t i n g at the o p p o s i t e point 0 in the r i n g , wi th a l l the

s p i n v e c t o r s a l i g n e d in the sane d i r e c t i o n , they w i l l be r e a l i g n e d together a f t e r one turn

in s p i t e of t h e i r d i f f e r e n t p r e c e s s i o n f r e q u e n c i e s . n>ey li.ivc tin- s a w pri-. t-ss i. >~. an.:'.(-

,-ind s p i n f i n e whatever t h e i r enerc i i / s a r e .

Now, another popular scheme i s a two-snake r i n g , i . e . a r ing equipped wi th two oppo­

s i t e S i b e r i a n Snakes which r o t a t e the s p i n v e c t o r

by n about two orthogonal and h o r i z o n t a l a x e s u,

v . I t has the t h e o r e t i c a l advantage of a more

s t a b l e s p i n c l o s e d s o l u t i o n , with an e n e r g y - i n d e ­

pendent v e r t i c a l o r i e n t a t i o n : downward i n one

h a l f - r i n g between the Snakes and upward in the

o ther ha l f - r i n g .

The o n e - t u r n mapping ( F i g , 7) of a s p i n d i r e c t i o n (1), l y i n g in the h o r i z o n t a l p l a n e ,

i s j u s t o b t a i n e d by adding a second n - r o t a t i o n about v 2 x i s which transforms d i r e c t i o n (4 )

i n t o d i r e c t i o n ( 5 ) . This f i n a l d i r e c t i o n (5 ) i s o p p o s i t e to the i n i t i a l d i r e c t i o n ( 1 ) ,

showing that the o n e - t u r n mapping i s e f f e c t i v e l y a a - r o t a t i o n and that the s p i n tune i s

F i g . 7 a) Spin motion in the h o r i z o n t a l plane of a two-snake l i n g . b) S u c c e s s i v e h o r i z o n t a l s p i n d i r e c t i o n s seen in the o r b i t frame.

( 1 ) I n i t i a l sp in d i r e c t i o n . (2 ) Spin d i r e c t i o n at I s Snake (SSI) e n t r a n c e . O) Spin d i r e c t i o n at I s , Snake (SSI) e x i t . ( 4 ) Spin d i r e c t i o n at 2 n Snake (SS2) e n t r a n c e . (5 ) F ina l s p i n d i r e c t i o n at 2 1 1 Snake (5S2) e x i t .

In a two-snake r ing one can study what happens when the p a r t i c l e e n e i g y c o i n c i d e s

w i th one of p r e v i o u s resonances (ya - k or k ± 0 Z ) , i - e . when a p e r t u i b i n g f i e l d component

i s resonant w i th s p i n p r e c e s s i o n in the arcs of the r i n g .

The one-turn mapping i s perturbed . The s p i n c l o s e d s o l u t i o n it s l i g h t l y d e v i a l e s i rom

the v e r t i c a l and the s p i n tune i s not e x a c t l y 1 /2 .

During resonance c r o s s i n g , i n s t e a d of be ing r e v e r s e d , the sp in c l o s e d s o l u t i o n n

on ly undergoes a t r a n s i e n t e x c u r s i o n away f i o c the v e r t i c a l . The o r i g i n a l v e r t i c a l d i r e c ­

t i o n (upward in one arc ) i s r e s t o r e d a f t e r c r o s s i n g . This i s in c o n t r a s t wi th the complete

l e v e r s a l in a r ing without s n a k e s .

v i [ b o u t Snakes v i t h t v o Snakes

I f t h e e v o l u t i o n o f n i s s u f f i c i e n t l y s l o v , t h e s p i n v e c t o r 5*, v h i c h r o t a t e s a b o u t n ,

v i l ] a c i a b a t i c a i l y f o l l o w i t i n i t s m o t i o n , ( S z f i n a l = - S ? i n i t i a l ) , and t h e r e i s no

d e p o l a r i z a t i o n . S i m u l a t i o n shows t h a t t h i s i s t h e c a s e f o r a t v o - s n a k e r i n g , e v e n f o r

s t r o n g r e s o n a n c e s , c o n f i r m i n g t h e i n i t i a l i d e a o f d e p o l a r i z a t i o n s u p p r e s s i o n by S n a k e s .

H o w e v e r , f o r v e r y s t r o n g i n t r i n s i c r e s o n a n c e s ( e > 0 . 2 ) l a c k o f a d i a b a t i c i t y has been o b ­

s e r v e d i n s i m u l a t i o n , i n d i c a t i n g t h a t such d e p o l a r i z a t i o n r e s o n a n c e s a r e s t i l l h a r m f u l .

F i n a l l y , how can a S i b e r i a n Snake be r e a l i z e d i n p r a c t i c e ? The s i m p l e s t i d e a i s t o

use a s o l e n o i d f o r o b t a i n i n g a i t - r o t a t i o n a o o u t t h e l o n g i t u d i n a l a x i s . A c c o r d i n g t o f o r -

m u l a ( 1 . 3 . 5 ) one n e e d s a f i e l d i n t e g r a l o f 3 . 7 5 2 To per G e V / c i n t h e s o l e n o i d . For i n s

t a n c e , t o r c r o s s i n g t h e f i r s t s t r o n g i n t r i n s i c r e s o n a n c e v = i}^ < 8 . 7 5 i n t h e AC5 w i t h a

S n a k e , t h e n e e d e d f i e l d i n t e g r a l amounts l o 1 6 . 8 Tm. O b v i o u s ' . y , vhen g o i n g t o h i g h e r

e n e r g i e s , t h e f i e l d i n t e g r a l becomes r a p i d l y too l a r g e . One i s f o r c e d t o c o n s i d e r a Sn.-ike

n a c f <jf t r a n s v e r s e f i e l d m a g n e t s . T h e r e i s a l a r g e v a r i e t y u f p o s s i b i l i t i e s . F i g u r e 8 s!i.!ws

o n e a t t r a c t i v e scheme v i t h s i x v e r t i c a l l y b e n d i n g magnets and s i x h o r i z o n t a l l y b e n d i n g

m a g n e t s . A c c o r d i n g t o f o r m u l a ( 1 . 3 - 7 ) , such a snake w o r k s a t a n e a r l y f i x e d f i e l d f o r

m a i n t a i n i n g t h e n - r o t a t i o n a t a l l e n e r g i e s . The 22 Tm o v e r a l l f i e l d i n t e g r a l f o r t h e

t w e l v e m a g n e t s i s m o d e s t , a s c o m p a r t o iO t h e b e n d i n g - f i e l d i n t e g r a l needed i n t h e a r c s a l

h i g h e n e r g y . T h e e x c u r s i o n of t h e v e r t i c a l and h n r i z o n t a l beam bumps i n s i d e t h e Snake

d e c r e a s e s when r a n p i n g i n e n e r g y . T h i s l e a d s t o a v a r i a b l e g e o m e t r y o f t h e bean l i n e . T h i s

e x c u r s i o n i s l a r g e s t a t t h e i n j e c t i o n e n e r g y and the magnet a p e r t u r e l i m i t s t h e l o w e s t

p o s s i b l e e n e r g y . on* on* side view

S k e t c h o f a S i b e r i a n Snake made o í s i x b e n d i n g m a g n e t s . Ai rows i n d i c a t e s p i n o r i e n t a t i o n and numbers i n d i c a t e s p i n r o t a t i o n a n g l e s .

I n c o n c l u s i o n S i b e r i a n Snakes a r e v e r y a t t r a c t i v e s o l u t i o n s f o t a v o i d i n g d e p o l a r i ­

z a t i o n of p r o t o n beams d u r i n g a c c e l e r a t i o n to h i g h e n e r g i e s . H o w e v e r , t h e r e c o u l d s t i l l be

a n upper l i m i t i n e n e r g y , a b o u t 1 - 1 0 T e V , where d e p o l a r i z a t i o n r e s o n a n c e s w o u l d becone so

s t r o n g t h a t S i b e r i a n Snakes become i n e f f i c i e n t .

- (vi -

POLARIZATION OF ELECTRONS I S STORAGE RINGS

The behaviour of e l e c t r o n beams in h igh-energy s t o r a g e r i n g s i s very tír*erent fron

proton beams. This i s due to synchrotron r a d i a t i o n vhich causes f l u c t u a t i o n s and damping

in p a n i c l e o s c i l l a t i o n s . For i n s t a n c e , the emi t tance of an e l e c t r o n beam i s f u l l y d e t e r ­

mined hy synchrotron r a d i a t i o n . On the c o n t r a r y , emi t tance of a proton beam depends on i t s

v a l u e a t i n j e c t i o n .

The same d i f f e r e n c e between e l e c t r o n s and protons appears in p o l a r i z a t i o n . Protons

must be i n j e c t e d p o l a r i z e d and f i n a l p o l a r i z a t i o n i s at most equal to i t s i n i t i a l v a l u e .

E l e c t r o n s become t r a n s v e r s e l y p o l a r i s e d in s i t u and do not need to be i n j e c t e d p o l a r i z e d .

Th i s i s due to a p o l a r i z i n g e f f e c t of synchrotron r a d i a t i o n . On the other hand, f l u c t u a ­

t i o n s and damping induced by synchrotron r a d i a t i o n a l s o causes d e p o l a r i z a t i o n vh i ch may

even o v e r c o c e the p o l a r i z i n g e f f e c t a t h igh e n e r g i e s . Correc t ion procedures are needed for

r e d u c i n g t h i s d e p o l a r i z a t i o n . F i n a l l y , l o n g i t u d i n a l l y p o l a r i z e d e l e c t r o n s are more i n t e ­

r e s t i n g in c o l l i d i n g beam e x p e r i m e n t s , and some s p i n manipulat ion i s needed for changing a

t r a n s v e r s e p o l a r i z a t i o n i n t o a l o n g i t u d i n a l one at inLerac t ion p o i n t s .

I I I . l Sokolov-Ternov p o l a r i z i n g e f f e c t

E l e c t r o n s , s t o r e d in a r ing , r a d i a t e in the magnetic f i e l d of bending magnets . The

s y n c h r o t r o n r a d i a t i o n power i s g i v e n by the c l a s s i c a l e x p r e s s i o n :

where r g i s the e l e c t r o n c l a s s i c a l radius and p the bending radius of t r a j e c t o r y in the

magnet ic f i e l d .

There are a l s o quantum a s p e c t s in synchrotron r a d i a t i o n . Their magnitudes depend on 3 c 1

the r a t i o Um^/E of the emi t ted-photon c r i t i c a l ¿netgy = 2 p T t o e l e c t r o n energy E.

Th i s r a t i o s c a l e s l i k e E2 / p and i s of the oi'Jer o t 10 6 for a l l the s t o r a g e r i n g s . Quantum

e f . ' e c t s are then s m a l l . However, t h e i r .ma l lnes s can be compensated by the very high ra ­

d i a t i o n r a t e . This happens for some spin e f f e c t ? .involved in quantum e m i s s i o n of s y n c h r o ­

tron r a d i a t i o n .

Vhen an e l e c t r o n e m i t s a photon, i t s s p i n s t a t e may e i t h e r not change ( n o n - s p i n - f l i p

e m i s s i o n ) or be reversed ( s p i n - f l i p e m i s s i o n ) . Moreover, the p r o b a b i l i t y ?f e m i s s i o n de ­

pends on the i n i t i a l e l e c t r o n s p i n s t a t e : e i t h e r "up" ( t ) , i . e . p a r a l l e l to the f i e l d , or

"down" ( 4 - ) , i . e . a n t i p a r a l l e l to the f i e l d . This g i v e s an asymmetry in the r a d i a t e d power

between t h e s e tvo s p i n s t a t e s .

The n o n - s p i n - f l i p asymmetry in the radiated pover oí the ivo s t a t e s i s s n a i l

u T T - u i J

h « c

On the o ther hand the s p i n - f l i p asymmetry i s q u i t e l a r g e :

a l t h o u g h i t s r e l a t i v e i n t e n s i t y i s very low :

-, U 4 r ~ — " 3

Consequently i the s p i n - f l i p t r a n s i t i o n from "down" s t a t e to "up" s t a t e i s very rare

and the "down" s t a t e g r a d u a l l y becomes more populated . The e l e c t r o n beam i s p o l a r i z e d i n

the d i r e c t i o n a n t i p a r a l l e l to the magnetic f i e l d which in fac t corresponds to the l o w e s t

energy s t a t e . (The e l e c t r o n magnetic moment i s then p a r a l l e l to the f i e l d ) . This p o l a r i ­

z i n g mechanise i s c a l l e d the Sokolov-Ternov e f f e c t . In the same way a p o s i t r o n beam i s

p o l a r i z e d in the d i r e c t i o n p a r a l l e l to the f i e l d . However the p o l a r i z a t i o n bu i ld -up r a t e

i s low a s compared to the t o t a l r a d i a t i o n r a t e , n e v e r t h e l e s s u s u a l l y much f a s t e r than the

r a t e of p a r t i c l e l o s s e s in the s t o r e d beam, and p o l a r i z a t i o n can be observed i f one w a i t s

a s u f f i c i e n t time a f t e r beam i n j e c t i o n .

I I I . 2 P o l a r i z a t i o n b u i l d - u p in magnetic f i e l d s

Let us s tudy the p o l a r i z a t i o n b u i l d - u p f i r s t in a s t o r a g e r ing with uniform magnetic f i e l d and then with a non-uniform magnetic f i e l d .

I I I . 2 . 1 Uriiforra_magnetic_f i e l d

According to formula (1.2.2) the popu la t ions N of the two s p i n s t a t e s can be

wr i t t e n :

N * = 5 0 ± P > . ( I I I . 2.1)

v h e r e N = N *• N i s the t o t a l number of s tored e l e c t r o n s and p ;c the degree of p o l a r i z a ­

t i o n .

The asymmetry A in the t r a n s i t i o n r a t e s X from t h e s e two spin s t a t e s i s w r i t t e n :

X - X_

from where :

A = j U±A) , ( I I I . 2 . 2 )

vi th A = X+ + X .

Now, the r a t e s of change in t h e i r populat ion are g i v e n by :

dN dN

U s i n g e q u a t i o n s ( I I I . 2 . 1 ) and ( I I I . 2 . 2 ) one o b t a i n s :

d N . NX

* ar • - r ™ • a n d o n e d e r i v e s t h e p o l a r i z a t i o n r a t e :

By i n t e g r a t i o n , a s s u m i n g no p o l a r i z a t i o n a t t i n e t = 0 , t h e e v o l u t i o n o f p o l a r i z a t i o n i s

P < t ) - A < e " M - 1 ) .

T h i s e q u a t i o n o f e v o l u t i o n g i v e s t h e u l t i m a t e d e g r e e o f p o l a r i z a t i o n :

| P ( " ) | - A . JL . 0 . 9 2 3 7 6 ( I I I . 2 . 3 ) 5 j 3

w h e r e X i s t h e Compton w a v e l e n g t h d i v i d e d by 2rt .

I n a s t o i a g e r i n g w i t h s t r a i g h t s e c t i o n s , i n w h i c h e l e c t r o n s do n o t r a d i a t e , t h e

p o l a r i z a t i o n t i m e i s l o n g e r by t h e r a t i o c£ t h e r i n g a v e r a g e r a d i u s R t o t h e b e n d i n g

r a d i u s p i n m a g n e t s . N u m e r i c a l l y :

t ( s ) = 9 8 . 6 6 p i ( m ) R ( m ) - ( I I I . 2 . 4 ) _ ^ E 5 ( G e V )

T h e u l t i m a t e d e g r e e o f p o l a r i z a t i o n i s h i g h , a l t h o u g h s l i g h t l y l o v e r t h a n 1 0 0 X d u e

t o t h e r e s i d u a l s p i n - f l i p p r o b a b i l i t y f rom "down" s t a t e t o " u p " s t a t e .

T h e p o l a r i z a t i o n t i m e d e c r e a s e s v e r y r a p i d l y when e n e r g y i s i n c r e a s e d . T h i s i s d u e t o

t h e v e r y f a s t i n c r e a s e o f r a d i a t i o n r a t e w h i c h c o m p e n s a t e s t h e s m a l l i n t e n s i t y o f s p i n -

f l i p t r a n s i t i o n .

T h i s p o l a r i z a t i o n b u i l d - u p by t h e S o k o l o v - T e r n o v e f f e c t has been o b s e r v e d i n a l l t h e

e l e c t r o n s t o r a g e r i n g s w h e r e i t has been s o u g h t . T h e f o l l o w i n g t a b l e g i v e s f o r some o f

t h e s e r i n g s t h e maximum e n e r g y E a t w h i c h p o l a r i z a t i o n has been o b s e r v e d , t h e p o l a r i z a t i o n

I S I I CESR PETHA

5 4 . 7 1 6 . 5

t ( m i n ) 7 0 1 6 0 15 AO 4 3 0 0 18

P X 9 0 9 0 > 7 0 8 0 80 3 0 * 6 0 - 8 0 *

VEPP2-M AC0 SPEAR VEPP4

. 6 2 5 . 5 3 6 3 . 7 5

7 0 1 6 0 15 4 0

9 0 9 0 > 7 0 8 0

* a f t e r 120 m i n . * * a f t e r o p t i m i z a t i o n

I l l . 2 - 2 _ N o n - u n i f o r m _ m a g n e t i ç _ f i e l d

T h e p o l a r i z a t i o n b u i l d - u p h a s been t h e o r e t i c a l l y s t u d i e d i n a g e n e r a l m a g n e t i c f i e l d

c o n f i g u r a t i o n . T h e k e y - p o i n t i s t h a t t h e S o k o l o v - T e r n o v e f f e c t i s a v e r y s l o u p r o c e s s com­

p a r e d t o s p i n p r e c e s s i o n a b o u t t h e f i e l d , and a l s o c o n p a r e d t o f l u c t u a t i o n s i n s p i n p r e ­

c e s s i o n i n d u c e d by quantum e m i s s i o n s and s p i n - o r b i t c o u p l i n g . T h e r e f o r e s p i n a l i g n m e n t by

t h e S o k o l o v - T e r n o v e f f e c t c a n o n l y , on a l o n g t i m e i n t e r v a l , a p p e a r a l o n g a d i r e c t i o n

w h i c h i s s t a b l e a g a i n s t f a s t s p i n p r e c e s s i o n and i t s random f l u c t u a t i o n s . T h e o n l y such

d i r e c t i o n i s t h e s p i n c l o s e d s o l u t i o n ñ ( s ) . P o l a r i z a t i o n i s b u i l t up a l o n g n ( s ) v h i c h

t h e n i s t h e e q u i l i b r i u c s p i n d i r e c t i o n .

A t f i r s t , one can c o n s i d e r o n l y t h e c a s e where s p i n p r e c e s s i o n f l u c t u a t i o n s g i v e n e ­

g l i g i b l e eifects on a v e r a g e , i . e . w h e r e t h e d e p o l a r i z i n g e f f e c t s p r o d u c e d a r e n e g l i g i b l e .

T h i s s i t u a t i o n i s e n c o u n t e r e d e i t h e r a t low e n e r g i e s , f a r away f rom d e p o l a r i z a t i o n r e s o ­

n a n c e s , o r a t h i g h e r e n e r g i e s when t h e s t o r a g e r i n g i s s u f f i c i e n t l y s p i n - t r a n s p a r e n t . A

q u a n t i t a t i v e c a l c u l a t i o n g i v e s t h e n t h e e x p r e s s i o n s o f t h e u l t i m a t e p o l a r i z a t i o n d e g r e e

P ( Π) and o f t h e p o l a r i z a t i o n t i m e T :

5J3 1"'6> (III.2.5)

^ - ^ ' - c ' . ' 5 < I - - J | [l- S <»->']> (III.2.6)

- {i~S -

where $ i s a uni t vec tor along the re ference o r b i t and £ a unit vector along the t r a n s ­v e r s e f i e l d component. The brackets < > i n d i c a t e an average along the ring c i rcumference , and the a b s o l u t e value of bending radius p i s taken for inc lud ing a l l p o s s i b i l i t i e s of f i e l d o r i e n t a t i o n .

One could imagine i n c r e a s i n g the p o l a r i z a t i o n by

i n c r e a s i n g fi-n in tht denominator of formula ( I I I . 2 , 5 ) .

Hovever, n cannot be s imul taneous ly p a r a l l e l to B and 6 which are or thogona l . Therefore maximum p o l a r i z a t i o n i s

obta ined when n.Ê i s l a r g e and B.n sma l l . The l a t t e r can­

not g i v e core than a very few percent i n c r e a s e of p o l a r i z a t i o n - In most cases non-uniform

f i e l d s w i l l then g i v e a lower p o l a r i z a t i o n than the maximum 92 .4 X expected in a planar

r i n g , as they are not p a r a l l e l to the equi l ibr ium spin d i r e c t i o n n everywhere.

One n o t i c e s in formula ( I I I . 2 . 5 ) that magnets in which the f i e l d i s orthogonal to ñ

do not c o n t r i b u t e to the p o l a r i z a t i o n . This i s s o because the s p i n - f l i p asymmetry in syn

chrotron r a d i a t i o n v a n i s h e s for a sp in d i r e c t i o n orthogonal to the f i e l d . However, s p i n -

f l i p p r o b a b i l i t y does not vanish and becomes d e p o l a r i z i n g in these magnets.

U i g g l e r s with a l t e r n a t i n g s i g n of magnetic f i e l d a l s o g i v e no c o n t r i b u t i o n to p o l a r i ­

z a t i o n . However, asymmetric w i g g l e r s , with higher f i e l d of one s i g n , can g i v e a l a r g e con­

t r i b u t i o n due to the s t r o n g p o l a r i z a t i o n dependence on the f i e l d i n t e n s i t y ( the p ^ f a c ­

tor in formula I I I . 2 . 5 ) . They w i l l a l s o speed-up the p o l a r i z a t i o n (formula I I I . 2 . 6 ) a s

f ore seen in LEP at 50 GeV where the p o l a r i z a t i o n time i s too long due to the small f i e l d

i n t e n s i t y in the arc magnets ( F i g . 10 ) .

I I I . 3 Resonant sp in d i f f u s i o n

The energy jump caused by a quantum emiss ion of synchrotron r a d i a t i o n in a bending

magnet e x c i t e s a synchrotron o s c i l l a t i o n and a l s o a h o r i z o n t a l or v e r t i c a l betatron o s c i l ­

l a t i o n i f the h o r i z o n t a l or v e r t i c a l d i s p e r s i o n does not vanish in that magnet.

In an i d e a l l y planar r ing , only hor izonta l o s c i l l a t i o n s are produced, n e g l e c t i n g angular d i s t r i b u t i o n of the emitted photons at high e n e r g i e s . The s tored beam i s f l a t , p o l a r i z e d in the v e r t i c a l d i r e c t i o n , and exper iences only v e r t i c a l l y bending f i e l d s . There i s no per turbat ion of sp in motion and the ring i s complete ly s p i n - t r a n s p a r e n t .

Hovever, in a rea l r ing with imperfec t ions and p o s s i b l y with v e r t i c a l bends, v e r t i ­ca l o s c i l l a t i o n s are produced a l s o and the equi l ibr ium spin d i r e c t i o n n i s not v e r t i c a l a l l around the r i n g . Then, the quantum e x c i t a t i o n of v e r t i c a l and h o r i z o n t a l betatron o s c i l l a t i o n s , as we l l a s synchrotron o s c i l l a t i o n s , perturbs the sp in-mot ion , v ia the s p i n -o r b i t c o u p l i n g . The sp in- transparency of the r ing i s des troyed .

Within a few damping times fo l lowing a quantum e m i s s i o n , the e x c i t e d synchrotron and be ta tron o s c i l l a t i o n s w i l l disappear aga in , and the p a r t i c l e w i l l be l e f t with a modif ied s p i n o r i e n t a t i o n . This perturbat ion becomes very s i g n i f i c a n t on resonance, i . e . when the s p i n tune f u l f i l s a resonant c o n d i t i o n (formula 1 . 6 . 1 ) corresponding to a perturbing f i e l d d r i v e n by the e x c i t e d o s c i l l a t i o n s .

The r e s u l t of s u c c e s s i v e random quantum e m i s s i o n s i s a resonant random d i f f u s i o n qf

s p i n v e c t o r s away from the e q u i l i b r i u m spin d i r e c t i o n . This sp in d i f f u s i o n competes with

the s p i n a l ignment of the Sokolov-Ternov e f f e c t and l eads to an e q u i l i b r i u m degree of

p o l a r i z a t i o n l o v e r than the u l t i m a t e va lue g iven by formula ( I I I . 2 . 5 ) .

I t i s v o : t h n o t i n g that s p i n d i f f u s i o n i s analogous to p a r t i c l e d i f f u s i o n in phase

s p a c e induced by quantum e m i s s i o n s . Fo l lowing t h i s point of view, the Sokolov-Ternov

e f f e c t l ooks l i k e a damping mechanism analogous to the damping of p a r t i r l e o s c i l l a t i o n s .

Tne e q u i l i b r i u m p o l a r i z a t i o n i s then analogous to the equ i l ibr ium beam e m i t t a n c e .

I t must be emphasized that the d i s c r i m i n a t i o n between sp in d i f f u s i o n and s p i n a l i ­

gnment i s on ly made p o s s i b l e by the very d i f f e r e n t c h a r a c t e r i s t i c times of quantum emis ­

s i o n ( < 1 0 ~ l J s ) , o s c i l l a t i o n damping ( 1 0 - î - l(T"'s) and p o l a r i z a t i o n bu i ld -up ( > 1 0 ' s ) .

I I I . 4 S p i n - o r b i t c o u p l i n g v e c t o r and e q u i l i b r i u m p o l a r i z a t i o n

Spin d i f f u s i o n i s c h a r a c t e r i z e d by the rate N of

quantum e m i s s i o n s and by the d e v i a t i o n :

d r ,

of the s p i n v e c t o r S* away from tfie spin c l o s e d s o l u t i o n n,

produced by one energy jump ^ and measured a f t e r danping

of the e x c i t e d o s c i l l a t i o n s . In a f i r s t order and l i n e a r

approx imat ion , t h i s d e v i a t i o n i s proport iona l to and

the d e v i a t i o n d*(s) per un i t of energy v a r i a t i o n i s c a l l e d t\

v e c t o r " • I t depends on the d e t a i l s of r ing o p t i c s and, in genera l ,

s of the quantum emiss ion around the l i n g .

I f

e "sp in -orb i t

v a r i e s v i i h t'

up l ing

azinjuth

Due to s p i n d i f f u s i o n , the average decrease of sp in component a long the e i u f l i b r i u

d i r e c t i o n n i s per un i t time :

1 n _ 1 S~'dT~ = 2

^ • a t ^ T o V 1 ^ '

vhere the average < > i s taken over the energy jump 6E/E and the a - ' n u l h s around the

c i r c u m f e r e n c e . This i s a l s o the rate of d e p o l a r i z a t i o n by s p i n d: . i s ion, vh ich i s in

ba lance with the p o l a r i z a t i o n rate T 1 of the Sokolov-Ternov e f f e c t eading to an appro­

ximate p o l a r i z a t i o n d e c r e a s e :

s p / p - W TS < I 3 I ! >

More p r e c i s e l y the e q u i l i b r i u m degree ? of p o l a r i z a t i

Kondratenko formula :

P = J L < | P " : I o.(r)-d*)>

= 5 p ' ÜTJip|(».Ä)' . » |í|'j> ' v h i c h r e p l a c e s formula ( I I I . 2 . 5 ) when sp in d i f f u s i o n i s ta'-

ven by the Derbenev-

in to account .

T h e o r i g i n o f t h e l i n e a r d* t e r m i n t h e n u m e r a t o r o f t h i s f o r m u l a ( I I I . 4 . 1 ) i s n o t

d i s c u s s e d h ^ r e . I t i s n o r m a l l y a s m a l l t e r m a s t h e s p i n - o r b i t c o u p l i n g v e c t o r must be much

s m a l l e r t h a n u n i t y f o r o b t a i n i n g a h i g h d e g r e e o f p o l a r i z a t i o n .

I n a s m a l l - a a p l i t u d e l i n e a r m o d e l o f o r b i t a l m o t i o n , t h e r e s u l t o f a f i r s t - o r d e r c a l ­

c u l a t i o n o f t h e s p i n - o r b i t c o u p l i n g v e c t o r d*(s) i s :

3 ( s ) = - I m [ ( in + i ? > * ( i x , û x t û ? * £ 2 • 4 S » o _ s > ] , ( I I I . 4 . 2 )

w h e r e m a n d (' a r e two o r t h o g o n a l u n i t v e c t o r s , s o l u t i o n s o f t h e s p i n m o t i o n and o r t h o g o n a l

t o t h e s p i n c l o s e d s o l u t i o n . Ä r e p r e s e n t t h e cot ± x , » z , + s

c a l b e t a t r o n a n d s y n c h r o t r o n o s c i l l a t i o n s r e s p e c t i v e l y

, < s ) ( r a * 1 ) e T » l t ( v 1 U 7> ' 6 „

( I I I . 4 . 3 . a )

. _ ( r a - 1 ) e

w h e r e D , D ' a r e t h e d i s p e r s i o n and i t s s l o p e , and * x

c h r o t i o n o s c i l l a t i o n s -

n n . i . i b )

i s t h e p l iase o f b e t a t r o n o r s y n -

T h e f i r s t f a c t o r i n û i s a r e s o n a n t f a c t o r w h i c h f o l l o w s f r o m p i l i n g - u p o f s u c c e s s i v e

p e r t u r b a t i o n s o f s p i n m o t i o n , t u r n by t u r n d u r i n g o s c i l l a t i o n s . I t c a u s e s t h e s p i n - o r b i t

c o u p l i n g v e c t o r t o become v e r y l a r g e w h e n e v e r t h e beam e n t r g v i s such t h a t t h e s p i n t u n e v

a p p r o a c h e s any o f t h e v a l u e s :

k t Q ( k i n t e g e r ) ,

D e p o l a r i z a t i o n m a i n l y o c c u r s i n t h e v i c i n i t y o f t h e s e b e t a t r o n o r s y n c h r o t r o n r e s o n a n c e s .

T h e s e c o n d f a c t o r ( i n b r a c k e t s ) i n f o r m u l a ( I I I . 4 . 3 . a ) e x p r e s s e s t h a t b e t a t r o n

o s c i l l a t i o n s and t h e i n d u c e d p e r t u r b a t i o n s o f s p i n m o t i o n a r e p r o p o r t i o n a l t o t h e

d i s p e r s i o n i t t h e a z i m u t h w h e r e quantum e m i s s i o n o c c u r s . T h i s q u a n t i t y i s f a m i l i a r f o r

c a l c u l a t i n g t h e beam e m i t t a n c e . For v e r t i c a l b e t a t r o n o s c i l l a t i o n s , i t d i f f e r s f r o n z e r o

o n l y i n p r e s e n c e o f v e r t i c a l o r b i t d i s t o r t i o n s .

But t h e r e a l k e y s f o r o b t a i n i n g h i g h p o l a r i z a t i o n a r e t h e s p i n - o r b i t c o u p l i n g i n t e ­

g r a l s t h a t a p p e a r i n ( I I I . 4 . 3 ) as t h e l a s t f a c t o r J z r ( s ) :

s + c

(m i i §

ih • è, ï R e * às'

s-tc

( n • it) K Jß

- h?8 -

* In the i n t e g r a l J&t the synchrotron phase fac tor e has been n e g l e c t e d , assuming a

very smal l synchrotron tune (Q £ « 1 ) .

where c i s the r ing c i rcumference , K ( s ' ) i s the quadrupel* s t r e n g t h and e the un i t vec

tor in r a d i a l or v e r t i c a l d i r e c t i o n . For each type of o s c i l l a t i o n , the corresponding i n t e ­

g r a l i s p r o p o r t i o n a l to t^e e f f e c t i v e s p i n r o t a t i o n away from the e q u i l i b r i u m d i r e c t i o n .1

dur ing one r e v o l u t i o n around the r ing , s t a r t i n g a ' the azimuth s of the quantum e m i s s i o n .

The s c a l a r product * m * i / ) . e z v a n i s h e s in a

planar r ing without magnet e r r o r s , where n ( s ' ) = & z

everywhere . In a r e a l r i n g with v e r t i c a l al ignment

e r r o r s of quadrupo les , however, the beam w i l l be sub­

j e c t e d to small v e r t i c a l k i c k s which may cause the

e q u i l i b r i u m s p i n d i r e c t i o n to d e v i a t e from the v e r t i ­

c a l . In t h i s c a s e , the c o u p l i n g i n t e g r a l s and J g do

not v a n i s h and g i v e a f i n i t e c o n t r i b u t i o n from r a d i a l

b e t a t r o n and synchrotron o s c i l l a t i o n s to d e p o l a r i ­

z a t i o n .

S i m i l a r l y , the v e r t i c a l d i s p e r s i o n caused by v e r t i c a l o r b i t d i s t o r t i o n s v i } , ?

f i n i t e c o n t r i b u t i o n from v e r t i c a l be ta tron and synchrotron o s c i l l a t i o n s to d e p o l a i ! . -i'>n,

s i n c e the coupl ing i n t e g r a l s J ? and J g c o n t a i n i n g the s c a l a r product ( • +• i i ) - ^ do not

v a n i s h in g e n e r a l .

A computer code <named SLIH) has been w r i t t e n for c a l c u l a t i n g the s p i n - o r b i t c o u p l i n g

v e c t o r d* and the e q u i l i b r i u m p o l a r i z a t i o n in a r ing with g i v e n i m p e r f e c t i o n s and v e r t i c a l

bends . I t uses an 8 x 8 matrix formalism vhich i n c l u d e s s p i n motion in a d d i t i o n to t r a n s ­

v e r s e and l o n g i t u d i n a l mot ions . The matrix formalism i s based on l i n e a r i z a t i o n of the

o r b i t a l motion for smal l ampl i tudes . This l i n e a r formalism cannot account for n o n - l i n e a r

r e s o n a n c e s .

l u c o n c l u s i o n , the s p i n - o r b i t coupl ing v e c t o r d* becomes l a r g e on resonance where po-

l a t i z a t i o n i s s t r o n g l y reduced. It s c a l e s l i n e a r l y with beam energy , s i n c e the s p i n pre­

c e s s i o n i n c r e a s e s f a s t e r than the p a r t i c l e v e l o c i t y r o t a t i o n when ei.ergy i s increased ( s e e

the ya+1 f a c t o r in formulae I I I . 4 . 3 ) . F i n a l l y i t s amplitude i s determined by the s p i n -

o r b i t c o u p l i n g i n t e g r a l s .

Figure 11 shows an exper imenta l scan of p o l a r i z a t i o n as a func t ion of bean energy in

the SPEAR s t o r a g e r i n g , vh ; ,ch shows s e v e r a l d e p o l a r i z a t i o n r e s o n a n c e s , in p a r t i c u l a r non­

l i n e a r o n e s . Between t h e s e resonances p o l a r i z a t i o n near ly reaches the maximum p o l a r i z a t i o n

( 9 2 . ¿ X) for a planar r i n g .

8 8-0, -Q, 3.Q, 3 -0 , 2 . 0 , - 0 ,

j .Q s . 2 Q t | j .20 , .0 , j -Qs .2Q t í P/P0

F i g . 11 R e l a t i v e p o l a r i z a t i o n {P„ = 9 2 . v e r s u s beam energy in SPEAR. Occurrence of resonance tune v a l u e s (R +k Q +k 0 +k 0 ) are i n d i c a t e d above the f i g u r e , in p a r t i c u l a r synchrotron sSteïlïtes f±0^ tnd ± 2 0 g ) of v*8 and v=3+Q x r e s o n a n c e s .

I I I . 5 Spin matching

In a r e a l r i n g , at high e n a r g i e s (> 5 CeV), the s p i n - o r b i t c o u p l i n g v e c t o r d* i s never n e g l i g i b l e , even o u t s i d e r e s o n a n c e s , and p o l a r i z a t i o n i s lower than a l lowed by the Sokolov -Ternov e f f e c t . The main problem i s to reduce t h i s d e p o l a r i z a t i o n for o b t a i n i n g a u s e f u l l y h igh d e g r e e of p o l a r i z a t i o n . Procedures , g e n e r a l l y named s p i r matching, have been invented for d o i n g s o , and one of them has been s u c c e s s f u l l y a p p l i e d in the PETRA s t o r a g e r i n g .

I I I . 5 . 1 G l o b a l _ s p i n - m a t c h i n g

In p r i n c i p l e , accord ing to formulae ( I I I . 4 . 2 ) and ( 1 I J . 4 . J J , d* v a n i s h e s i f the f i v e .pin-orbit coupl ing i n t e g r a l s J + } { , J i Z f **s

a r e made to v a n i s h in every magnet. These i n t e ­g r a l s are p r o p o r t i o n a l to s p i n r o t a t i o n away from e q u i l i b r i u m spin d i r e c t i o n n as produced by r a d i a l , v e r t i c a l and synchrotron o s c i l l a t i o n s r e s p e c t i v e l y in one r e v o l u t i o n around the r i n g . When these i n t e g r a l s v a n i s h , the s p i n o r i e n t a t i o n i s again a long ñ a f t e r each turn. The r i n g i s p e r f e c t l y s p i n - t r a n s p a r e n t and no s p i n d i f f u s i o n can occur . A i l d e p o l a r i z a t i o n resonai c e s are suppressed at the same t ime.

However, such a g l o b a l spin-matchir .g cannot e a s i l y be achieved in any s i t u a t i o n . When s p i n - t r a n s p a r e n c y i s l a c k i n g due to i m p e r f e c t i o n s d i s t r i b u t e d a l l around the r i n g , one has ten c o n d i t i o n s to s a t i s f y at each magnet in the r ing ( t h e r e a l and imaginary par t s of each i n t e g r a l must be c a n c e l l e d out). The total number of c o n d i t i o n s ¡s ten times the number of

- MO -

m a g n e t s . M o r e o v e r , t h e v a l u e o f t h e J ^ and i n t e g r a l s depend on t h e a m p l i t u d e s o f ic i -

p e r f e c t i o n s w h i c h a r e n o r m a l l y unknown. For a l l t h e s e r e a s o n s g l o b a l s p i n - m a t c h i n g i s n o t

p r a c t i c a b l e i n g e n e r a l .

N e v e r t h e l e s s , i f l a c k o f s p i n - t r a n s p a r e n c y i s o n l y d u e t o a few v e r t i c a l b e n d s , s u c h

a s t h o s e o f s p i n r o t a t o r s ( s e e s e c t i o n M I . 6 ) , t h e s p i n - m a t c h i n g c o n d i t i o n s d e g e n e r a t e

i n t o a much s m a l l e r n u m b e r . T h e y can be a d d e d t o t h e o t h e r m a t c h i n g c o n d i t i o n s o f t h e beam

o p t i c s a n d o n e o n l y n e e d s a f e w more f o c u s i n g e l e m e n t s f o r s a t i s f y i n g t h e m .

T h e s p i n - o r b i t c o u p l i n g i n t e g r a l s depend on beam e n e r g y , s i n c e t h e (m if) c o m p l e x

v e c t o r an f o r m u l a e ( I I I . 4 . 4 . ) p r e c e s s e s p r o p o r t i o n a l l y t o beam e n e r g y . T h e r e f o r e s p i n -

m a t c h i n g c o n d i t i o n s a r e e n e r g y - d e p e n d e n t and s p i n - m a t c h i n g p r o c e d u r e must be r e p e a t e d a t

e a c h o p e r a t i n g e n e r g y .

I I 1 . 5 . 2 H a r m o n i c s p i n - m a t c h i n g

I n f a c t , t h e l a r g e s t c o n t r i b u t i o n t o d e p o l a r i z a t i o n comes f r o m t h e e x c i t a t i o n o f a

few r e s o n a n c e s c l o s e s t t o t h e s p i n t u n e , a s t h i s c o n t r i b u t i o n i s i n v e r s e l y p r o p o r t i o n a l t o

t h e s q u a r e d d i s t a n c e f r o m s p i n t u n e t o t h e top o f t h e r e s o n a n c e . I t w o u l d be e n o u g h t o

c o m p e n s a t e t h e s e n e a r b y r e s o n a n c e s , and t h e number o f c o n d i t i o n s t o f u l f i l w o u l d t h e n be

r e d u c e d . T h i s i s t h e o n l y p r a c t i c a l p o s s i b i l i t y f o r r e d u c i n g d e p o l a r i z a t i o n due t o r i n g

i m p e r f e c t i o n s and t h i s i s e s s e n t i a l a t h i g h e n e r g i e s w h e r e t h i s d e p o l a r i z a t i o n i s l a r g e .

S i m i l a r l y t o t h e c a s e o f p o l a r i z e d p r o t o n s ( s e e S e c t i o n I I . 2 ) , t h e s t r e n g t h o f a d e ­

p o l a r i z a t i o n r e s o n a n c e i s p r o p o r t i o n a l t o t h e s p i n r o t a t i o n c a u s e d by a r e s o n a n t component

o f t h e p e r t u r b i n g f ; e l d . I n t h e c a s e o f an e l e c t r o n r i n g w i t h i m p e r f e c t i o n s , t h i s p e r t u r ­

b i n g f i e l d i s p r o d u c e d by t h e o s c i l l a t i o n s f o l l o w i n g q u a n t u m e m i s s i o n s and i s p r o p o r t i o n a l

t o t h e i n t e g r a n d « y ( s > o f t h e s p i n - o r b i t c o u p l i n g i n t e g r a l J y ( s ) f o r e a c h t y p e o f

o s c i l l a t i o n ( y = + x , + z , s ) :

s + c

J ( s ) = w y ( s ' ) d s ' .

J s

T h e r e s o n a n t component i s t h e n o b t a i n e d by a f r e q u e n c y a n a l y s i s o f t h e p e r t u r b i n g

f i e l d a m p l i t u d e « y ( s ) . A c c o r d i n g t o f o r m u l a e ( I I I . 4 . 4 ) , w y has a 2 n ( v + Q y ) phase a d v a n c e

p e r t u r n * and i t s f r e q u e n c y d e c o m p o s i t i o n i s :

2 i f i ( \ > - p : Q ) s / c

« y ( s ) = I e p y e y ( p i n t e g e r )

- 2 i n ( v - p ± 0 ) s / c £ = ( ) w ( s ) e ' d s / c

p . y y

- T x z s

* (m + it} and e ' ' h a v e 2ir\) and ± 2r tQ x _ phase a d v a n c e r e s p e c t i v e l y ( w i t h o u t

n e g l e c t i n g t h e e phase f a c t o r i n J ) .

- t'S] -

Compensation of the v - p r resoné .ice c o n s i s t s of c a n c e l l i n g i t s s t r e n g t h ^. One has

on ly tvo c o n d i t i o n s to f u l f i l for each resonance to be compensated, as i s a complex

q u a n t i t y r e p r e s e n t i n g a perburbing f i e l d or thogonal to the e q u i l i b r i u m s p i n d i r e c t i o n n.

Th i s compensation can be r e a l i z e d e x p e r i m e n t a l l y by us ing c o r r e c t o r s vh ich counterac t r ing

i m p e r f e c t i o n s and vh ich are chosen to act on the nearby d e p o l a r i z a t i o n re sonances .

The concept of harmonic sp in-matching has been extended to any type of harmonic c o r ­

r e c t i o n used to o p t i m i z e the degree of p o l a r i z a t i o n . For i n s t a n c e the procedure s u c c e s s ­

f u l l y a p p l i e d in PETRA i s based on p o l a r i z a t i o n o p t i m i z a t i o n by varying some harmonics of

the v e r t i c a l c l o s e d - o r b i t d i s t o r t i o n . As exp la ined in s e c t i o n I I I . 4 , the s p i n - o r b i t cou­

p l i n g i n t e g r a l s and J g are s e n s i t i v e to any d e v i a t i o n of the e q u i l i b r i u m s p i n d i r e c t i o n

from the v e r t i c a l . Such a d e v i a t i o n i s due to v e r t i c a l c l o s e d - o r b i t d i s t o r t i o n , and in

p a r t i c u l a r to the d i s t o r t i o n harmonics c l o s e s t to the sp in tune , s i n c e the p e r t u r b i n g

f i e l d corresponding to t h e s e harmonics i s near ly in phase v i t h s p i n p r e c e s s i o n .

Exper imenta l ly e i g h t o r b i t c o r r e c t o r s have been used for vary ing the s i n e or the

c o s i n e component of the c l o s e s t harmonics (37 and 38 harmonics at 1 6 . 5 G E V ) . The

degree of p o l a r i z a t i o n was measured with a Po lar imeter and opt imized by vary ing the

ampl i tude of these harmonics components ( F i g . 1 2 ) .

pr

F i g . 12 P o l a r i z a t i o n P versus s i n e and c o s i n e compo­nents of the 38 v e r t i c a l c l o s e d - o r b i t har­monics in PETRA at 16 .5 GeV ( a l l q u a n t i t i e s in a . j i t r a r y u n i t s ) .

14ÜZ 61. Qz

6-3-Q,

16 4 16 5 16 6

This v a r i a t i o n in harmonic amplitude has no v i s i b l e

e f f e c t on the v e r t i c a l c l o s e d - o r b i t d i s t o r t i o n , as the

37**1 and 3 8 t harmonics are far away from the v e r t i c a l

b e t a t r o n tune (Q z = 2 3 . 3 ) , and have a very smal l ampli­

tude .

Th i s method needs a measurable degree of p o l a r i z a t i o n

a t the beg inn ing . For t h i s reason the c l o s e d o r b i t must

have p r e v i o u s l y been c a r e f u l l y correc ted by the usual cor ­

r e c t i o n procedures .

F i g . 13 P o l a r i z a t i o n P ( i n a r b i t r a r y u n i t s ) v e r s u s beam energy in PETRA a f t e r harmonic s p i n - m a t c h i n g . Maximum po lar i z a t ion i s about 60-80X. Arrows i n d i c a t e the energy l o c a t i o n of some d e p o l a r i z a t i o n r e ­sonances .

After a p p l y i n g harmonic s p i n matching at a c e r t a i n energy , the p o l a r i z a t i o n i s l a r ­

g e s t ( 6 0 - 8 0 Ï ) in a smal l range around t h i s energy ( F i g . 1 3 ) , and d e c r e a s e s vhen the energy

i s s h i f t e d , s i n c e the c o u p l i n g i n t e g r a l s J z s are energy-dependent through the s p i n

t u n e . Horeover, vhen approaching d e p o l a r i z a t i o n resonances r e s i d u a l va lues of t h e s e

i n t e g r a l s s t i l l l ead to l a r g e d e p o l a r i z a t i o n .

I l l - 6 Spin r o t a t o r s

The exper imenta l study of e l e c t r o v e a k e f f e c t s in e + e and e*p c o l l i s i o n s c a l l s for

l o n g i t u d i n a l l y p o l a r i z e d e l e c t r o n s of both h e l i c i t i e s at i n t e r a c t i o n point*. The impor­

tance of p o l a r i z a t i o n for t e s t i n g i n t e r a c t i o n models at very h igh e n e r g i e s (Ej> e am >

lb GeV) enhances the i n t e r e s t for p o l a r i z e d beams i n e l e c t r o n s t o r a g e r i n g s , which «^as

poor a t lower e n e r g i e s . However, e l e c t r o n s become t r a n s v e r s e l y p o l a r i z e d . i n s t e a d of

l o n g i t u d i n a l l y , by the Sokolov-Ternov e f f e c t . Several schemes {90° s p i n r o t a t o r s ) have

been proposed for r o t a t i n g the v e r t i c a l p o l a r i z a t i o n in the arcs of a s t o r a g e r ing i n t o a

l o n g i t u d i n a l p o l a r i z a t i o n a t the i n t e r a c t i o n p o i n t s -

Let us d e s c r i b e s c h e m a t i c a l l y the s p i n r o t a t o r which w i l l be b u i l t for the ep HERA

c o l l i d e r , i t f o l l o w s from a compromise between the geometr ica l and o p t i c a l c o n s t r a i n t s of

t h i s r i n g in one hand, and the need for a h igh degree of p o l a r i z a t i o n on the - i ther. Again ,

the e x i s t e n c e of a good compromise i s an i l l u s t r a t i o n of the r e l a t i v e freedom in s p i n

m a n i p u l a t i o n s a l lowed by the s p i n - o r b i t coupl ing ( s e e S e c t i o n 1 - 4 ) .

F i g . 14 Sketch of a HERA mini r o t a t o r pair (arrows i n d i c a t e s p i n d i r e c t i o n )

A p a i r of 90° s p i n r o t a t o r s , of a s o - c a l l e d "mini r o t a t o r " type , w i l l be i n s t a l l e d

around each of the i n t e r a c t i o n r e g i o n s , which turns th« s p i n v e c t o r , a f t e r l e a v i n g the

a r c , i n t o the beam d i r e c t i o n and then back i n t o the v e r t i c a l before e n t e r i n g the f o l l o w i n g

a r c . Each mini r o t a t o r c o n s i s t s of t h r e e h o r i z o n t a l l y bending magnets, i n t e r l e a v e d with

t h r e e v e r t i c a l l y bending magnets which superimpose a v e r t i c a l beam bump to the h o r i z o n t a l

beam d e f l e c t i o n ( F i g . 1 4 ) . A pair of such mini r o t a t o r s i s symmetric in the h o r i z o n t a l

p lane v i t h r e s p e c t to the i n t e r a c t i o n p o i n t , but ant isymmetric in the v e r t i c a l p l a n e . Both

h e l i c i t i e s are obta ined by i n v e r t i n g the s i g n of the v e r t i c a l beaa bump in each mini

r o t a t o r .

- b85 -

The mini r o t a t o r i s l a i d out for a chosen d e s i g n energy ( 2 9 . 8 GeV) at which i t s i m u l ­

t a n e o u s l y p r o v i d e s the c o r r e c t h o r i z o n t a l beam geometry and s p i n r o t a t i o n ( F i g . 1 5 ) . When

v a r y i n g the bean energy , h o r i z o n t a l bean geometry i s « a t n t a í n e d by ramping the h o r i z o n t a l

r o t a t o r magnets in s y n c h r o n i s e wi th a l l the o ther r ing magnets. For keeping the s p i n d i ­

r e c t i o n v e r t i c a l in the a r c s and l o n g i t u d i n a l at i n t e r a c t i o n p o i n t s , one needs two para­

meter s to vary'- une i s the ampl i tude of the v e r t i c a l bean bump i n the mini r o t a t o r ; the

o t h e r i s the ampl i tude of a superimposed h o r i z o n t a l beam bump vh ich v a n i s h e s at the d e s i g n

e n e r g y . The mini r o t a t o r can thus be operated in a range g o i n g from 27 GeV to about

35 GeV. The i n c r e a s e o f energy from the l o v e r to the upper l i m i t of t h i s range has the

advantage of reducing the p o l a r i z a t i o n t ime from AO min to 12 min, and a l s o to i n c r e a s e

from 80% t o 8f>X the maximum degree of p o l a r i z a t i o n a l lowed by the Sokolov-Ternov e f f e c t .

56 m

F i g . 15 S i m p l i f i e d s k e t c h of the mini r o t a t o r geometry . H : h o r i z o n ­t a l l y bending magnets, V : v e r t i c a l l y bending magnets (arrows i n d i c a t e s p i n d i r e c t i o n ; m o d i f i c a t i o n s required by the head-on ep c o l l i s i o n scheme and superimposed h o r i z o n t a l beam bump are not shown)

In p r i n c i p l e r o t a t o r p a i r s , which are ant i symmetr ic i n both the h o r i z o n t a l and v e r ­

t i c a l p l a n e s , would have the advantage of r e s t o r i n g the v e r t i c a l s p i n d i r e c t i o n in the

a r c s a t any e n e r g y , s i n c e the s p i n t rans format ion , in an ant i symmetr ic r o t a t o r p a i r , i s

the i d e n t i t y whatever the energy . However, such ant i symmetr ic schemes have the drawback of

e i t h e r a s m a l l e r maximum degree of Sokolov-Ternov p o l a r i z a t i o n , or a more space-consuming

geometry , and were t h e r e f o r e not chosen for HERA.

The most important property c h a r a c t e r i z i n g a r o t a t o r i s i t s e f f e c t s on p o l a r i z a t i o n .

The mini r o t a t o r as chosen r e s u l t s from a min imizat ion of i t s d e p o l a r i z a t i o n , for an

a c c e p t a b l e l e n g t h o f about 56 m.

The f i r s t e f f e c t i s a reduc t ion of the Sokolov-Ternov degree of p o l a r i z a t i o n , as the

s p i n d i r e c t i o n i s not a n t i p a r a l l e l to the f i e l d in the r o t a t o r magnets ( s e e formula

I I I . 2 . 5 ) . For a g i v e n s p i n r o t a t i o n in t h e s e magnets, the reduc t ion i s i n v e r s e l y propor­

t i o n a l to the squared bending radius and s c a l e s l i k e the i n v e r s e of the squared r o t a t o r

l e n g t h . The sp in r o t a t i o n a n g l e s in these magnets and t h e i r l e n g t h s have been chosen for

min imiz ing t h i s d e p o l a r i z i n g e f f e c t , t oge ther with other minor a s p e c t s . D e p o l a r i z a t i o n

actounts to o n l y BX a t 35 GeV for e i g h t 56m-long mini r o t a t o r s in HERA, j u s t i f y i n g the name

of "mini r o t a t o r " .

- 68-J -

The second e f f e c t i s a breakdown of s p i n - t r a n s p a r e n c y . The r o t a t o r s g e n e r a t e non-

v a n i s h i n g s p i n - o r b i t c o u p l i n g i n t e g r a l s and lead to l a r g e d e p o l a r i z a t i o n by s p i n d i f f u ­

s i o n . A g l o b a l s p i n - m a t c h i n g procedure has been a p p l i e d for r e s t o r i n g s p i n - t r a n s p a r e n c y

( s e e S e c t i o n I I I . 5 . 1 ) . The r e l a t i v e l y smal l l e n g t h of the mini r o ' a t o r s permits having no

f o c u s i n g e l e m e n t s in them. Thus the a r c s and the s t r a i g h t s e c t i o n s between r o t a t o r s can

s e p a r a t e l y be made s p i n - t r a n s p j r e n t . Only two c o n d i t i o n s for the a r c s and t h r e e c o n d i t i o n s

for the s t r a i g h t s e c t i o n s are r e q u i r e d . For i n s t a n c e , one c o n d i t i o n e x p r e s s e s the s p i n -

transparency for h o r i z o n t a l b e t a t r o n o s c i l l a t i o n in the s t r a i g h t s e c t i o n between r o t a t o r s

of a p a i r where the s p i n v e c t o r i s l o n g i t u d i n a l . The o v e r a l l s p i n r o t a t i o n about the ver

t i c a l , as produced by t h i s o s c i l l a t i o n , must vani sh in t h i s s e c t i o n . As only r o t a t i o n s

about the v e r t i c a l are i n v o l v e d h e r e , sp in and v e l o c i t y r o t a t i o n s are g l o b a l l y propor­

t i o n a l , and the s l c p e of a b e t a t r o n t r a j e c t o r y must be the same at the ends of t h i s s e c ­

t i o n . Th i s c o n d i t i o n i s a u t o m a t i c a l l y r e a l i z e d for a s i n e - l i k e b e t a t r o n t r a j e c t o r y due to

the o p t i c a l symmetry about the i n t e r a c t i o n po int ( IP) when the s t r a i g h t s e n i o n i s sym­

m e t r i c v i t h r e s p e c t to t h i s p o i n t . I t s u f f i c e s to impose t h i s c o n d i t i o n for a c o s i n e - l i k * 7

t r a j e c t o r y .

rotator cosins.like rotator

t r a j e c t o r y

A c a l c u l a t i o n ( w i t h the SLIM code) of the s p i n - o r b i t c o u p l i n g v e c t o r d* shows that the

e q u i l i b r i u m p o l a r i z a t i o n reaches obX at 35 GeV for HERA with four r o t a t o r p a i r s and s p i n -

t r a n s p a r e n t o p t i c s , and without i m p e r f e c t i o n s .

I I I . 7 Nonl inear d e p o l a r i z i n g e f f e c t s

Two n o n l i n e a r d e p o l a r i z i n g e f f e c t s are expected to be important a t high e n e r g i e s .

However, the in format ion a v a i l a b l e on them i s s c a r c e .

T i l . 7 . 1 D e p o l a r i z a t i o n enhancement by energy spread

Beam energy spread i n c r e a s e s q u a d r a t i c a l l y with energy and reaches a r . m . s . va lue of

about 50 HeV at 50 GeV. On the o t h e r hand, the spac ing between d e p o l a r i z i n g resonances i s

c o n s t a n t w i th energy and each p a r t i c u l a r type of resonance occurs r e p e a t e d l y wi th a

440 HeV s e p a r a t i o n .

P a r t i c l e s v i t h s u f f i c i e n t l y large energy d e v i a t i o n , i . e . with la synchrotron am

p l i t u d e , may ..ien approach a resonance energy even i f '.ne c e n t r a l beam energy i s s e t as

far a s p o s s i b l e away from nearby r e s o n a n c e s . Larger d e p o l a r i z a t i o n than e x p e c t e d for smal l

o s c i l l a t i o n s in l i n e a r theory can then occur .

However, t h i s p i c t u r e of a p a r t i c l e approaching a resonance i s not very c o n s i s t e n t a s

i t x i x e s the time domain and the frequency domain. A more c o r r e c t way of c o n s i d e r i n g t h i s

energy spread e f f e c t i s to take in to account the frequency modulation of s p i n p r e c e s s i o n

- Ö85 -

produced by synchrotron o s c i l l a t i o n s , as s p i n tune i s p r o p o r t i o n a l to energy . Th i s s i t u a ­

t i o n i s very s i m i l a r to the well-known e f f e c t of frequency modulation in RF-vaves . I t

l e a d s t o the appearance of s a t e l l i t e s around the c e n t r a l frequency. S i m i l a r l y for the s p i n

mot ion , synchrotron s a t e l l i t e s are generated around any d e p o l a r i z i n g resonance . These

s a t e l l i t e s are r e g u l a r l y spaced by the synchrotron tune 0 g . If the energy spread i s su f ­

f i c i e n t l y l a r g e , i n v o l v i n g l a r g e synchrotron a m p l i t u d e s , then s e v e r a l s a t e l l i t e s are

e x c i t e d and the energy range i n which d e p o l a r i z a t i o n by a p a r t i c u l a r resonance occurs i s

widened. This i s e q u i v a l e n t to say in g that p a r t i c l e s with l a r g e energy spread are

approaching resonance .

Already at 3 . 7 GeV in SPEAR h i g h r r - o r d e r synchrotron s a t e l l i t e s of a b e t a t r o n r e s o n ­

ance ( v = 3 * Qy) have been observed ( s e e F ig . 1 1 ) . More s a t e l l i t e s and larger d e p o l a r i ­

z a t i o n are expec ted at h igher e n e r g i e s . A n a l y t i c models p r e d i c t t h a t , at 50 GeV in LEP,

d e p o l a r i z a t i o n w i l l be enhanced by about a f a c t o r f i v e due to the energy spread e f f e c t .

The o n l y cure would be to reach higher e f f i c i e n c y in a c h i e v i n g s p i n - t r a n s p a r e n c y by - = r -

monic s p i n - m a t c h i n g .

I I I . 7 . 2 B e a m - b e a m d e p o l a r i z a t i o n

In s t o r a g e t i n g c o l l i d e r s , beam-beam i n t e r a c t i n i s r e s p o n s i b l e for beam blow-up

which l i m i t s the performances . Th i s beam-beam i n t e r a c t i o n should a l s o perturb s p i n mot ion ,

v i a s p i n - o r b i t c o u p l i n g . Due to the n o n - l i n e a r i t y of the space charge f i e l d , n o n - l i n e a r

d e p o l a r i z a t i o n resonances are expected to be e x c i t e d . Beam-beam d e p o l a r i z a t i o n should

p a r t i c u l a r l y be important a t the beam-beam U n i t .

E x p e r i m e n t a l l y , over 707. p o l a r i z a ­

t i o n has been observed in h i g h - l u m i n o ­

s i t y e*e" c o l l i s i o n s a t 2 x 3 . 7 GeV in

SPEAR, and has been used for h i g h - e n e r g y

physi cs exper iments at t h i s energy .

However, t h i s p o l a r i z a t i o n in c o l l i s i o n

mode could not be reproduced a t a l a t e r

s t a g e o f SPEAR development .

P o l a r i z a t i o n has a l s o been observed

in c o l l i s i o n mode at 16 .5 GeV i n PETRA.

Figure 16 shows the d e c r e a s e of p o l a r i z a ­

t i o n near tne beam-beam l i m i t where the

beams a r e blown-up.

Although the beam-beam d e p o l a r i z a ­

t i o n mechanism i s not v e i l unders tood ,

t h e s e s c a r c e exper imenta l r e s u l t s sup­

port a moderate optimism for the future

of e e or ep exper iments with p o l a r i z e d

oeams. p o i n t s .

R

10

D.5

Lz 2.6x10 L = 3.9x10

5 10 KmA)

Fig . 16 Rat io R of two-beam p o l a r i z a t i o n to s i n g l e - b e a m p o l a r i z a t i o n (= Q0Z) versus beam i n t e n s i t y I at 16 .5 GeV in PETRA. Luminosity L(cm~ s ) i s i n d i c a t e d for two

- 686 -

ACKNOULEDGEHENTS

1 a n i n d e b t e d t o J . P - K o u t c h o u k , B. V . Hont a g u e and K. G. S t e f f e n f o r c a r e f u l l y r e a ­

d i n g t h e f i n a l d r a f t o f t h i s p a p e r and f o r m a k i n g v a l u a b l e comments .

BIBLIOGRAPHY

For an e lementary i n t r o d u c t i o n to the concept of p o l a r i z a t i o n in p a r t i c l e b e a o s , s e e : J - S . BELL, Report CERN 75-11 ( 1 9 7 5 ) .

For a g e n e r a l review on p o l a r i z a t i o n in h igh-energy r i n g s , s e e : B.U. Montague, P h y s i c s Report s , Vol . 113 ( 1 9 8 4 ) .

For the s tudy of the a c c e l e r a t i o n of p o l a r i z e d protons in s y n c h r o t r o n s , s e e : E.D. Courant and R.D. Ruth . .Report BNL-51270 ( 1 9 8 0 ) . R.D. Ruth, Proc. of the 12 I n t . Conf. on High-Energy A c c e l e r a t o r s (Fermi lab , 1983) p. 286 .

For the s tudy of e l e c t r o n p o l a r i z a t i o n in s t o r a g e r i n g s , s e e : A.U. Chao in P h y s i c s of High-Energy P a r t i c l e A c c e l e r a t o r s (Fermilab Summer Schoo l , 1981) AIP Conf. Proc . n° 8 7 , p. 395 . A.V. Chao i n 19fi3 P a r t i c l e A c c e l e r a t o r Conference, IEEE Transac t ions on Nuclear S c i e n c e , Vol . 30, p . 2383 and SLAC-Pub 3081 ( 1 9 8 3 ) .

APPENDIX

A . 1 . 1 The p o l a r i z a t i o n ^ e c t o r b e h a v e s c l a s s i c a l l y .

In Quantutt Mechanics any pure s p i n s t a t e * of one p a r t i c l e i s c o n p l e t e l y s p e c i f i e d by a ket v e c t o r [<(«>, f o l l o w i n g the Dirac n o t a t i o n . The two "up" and "down" s p i n s t a t e s a long

the Oz a x i s form a b a s i s for the ket v e c t o r s of a sp in 1/2 p a r t i c l e - On t h i s b a s i s a kei

v e c t o r i s r epresented by a column v e c t o r with two components F and g, c a l l e d a s p i n o r :

I * - 0 where f and g are complex numbers, normalized such that [FL* + j g | = 1.

The two b a s i c "up" and "down" s t a t e s are represented by the s p i n o r s and r e s ­

p e c t i v e l y . According to the S u p e r p o s i t i o n P r i n c i p l e any s p i n s t a t e i s a l i n e a r s u p e r p o s i ­

t i o n of t h e s e b a s i c s t a t e s , represented by the l i n e a r s u p e r p o s i t i o n of t h e i r s p i n o r s :

Ü - «• 6 ) • « • ( ? )

I t i s shovn in Quantum-Mechanics textbooks that the three components of the uni

v e c t o r ? in the d i r e c t i o n o f the s p i n 3 for a pure s t a t e are g i v e n by :

P = fg* -» f*g

P . i ( £ g * - £*g) <*•'> P Z - If I1 - l e i 2

a s a f u n c t i o n of the sp inor components E and g. In p a r t i c u l a r one can e a s i l y check that

t h i s formula g i v e s the c o r r e c t s p i n d i r e c t i o n in the case of the "up" and "down" s t a t e s .

Now, from quantum Indeterminacy the measurement of the s p i n components S^, S^, $ z

a l o n g the t h r e e axes Ox, Oy, Oz w i l l not always g i v e the same r e s u l t in g e n e r a l . I f one

c o n s i d e r s t h e i r average v a l u e s over many measurements, they are g i v e n by the three compo­

n e n t s of the s p i n operator S* o p on the s p i n s t a t e * :

<+l? o p l*> .

which i s named the quantum average of the spin operator S*^.

Un the p r e v i o u s b a s i s t h i s operator i s represented by three 2 x 2 m a t r i c e s , propor­

t i o n a l to the P'.uli m a t r i c e s a^, a^, o^, and a c t i n g on the sp inor :

S o p = ~T n ( A . 2 )

where a •= (a

x>ay>c

z) ant* :

°x • (J J ) "y - G "o) °z • 6 - Î ) •

With the use of t h i s b a s i s the c a l c u l a t i o n of the quantum average <*fS*0pf+> i s

s t r a i g h t f o r w a r d and the r e s u l t i s :

<A. 1)

F i n a l l y , accord ing to the Ehrenfest theorem, such a quantum average must behave c l a s s i ­

c a l l y . I t means that the e v o l u t i o n of the p o l a r i z a t i o n v e c t o r P* i s governed by a c l a s s i c a l

e q u a t i o n of mot ion, l i k e the Thomas-BMT equat ion introduced l a t e r in s e c t i o n A . I.J.

- 6 8 8 -

For a bunch of p a r t i c l e s , the p o l a r i z a t i o n v e c t o r , vhich i s the s t a t i s t i c a l average

of i n d i v i d u a l p o l a r i z a t i o n v e c t o r s , has the sane property .

For r a d i a t i n g e l e c t r o n s , which are in a mixed s t a t e , the p o l a r i z a t i o n vec tor behaves

c l a s s i c a l l y t o o . However, i t can only d e s c r i b e the s p i n s t a t e e v o l u t i o n a f t e r averaging

over the p o s s i b l e s p i n - f l i p t r a n s i t i o n s . I t cannot d e s c r i b e i n d i v i d u a l t r a n s i t i o n s .

k.1.2 Any s p i n s t a t e i s f u l l y determined by_the p o l a r i z a t i o n y e c t o r .

This s ta tement means that the average va lue <A> of any observable q u a n t i t y A, a s

obta ined in a measurement, i s complete ly determined by the p o l a r i z a t i o n v e c t o r o n l y . This

average va lue <A> i n v o l v e s a quantun average for each p a r t i c l e and an ensemble average

over a l l p a r t i c l e s of a bunch.

F i r s t l y , the quantum average «Cif ¡A|^ k > for a p a r t i c l e fk in a pure s t a t e \i>^> can be

c a l c u l a t e d with the b a s i s used i n the prev ious s e c t i o n , knowing the matrix e l ements A ,

A , A ^ and A of the corresponding operator A Q p between the bas ic s t a t e s :

<«k|AlV = \\\* K . » K l ! * - * * W A . - * £ A •

The sp inor components a t e then expressed in terns of the p o l a r i z a t i o n v e c t o r compo­

nents of the p a r t i c l e , g iven by formulae ( 1 . 2 . 3 ) :

< < V | A | V . \ | ( l . P l k ) A ^ . ( l - P i k > A _ _ . ( P K k . i P y k ) K _ < < P x k - i P y k ) A _ J .

Secondly , one has to take the ensemble average over a l l the p a n i c l e s in a bunch (or

among a l l the p o s s i b l e s t a t e s of a mixed s t a t e ) :

<A> . -J- I < ^ |A| V .

The r e s u l t :

<A> = \ | ( 1 - P Z ) ( 1 - P S ) A__+ CPX* i P y ) A _ + ( P x - i P y ) A _ + | , (A.A)

shows that the average value <A> depends only on the components of the p o l a r i z a t i o n

v e c t o r .

This i s a c h a r a c t e r i s t i c f e a t u r e of sp in 1/2 p a r t i c l e s , which cannot be g e n e r a l i z e d

to p a r t i c l e s of h igher s p i n , a l though a p o l a r i z a t i o n v e c t o r can s t i l l be de f ined and p lays

an important r o l e in the d e s c r i p t i o n of p o l a r i z e d s t a t e s .

The r e s u l t presented here i s u s u a l l y derived from a s imple r e l a t i o n s h i p between the

p o l a r i z a t i o n v e c t o r and the d e n s i t y icatrix represent ing the sp in s t a t e of a p a r t i c l e en-

semble- For s i m p l i c i t y we have preferred to g i v e a more elementary proof .

A. 1 . 3 Eguiva lence of_a_guanturn-mechanical_description of s p i n _ p r e c e s s i o n .

Very o f t e n in the l i t e r a t u r e , sp in p r e c e s s i o n i s s t u d i e d in a quantum-raechanical f o r ­

malism i n s t e a d of s o l v i n g the Thomas-BHT equat ion (formula 1 . 3 . 4 ) . Then one has to s o l v e

the Schrödinr-Tr equat ion for the spinor vhich c h a r a c t e r i z e s the s p i n s t a t e of one

p a r t i c l e :

- 6 8 9 -

i l , íjf , H | * > ( A . 5 )

where H i s the H a n i l t o n i a n operator for s p i n motion.

The e q u i v a l e n c e between both methods i s obtained by v e r i f y i n g that the Thomas-BHT

e q u a t i o n f o l l o w s from the Schrödinger equat ion with a s u i t a b l e c h o i c e of the Hamiltonian H.

From the Schrödinger equat ion (formula A . 3 ) and from the conjugate equat ion :

- i t , a < í l = < * ¡ H ,

one d e r i v e s an e q u a t i o n for the quantum average of the s p i n operator S*^ :

i h 71 «^op!"» = < * ' | ä o p ' H ' I * 5 • < A - 6 >

where [S* , H] = !a H - H S* i s the commutator of these two o p e r a t o r s , op op op r

The Hamiltonian H i s a s c a l a r q u a n t i t y , de f ined up to an a r b i t r a r y a d d i t i v e c o n s t a n t , and i s an operator represented by a 2 x 2 m a t i i x . Such a a a t r i x must be a l i n e a r combination

of the three Paul i m a t r i c e s . The Hamiltonian H can then be w r i t t e n :

H = S* . J , op

where 3 is a v e c t o r to be determined.

The commutator [S* 0p, HJ i s e a s i l y c a l c u l a t e d by us ing the matrix r e p r e s e n t a t i o n ( f o r ­

mula 1 . 2 . 4 ) for S*0p and knowing the commutation r u l e s of Paul i matr i ce s :

|S* , S* .Ú] = ih Ú x 3 1 op' op ' op

The e q u a t i o n ( 1 . 3 . 1 0 ) becomes :

i h ar ^ l ^ o p l ^ - i* if * <+|3op|*> ,

which i s i d e n t i c a l to the Thomas-BHT equat ion ( 1 . 3 . 4 ) , when choosing ff = ^ » T

The Hamiltonian operator H for s p i n motion i s then :

One can e a s i l y check that t h i s Hamiltonian g i v e s the correct magnetic energy (formula

1 . 1 . 3 ) at the n o n - r e l a t i v i s t i c l i m i t , through the Correspondence P r i n c i p l e .

F i n a l l y , i t i s worth n o t in g that the equ iva l ence between the Schrödinger equat ion and

the Thomas-BMT e q u a t i o n , a s shown here, i s j u s t a proof of the Ehrenfest theorem, which

was invoked in s e c t i o n 1 . 2 . 2 , for the p a r t i c u l a r case of sp in motion.

PAK71CLE TRACKING

H. Mais , G. Ripken and A. Wrulich*)

DESY, I totkestraße 8 5 , 0-2QQ0 Hamburg 52

F. Scheldt

I I . P h y s i c a l . I n s t , der Univ. Hamburg, D-2000 Hamburg

ABSTRACT

After a b r i e f d e s c r i p t i o n of t y p i c a l appl i ca t ion s of p a r t i c l e

t r a c k i n g in s t o r a g e r i n g s and a f t e r a short d i s c j s s i c i of some

l i m i t a t i o n s , and problems r e l a t e d wi th t r a c k i n g we summarize some

concept s and methods developed in the q u a l i t a t i v e theory r-f dynami­

cal s y s t e m s . We show how t h e s e concent s can be appl ied to the

proton r ing HER--.

I . In troduct ion

The aim of t h i s c h a D t e r i s to d i s c u s s some a p p l i c a t i o n s and l i m i t a t i o n s of p a r t i ­

c l e tracking in storage rings 11 2*JK

Although c o l l e c t i v e phenomena, as for example i n s t a b i l i t i e s * are very important for

a c c e l e r a t o r s we r e s t r i c t o u r s e l v e s to the s i n g l e p a r t i c l e dynamics , i . e . we study the

e q u a t i o n s of -notion of a s i n g l e charged u l t r a r e l a t i v i s t i c (v = c i p a r t i c l e jnder the i n ­

f l u e n c e of e x t e r n a l e l e c t r o m a g n e t i c f i e l d s . In g e n e r a l , these e q u a t i o n s arc n o n l i n e a r , The

main n o n l i n e a r i t i e s are due to the bean-beam i n t e r a c t i o n , due to non l inear c a v i t y f i e l d s

or due t o t r a n s v e r s e m u l t i p o l e f i e l d s . These m u l t i p o l e f i e l d s are e i t h e r in troduced a r t i ­

f i c i a l l y e . g . by s e x t u p o l e s which compensate the natural c h r o m a t i c i t y or they occur natu­

r a l l y as d e v i a t i o n s from l i n e a r f i e l d s due to e r r o r s . S ince the- bean-uea i i n t e r a c t i o n w i l l

be t r e a t e d in e x t r a seminars we s h a l l not c o n s i d e r i t "ere , !.'o s h j l l d i s c no: cons ider e f ­

f e c t s which are induced by r a d i a t i o n sueii as r a d i a t i o n damping and quantum e x c i t a t i o n s

which are very important for l i g h t p a r t i c l e s l i k e e l e c t r o n s and p o s i t r o n s . In proton s t o ­

rage r i n g s t h e s e e f f e c t s can approximate ly be n e g l e c t e d . The r a d i a t i o n l o s s e s of a proton ; n HERA for example are a fac tor 1 0 ~ 7 l e s s than the l o s s e s of the e l e c t r o n .

Z. Hamiltonian d e s c r i p t i o n of the proton motion

T:ie s t a r t i n g po int for the proton dynamics i s the fol lowing r e l a t i v i s t i c '.agrar.gi \r\ for .i

charged p a r t i c l e under the i n f l u e n c e of an e l e c t r o m a g n e t i c f i e l d descr ibed by a vector po­

t e n t i a l M r , t ) °

L = - n 0 c J / 1 - r ' / c 2 ' + f rA(r_ , t ) . [1)

*) Present addres s : SSC, LBL, Univ. Res. A s s o c . , U n i v e r s i t y of C a l i f o r n i a , Berke l ey , USA.

- ï>9l -

U s u a l l y , one changes to a Hamilton,an d e s c r i p t i o n of motion and one i n t r o d u c e s the c u r v i ­l i n e a r c o o r d i n a t e system d e p i c t e d in F i g . 1 .

I t c o n s i s t s of three u n i t v e c t o r s _e T , r ( e z a t tached to the d e s i g n o r b i t of the

s t o r a g e r i n g , 5 i s the path length along t h i s t r a j e c t o r y . For s i m p l i c i t y , we have assumed a

plane r e f e r e n c e o r b i t wi th h o r i z o n t a l curvature * o n l y . Using s as an independent v a r i a b l e

and in troduc ing d i f f e r e n c e v a r i a b l e s wi th r e s p e c t to an e q u i l i b r i u m p a r t i c l e on the d e s i g n

o r b i t one o b t a i n s (v = c , 0 = 5 - c t , p c =

( 1 + k x ) • { ( 1 + y 0 ) J - (px . f - A x i 2

1/7 - ( P * - f - A 2 ) ) - d (2)

i e q u a t i o n s of motion

< dT

ALL dp,- • I H

" P X r< S X

A H O P J 9 H

A P 2 ds AZ

ALL D P 0 A H

3Po ds A O

(3 )

and A = ( A 7 , A x , A 2 ) s a t i s f y i n g Maxwel l ' s e q u a t i o n s .

By expanding the square roo t in e q u a t i o n {2) and the vector potential A i n t e J Taylor s e r i e s var ious examples for n o n l i n e a r motion can be i n v e s t i g a t e d .

Example 1: Nonl inear c a v i t y f i e l d s

+ j Hz xz ~ Hxpa + y ( s ) C O S O [4J

e f 3 M with g . = — —— , V ( s ) = c a v i t y v o l t a g e .

Introducing the d i s p e r s i o n f u n c t i o n D d e f i n e d by

D" = - (K* + g 0 ) D + H ( ' = ±) ( 5 )

v ia the canonica l t rans format ion

F2 = p x ( x - p 0 D ) + p 0 D ' x + p o 0 + p z z - i DD' p Q

a (6)

one o b t a i n s

H = £ P V + y ( g 0

+ +

- I H 0 P o

! + V(s) c o s ( ö + D p x - D ' 7 ) . (7)

If there i s no d i s p e r s i o n in the c a v i t y reg ion (V(s ) D(s) = 0) the synchrotron motion

( o , p Q ) i s c o m p l e t e l y decoupled from the be ta tron motion ( x , P K , z , p z ) 8 ) . In the c a s e of

a small d i s p e r s i o n one can w r i t e

* | P Z ! - | g o 7 ! -

- ¿ n D p 0 * + V(s ) cos ô

- v ( s ) • (Dp~x - D ' x ) s i n o . (8)

Example 2: As a second example of non l inear inotion we c o n s i d e r the i n f l u e n c e of t r a n s v e r s e

m u l t i p o l e f i e l d s wi th the f o l l o w i n g Hami1tonian:

The equat ions of motion are g iven by

x

z ' = p 2

Pz = (10)

w i th ( B z + i B x ) = B 0 I ( b n + i a n ) ( x + i zf n=2

The e q u a t i o n s o f motion in t h e s e two examples are h i g h l y n o n l i n e a r , and in genera l t 1 v

cannot be s o l v e d a n a l y t i c a l l y .

3 . Dynamic aperture

One of the most important t o p i c s in a c c e l e r a t o r p h y s i c s one lias to s tudy i s the dy­

namic a p e r t u r e . This i s an e f f e c t i v e aperture of p a r t i c l e mot ion , beyond which the p a r t i ­

c l e motion becomes u n s t a b l e due to the n o n l i n e a r magnetic f i e l d . Tigure Z shows the i a e a l

ca^e where the dynamic aperture i s almost rne same as the p h y s i c a l aperture d e f i n e d mainly

by the s i z e of the vacuum chamber.

Vertical dimension of vacuum chamber

Stable ^ {bounded)

motion

physical aperture

horizontal dimension of vacuum chamber

dynamic aperture

F i g . 2 Dynamic a p e r t u r e , p h y s i c a l aperture

Among the q u e s t i o n s for study a r e :

i ) Is i t p o s s i b l e to c a l c ú l a t e and p r e d i c t the dynamic aper ture

and how can t h i s be done?

- 0 Í M -

i i ) How dues i t depend on the non 1 inear i t i e s (rrcuUipole d i s t r i b u ­

t i o n , s p a t i a l d i s t r i b u t i o n ) ?

i i i ) How does i t depend on tunes? c l o s e d o r b i t d i s t o r t i o n s ?

Tracfcir.q codes have been w i d e l y used to i n v e s t i g a t e t h e s e prob l e n s .

4 . P a r t i c l e t r a c k i n g

The na in idea of t h e s e codes i s t o track p a r t i c l e s over many r e v o l u t i o n s in a r e a l i s ­

t i c ncdel of the s t o r a g e r ing and to observe the amplitude of the p a r t i c l e •U s s p e c i a l

p o i n t s 0 . Given the i n i t i a l amp 1 i tude ¿ ( s 0 ) = ( x ( s 0 ) , p x ( s 0 ) , z ( s 0 ) , p z ( s 0 ) , c ( s 0 ) , p 0 ( s 0 ) ) one

needs to know ^ ( s c + nL) (L = c i rcumference of the a c c e l e r a t o r ) fer n of the order of ] 0 9

(corresponding t o a s t o r a g e t ime o f a p a r t i c l e of about 10 hours in HERA). D i f f e r e n t me­

thods and codes have been deve loped to e v a l u a t e y_(s 0 + n l ) . Among o t h e r s there are

MARYL1E91, TRANSPORT 1 0 1, RACETRACK111 and PATRICIA 1 2 1 . The l a s t two codes are kick codes

where the n o n l i n e a r e l ements are r e p l a c e d by 6 -kicks according t o :

a n m ( s ) x n ? ' " > J n m x n 2 m - i ( s - s v ) . ( 1 1 )

In a l l c a s e s mentioned the problem i s reduced to the s tudy or n o n l i n e a r s y m p l e c t i c nap-

p ings of the form:

£ ( s 0 + nL) = Kits,, • (n - 1) • L)) (12)

or in shorthand n o t a t i o n

y_(n) = T(^(n - 1)) . (12a)

The dimension uf the mapping (dimension of y) can vary from two t o s i x according to

trie e f f e c t s one has inc luded (pure x- or z -mot ion , coupled be ta tron ( x - 2 ) mot ion , comple­

t e l y coupled synchro -be ta tron m o t i o n ) .

As an example for a kick code we b r i e f l y d e s c r i b e RACETRACK1 l\ which i s a f a s t com-

puver code t o t r e a t t r a n s v e r s e magnet ic m u l t i p o l e f i e l d s up to 20 p o l e s . Several addi ­

t i o n a l f e a t u r e s , such as l i n e a r o p t i c s c a l c u l a t i o n s , c h r o m a t i c i t y adjus tment , tune v a r i a ­

t i o n , o r b i t adjustment and i n c l u s i o n of synchrotron o s c i l l a t i o n s are a v a i l a b l e . A schema­

t i c f ¡ow diagram i s shown in Fin,. 3 .

Typical examples for the dynamic aperture of HERA o b t a i n e d with RACETRACK are shown in

F i g s . 4 and 5 5 > ( f our -d imens iona l couoled betatron c a s e ) .

The main problems with t r a c k i n g codes are the unavoidable rounding e r r o r s of the com­

puters and the l i m i t e d CPU-time. The rounding e r r o r s depend on the number system used by

- ö9o -

the compiler and they can d e s t r o y the s y m p a t i c s t r u c t u r e of the non 1 inr-ar mappings.

Thus, t h e s e rounding e r r o r s can s i m u l a t e non-pi iys ica 1 damping e f f e c t s J ) . In order to e s t i -

rr.ate the order of magnitude of t h e s e e f f e c t s nne ca-- swi tch to ^ higher p r e c i s i o n s t r u c t u ­

re in the computer hardware or so f tware and observe the d i f f e r e n c e s . Another way i s to

conpare the d i f f e r e n c e s Detween forward tracking of the p a r t i c l e and backward t r a c k i n g 1 5 ' .

The limited CPV-tme restricts the number of revolutions o.'.-v can track to about J D & ( 1 0 t

r e v o l u t i o n s in HERA with r u l t i p o l e e r r o r s requ ire a CPU-time in the order of days on an

IBM 3081 K) .

Bes ides t h e s e t e c h n i c a l problems t h e r e are a l s o some p h y s i c a l problems r e l a t e d with

the e v a l u a t i o n and i n t e r p r e t a t i o n of the t rack ing d a t a . Tor example , f a s t i n s t a b i l i t i e s

wi th an e x p o n e n t i a l i n c r e a s e of ampl i tudes beyond a c e r t a i n boundary can e a s i l y be d e t e c ­

ted whereas s l o w , d i f f u s i o n l i k e p r o c e s s e s which become dangerous o n l y a f t e r 1 0 s or 10 E

r e v o l u t i o n s are much mor.1 d i f f i c u l t to d e t e c t .

N e v e r t h e l e s s t rack ing i s the o n l y way t o o b t a i n r e a l i s t i c e s t i m a t e s for the dynamic

aperture up t o 1 0 5 - 1 0 G r e v o l u t i o n s , but i t i s very d i f f i c u l t t o e x t r a p o l a t e t h e s e data

tn longer t imes (1Ü 9 r e v o l u t i o n s and more) .

In order to ge t maximum informat ion nut of t h e s e numerical s i m u l a t i o n s and for a

• u t t e r understanding of the u n d e r l y i n g p h y s i c s ont should a l s o apply a n a l y t i c a l (per turba­

t i o n ) m e t h o d s 1 * 1 . To understand how n o n l i n e a r systems n i g h t d e v e l o p nne should a l s o know

s m e of the r e s u l t s of the q u a l i t a t i v e theory of dynar ica l s y s t a r s .

5 . Q u a l i t a t i v e theory of dynamical systems

Although t h e r e are e x c e l l e n t rev iew a r t i c l e s on t h i s f i e l d 1 * > 1 5 • 1 6 • 1 ' > 1 B ' we summari­

ze sonic important r e s u l t s in order to make t h i s Wi;ir.itr MS -u-1 í'-oi:i::i i IK\! ny > M S ^ ¡ hl -

The r e d u c t i o n of a HamilIonian system to a n o n l i n e a r mapping as done by t r a c k i n g c o ­

des Mas been a we 11-known procedure s i n c e Poincar^ ( 1 8 9 0 ) . Consider for example a t w o - d i ­

mensional HamiIton i an system wi thout e x p l i c i t time ( s - ) dependence H(qj , q ¡ , p j , p a ) . The

corresponding phase space i s four d i m e n s i o n a l , and s i n c e H i t s e l f i s a c o n s t a n t of the mo­

t i o n the p h y s i c a l l y a c c e s s i b l e phase space i s t h r e e d i m e n s i o n a l . Consider a s u r f a c e Z in

t h i s t h r e e - d i m e n s i o n a l space as d e p i c t e d for example in F i g . 6 .

The bounded p a r t i c l e motion induced by the Ham i H o n i a n H w i l l g e n e r a l l y i n t e r s e c t

t h i s s u r f a c e in d i f f e r e n t p o i n t s (P 0 . . . P3 . . . ) . If one is not i n t e r e s t e d in the fine de­

t a i l s of the o r b i t but on ly in the behaviour over longer time s c a l e s i t i s s u f f i c i e n t t o

c o n s i d e r the c o n s e c u t i v e p o i n t s —> P¡—^Pj-*... of i n t e r s e c t i o n . These c o n t a i n enri­

p í e t e informat ion on the Hamiltonian s y s t e m . In t h i s sense one lias reduced thç Hamiltonian

dynamics to a mapping of ï. to i t s e l f which is in general n o n l i n e a r ( P o i n c a r t s u r f a c e of

s e c t i o n t e c h n i q u e ) . S i m i l a r mappings can a l s o be der ived for Hamiltonian sys tems with ex ­

p l i c i t p e r i o d i c t i m e - ( s - ) d e p e n d e n c e ( t h i s i s normally the c a s e in s t o r a g e r i n g s ) .

F i g . 6 Poincarfr s u r f a c e - o f - s e c t i o n method

Another important f a c t and, a f t e r the work of C l i i r ikov 1 ö ) , one of the few beacons

among an o t h e r w i s e s t i l l dense mis t of d i v e r s e phenomena i s the KAM-theorem (KÛLVÛG0R0V,

ARNOLD, MOSER; s e e for example Ref. 1 4 ) . We w i l l o n l y i l l u s t r a t e t h i s theorem in the

two-dimens ional case and i n s t e a d of c o n c e n t r a t i n g on n a t h e n a t i c a l r i gour we w i l l d i s c u s s

i t s p h y s i c a l i m p l i c a t i o n s . Consider f i r s t the bounded n o t i o n of a two-d imens iona l autono­

mous (no e x p l i c i t t - ( s - ) dependence) han i 1 ton i an System which i s i n t e g r a b l e . Roughly

s p e a k i n g , an n-dimensional system H(q, . . . q n > p, . . . p n ) i s i n t e g r a b l e i f t h e r e e x i s t s a

c a n o n i c a l t rans format ion to a c t i o n - a n g l e v a r i a b l e s (I j . . . I n , 6 ; . . . 6 n ) such tha t the

transformed Harniltcnian depends on ly on the n ( c o n s t a n t ) a c t i o n v a r i a b l e s I x . . . I n . For

the cons idered two-dimens ional c a s e t h i s impl i e s that the motion i s r e s t r i c t e d t o a

two- torus parametr ized by the two angle v a r i a b l e s 5, and Q 2 as d e p i c t e d in F i g . 7.

I , In)

F i g . 7 S u r f a c e - o f - s e c t i o n t echn ique for an i n t e g r a b l e system

- 69.S -

As s u r f a c e of s e c t i o n one can choose the { l j - 5 j ) - p l a n e f o r s 2 = c o n s t . In t h i s sur ­

f a c e of s e c t i o n which may he chosen t o be j u s t the p lane of the page the motion of t h e

i n t e g r a b l e two-dimensional system looks very s i m p l e .

During the motion around the torus from one c r o s s i n g of the plane t o the next the

rad ius of the torus ( a c t i o n v a r i a b l e ) does not change,

I i ( n ) = I, ( n - 1 ) ,

and the angle B¡ changes according t o

8j (n) = e ^ n - 1 ) + L!1 • T

where T i s j u s t the r e v o l u t i o n time in 9 , - d i r e c t i o n from one i n t e r s e c t i o n of trie p lane t o

the next

T = ll U ) 2

Thus one o b t a i n s for an i n t e g r a b l e system

I , ( n ) = M n - 1 )

B , ( n ) = S j i n - l ) + 2*0 (1 , (0 ) ) - (13)

The term a i s the s o - c a l l e d winding number. I t i s the r a t i o of the two f r e q u e n c i e s o f the

system and i t g e n e r a l l y depends on I t . If a is i r r a t i o n a l the S ^ n ) form a dense c i r c l e

w h i l e i f o. i s r a t i o n a l the 8 j ( n } c l o s e a f t e r a f i n i t e sequence of r e v o l u t i o n s ( p e r i o d i c

o r b i t ) . Thus, t h e r e are invar iant curves ( c i r c l e s ) under the mapping which belong to

r a t i o n a l antl i r r a t i o n a l winding numbers. What happens now if a per turbat ion i s swi tched

on , i . e . i f

Ii (n) - I , ( n - 1 ) + c f ( I , ( n ) , e j n - l ) )

S i i n ) = B . i n - l ) + 2 n a [ I 1 ( n ) " ) + e g ( 11 (n ) , 8 , (n - l ) ) ? (14)

In p a r t i c u l a r , can one s t i l l f i n d i n v a r i a n t turves ' ! The KAM-theorem says t h a t t h i s i s i n ­

deed the c a s e i f the f o l l o w i n g c o n d i t i o n s are f u l f i l l e d ( t o g e t h e r w i l l some requirements

of d i f f e r e n t i a b i l i t y and p e r i o d i c i t y for f and y; for more d e t a i l s s e e f o r example

Ref. 1 4 ) :

i ) The per turbat ion must be weak

i i ) a = — must be s u f f i c i e n t l y i r r a t i o n a l , i . e . ,ct - i - \ > —1—¿ . " 2 q q2+6

Under t h e s e assumptions most of the unperturbed t o r i s u r v i v e the p e r t u r b a t i o n a l though in

d i s t o r t e d form.

The r a t i o n a l and some nearby t o r i however are d e s t r o y e d , on ly a f i n i t e number of

f i x e d p o i n t s of the r a t i o n a l t o r i s u r v i v e - ha l f of them are s t a b l e ( e l l i p t i c o r b i t s

- b 9 9 -

around t h i s f i x e d p o i n t ) , h a l f of them are u n s t a b l e ( h y p e r b o l i c o r b i t s ) . The h y p e r b o l i c

f i x e d p o i n t s are the source of c h a o t i c motion in phase s p a c e , i . e . motion which i s e x t r e ­

mely s e n s i t i v e t o the v a r i a t i o n of i n i t i a l c o n d i t i o n s . The motion around the e l l i p t i c

f i x e d p o i n t s CAN BE cons idered as motion around 3 TORUS w i th s m a l l e r r a d i u s and the argu­

ments used t i l l now can be repeated on t h i s smal l er s c a l e g i v i n g r i s e ta the schemat ic

p i c t u r e shown below.

F i g , 0 P e r t u r b a t i o n of an i n t e g r a b l e system

Thus, the p h a s e - space pat tern of a weak 1 y perturbed i n t e g r a b l e two-dimer-s ion a I s y s t e n

looks e x t r e m e l y c o n p l i c a t e d . There are r e g u l a r o r b i t s conf ined to t o r i and among them are

d i s t r i b u t e d c h a o t i c t r a j e c t o - i e s in a d e l i c a t e marner'. One should po in t out a t t h i s s t a g e

that t h e r e are no a n a l y t i c a l methods for c a l c u l a t i n g t h e s e c h a o t i c o r b i t s - p e r t u r b a t i o n

t h e o r i e s d i v e r g e .

6 . S t u d i e s o f c h a o t i c behaviour in HERA caused by t r a n s v e r s e magnet ic m u l t i p o l e f i e l d s

Mow we would l i k e t o p r e s e n t numerical r e s u l t s us ing RACETRACK wit^> s p e c i a l emphasis

on f i n d i n g and i n v e s t i g a t i n g c h a o t i c t r a j e c t o r i e s in phase Fpace lv<20\ The c a l c u l a t i o n s

have been performed on a 370 E Emulator and the IBM 3081 K. The number of r e v o l u t i o n s was

var ied between 3G0G0 and 300000 us ing a HERA proton o p t i c s wi th a f i x e d r e a l i s t i c m u l t i p o -

le d i s t r i b u t i o n o f the kind r e s u l t i n g from n o n l i n e a r f i e l d e r r o r s in the superconduct ing

magnets .

At f i r s t , we have s t u d i e d p u r e l y h o r i z o n t a l motion ( i . e . wi thout c o u p l i n g to the

v e r t i c a l b e t a t r o n mot ion) which of course l e a d s to a two-dimensional n o n l i n e a r mapping.

F i g . 9 shows a p x - x p l o t of a p a r t i c l e t r a j e c t o r y near the dynamic a p e r t u r e .

In an en larged s c a l e one c l e a r l y s e e s the i s l a n d s t r u c t u r e around e l l i p t i c f i x e d p o i n t s

jnd the c h a o t i c (area f i l l i n g ) behaviour near the hyperbo l i c f i x e d p o i n t s ( s e e F i g . 1 0 ) .

In t h i s two-dimensional c a s e the dynamic aperture could be i d e n t i f i e d with the

l a r g e s t e x i s t i n g KAM-circle. There e x i s t wel l -known methods for i n v e s t i g a t i n g the break-up

of t h e s e border l i n e s 2 1 ' 2 2 1 whose d i sappearance with i n c r e a s i n g p e r t u r b a t i o n would lead to

3.0

- 1 0 . 0 - 5 . 0 0 .0 5. C 10.0 Fig . 9

0.3333 . • r — • -r-i. • • • : ' • ]

0.J3SO • '.( " ' • • . -

0 . ! 385 r • '• . -

0. 0M16 - :

. . .

- 0 . 3555 i- J r " ' î

-0.2500 Í • • • ' • • • — ä 5 .30 3 . 5 5 5.1,0 S.M5 5 .50

Fig. 10

a kind of g l o b a l chaos , a s i t u a t i o n one n a t u r a l l y wants to avoid in s t o r a g e r ing p l i y s i c s .

In a d d i t i o n , two-dimensional systems are s p e c i a l in that the e x i s t e n c e ùf KAM-circles im­

p l i e s exac t s t a b i l i t y . S ince c h a o t i c t r a j e c t o r i e s cannot e scape without i n t e r s e c t i n g t h e s e

i n v a r i a n t s u r f a c e s , they are f o r e v e r trapped between t h e s e t o r i i f they indeed e x i s t .

This i s not true for h igher dimensional systems where the KAM-theorem p r e d i c t s

t h r e e - t o r i ( S , x S 1 x S , ) in s i x - d i m e n s i o n a l phase s p a c e , f o u r - t o r i in e i g h t - d i m e n s i o n a l

phase space e t c .

- 7Ü1 -

Here c h a o t i c t r a j e c t o r i e s can in p r i n c i p l e always escape a l though t h e i r motion can be

o b s t r u c t e d s t r o n g l y by these t o r i . Chaot ic r e g i o n s can even form a connected web along

which the p a r t i c l e can d i f f u s e as has been demonstrated by Arnold for a s p e c i a l example

(Arnold d i f f u s i o n , s e e for example 1 ** ' ) .

As a next s t e p we c o n s i d e r the f u l l y coupled x - z motion in HERA under the i n f l u e n ­

ce of the n o n l i n e a r m u l t i p o l e f i e l d s . There are s e v e r a l p o s s i b i l i t i e s for d i s p l a y i n g

four -d imens iona l phase space t r a j e c t o r i e s . The s i m p l e s t way i s to draw p r o j e c t i o n s on to

the d i f f e r e n t p l a n e s ( x . p , . ) , ( z , p z ) , ( x , z ) , ( p x , P 2 ) > ( x , p 2 ) and (z .pjj) but one can a l s o

use t h r e e - d i m e n s i o n a l p r o j e c t i o n s and c o l o u r to r e p r e s e n t the four th v a r i a b l e 20K

In t h i s h i g h e r - d i m e n s i o n a l c a s e one cannot s imply use the area f i l l i n g proper ty f o r

d i s t i n g u i s h i n g c h a o t i c t r a j e c t o r i e s from r e g u l a r o n e s , one rends so^d o t h e r c h a r a c t e r i s t i c

f e a t u r e s . One property of c h a o t i c motion i s the e x p o n e n t i a l s e p a r a t i o n of two p h a s e - s p a c e

p o i n t s which i n i t i a l l y have been c l u s e t o g e t h e r . Formally t h i s can be d e s c r i b e d by the

c h a r a c t e r i s t i c Lyapunov e x p o n e n t 1 4 i

d ( o ) -t —

| d ( o ) |

where d ( t ) d e s c r i b e s how the ( E u c l i d e a n ) d i s t a n c e between two adjacent phase space p o i n t s

e v o l v e s wi th t ime and d ( o ) i s the i n i t i a l d i s t a n c e . Non-zero Lyapunov e x p o n e n t s are a quan­

t i t a t i v e measure for s t o c h a - s t i c i t y of the cons idered t r a j e c t o r i e s .

Typical examples for r e g u l a r and c h a o t i c t r a j e c t o r i e s for HERA are shown in F i g s . U

t o 2 2 . He show the p r o j e c t i o n s of these o r b i t s onto the d i f f e r e n t p l a n e s .

F i g . 11 Pz versus z ( r e g u l a r t r a j e c t o r y ) F iu . \? P2 versus z ( c l i - i o tk t r a j e c t o r y }

Fig . 17 x versus Pz ( regular t r a j e c t o r y ) Fig . 18 x versus P, ( c h a o t i c t r a j e c t o r y )

F i g . 19 P x versus P r ( r e g u l a r t r a j e c t o r y ) F i g . 20 PK versus ? 2 ( c h a o t i c t r a j e c t o r y )

F i g . 21 z v e r s u s P x ( r e g u l a r t r a j e c t o r y } F i g , 22 z versus P y { c h a o t i c t r a j e c t o r y )

F i g u r e s 23 and 24 show how the d i s t a n c e between two adjacent phase space p o i n t s e v o l v e s

wi th t i m e , f i r s t f o r a r e g u l a r t r a j e c t o r y ( l i n e a r i n c r e a s e ) n d second for a c h a o t i c o r b i t

( e x p o n o n t i o l i n c r e a s e ) .

E-W I t" 'Î-SÏ

ig.« 1.03» i Oí

f i ' i . ?3

7. Summary

Thus, hl.ïA shows a l l the f e a t u r e s which are c h a r a c t e r i s t i c for noruntegrable l î a x i U o -nian s y s t e m s . However, because of the p o s s i b i l i t y of Arnold d i f f u s i o n the e x i s t e n c e o f t o r i does not imply g l o b a l s t a b i l i t y in the four-d imens ional c a s e (coupled be ta tron mo­t i o n ) contrary to the uncoupled c a s e . Unt i l now, t h e s e c h a o t i c t r a j e c t o r i e s have been o b ­served o n l y near the dynamic a p e r t u r e , However, i t i s not c l e a r whether t h i s i s a l s o true for the c a s e of coupled synchro-be ta tron motion ( s i x - d i m e n s i o n a l mappings) and how r e l e ­vant t h e s e c h a o t i c r e g i o n s are in p r a c t i c e . Further i n v e s t i g a t i o n s in t h i s d i r e c t i o r and more computer experiments are c e r t a i n l y needed for a b e t t e r u n d e r s t a n d i n g . In a d d i t i o n , the a p p l i c a t i o n of p e r t u r b a t i o n methods might be he lp fu l in s u g g e s t i n g d i r e c t i o n s f o r f u r t h e r i n v e s t i g a t i o n s and how to des ign t h e s e numerical exper iments 2 3 J ,

Recent ly i n t e r e s t i n g attemps have a l s o been made to compare the t h e o r e t i c a l and

t r a c k i n g p r e d i c t i o n s wi th machine e x p e r i m e n t s 2 ú • 2 5 } ,

For future work i t i s a l s o d e s i r a b l e tc extend t h e s e i n v e s t i g a t i o n s t o i n c l u d e c o l ­

l e c t i v e e f f e c t s and spin e f f e c t s . Promising a t t e n p s have been -nade a l r e a d y Ï É ' Z 7 > 2 B ) but

many q u e s t i o n s are s t i l l open.

Acknowledgements

I t i s a p l easure to thank our c o l l e a g u e s at DESY Dr. O.P. Barber, Dr. R. Brinkmann

and Dr. F. Wil leke for many he lp fu l d i s c u s s i o n s .

Re ferences

1) R,V. Servranckx, Proc . P a r t . A c c e l . Conf . , Vancouver 1985, IEEE Trans . Nucl. S e i . HS-32. 2186 ( 1 9 8 5 ) .

2) E. K e i l , CERfí 8 4 - 0 1 , 1984.

3) A. w r u l i c h , DESY HERA 8 4 - 0 7 , 1984.

4 } G. Ripken, DESY 8 5 - 0 8 4 , 1985.

5) H. Mais , G. Ripken, DESY Report to be p u b l i s h e d ,

6) C .J .A . Cars ten , H.L. Hagedoom, lue J, I n s t r . Meth. 212 , 37 ( 1 9 8 3 ) .

7) T. Suzuki , Par t . A c c e l . 12, 237 ( 1 9 8 2 ) .

8) A. P i w i n s k i , t h e s e p r o c e e d i n g s .

9) A . J . Dragt , O.P. Douglas E. F o r e s t , L.M. Healy , F. Ner i , R.D. Ryne, Proc, Par t . A c c e l . Conf, Vaocouve- }r';5» IEEE Trans. Nucl . S e i . MS-32 , 2311 ( 1 9 8 5 ) .

10) ,<.L. Brown, D.C. Carey, C. I s e l i n , F. Rothacker, CERN 8 0 - 0 4 , 1980.

11) A. Wrulich, DESY 84-026 , 1984.

12) H. Wiedemann, PEP Bote 220 , S I X , 1976.

13) P. Ui lhe lm, Diploma Thes is Univ. of Hamburg, 1985.

14) A . J . L ichtenberg , M.A, Lieberman, Regular and s t o c h a s t i c mot ion, Spr inger , Hew Vork, B e r l i n 1983 .

15) M.V. Berry , American I n s t i t u t e of P h y s i c s , Conf. Proc. Uo. 46 , 1978.

16) R.H.G. Helleman, Fundamental problems in s t a t i s t i c a l mechanics V, North Holland Publ. Co. 1980.

17) H. Hênon, Chaotic behaviour of d e t e r m i n i s t i c sys tems , North Holland Publ . Co. 1983.

18) B.V. Chir ikov , Phys i c s Reports 5 2 , 263 ( 1 9 7 9 ) .

19) H. Mais, F. Schmidt, A. Wrutich, Proc. Par t . A c c e l . Conf. Vancouver 1985, IEEE Trans. Nucl . S e l . H5-32, 2252 ( 1 9 8 5 ) .

20) F. Schmidt, p r i v a t e communication and PhD-thes i s Univ. of Hamburg, to be p u b l i s h e d .

21) J.H. Greene, J . Hath. Fhjrs. 2 0 , 1183 ( 1 9 7 9 ) .

22} U.S . MacKay, Renormal isat ion in area preserv ing maps, D i s s e r t a t i o n , Pr ince ton u n i v e r s i t y , 1982.

23) r . W i l l e k e . FERMILAB FN-422. 1985.

24) D.A. Eduards, R.P. Johnson, F. W i l l e k e , FERMILAB-Pub-85/59, 1985.

25) P.L. Morton, J.H. P e l l e g r i n , T. Raubenheimer, L. Rivkin, M. Ross , R.D. Ruth, U.L. Spence, Proc . P a r t . A c c e l . Conf. , Vancouver 1905, IEEE Trans. Hue 1. S e i . NS-32, 2291 ( 1 9 8 5 ) .

? 6 ) D.H. Siemann, h. ierican I n s t i t u t e of P h y s i c s , Conf. Proc . No. 127, 1985.

27) J. Kewisch, DESY 8 5 - 1 0 9 , 1985.

THE RADIOFREQUENCY qUAORUPOtE LINEAR ACCELERATOR

H. Puglisi

University of Pavi«, Department of Theoretical and Nuclear P h j i i c s , Pavía, Italy

ABSTRACT

The seminar i s aimed to give a comprehensive picture of an KFQ. After a

short descr i pt i on of the acce1erat i nq structure the T-K expans i on is

treated and the fundamental formula for the potent ial i s derived. The vane

t ip s shaping, completed to f i r s t order is followed by the physics of the

machine where the most important parameters are l i s t ed and i l l u s t r a t e d .

Since the RFQ i s e s s e n t i a l l y a cavi ty resonator th i s topic has been qWen

particular a t tent ion . Design and other technical considerations complete

the p i c ture , while in the la s t s e c t i o n the new ideas are br i e f l y

out l ined .

I. INTRODUCTION

The RFQ is a linear accelerator for ions that uses e l e c t r i c f i e l d s to simultaneously focus ,

bunch and accelerate a beam of heavy p a r t i c l e s .

While, in pr inc ip le , the RfQ can accept, focus and accelerate to the decided energy any kind of

charged par t i c l e th i s machine is part icular ly convenient for accepting and accelerating an intense,

low ve loc i ty beam from a continuous dc injector . In t h i s case the main advantaqes of the RFQ can be

summarized as fo l lows: small s i z e , low voltage dc inject ion , compatibi l i ty with complex ion sources,

bunching with high e f f i c i e n c y , high beam current capacity, high output beam quality* easy operation.

Conversely, at high energy (2-3 MeV/PWJ) most of the above advantaqes become scarcely s i an i f i cant and

the standard l inacs are preferable.

Before going further we should recal l that the RFQ was invented by Kapchinskii and Tepliakov in

197Q 1 ' , the f i r s t Russian t e s t was in 1975. Later on the work began at Los Alamos (1978) and

subsequently in meny other places as for instance Berkeley, Brookhaven, CERN, Chalk-River, GSI,

Frankfurt, Saclay, Tokyo. The so -ca l l ed Proof of Principle "POP" was given at Los Alamos in 1980.

Since that time many RFQ have been success fu l ly realized and the interest in th i s machine is no

longer limited only to the high e n e ^ y phys ic i s t because many industries are now planning t o use the

RFQ for ion implantation, t i j L i n q of mater ia ls , medical purposes. Final ly the RFq can play an

important role as a heavy-ion accelerator for the inert i al fus ion.

2. THE ACCELERATIHG STRUCTURE

Basical ly an RFQ (Radio Frequency Quadrupole) i s made by four equal e lectrodes symmetrically

placed around the beam ax i s , excited hy an appropriate radio frequency voltaae and contained in a

vacuum tank with highly conducting wal l s . Depending upon the ir shape the e lectrodes are named as

- 707 -

Fig. 1 Tne BNL RFTJ

"vanes" or "rods" and in Fig. 2 a group of four idealized vaner is sketched in order to show the

geometry of the arrangement and to give a rough idea of the vane t ip shaping that is needed for

creating the appropriate accelerating f i e l d s .

Actually the whole beam dynamics of the machine depends upon the shape of the vane t ip s and i t

i s rather evident that the vane t ip s shaping of a physical machine wil l be determined by a compromise

among many conf l ic t ing requirements. Nevertheless for the sake of c l a r i t y the physics of the machine

wil l be discussed on the basis of the sketch already seen.

Fig. 2 Schematic view of the four vanes assembled for creating the longitudinal f i e ld

In Fig. 3 only a horizontal and a vert ica l vane have been sketched toqether with the appropriate

reference axes. The peaks for each t i p are the nearest ponts to the z axis and the val leys are the

points that , being on the coordinate planes, are the most distant from the z ax i s . The distance

between two adjacent peaks or va l leys changes alnnq the beam axis ac:n-dinq to the beam dynamics

and the structure i s non-periodic.

Fiq. 3 Schematic view of two adjacent vanes

We consider now the assembly of four vanes and two planes normal to the z axis passing throuqh

two adjacent peaks of the horizontal vane. The space limited by the two planes i s ca l led an elemen­

tary unit each made up of two c e l l s . In Fig. 4 again one horizontal and one vert ica l vane are repre­

sented with the vert ica l vane rotated 90° into the same plane as the horizontal vane. The horizontal

vanes are supposed to be held at a dc potential equal to +V/2 while the vert ica l ones are supposed to

be held at a potential equal to -V/2.

Fig. 4 Two adjacent vanes are shown on the same horizontal plane. The potential between the vanes i s sketched.

It i s now rather evident that a p o s i t i v e l y charged p a r t i c l e passing throunh the f i r s t c e l l (with

length equal to CL) wi l l gain energy, while the same par t i c l e wi l l lose energy in going through the

second due to the potential along the beam a x i s .

If now we imagine exc i t ing the four vanes with an RF vol tage then the whole structure can be

accelerat ing i f the p a r t i c l e takes a half period of the RF voltage to pass through each c e l l . It

should be noted that under the previous hypothesis when one c e l l i s focusing on the ver t i ca l plane

(and neces sar i ly defocusing on the horizontal) then the fol lowing c e l l wi l l be focusing on the hor i ­

zontal plane while defocusing on the v e r t i c a l . The net re su l t can be focusinq as predicted by the

theory for the al ternate gradient machines.

Since both the transverse and the longitudinal focusinq c r i t i c a l l y depend on the shape of the

vane t i p s , a deta i led knowledge of the f i e l d s in the beam region i s required, unfortunately, employ­

ing the geometry already indicated, the ca lculat ion of the electromagnetic f i e ld d i s tr ibut ion is very

coirvlicated and some simplifying hypotheses need to be sought. Actually, if one attempts to design a

phys ica l ly rea l i zab le (and useful) structure then any kind of calculat ion (even the simplest one)

shows that the important part of the RFQ cross sec t ion has very small dimensions '" compared with the

free-space wavelength of the acce lerat ing f i e l d . For th ;it reason M. Kapcninskii and V.A. Tepliakov

did the ca lcu la t ions assuming the e l e c t r o s t a t i c d is tr ibut ion for the f i e ld inside the beam region.

The subsequent experience proved that th is analysis is s u f f i c i e n t l y accurate.

3. OUTLINE OF THE T-K EXPANSION

The d i s t r ibut ion of the e l e c t r o s t a t i c f i e l d due to the vanes i s normally known as the T-K expan­

s i o n 1 ' 2 ) . This is e a s i l y obtained solving the Laplace equation, written in cy l indr ica l coordi­

nates , with the technique of the separation of variables in our cyl indrical reference frame. The z

axis is the beam axis and the origin is located as in Fig. 3. The <i> coord'^ate i s equal t o zero on

the p o s i t i v e sect ion of the x a x i s .

If U = U (r,z,iL-) is the unknown potential then the Laplace equation is as fo' lows:

" D ' U i -DU i T ) 'U JÛHJ_ " O r 2 * r t i r * r 2 3 y ' ^ z 2 ' ( 1 )

In o»*der to separate the variables we assume that:

U • n r , Y > • $ l z ) •

Upon subst i tut ing Eq. (2) into Eq. (1) and separating we obtain the system:

"Dr * r "Dr r"

•Q ' f t l z ! - h " $ ( z )

(2)

(3)

(4)

where h 2 is an arbitrary constant to be determined using the boundary condit ions .

- 710 -

where p and q depend neither on z nor on g..

If we assume that h 3 1s a pos i t ive number and that A and B do net depend upon z then

d)lz) - Acoslhz) * B s i n l h z ) ( 5 )

i s a solution of Eq. (4 ) . Because of the l inear i ty of the previous equation then any linear combi­

nation of functions l ike (5) i s a solut ion cf Eq. Í4) that could obey the boundary condit ions .

At th i s po'nt we assine that our structure i s periodic. In th is case the potential we are look­

ing for should be a periodic even function of z with period equal to L. It follows that:

F ( Z ) w Çn A n C O S I 2TtnZ_ j n - 1 , 2 . 3 (6)

i s a solution of Eq. (4) that can f i t the actual boundary conditions of the particular problem if the

appropriate values are given to An = An {r.^J.

From Eq. (6) the values of the separatrix constant are a lso known because h should be equal t o

Zn(n/L). In th i s continuation the quantity Z-a/l will be s e t equa 1 to k (normally known as the phase

constant ) . Now for each value of h we have a solution of Eq. ( 3 ) . In order to s a t i i f y the fundamen­

tal re lat ionship already assumed, U = f (r, ,p). $(z) each solut ion of Eq. (3) should be multiplied by

the corresponding $(2). Consequently the quanti t ies appearing in Eq. (6) should be those functions

which are so lut ions of Eq. (3 ) ; each solut ion A being determined by the eigenvalue nk that pertains

to the corresponding (* (z ) .

The method used above can be employed for solving Eq. ( 3 ) . Aqain we seoarate the variables

assuming the product so lut ion:

f(r,-f) = Rir) • Giw)

where R a.id 9 are respect ive ly functions OF r and <j, only. Substituting and separating the variables

we obtain:

G ' V 1 - - m l Ç » Y ) (?)

R"ïr) i R'ír) rínk)* ( m H Rtr) (8)

'where m is a new arbitrary constant to be determined by the boundary conditions. If we assume that m

i s a pos i t ive number a solut ion of Eq. (7) is as fo l lows:

9 Í Y ^ p c o s t m ^ ) + q s i n t m Y ) (9)

The e l e c t r i c a l exc i ta t ion of the structure (the vert ica l vanes are in paral le l as che horizontal

ones) requires that the potential U should be a periodic even function of the variabie ¿ with period

equal to n. This means that to meet the above requirements we should have:

q = 0 , m = 2s , s = 1,2,3

and, consequently, the e functions should have the form*

0(\|/J = A s cosUs-y).

Agair each value of m must be subst i tuted into Eq. (8) in order to obtain a radial function that

depends upon both n and m. The general so lut ion wi l l be written term by term assumino that m ranqes

from 0 to » and that for each value of n the index s can assume all the »alues that f i t the boundary

condit ions .

In general the boundary condit ions mentioned above can be sumiarized as fo l lows:

1) For r = 0 the potential should remain f i n i t e .

2) For kz = and kz = 3n/2 each unit should exhibi t a four pole symmetry. (The potential on the

axis i s equal to zero . )

Taking into account the above conditions for n = 0 the contribution UQ to the tota l potential 'J

is as fo l lows:

U„ = S, A,r"cosl2jY> l10>

where because of the four pole symmetry (independently of z) we must have:

J = 2s + 1 s = 1, 2 , . . .

For n * 0, Eq. (8) is solved by the modified Bessel functions of order 2s and as a contribution

to the tota l potent ia l we obtain:

= j í 9 A 3 ( n k r ) • cos Í2SY'| cosinkz) (ii)

where (n * s) should be odd in order to f i t the four pole s>mmetry that the structure per iod ica l ly

e x h i b i t s . (The Neumann functions are excluded because » as already sa id , the potent ial must remain

f i n i t e for r = 0 . )

Adding the various contributions we obtain the well known T-K expansion:

and the e l e c t r o s t a t i c problem i s v i r t u a l l y solved

4. THE VANE TIPS SHAPING

As was demonstrated in the previous paragraph the T-K expansion i s very complicated and conse­

quently if the vane t i p geometry is assigned then a s u f f i c i e n t l y accurate description of the e l e c t r o ­

s t a t i c f i e l d in the beam region could be a very d i f f i c u l t problem. Un the other hand an adequate

study of the beam dynamics inside the machine can be done only if the f i e ld in the beam reqion i s

well known. A reasonable procedure 3) for overcoming the problem can be to shape the vane t ip s in

such a way as to coincide with the equipotent ia ls described by a few terms of the T-K expansion. In

other words we s e l e c t a reasonable form for the potential in the beam reqion. When the potential i s

known the f i e l d i s known, and the beam dynamics associated with the se lec ted potential can be defined

completely. If the calculated beam dynamics i s sa t i s fac tory then the vane t ips must have the form of

the equipotent ia ls that l imit the beam region.

The simplest function that describes a potential consis tent with a boundary condition of the

beam region i s obtained maintaining only the lowest-order terms of the T-K expansion. Accordingly,

i f we s e t n = 1 and s = 0 we find that the simplest form for the potential in each unit i s as

fol lows:

Fír,"z,ví = A^cosízy/ •* AJ/íkrícoslkz) (13)

where í<l and A., are two constants to be defined and k = 2n/ l i s the phase constant. Since the four

vanes are powered by an RF voltage with period T (radian frequency w equal to 2n/T) the complete form

of the quas i - s ta t i c potential i s :

Utr.z.Y.U = F(r,z,\W • sin(ut + (14)

where $ is the phase of the RF voltaqe when the charqed p a r t i c l e enters the unit .

From the physical picture or the accelerator we know that the "synchronous" par t i c l e should pass

through the unit exact ly in one period of the radio frequency vo l tage . Consequently i f ß is the

average normalized ve loc i ty of the synchronous par t i c l e we can write:

k __?I_._2JL_. (15) i r>cT r>A

where x i s the free-space wavelength of the applied f i e l d .

A t and A 2 indicate two constants which depend on the geometry of the unit and th i s means that

with th i s choice we can input only two boundary condit ions . It should be noted that those boundary

condit ions are not as arbitrary as could be thought because once Aj and A 2 are given the result inq

structure should be physical ly rea l i zab le and e l e c t r i c a l l y compatible.

Accordingly, with the scheme given in Fig. 3 our boundary conditions at z = 0 are as fol lows:

Y . o

V .ZL • f".ma 2

U . V _ . ^ _ sin[uL<j>l

2 2

U_ _ V_ . _X_ sin (ut •,(,).

2 2

Upon subst i tut ing the boundary conditions in Eq. (14) we obtain the system:

f A a 2 A, I l k a ) V

A , l m a ) !

+ A 4 I 0 (mka)_ _ V_

2

Solving with the Kramer rule we obtain:

A, L(mka) + í (ka)

a ' U m k a M m a l ' I J k a ) 2

Llmka) + m* Ltka)

Now A1 and A 2 are dimensional quanti t ies and t h i s may create problems in the subsequent manipula­

t i o n s . For t h i s reason two new dimensionless parameters, A and X, are defined as fol lows:

2 a'A,

V

A.

- A L ( k a )

(16)

: imka).m"l lka)

Upon subst i tut ing in Eq. (14) we obtain:

U _Vj^ X | ^ j 2 cos(2Y> + A l . lkr ) cos(kz) (17)

and i t should be remembered that the inter-vane voltage V is equal to VQ s in {J. +

Now we assume that the potential U is the potential actually existinq in one unit and we want ic shape the vane tips in such a way as to realize this potential distribution. Once a, m and k are given, then the shape of the unit is uniquely determined and one of the simplest ways to arrive at the vane profile is to determine a rea^onaole number of vane cross sections along the z axis. These cross sections can be determined by solving, numerically, Eq. (17) in which U is made equal to the potential of the considered vane, the 2 coordinate is an input for each cross section and a series of values is given to <$,. For each value of 4. the corresponn'inq value of r is calculated. To calculate the profile of the cross section of a horizontal vane the potential must be set equal to +V/2. For a chosen value of z = we select a series of values for 4,(0 < 4, < r/4) and find the corresponding values of r solving Eq. (18):

1 _ X f_ c o s ^ ) AIo(kr) cos^kan) (1R a1

An identical procedure has to be followed to find the cross section of a vertical vane. The potential must be set equal to -V71 and s/4 < 4, < n/2. Consequently Eq. (19) is the one to be solved.

- 1 m X j ^ cos(2yJ + AIo(kr)cos(kznJ ( W )

a"

The procedure outlined above cannot be followed blindly and some remarks are in order. A unit entirely generated using tqs. (18) and (19) might lead to a physically unrealizable structure. This means that once the pole tip profiles are found then the remainder of the vanes have to be determined using different criteria. Moreover for z = 0 and 2 = $\ the unit has identical cross sections while in an accelerating unit the initial and final sections are different. If the particles are not rela­tivistic their velocity increases during the acceleration. This means that the distance travelled by the synchronous particle during one first half-cycle of the accelerating field should be shorter than the one travelled during the second half of the same cycle. The distance travelled by the synchron­ous particle during a half cycle of the accelerating voltage is called the unitary cell length and is obviously equal to ßx/2 where p is the averaqe normalized velocity of the transit particle.

A portion of a horizontal vane that contains two adjacent cells is sketched [exaqqeratinq for the sake of clarity) in Fig, 5. The very nature of the RFQ accelerator requires that if, for instance i the cell "n" begins at a peak of a horizontal vane then the "n +• 1" cell beqins where a peak on the vertical vane occurs. This means that we could make use of the symmetrical expansion of a function (the vane profile) pretending that the distance $\ is just twice the spatial period of a non-physical ly existing cell (with evert-syrmetry properties) that as regards the first half does coincide with the actual one. Consequently the shape of an actual cell can be determined with the same procedure outlined above but a new cell begins every time z = fix/2. In other words once each value is assigned to the four parameters k, a, m, ß, then the detailed procedure for findinq the profile of the vanes is listed below.

X

Fig . 5 Portion of a horizontal vane that i s part of two adjacent c e l l s

1. The P T N cross sect ion of the horizontal vane of the J ce l l is obtained by solvinq Eq. (18)

written as fo l lows:

where Zp i s the p portion of the cell length lj and the 4, coordinate is varied from zero to

2 . The P l n cross sect in 0 f the vert ica l vane of the same J ce l l i s obtained from the same

equation where +1 i s subst i tuted by - 1 , and the ^ coordinate varies from it/4 to TJZ (again k¿

z wi l l vary from P to n).

(U = +V/2, <p = 0). Ch the other hand the c e l l length is now 1 J + 1 and th i s length 'Id be

divided into many intervals as above but the argument of the cosine should vary from r. t>- 2n.

Thi^ means that the quantity * should be added to the argument of the cos ine .

Fron the above arguments i t i s evident that there is no cont inuity between adjacent c e l l s and

the previous procedure should be modified in such a way as to obtain continuity between adjacent

c e l l s as shown in Fig. 5. For instance a and m can be made l inear functions of z. On the other hand

even the modified procedure could generate c e l l s in which the curvatures along z and in the X-Y

planes may create serious mechanical and e l e c t r i c a l problems. In Fig. 6 the assembly of four

physical vanes is sketched together with some cross sect ions of the whole machine.

The above procedure can be followed and the vane t i p prof i l e s exhibit a very complicated shape

that should be machined with a high degree of accuracy. Moreover, e spec ia l l y at the low energy end,

the e l e c t r i c f i e l d may be too much enhanced. A better mechanical solut ion i s obtained if the pole

t i p cross - sec t ion may have a constant curvature radius . This can he achieved by introducing higher

order multipoles into the potential function. Starting from an optimized two-term potential

structure at every ce l l one can try a formula with more than two terms in order to minimize the

Tl/4.

3. The next c e l l (J + 1) now begins with a radius a . . - m . on the horizontal vane

Fig. 6 Possible shape for the four vanes. Assembly in the container and cross sections.

deviation of the pole cross section from a circle of constant radius. Obviously many other important manipulations can be done to cope with the particular performance required, but the procedure outlined above remains substantially the same.

5. PHYSICAL CONSIDERATIONS

The lowest order potential function depends upon three parameters: a, m and k and we have seen that each cell is completely determined whenever the value of those parameters is specified. From the gradient (changed in sign} of the potential function (17) we obtain the fields inside the beam region as follows.

r E XV rcosiz\|f) kAV I.(kr)cosikz)

XV rsin Izy) a' (20)

kAV ijkrisinlkz)

where V = V0 sinfut + $) is the intervane voltage3).

It is immediately clear, by inspection, that a particle travelling on the z axis does not experience any transverse force while for this particle (charge a and mass m 0) the accelerating force f ? becomes:

. qMVsin(kz)_ AV^ sin/_27t_z) sinfut <j> 2 (3A l|3A / \ +

(21)

This means that the quantity X is related to the transverse focusing force while A is connected with the voltage gain per cell. In fact the potential difference ¿U that exists on the axis between the beginning and the end of each cell is equal to AV as can be easily verified upon substitution.

U0 _ U|0Aj . V j A _ A j ^ c o s ^ pAjj j _ AV

Another important element far the design of the accelerator is the dynamic gain of energy per cell, ¿E. Taking into account that in an R.FQ the accelerated particles are not relativistic, the motion of a particle travelling on the z axis is as fallows:

"AqV. ßAm„

sin(kz) sin l u i • i/) (22)

where ß is the normalized average velocity of the transit particle.

- 7 1 8 -

* -. i L - _ ^ r ^ V _ X _ _kW_ I ^ k x J c o s t k z J l . (25) I T , o m 0 L a a z J

The modified Besssel function I L {kx) can he expanded and for small values of x we obtain J ¡ ( I Í X ) : kx/2 . Substituting into Eq. (25) and using the e x p l i c i t formula for V and k we obtain:

X = I* Ç^X» s i r . t u t - * -< f r ) " | x f u 1 A V Q q c o s f Z T C S \ s i n l u t + fr) 1 x . " i a ' m , J : j3*Aa \(Ï?J j

Again we can make the hypothesis k2 = ujt and after a i i t t l e algebra we obtain:

I" q X V . s i n l u U ^ J l x L a a m „ J

L 2 m op ! A 2 J

P n J AV„g s i r . 12ui.* 9>)"j x .

!_ 2 mß'tf \

(26)

It has been demonstrated (3) that under very broad conditions the above equation can be soWed

ana ly t i ca l ly and consequently the quantity ÛE can be calculated with a high degree of accuracy.

Nevertheless in a "normal" RFQ the r e l a t i v e variation of v e l o c i t y per e e l ' Aß/e is always small and

th i s .neans that the average and the instantaneous ve loc i ty of the t rans i t part ic le are rather c l o s e .

In th is case we can write u. t = kz and consequently the force f ; becomes;

f z _ TtgAV0_ f c o s i t j ) c o s ( z k z * ( p i 2ßA [

Integrating over the c e l l length we obtain:

4

The numerical computations and the experience on the actual ly ex i s t ing RFQ have proved that (24) i s

s u f f i c i e n t l y accurate (in the above formula r]b i s the value of the t rans i t time factor for a lonqi-

tudinal f i e l d with space variat ions equal to s in kz) .

In addition to Eq. (22) , which describes the motion of a par t i c l e travel l ing on the z a x i s , we

need the equations for the motion in the transverse planes. While the general theory of the trans­

verse motion i s very complicated, i t is rather easy to define the parameters which determine the

s t a b i l i t y of the beam on the transverse planes. For instance, using the expression of the gradients

[Eq. ( 2 0 ) ] , we can write the d i f ferent ia l equation of the displacement alonq * as fol laws:

Now in order to obtain dimensionless coe f f i c i en t s we multiply by T 3 / \ both s ides of Ea. (26) and

change the variables as fo l lows:

AX -\ «Li

Upon subst i tut ing into Eq. (26) we obtain:

djj L J where, fol lowing the nomenclature used at Los A l a r o s 3 ) ,

r - "5 X C O S V 4 K j * $) (27)

¿ "A2 AVQ SIN 4» " 7 (IZM0C2 (?«)

and y = i / s i n

It should be noted that B and ¿ can be interpreted as normalized forces . B i s responsible for

the focusing e f f ec t while a defocusing e f f e c t corresponds to Û when <t> is negative, as is the case in

a l inac .

The so lut ions of the above equation can be convergent or divergent depending upon the numerical

value of the parameters 8 and ¿. Since t is always very small ( t yp i ca l l y * 0.05 at the inject ion)

then a good degree of s t a b i l i t y is obtained i f B i s larger than a few units and smaller than = 15.

A more general analysis of the transverse s t a b i l i t y wi l l not be undertaken here because i t

requires the use of techniques 4 ! too spec ia l i zed for a general seminar on the RFQ. Nevertheless i t

should be anphîsized that the general theory of the radial s t a b i l i t y , val id for a l inear acce lerator ,

i s applicable to the RFQ.

Before leaving the problem of the radial s t a b i l i t y i t is important to consider the r o l e of the

radius r Q ( r being the distance of the vane t ip s from the axis ) that occurs for any c e l l when cos (kz)

= 0. In fact if the above condition is f u l f i l l e d then Eq, (17) reduces to :

It fo l lows that both for <[, = 0 and 4, = rJ2 the distance r form the axis of the pole t i p s is as

fo l lows:

(29)

r r a (30)

4x

This particular value of the radius i s the so-ca l led four-pole radius because on the planes for

which kz = it/2 the vanes show perfect four-polar syrrnetry with hyperbolic cross sect ions defined by

the equations:

r horizonal vane.

r r. t Ti/4 < Y - xi2^ vert ica l vane .

"J -cos Izy^ Moreover a very simple calculat ion can show that when the radius i s equal to r Q , the radius of curva­

ture at the pole t i p s i s a lso equal to r 0 (on the X-Y plane) .

Returning to the transverse focusing we observe that i f V is constant, keeping the focusing

strength at a fixed value requires that the quantity X/a 2 remains constant alonq the machine and t h i s

means [Eq. ( 3 0 ) ] that the radius r should remain constant . Moreover a f ixed value of r Q can be

expected to minimize variations in the vane-to-vane capacitance and should f a c i l i t a t e the design pf

an RFQ in which the pole t ip voltage d is tr ibut ion i s required to be f l a t over i t s ent ire length, For

the above reason the quantity r 0 can be regarded as a character i s t i c averaqe radius of the RFQ pole

t ip s that a f fects a l l the design of the machine.

6. THE STRUCTURE Of AN RFQ

The accelerat ing and focusing f i e l d s depend upon the voltage applied to the four vanes. This i s

normally obtained via a cav i ty resonator where the vanes are a fundamental part of the whole s t ruc ­

ture . More s p e c i f i c a l l y the RFQ cavity(vanes and container) should be designed for resonating, at

the working frequency, in such a way as to create the desired voltaqe on the vanes. This i s a prob­

lem that requires some knowledge of the microwave technique. Because the RFQ i s e s s e n t i a l l y a radio

frequency device where the microwave techniques play a major ro le i t may be useful to g i v e , in the

fol lowing, a short out l ine of th is top ic .

An electromagnetic f i e l d that depends upon the time as a sine wove can e x i s t and propagates inside a hollow cyl indrica l pipe with a perfect ly conductinq wall if certain condit ions are met. A short way to permit ca lcu lat ions to be made i s to assume that both the f i e l d s E and H depend upon z and t as fol lows:

F( r ,Y ,2 , t ) = f lr,Y> e jut -32

(31)

where F and f stand both for E and H and w i s the radian frequency of the f i e l d .

Moreover assuming that no currents are contained in the bounded volume then the Maxwell equation

that we need can be written i s fol lows:

Solving with the Kramer rule and using the normal notation:

we obtain the transverse components of the f i e l d as functions of the longitudinal components E z and

E r j _ r »

K ; L r O Y

Kc! [ " r O Y

H Y J T J U E O J ^ T

K E * L Or

This means that i f the longitudinal components of the f i e ld are known then the transverse ones

can be obtained by derivat ion. Moreover i f E 2 i s always zero then Hz * (1 (because otherwise tha

whole f i e l d i s zeroj and we have the family of the so -ca l l ed TE modes, where TE i s an abbreviation

for transverse e l e c t r i c mode. If Hz = 0 and consequently E z * 0 we hrve the family of the TM

modes (transverse magnetic).

Since the above system i s l inear then the superposition principle applies and any sinusoidal

f i e l d can be reduced to a l inear combination of r and TM modes*). I t i s now important t o

*) In the l i t era ture concerned with part ic le accel ators the TM modes are ca l l ed "accelerating modes" while the TE modes are ca l l ed "deflecting modes".

r O Y i

j U M •PH."] Or j

1 DH, Or

ii OH. ] r O Y J

(32)

recognize that the structure containing the vanes should De excited with a TE mode. In fact only a

TE mode can create the four-polar focusing f i e ld that , on the other hand, cannot be acce lerat ing .

The modulation on the vane t ip s introduces the local perturbation that is adequate for creat i rg ,

l o c a l l y , the accelerat ing f i e l d . Consequently we are natural ly led to finding a poss ible so lut ion

far H z in the structure already described.

Under the previous hypotheses the Maxwell aquations are as fo l lows:

y . E = o V • H = 0 V X E : - j U U r i

V x » = - I U E E .

Taking the curl of the last equation, subst i tut ing Ï «E and reca l l ing that:

V x V x H : V ( V - HJ - y 2 H

we obtain the familiar wave equation for the vector H. Since the same procedure applied to the curl

of E g ive s , formally, the same resu l t we can write:

V ' I E 1 L . V (El ,33)

I H | I H J

Expanding the above equation and retaining the longitudinal z component of H we obtain:

O ' H , , i O H , , i O ' H , + V U , . 0 m ( 34)

O r * r "Or r2 " D y 2

Equation (34) can be solved with the same technique that has been used for the T-K expansion.

We can assume that H z = R(r).e(u>) where R = R(r) is a function of r and e = e U ) i s a function of

4,. Substituting and manipulating we obtain:

r ' Í T r R l K C V < T (35)

R * R e

The le f t side is a function of r alone, the right of <\, alone. Consequently i f both s ides are to

be identical for al l values of r and 4, then both s ides must be equal to the same constant: for

instance v 2 (assumed p o s i t i v e ) .

By subst i tut ion we obtain:

R" T_L R' + (K / . v l^R = 0

The f i r s t equation is solved with the Bessel and the Neumann functions of order v whereas the

second i s solved with s inusoids .

For r = 0, Hz cannot be i n f i n i t e and th i s means that the Neumann function does not f i t t h i s

boundary condit ion, on the other hand the f i e l d should be the same every time we vary ^ by a multiple

of 2n. This means that v must be an integer. Moreover a proper se l ec t ion of the or ig in for the c o ­

ordinate wi l l allow us to use e i ther the s ine or the cos ine in the trigonometric part of the

s o l u t i o n .

Consequently we obtain:

HZ = HDJUIKCR)-COSIVY). <36> The f i e l d described by Eq. (36) i s para l le l to the conducting wall and automatically obeys the

boundary condit ions . Conversely, from the second of the Eqs. (32) and using (36) , we obtain E that ,

being paral le l to the boundary, must be zero on the perfec t ly conducting wal l s . This means that on

the boundary (r = a) the derivat ive of J v (K c -r) must be zero and we obtain:

J; (KCA) - O (37)

where a i s the inner radius of the cylinder while J v ' ( k c > a ) indicates the value of the derivat ive

of the Bessel function of order v for r = a. Equation (37) determines the i n f i n i t e s e r i e s of the

K c , and for each we have a particular E l u t i o n indicated as the T E v i mode. S p e c i f i c a l l y , v

indicates the number of variations along <|i and a indicates the order of the zero which determines the

part icular so lut ion .

If we are looking for a transverse f i e l d with four-pole symmetry and no variat ions along z {as

required from the assumed form of the potent ia l ) we have to s e t :

Y = 0 ; v = 2

and from Eq. (37) , s e l ec t ing the f i r s t zero, we obtain:

KCD - U</7j7a - 3 . 0 5 4 2 4 OR FC „J5Ü. MH2 (33)

A where f c is the so-ca l led cut-off frequency of the se lected mode (Fig. 7) . (In a waveguide the

cut-off frequency is always the one for which _ 0 . )

- 724 -

+

+ Fig . 7 E lec tr ic u n e s of force for TE mode in the cross sec i ton of a uniform cyl indrical waveguide

This means that an i n f i n i t e l y long cyl indrical l o s s l e s s pipe, with inner radius equal to a, can

support the a x i a l l y uniform four-pole mode. The re lat ionship between frequency and mode being

defined by Eq. (38) .

An i n f i n i t e l y long wave guide i s not a practical device but it is possible to tn1 :1d a physical

structure where, for a long portion of the a x i s , the f i e l d has four-pole symmetry and is adequately

uniform ( th i s structure i s the RFQ resonant cavi ty where the cyl indrical wal l , the vanes and the end

sect ions are fundamental parts of the whole s tructure) .

In order to have some ideas about the cyl indrical cavi ty resonators we imagine short c i r c u i t i n g ,

with a conducting wall normal to the a x i s , both ends of our hollow pipe leaving a clearance equal t o

L between the short c i r c u i t s . Now, as well as the above conditions on tbe cy l indrica l wall

[Eq. (37)], the e l e c t r i c f i e l d of a TE mode should be zero on the short c ircui t ing surfaces (which

are parallel both to ty and to E^) and we have a third condition that enters into the determina­

tion of the cavi ty resonant frequency. It i s nearly obvious that th i s condition i s f u l f i l l e d i f the

distance L i s an integer multiple of the half wavelength of the f i e l d as measured ins ide the pipe.

Let us cal l R v A the value of the argument that s a t i s f i e s Eq. (37). Consequently we have:

V + u > e = [^-J (39)

and we see that w and Y can be given any value consistent with Eq. (39).

In order to build up a stat ionary f i e l d we need propagation in both direct ions of the z a x i s .

This means that y should be imaginary and we put > = j ö . If Xg is the wavelenqth inside the pipe

(the so-cal led guide wavelength) then i t is rather obvious that 6 = 2-n/\g. In fact when we pass

through a distance equal to \g the f i e ld has to repeat i t s e l f because the argument of & i 6 * g

changes by 2 n .

- 7 2 5 -

F i g . 8 Three d i f f e r e n t nodes e x c i t e d i n t h e same c y l i n d r i c a l r e s o n a t o r

The degeneracy already seen allows zero value for the l a s t index of a TM mode. This cannot

happen for a TE mode because the transverse e l e c t r i c f i e l d must be always zero on the short c i r c u i t ­

ing wal ls at the end of the cav i ty . Therefore if no variat ions are allowed along z ( l a s t index equal

to zero) then the whole f i e l d should go to zero.

Up t o th i s point we considered only the elementary cy l indr ica l resonator where the f i e l d s E z

or Hz are completely described with only the ir eigenfunctions and the resonant frequency is the

corresponding eigenvalue. However the cavi ty resonators used as acce lerators , even maintaining the

cy l indrica l symmetry, are often much more complicated and, in order to s a t i s f y the boundary condi­

t ions dictated by a technical resonator, the complete se t of the cy l indrica l eigenfunctions is

normally required.

Substituting in Eq. (37) and reca l l inq that = (2-n/\)2, where \ is the free space wavelength

of the f i e l d , we obtain:

Rearranging and introducing the third condition that the resonator length I can be equal only to an

integer number, say p, of half guide wavelengths we obtain.'

A b z l

where \ is the free space wavelength fo a cyl indrical resonator of radius a and lenqth L operating in

the TE v ip mode.

At this point a very short outline of th<î TN modes for a cylindrical cavity seems in order. Equation (33) can be solved for E z and following step by step the outlined procedure we obtain:

E 0J v(K cr) • cos Ivy) . (42)

The E z component is, by definition, parallel to the perfectly conducting wall and consequently E z

must be zero for r = a. This condition is verified if:

Now it is rather evident that since E 2 is always normal to the short circuit at the ends of the cylindrical cavity then an infinite series of modes can exist with no variations along z (TM v K I

modes). As a consequence it happens that a cylindrical cavity can support any TH v j [ 0 mode indepen­dently of its length.

In addition to the above degenerate modes, a cylindrical cavity can exhibit a TM resonance if the cavity length L is equal to an integer number of half wavelengths measured inside the cavity. Again, following the procedure outlined for the TE modes, we find that the resonant wavelength of a TM mode is given by the formula (41) where now RvJL is the zero of order 1 of the Bessel function of order v. In Fig. 8 examples of resonant modes are illustrated.

Even a simple outline of the general theory would go beyond the purposes of this rather intui­tive treatment. Many powerful computer programs are now available for analyzing, with qood accuracy, practically any useful cylindrical resonator5).

The cavity for an RFÛ. originates from a TE z l l cylindrical resonator which is loaded with four V-shaped vanes symmetrically connected to the cylindrical wall as shown in Fig. 9. The vanes termi­nate at some distance from the short circuiting wall and consequently the central vane section is symmetrically coupled to the two end sections. (We should observe, in passing, that this resonator is no longer uniform along the abscissa.) The boundary conditions provided by the end sections allows the whole cavity to resonate in a very complicated manner where the fields are nearly uniform inside a large portion of the vane section. More specifically this condition is obtained if the TE 2 1

cut-off frequency of the uniform guide represented hy the vane section is slightly below the "perating frequency of the whole cavity.

J„(Kca) = o.

Fig. 9 Simplified axial and longitudinal cross sections of an RFQ

Figure 10 shows one of the four pole sect ions of the BNL RFQ. Since in th i s machine the focus­

ing fo=-ce i s held constant then i t fo l lows that al l the four pole sect ions should be equal.

Fig. 10 Symmetric cross sectiun of the BNL RFQ

As explained above, the choice of "constant r 0 " minimizes the difference of the e l e c t r o s t a t i c

capacity between di f ferent portions of the same structure but those di f ferences are r.on-vanishing.

This means that some "distributed tuning" along the structure would help in obtaining the qood uni­

formity of the f i e l d that is r e a l l y needed. For th i s reason each vane has been loaded with two bars

tapered along the z axis and, in f a c t , a very good uniformity of the f i e l d has been achieved after a

careful adjustment of the bars.

The "distributed" tuning mentioned above el iminates the f i e l d d i s tor t ions that a ser i e s of

lumped tuners would certa in ly introduce. Nevertheless the vane t ip s modulation, the unavoidable

tuners at the end s e c t i o n s , the devices for feeding the power, and many other mechanical

complicat ions, always make the spurious modes which are near the wanted T E 2 ] 0 very strong. Fiqure 11 shows the e l e c t r i c l ines of force between two adjacent pole t i p s of an RFQ for the quadrupole ( T E 2 L 1 )

and dipole ( T E m ) modes.

In a uniform cavi ty the dipole mode is always below the quadrupole mode and the same should

happen in a well balanced RFQ cav i ty . In th is case , as the whole cavi ty should be tuned jus t above

the cut -of f frequency of the guide corresponding to the vane sect ion of the cav i ty , then i t follows

that dipole mode is enhanced. For t h i s reason mode suppressing special techniques are required.

PROB. NAHE = BNL RFQ 2A 1 0 1 0 RBCT PROB. NAUB = BNL KTQ ZA 1 0 1 1 RBCT

Fig. 11 Lines of force of the e l e c t r i c f i e ld between adjacent pole t ip s for the T E 2 ; 1 and T E u l mode

7. DESIGN AND TECHNICAL CONSIPERATIONS

A ful l technical description of the machine, together with practical design considerations would

go beyond the purposes of th i s seminar. Nevertheless some of the problems concerning the whole RFC,

wil l be i l l u s t r a t e d in order to improve the general picture of the machine.

7.1 Tuning and exc i ta t ion of the cavi ty

Figure 12 shows an idealized sect ion of an RFQ where the horizontal vanes have been removed for

s impl ic i ty . From the drawing i t is evident that the e lectrodes placed on the end sect ions load both

ends of each vane with an adjustable capacity to ground that great ly helps in balancing the vanes and

tuning the whole cav i ty .

T E R M I N A T I O N O F U N I F O R M F I E L D R E G I O N

• •

d q

F R I N G E

F I E L D

R E G I O N

U N I F O R M

F I E L D R E G I O N

F R I N G E

F I E L D

R E G I O N

Fig. 12 Schematic axial section of an RFQ. The tuners placed on the end sect ion are shown.

- 7 2 9 -

Fig. 13 Example of coaxial manifold (Los Alamos)

It should be noted that th i s technique allows the RFQ cav i ty to be excited from many posi t ions

uniformly distr ibuted along the outer wall of the machine. Moreover th i s distributed exc i ta t ion is

obtained without introducing e lectrodes in the regions between the vanes as shown in Fiq. 13.

Any cavi ty resonator can be fed in many ways. Electrodes capaci t ive ly coupted with the vanes

and connected to the RF generator are not favoured because they tend to arc in case of temporary mis­

match (poor vacuum, mult i pacting, detuning . . . ) . The loop coupling, with one or more exc i tat ion

loops placed near the end sec t ions and coupled with the magnetic f i e l d s which e x i s t between the

vanes, i s much more used.

In addition to the "lumped" devices for coupling to the RF power source, many other " d i s t r i ­

buted" coupling methods can be used. These methods, which are well known and widely used in micro­

wave techniques, have been used at Los Alamos s ince the beginning and later on were adopted in many

other laboratories . The solut ion proposed by Los Alamos i s for a large portion of the RFQ cavity to

be s_ymmetrical ly inserted into a shorter cyl indrical cavi ty so that the new structure can be con­

sidered ds a coaxial cable shorted at both ends, where the surface of the inner conductor coincides

with the outer boundary of the RFq cav i ty . A coaxial cable shorted at both ends resonates , in a

transverse electromagnetic mode, when i t s length i s equal to half the free-space wavelength of the

exc i t ing RF f i e l d . If t h i s cavi ty i s made exact ly equal to A/2 and some coupling s l o t s are opened on

the outer wall of the RFQ cav i ty , then the exc i ta t ion of the coaxial resonator a lso e x c i t e s the RFQ

cav i ty . This coaxial cavi ty which matches the RF power generator to the RFQ cavi ty i s known as the

coaxial manifold.

- 730 -

7.2 Suppression of the spurious modes

As was seen in the previous paragraph, an ideal resonant cavi ty can o s c i l l a t e in an in f in i t e

number of modes. Actually an RFQ always exhibi ts a large number of strong resonances th¿* very often

are randomly bunched into very small intervals of frequency. Those modes reduce the amount of power

that could e x c i t e the requirea one (the T E 2 I 1 ) and, by d i s tor t ing severely the f i e l d , impair the c a l ­

culated beam dynamics. Part icularly dangerous i s the dipole mode already seen.

For the above reasons many useful devices have been invented in order to eliminate as many

spur ; ous modes as p o s s i b l e , at l eas t in the neighbourhood of the working frequency. Two di f ferent

methods wi l l be quoted here to give an idea of the problem.

With the f i r s t method 6) the vanes of the same polari ty are e l e c t r i c a l l y connected wi .h con­

ducting rings as shown in Fig. 14. It is interest ing to note that the same technique was success ­

f u l l y used at the dawn of the microwave tubes when the eight resonant c a v i t i e s of the magnet!on were

synchronized by connecting "with a conducting wire" the homologous edges of two adjacent c a v i t i e s

(the so -ca l l ed strapped magnetron).

The second method proposes the insertion of loops coupled with both modes T E 2 1 J and T E j ^ . The

loops are connected in such a way as to short c i r c u i t the T E t l l mode while allowing the ex is tence of

the T E ; i l mode. A practical device based on th i s cr i t er ion was real ized at BNL6) where the RF

power is fed to the RFQ through two groups of loops (four for each group) placed inside the two end sect ions of the machine. The eight loops are exicted in parallel and are coupled to the H f i e l d that

e x i s t s among the vanes. By se lec t ing the proper orientat ion for each loop i t is possible to short

c i r c u i t the T E | U mode. Figure 15 shows a picture of the Dower s p l i t t e r connected with the eight

coaxial c a b l e s .

7.3 Design considerations

The operating frequency is a v-'y important design parameter. Since the ce l l length i s equal to

ß\/Z, i t follows that the higher is the frequency the shorter is the machine. On the other hand for

Fig. 14 Technique for mode suppression (Berkeley)

Fig . 15 The power s p l i t t e r used at BNL for feeding the RFQ from eiqht places and simultaneously suppressing the T E m mode

very high frequencies the length of the eel Is hocomcstoo short at the low energy end of the machine.

Moreover the working frequency determines the radius of the RFQ cavity and too low frequencies demand

a very large diameter.

Another important parameter i s the voltage between the adjacent vanes. As a general rule th i s

voltage should be as high as poss ib le , obviously avoiding the risk of sparking.

If the ion spec ies with the ir i n i t i a l and f inal energies are spec i f i ed , and f, the frequency,

and intervane vol tage are given, then the RFQ design i s determnied when the three independent func­

t ions a ( z ) , m(z) , (,(z) are given, where 2 i s the axial distance along the accelerator . Two di f ferent

ways for arriving at the above functions are indicated in Ref. 3. The methods used at Los Alamos can

be better understood with the aid of Fig. 16, which shows a functional block diagram of an RFQ where,

beside the acce lerat ion , the greatest attention was paid to l imit the growth of the radial emittance

of the beam.

RADIAL GENTLE ACCELERATOR —* MATCHING SHAPER BUNCHBR ACCELERATOR SECTION

BUNCHBR

Fig. 16 Functional diagram of an RFQ

As indicated in the f igure , the f i r s t secion accomplishes the trans i t ion from a beam havina time

independent character i s t i c s to one that has the proper variat ions with time (in th is sect ion the pro­

f i l e of the vanes is smooth). In order to obtain high capture e f f i c i e n c y the bunching and the energy

of the beam should be slowly varying functions of z . This is achieved in two different sect ions o*

the machine. Typically the quantity A increases ve-y l i t t l e in the shaper, while i t undergoes a s i g ­

ni f icant change in the gent le buncher. In the la s t s e c t i o n , s ince the bunching is nearly completed,

both the synchronous phase and the value of A are held constant, f igure 17 shows the suggested

variat ions of the parameters along the machine. It is important to recognize that when x, $ and k

are given then a and m are consequences of the assigned values for A and X.

R FQ ACCELERATION C Y C L E

4 " RADIAL GENTLE SHAPER GENTLE ACCELERATOR MATCHING BUNCHER

Fig. 17 A poss ible choice for the function A(z) , $.(z) and B(z)

Another important parameter is the maximum value of the e l e c t r i c f i e ld E s that should always

be kept below the sparking l imi t . Analytical and numerical ca lculat ions show that the maximum f i e l d

E s occurs around the middle of each separation. A good approximation for E s can be as fo l lows:

(41) r,

where a, the enhancing factor, i s near 1.4 and obviously depends on the pole t ip shaping.

If the operating wavelength x, the normalized focusing force B and the maximum f i e l d E s are

assigned, then combining the equations (28) , (39) and (44) we obtain the value for rQand VQ as

f o l l o w s 3 ) :

E.À qAE. (45)

If we ca l l E 0 the average wave of the amplitude of the e l e c t r i c f i e ld in each c e i l we can wr:te:

Ea a AV0 . E 3 2 A q . \ L , (46)

and th i s means that once the value of E 0 i s assigned (according to the se lec ted energy qain per c e l l )

then the value of A is determined. Prom Eq. (16) and taking into account that X and rQ are corre­

lated [Eq. (30 ) ] we can ca lcu la te (numerically) the values for m and a. Values for m equal to one at

the low energy end, and near to two at the high energy end, normally produce a good compromise

between acceleration and focusing e f f i c i e n c y .

8. RECENT DEVELOPMENTS

The RFQ described i s a very complicated radio frequency resonator which has the purpose of

creat ing the special RF f i e l d s capable of focusing and accelerat ing a beam. Following the f i r s t pro­

posal from Kapchinski and Tepliakov, i t was clear that any device capable of exc i t ing four su i tab ly

shaped e lectrodes could be used; the outstanding so lut ion studied and real ized at Los Alamos was

success fu l ly adopted in many laboratories and lasted unt i l new development: were presented at the

Santa Fe conference on "Particle Accelerators" in 1982.

The leading i d e a 5 ' that was very simple was for a non-uniform transmission l i n e , made with

four bars with c i rcu lar cross s e c t i o n , be used for creating the special f i e l d needed in an RFQ.

Figure 18 shows a very simple (.nd e f f e c t i v e arrangement. It is rather evident that each bar can be

turned on a lathe, while for shaping the vane pole t ip s the very complicated and expensive tridimen­

sional mi l l ing machine was mandatory. Moreover the reciprocal pos i t ion of the tars can be e a s i l y

adjusted without interfering with the container tha t , on the other hand, can have a cross sect ion

independent of the working frequency.

Fig. 18 Transmission l ine formed with four bars. The indicated shaping produces the focusing and accelerat ing Field (Frankfurt Univ.)

Wnile the mechanical advantages obtained with the four bars are r e a l l y enormo.js, there are some

doubts about the e l e c t r i c a l e f f i c i ency of th is structure. The surface offered by the four bars to

the RF currents is always smaller than the one offered by the equivalent vanes and the corresponding

four vane RFQ exhib i t s a larger shunt impedance. The choice of the best way for designing an RFQ

- 73.1 -

cannot be decided on theoret ical b a s i s . Oily the purpose for which each machine is designed can

indicate what i s more important; the mechanical s impl ic i ty or the RF power consmption. Figure 19

shows a sketch of the fundamental structure of an RFQ real ized with the bars. Only two bars, of

opposite polar i ty are shown for the sake of c l a r i t y . The U-shaped support can be considered as a

piece of uniform transmission l ine made from two paral le l metal l ic tapes . One end of the transmis­

sion l ine {the bottom) is short c ircui ted while the other end i s loaded with the four bars tha t , as a

f i r s t approximation, behave as a "distributed" capacity.

Let ZQ be the character i s t i c impedance of the transmission l ine and c the loading capacity of

the bars. Then, i f losses and radiation are neglected, the structure wil l exhibit an in f in i t e

impedance at the open end if the length of the support, the radian frequency u and the loading

capacitance obey the well known re la t ion :

' = z 0 Lanf u M ( 4 7 )

where Vf is the phase v e l o c i t y tha t , in our case , can be set equal to the speed of l ight in vacuum.

w P H Y S I C A L S C H E M E

OJ oj cZQ Tan ~- P = 1 E L E C T R I C A L S C H E M E

Fig. 19 The resonant support (foreshortened quarter-wavelength support)

If the above condition i s ver i f i ed then the metal l ic support does not perturb the bars (the so -ca l l ed

X/4 support). The whole structure of the machine can be real ized by supporting, per iod ica l ly , the

ba.'S with resonating supports. From the f i r s t proposal many different resonating supports have been

i n v e n t e d 7 - 8 ) and a large variety of devices have been te s ted . It is important to note that each

resonating support is magnetically coupled, at l e a s t , with the neighbouring one. Taking advantage of

t h i s s i tuat ion i t i s poss ible to arrange that al l the elements of the structure resonate in phase,

independently of the physical length of the bars. Consequently the amplitude of the voltage which

e x c i t e s the bars is constant.

- 73S -

The four bars and the supports should be contained m an appropriate meta l l i c tank in order to

prevent radiaion escaping from the structure but, in t h i s case , the container is not part of the

fundamental s tructure as for the vane RFQ.

A£KNOWLEDC' €NTS

The author i s indebted to C. Rossi for h i s coup, ation and to C. Guida and G. Bonaschi for

correct ing the present work. The drawings and the photographs obtained from my friends in Los Alamos

and Brookhaven have been essent ia l for t h i s seminar.

REFERENCES

1) I.H. Kapchinskii and V.A. Tepliakov, Linear ion ac e lerator with s p a t i a l l y homogeneous strong focusing. Translated from: Pribory i Tekhnika e sperimenta N.2 pp. 19-22 March-April 1970.

Los Alamos S c i e n t i f i c Laboratory Collection of Pap r s on the Radiofrequency Quadrupole (RFQ) presented by accelerator technology div is ion pers inel March 79 tc May 81.

2) J.L. Laclare and A. Ropert, The Saclay RFQ, LNS. 06: 1 June 1982, Laboratoire National Saturne.

3) K.R. Crandall, R.H. Stokes and T.P. Wangler, RF qutdrupole beam dynamics design s t u d i e s , Proc. Linear Acc. Conf. Los Alamos c o l l e c t i o n of papers (see Ref. 1 ) .

<*) J. Le Duff, Dynamics and accelerat ion in 1inear structure , Proc. CERN Accelerator School, General accelerator physics , Gif-sur-Yvette , 1984 (CERN 85-19, 1985), p. 176-7.

5) H. Klein, Development of the d i f ferent RFQ accele it ing structures and operation experience, Proc. Part ic le Acc. Conf., Satna Fe, 1983 (IEEE 1 ins . Nucl. Se i . NS-30, No. 4 (1993).

6) S. Abbott e t a l - , Lawrence Berkeley Lab. LBL 14624 (1982).

7) A. Schempp, H. Deitinghoff, M. Ferch, P. Junior and H. Klein, Four-rod >J2 RFQ for l ight ion acce lerat ion , Proc. Eighth Conference on the applicaiton of accelerators in research and industry, Denton, Texas, 12/14 Nov. 1984.

8) S.O. Schriber, Present s ta tus of RFQ, Los Alamos ational Laboratory, At-fto Ms HfiU - Los Alamos, W 87545.

FUNDAMENTAL FEATURED OF SUPERCONDUCTING CAVITIES FOR HIGH ENERGY ACCELERATORS

H. Piel

Dc| .jr uii'-nt of Physics, University üf Wupper Lai, Wuppertal, West C'-m.-iry

ABSTRACT

Super conducting accel era t j ng systems arc presently under design, 'est

or construction for electron positron storage rings and ¡ m e a r /ict «-

lera tors for nuclear phys i es research . Th j s seit i r.ar tries tc -j i v- an

i ntroduct ior. tc super conduct i r:<3 acce 1er at i r.g cav : r i f-s wh i ch. -in- the

muir, elements of these instruments, The fundamental features of super­

conductors in rf fields namely the surface resistance and thi_- funda­

mental limits of the accelerating field are discussed and thr- design

[.I inci [,1 es of super conduct ing cav i ti es for high energy acce 1orater s

are out J ined. Speci aJ a ttent ion is (,, j,1 ; to the a noma 1 ous lesses i r,

these resonaters which are responsible f<"-r the performance '. i mi ; a r k o s

observed today. Diagnostic techniques, defects, therma; stability,

hifjh purity niubiaii and, lasL but not loas', e Lectrc-. loadir.y a m the

Key words in this context. Cutr^.'it pre jeets i r. different labor -t ror i es

and the important parameters and achievements of expe.iments directed

towards the application of supercenduetinq cavities in high energy

accelerators are reviewed brirfly.

I . [ NTft OD L'CT 11 ) N

II is now more than 20 yea r s aero since the first el cet runs were uccel er ateo in a

superconducting lead-plated resonator at Stanford ''. Between l'JGfi and 197o very successful

experiments with X-band rescnaLors fabricated from bulk niobium ^' laid the ground for

1 arge sea le systems bu i 11 theroaf ter .

Iti the lieg inning of the 7Q's construct icr. of the Stanford Superconducting Recyclotron^,'

tht- Illinois M.icrctro:i using a superconducting accelerating section and the CERN-Karls-

rur.e s.c. Particle Seperator J' was started. In l'J74 a superconducting resonator success­

fully accelerated ar. electron bean to 'I ÇeV in the CORNELL Synchrotron. ^ and in 1976 the

construction of the Argonne s.c. Heavy Ion Post-Accelerator ^ was beg'in. In 1977 the first

Free Flùctrt.i Laser was operated using the high brightness heam of the Stanford Supercon­

ducting Accclerator ^ . Several of these devices have now been operated for many thousands

of huurs reliably and under routine conditions. It was shown that the drastic reduction of

the rf surface resistance in s.c. cavities could be achieved even in complex resonators.

The early expectations however, tc reach the very high electric accelerating or deflecting

fields premised by the elementary theory cf superconductors in radio frequency fields were

net Fulfilled, In analysing the performance of the s.c. rescndr.ors it is necessary to con­

sider their geometry. Superconducti ny structures for proton or heavy ion accelerators

therefore have to be discussed separately from accelerating structures for electrons.

According tc the title- of this seminar I want t(- focus r n velocity ''-f li'iht s'r c'jri'í.

This is done to concentrate on one important aspect of the application f '!"* s . - cavL tic-

Other important applications of s.c. c_viti.es aro in the field r[ heavy i<"-n aciii-Tdrcrs

{for a review see Reí. 9j . The exper i pruts with the 3 i noli- Ateo: Maser and the- Siiterccn

ducting Cavity Stabilized Osciliatcr are examples ef the successful application <f s.c

cavities outside accelerators i r. atomic physics ar.d metre Logy.

Although the accelerating fields of 2 to J KV/'n achieved in the first operating s.c.

accelerators were about 10 to 3Ü tiroes lower than expected from BCS-th^ory, the early re­

sults at X-band ar.d recer.t experiments at L- and S-band frequencies shew that there are nc

o'.ner fundamental limitations. Research and development work in rf superconductivity shoLil

therefore be rewarding.

This seminar tries tc give an ir.tro.1uc t ion to the fundamental features cf supereonduc

t m y cavities and is organized as follows: Jn the following section the concept of coupled

resonators which form an accelerator module and important quantities like the rf surface

resistance, the cavity Q and the shunt impedance arc introduced. Section 3 discusses the

fundamentals of rf superconductivity. A short introduction to superconductivity is given.

The surface resistance of a superconductor in an rf field is explained in the frame of a

two-fluid model and the critical rf magnetic surface field is introduced. The fourth

section gives design considerations for superconducting cavities and addresses the problem

of electron multipacting. In section five the importance of anomalous losses in s.c. cavi­

ties is outlined. The diagnostic i.ethod, microscopic defects, thermal stability and high

purity niobium as weLl as the progress in electron field emission studies are described.

Section six deals with cavities covered with superconducting thin films. Niobium sputtered

onto a copper cavity and niobium cavities with a Nb3Sn surface are the two subjects. The

last section gives a brief review of achic^em' nts i n pr'sent experiments diiected towards

the application of rf superconduct .vity to high er.ergy iccolyrators.

For additional reading or; tne subject of this seminar the references; '}, 15 and If' are

suggested.

SOME CAVITY FUNDAMSÍTALE

2.1 Coupled cavities

The heart oí each high energy accelerator is the rf accelerating section which gene­

rally is composed of a r.unbet of accu I er a t i rig .-nodulos each of which is a chrtin of coupled

ri resonators. For educational purposes we want tc assvinv.* that such a module is a string of

weakly-coupied pill-bcx fdvitirs as shov.n in Fiq. ¡a each of which is excited in the ™ Q I Q ~

mode. This mode has a longitudinal r-l^ctrjc field on the axis of the cavity which is

surrounded by the circular field h i w s ot the magnetic field which reaches its maximum at

- 738 -

J7070 Fig, la Chain of weakly-coupled pill­

box cavities representing an accelerating module

Fig, lb Chain of coupled pendula as a mechanical ri. logue to Fig. 'a

the cylindrical wall of the cavity. The accelerating module of Fig. la is a chain of coupled oscillators very much like the coupled pendula shown in Fig. lb. The; resonant fre­quency of the free pendulum corresponds to the resonant frequency <U q = ¿nf^!

u =2.405 c/a D

c = velocity of light (I) a = radius of pill-box cavity

of the TM o l o-mode of the pill-box cavity. The coupling spring between the pendula is equi­valent to the coupling electric flux through the small iris openings connecting the indi­vidual cavities.

In classical normal conducting linear accelerators such a module consists of many cavi­ties and is generally operated in a travelling wave mode. Tlie rf power is coupled into the first cavity of the string, travels down the structure and is absorbed strongly by the rf losses in the cavity walls. In superconducting accelerating modules these losses are re­duced by many orders of magnitude and a travelling wave operating mode is inappropriate. A superconducting accelerator module is therefore operated as a chain of N coupled resonators. Such a module is then excited in one of its N eigenmodes. By solving the characteristic equations of such a coupled oscillator system one obtains for the resonant frequencies u> of the eigenmodes and the axial electric field E n iq,t) of the n-th cavity the following relations

oiqa = u 2 (1 + K(l-cosaq)) [2)

E n ( q , t l = E o s i n ' n p - a q > c o s "q* O)

E = maximum axial electric field

N = number of coupled cavities

K = coupling factor between cavities.

- "739 -

2.2 Surface resistance, cavity Q and shunt iir.pedance

In normal conducting cavities fabricated from high conductivity copper the electro­magnetic field penetrates into the cavity wall by the skin depth & with

o = electrical conductivity [for copper at room temperature b. Bo* 107/!"!m)

u = magnetic perneability of the cavity wail.

At 5O0 MHz this skin depth is about 3.0 um. The rf losses per unit surface area P g produced in this thin layer can be expressed as

where H s is the magnetic surface field and R-; is the surface resistance. R s has the dimen­sion of Otims and for a normal conducting cavLty is given by

» _ fJUÏ, 1/2 _ 1

s ~ (2o' " aT * <e>

This gives at 5CX3 MHz a surface resistance cf 5 - 8 mí!. In very pure metals, a which is pro­portional to the mean free path I of the conduction electrons can be increased by more than four orders of magnitude if the conductor in cooled to the temperature of liquid heliu-n. The rf surface resistance however decreases only by a factor of about five. This behaviour is not explained by (6) and is due to the anomalous skin effect which has to be considered when i becomes comparable to the classical skin depth 6. In the limit of l»$ the surface resistance is given by

In the so callee? n-inode (q = N or = -) the decelerating module oscillates m its highest frequency and normalized to the acc. field has the smallest rf losses in its walls. This is the reason why the TI-raode is a favourite mode of operation for accelerating modules. In t-his mode the accelerating fields are equal in magnitude and opposite in direction in each pair of cavities as shown in Fig. la and seen from Eq. (3). A velocity of light electron which enters the first cavity at time O will enter the second cavity after a time T = d/c. If this time equals half the rf period (n/w^), then the electron will receive a maximum of acceleration in the accelerating module. The length d of one cavity of the module is then equal to 7[c/U]^ where U)^ equals the n-mode frequency of the module according to Eq. (2). ft disadvantage of the ir-mode is its sensitivity to mechanical tuning errors af the individual cells of a module which scales with N 2. The average accelerating field E a referred to fre­quently in this seminar is given as E a = V / 2 , where V is the voltage gain of the electron after traversing an accelerating module of length Í = N*d. E a is directly proportional to E .

740 -

S sca.es like ui and becomes independent of t.. It is there-fore of no benefit tc c-joi a

normal conducting cavity to low temperatures.

The guality factor Q of a cavity is directly related to its surface resistance. O is

defined as the ratio of the energy stored in a cavi'y to the energy lost fer rf period.

Energy can re transferred to the particle beam., it can be radiated cut of the cavity

through oper.ir.gs or antennas and it is dissipated and converted tc heat by t.ie rf losses

ir. the cavity wall. If only the losses P from the unavoidable Joule heating of the cavity-

wall are taker, into account one arrives at the ur.loaded Q of o cavity:

Q = Ï Ï (3) P w *

The stored energy U is proportional to the cavity volume and to the square of the average

accelerating field. scales with the surface resistance?, the area of the c w i t v wall and

is also proportional to E.,"- I" therefore can be shown that (8) reduces to

2 = G/R (0) o s

.•here Ci is the sc called geometry constant of the cavity. It is ir.dej er.i'-ni. of the cavity

frequency and for resonators like the ones shown in Fiqs. la and 4 is approximately 270 tc

iOO ÍL.

If P is the rf power per unit leng'.h necessar -' to i.aintain an accelerating f.eld

in an unleaded cavity, then the shunt impedance (per unit length) r of thv accelerator

cavity it, defined by

P = E Vr. d o )

The shur.t impedance is proportional to Q n . Tht: specific shunt impedance is defined as the

ratio i/Q . For a single cell cavity, shaped like the cells cf the accelerator structure

in Fig. '1, r/Q^-d is about 150 P.. The length d of one cell in an accelerator structure for

highly rolativistic electrons operated in the Ti-mode is r.c/u as already mentioned in

section 2.1. One therefore obtains from Eqs. C O and (10)

P - tr c R E 2 /1 G . (Ill s a

Froir. this cr.e concludes that the rf newer necessary to maintain a giver, accelerating field

per uni t length is, loi normal conducting cavities, proport ionaï tc 1 ' *". High frequen­

cies are therefore favoured for the operation of normaÏ conducting linear accelerators. At

SOO M H z , which is a frequency typical for storage ring cavities. Eg. (11) gives for copper

at room temperature and for = S MV/m a dissipated rf power of 1.0 MW/m.

This very high power dissipated in normal conducting accelerator cavities is the main

rea son i'or the interest in r f super conduct! v i ty .

3 . S U P E R C O N D U C T I N G C A V I T I E S

3 . 1 S h o r t i n t r o d u c t i o n t o s u p e r c o n d u c t i v i t y

I t i s w e l l known t h a t many m e t a l s a n d a l l o y s b e c o m e s u p e r c o n d u c t i n g n r l o w a c e r t a i n

c r i t i c a l t e m p e r a t u r e w h i c h i s c h a r a c t e r i s t i c f o r t h e s p e c i f i c m a t e r i a l . T h e h i g h e s t T ^

known t o d a y i s a b o u t 23 K e i v i n a n d t h e r e f o r e a l l s u p e r c o n d u c t i n g d e v i c e s h a v e t o b e c o o l e d

b y l i q u i d h e l i u m . O n l y t h r e e y e a r s a f t e r t h e f i r s t l i q u e f a c t i o n o f h e l i u m . H e i k e

K a m m e r l i n ç h O n n e s f o u n d i n L e y d e n 191] t h a t m e r c u r y l o s t i t s r e s i s t i v i t y c o m p l e t e l y

b e l o w 4 . 1 5 K . A g r e a t many b u t n o t a l l m e t a l s b e c o m e s u p e r c o n d u c t i n g a n d i t t o o k a l m o s t 19)

5 0 y e a r s b e E c r e B a r d e e n , C o o p e r a n d S c h r i e f f e r c o u l d e x p l a i n t h e m e c h a n i s e b e h i n d t h i s

p h e n o m e n o n i n t h e i r t h e o r y o f t e n r e f e r r e d t o a s t h e B C S t h e o r y , i t w o u l d b e b e y o n d t h e

s c o p e o f t h i s s e m i n a r t o g i v e a n a c c o u n t o f t h i s b e a u t i f u l t h e o r y b u t i t may b ^ u s e f u l t o

e x t r a c t s o m e i n g r e d i e n t s i n o r d e r t o e x p l a i n t h e t w o - f l u i d m o d e l o f a s u p e r c o n d u c t o r g i v e r ,

b y H . L o n d o n a l r e a d y i n 1934 . T h i s m o d e l i s v e r y u s e f u l f o r u n d e r s t a n d !r.q t h e b a s i c

f e a t u r e s o f a s u p e r c o n d u c t o r i n a n r f f i e l d .

I t t u r n s o u t t h a t d u e t o t h e i n t e r a c t i o n o f t h e c o n d u c t i o n e l e c t r o n s i n a m e t a l w i t n

t h e v i b r a t i o n s o f t h e a t o m s i n t h e l a t t i c e t h e r e i s a v e r y s m a l l n e t a t t r a c t i o n b e t w e e n

e l e c t r o n s . A s a r e s u l t o f t h i s c o n d u c t i o n e l e c t r o n s c a n f o r m i n t o p a i r s , t h e s o c a l l e d

C o o p e r p a i r s . T h e e n e r g y o f p a i r i n g 2 A ( T J | i ( T ) w o u l d b e t h e p a i r i n g e n e r g y p e r e l e c t r o n )

i s v e r y weak a n d i n t h e B C S t h e o r y g i v e n a t T = 0 t o b e

M o l = a k T ( 1 2 : ç

a = 1 . 7 5

k = B o l t z m a n n c o n s t a n t .

O n l y a v e r y s m a l l t h e r m a l e n e r g y i s n e e d e d t o i o n i z e a C o o p e r p a i r b a c k i n t o two

" n o r m a l " e l e c t r o n s . A t T = 0 a l l c o n d u c t i o n e l e c t r o n s a r e p a i r e d b u t a t f i n i t e t e m p e r a t u r e s

t h e r e i s a l w a y s a p r o b a b i l i t y t h a t a p a i r i s b r o k e n u p . T h i s p r o b a b i l i t y i s g i v e n b y t h e

B o l t z m a n n f a c t o r e x p ( - á ( T ) / k T ) a n d f o r t h e r a t i o o f t h e d e n s i t i e s o f n o r m a l e l e c t r o n s ( n e )

a n d C o o p e r p a i r s ( n c l we f i n d ;

_ - à(T)

V n c - e IT * i l 3 )

A t t e m p e r a t u r e s b e l o w T c / 2 , n a n d ¿ a r e v e r y c l o s e t o t h e i r v a l u e s r. a n d <Mo) a t T ^ O .

T h e " t w o f l u i d s " t h e r e f o r e a r e t h e s u p e r f l u i d o f C o o p e r p a i r s o f d e n s i t y n ^ a n d t h e n o r m a l

f l u i d o f c o n d u c t i o n e l e c t r o n s o f d e n s i t y n (TÎ w i t h

T

(14)

f o r T <

H . P . F e y n m a n 2 I ' g i v e s a v e r y i n s t r u c t i v e e x p l a n a t i o n a s t o why t h e fluía o f C o o p e r p . i i r s

c a n c a r r y an e l e c t r i c c u r r e n " , w i t h o u t a n y l o s s e s . C o n t r a r y t o n o r m a l e l e c t r o n s C o o p e r p a i r s

are Bose par ticies. When there are many Basons in a given state then there is an especially

large probability for the other Bosons to go into the same state. So nearly all Cooper

pairs will be locked down at the lowest energy in exactly the same state and it will not he

easy to get one of them out of this state. The probability to go into this state is by a

factor • n_ higher than into any other stete and n Q is a very large number. Therefore --ill

Cooper pairs move in the same quantum state. Resistivity comes from knocking on electrons

and transferring energy to the lattice but this becomes impossible because they are all

bound into Bosons.

Cooper pairs can be ionized by electromaqnetic radiation if the frequency is high

enough. Tne energy of the photons has to be

Ti ui 2 û (T) [15)

which in the case of niobium (2A(o) = 3.12 meVi results in a frequency of about 700 G H z .

It sht-'uld be noted that Cooper pairs are not closely bound like, for example, the

nucleus airi its i'lectrons in an atom. Cooper pairs are ordered states in momentum space

with the two elec trons having oppos i te bu t equal momenta and oppos i'e spins. For our

purpose however it is qualitatively acceptable, although somewhat superficial, to consider

a Cooper pair as d bound state with a rather large extension for which the coherence length

•!, gives a good mi.'Osure. -;, is a material constant and ranges typically between ?fl nm (nio­

b i u m and lf.OO um [aluminium). The distance between Cooper pairs is therefore considerably

smaller than their "size".

A sufficiently strong magnetic field will destroy superconduct tv i ty. The critical value

o: trie dp: J lied fielu is der-oted by H„ f Ti and exhibits a temperature dependrr.ee given by

K (T) = H (o) (l-(T/7 ) 2 ) - (i0) c c c

Meissner and Ochsenfelu found that i.-, a superconductor which is c o d e d in an external

field smaller than Il c, celr-w T r the magnetic field is completely expelled. The interior cf

the superconductor is screened by curren ts which flow in a very thi n skin layer. The

external magnetic field exponentially decays in this surface layer and its decay 1 e n g t h is

calied the London pen-'tration depth >• It ranges between 15 and 110 nm nr.d is material

dependent.

There are two classes of superconducting materials denoted as type I and type I I

superconductors. There is no difference in the fundamental mechanism of superconductivity

between them. They differ from each other only by a completely different Meissner effect.

A good "vpe 1 superconductor excludes a magnftic Field until superconductivity is

destroyed abruptly at and then the magnetic field penetrates completely. A good type I I

superconductor expells the field only for relatively weak external fields smaller than .

Above H c ¡ the fielt: partially penetrates into the superconductor which remains superconduc­

ting. At a much higher H field, sometimes lOO kûe or m o r e , the flux penetrates completely

and the superconductivity vanishes. The so called thermodyn-ïmical critical fiel-i is ther.

approximately the geometric mean of the lower and upper critical magnetic field:

H = (H • H . ) 1 / 2 . í c cl c2

3.2 Basic characteristics of a superconducting cavity

3.2.1 The rf resistance of a superconducting surface

In the case of a normal conducting rf resonator the electromagnetic field penetrates

by the skin depth into the cavity w a l l . In a superconducting cavity the equivalent "super­

conducting skin depth" is approximately equal to the London penetration depTh and there­

fore about two orders of magnitude smaller than 6. In contrast to the zero resistivity for

dc electric currents there are losses if the superconductor is exposed to a high frequency

field. This can be explained by the two-fluid model. The time varying magnetic surface

field, H £ cosidt, penetrates into the superconductor and induces in the ":;.c.skir. depth" .in

electric field. The amplitude of this field will therefore be proportional to ^P-s- The

electric field accelerates the Cooper pairs which transport this part of the surface

current without losses. It will also accelerate the normal electrons which can inteiact

with the lattice and produce losses according to the anomalous skin effect. The power dissi

pated in the wall of the s.c, cavity per unit area (the index t denotes the two fluid

model) can therefore be expressed as

P L - n (T) ^ H 2 . 118) s e s

Using Eq. (14) one arrives at

Comparing Eqs. 15) and (19) one yets for the surface resistance in the two-flu

frequencies well below 'he ionization limit and for T < T /?.-.

where A may depend on material parameters like X,l,,l and v^. For frequencies below lo GHz

and for T * T ¿2 the experimental data are in fact described well by the relation

The residual resistance R which is temperature independent and not related to the super­

conducting surface is easily separated. The first term in Eq. ( 2 1 ) , which is often referred

to as the BCS resistance, agrees remarkably well with tin -«='ilt of the two-flu,J -nodel.

- 7JJ ~

expressions for the S U T fact* osistance which *ro based the BCS theory ha'.-e bc-er, derived by Mortis and ääitleeii ¿ 3 iud Abrikosov, Corkcv and Khilatnikriv " '• Computations >-•* the saifàCB resistance based c- these rather complex expressions h$ve t>e<?n performed fry

H a l b n t t u r ""^ and Turn^aure , .\ further refinement of the BCS theory i n regard to rf superconductivity has been achieved by R. Blaschlte by including the 3nis°trupy of tin-

päjri n (, (jnc-igy which is induced fcv the anisotropy of ¿ crystal Jatrico. This modification

remtv^d a Long existing discrepare/ between experiment and theory in respect ta the frequen­cy (Impendence of the suïfact íesi la.ice. The quadratic dependtr.ee reflected by the two-fluid

2/ s ixodoi has to g^i into a v. b*?hav nr as tfie frequency approaches t^e irúim* C; ?ti l : m c .

t'ií(. ^ Fireriiioncy dependence of t : ;> surface Fiq. .1 Temperature <Jependor.£ç the s Ut-

resistance of niobiuw: at .2 K Faco r e s i s t a ^ cf niobium at 3

Fiqur L i ¿ compares experintentai d, i on the surface resistance of niobium at 4.2 K by

li- Klein <md G. Hujier w¡-j¡ the computational results of SlfäRC/-,*re. THi? agreement is

estcei lynt. Tin» two-fluid model d e r r i b e s the Er^guency dependence bel¡,w lo G!'* quite w e n

bu! cannot account for the chang, i f slojip at very hi<ih frequencies.

Fujure J shows tho tenperütu-dependence oí the sacíate resistance of a sir.qle ce\\

niobiua! cavity at a frequency of GHz. The exponent i <i 1 temppi.u.Lîrc i3f.perK:cnc<? explained by

the two-fluid model ¿5 nicely d<v strared as well as the existence of a residual resist­

ance K h i r i i is c h a tac tec ¡ zed by »• teinrOfatgrc i tidf pende net-, Extracting a Eroft( t_he data of

Fig. 3 and many othpr experiment it uthf>r f r eguenc i OS , ono finds a verV » ^ a r to 1 (or

¿ requeue t--s i-^iow J(JGf¡/.. JJîi-s value J S . n.sc to t]if-- pi L di 1.1 i un ut . i , , . . .-; U.-v.ry.

- 745 -

A t '>CO MHz a n d a t a t e m p e r a t u r e o f 4 . 2 K t h e B C S s u r f a c e r e s i s t a n c e o f n i o b i u m i s

7o nii c o m p a r e d t o t h e 5 . 8 mil o f c o p p e r a t r o o m t e m p e r a t u r e . F o r an a c c e l e r a t i n g f i e l d o f

5 MV/m ( e x a m p l e i n s e c t i o n 2 . 2 ) t h e d i s s i p a t e d p o w e r i n a s u p e r c o n d u c t i n g a c c e l e r a t i n g

m o d u l e w i l l b e o n l y 12 W. T h i s p o w e r i s a b s o r b e d a t 4.2 K a n d h a s t h - j r e f o r e t o b e c o r r e c t e d

f o r t h e C a r n o t a n d t e c h n i c a l e f f i c i e n c y o f a A.2 K r e f r i g e r a t o r . T h i s b r i n g s t h e 12 W t o

5 . 5 kW w h i c h i s t w o h u n d r e d t i m e s l o w e r t h a n t h e p o w e r d i s s i p a t e d i n a n e q u i v a l e n t c o p p e r

s t r u c t u r e .

A n o t h e r i m p o r t a n t d i f f e r e n c e b e t w e e n a s u p e r c o n d u c t i n g a n d n o r m a l c o n d u c t i n g c a v i t y

becomes apparent iE one combines t h e é q u a t i o n s 111) ana 121) n e g l e c t i n g t b e r e s i d u a l r e s i s ­

t a n c e R . O n e o b t a i n s t h e n f o r t h e p o w e r d i s s i p a t e d i n a s u p e r c o n d u c t i n g c a v i t y :

r e s

' . s - f ü r * " ° K 2 •

One = e e s t h a t , c o n t r a r y t o t h e c a s e o f n o r m a l c o n d u c t i v i t y , l o w f r e q u e n c i e s a r e p r e f e r r e d

i n s u p e r c o n d u c t i n g c a v i t i e s . P r e s e n t l y t h e v a l i d i t y o f t h i s s t a t e m e n t i s l i m i t e d t o s u r f a c e

r e s i s t a n c e s l a r g e r t h a n 50 t o 10O n i i . T h e r e a s o n f o r t h i s i s t h e r e s i d u a l r e s i s t a n c e Rr e s

w h i c h , t o o u r p r e s e n t k n o w l e d g e , i s n o t a p r o p e r t y o f a s u p e r c o n d u c t i n g s u r f a c e i n a n r f

f i e l d . I t i s c a u s e d b y a n o m a l o u s l o s s e s w h i c h a r e d e s c r i b e d i n m o r e d e t a i l i n s e c t i o n 5 .

T h e y a r e c r i t i c a l l y d e p e n d e n t o n t h e p u r i t y o f t h e c a v i t y s u r f a c e . C h e m i c a l e t c h i n g ,

e l e c t r c p o l i s h i n g , r i n s i n g w i t h u l t r a p u r e w a t e r a n d m e t h a n o l a n d v e r y h i g h t e m p e r a t u r e

t r e a t m e n t (up t o a b o u t 1 8 0 0 ° C ) i n a UHV f u r n a c e a r e f i n a l p r e p a r a t i o n s t e p s f o r s u p e r c o n ­

d u c t i n g c a v i t i e s f a b r i c a t e d f r o m n i o b i u m . N o r m a l c o n d u c t i n g r e s i d u e s l e f t on t h e c a v i t y

s u r f a c e b y t h e s e p r o c e d u r e s c a n c o n t r i b u t e s i g n i f i c a n t l y t o t h e r e s i d u a l r e s i s t a n c e .

A c h i e v e d r e s i d u a l r e s i s t a n c e s o f 1 nïï o r , m o r e t y p i c a l l y , 10 nfi c o r r e s p o n d t o o n l y a b o u t

0 . 1 t o 1 ppm o f n o r m a l c o n d u c t i n g s u r f a c e a r e a . I t i s t h e r e f o r e o b v i o u s t h a t s u p e r c o n d u c ­

t i n g c a v i t i e s h a v e t o r e c e i v e t i . e i r f i n a l s u r f a c e p r e p a r a t i o n a n d a s s e m b l y i n a c l e a n

room e n v i r o n m e n t .

3 . 2 . 2 F u n d a m e n t a l f i e l d l i m i t a t i o n s

A l l t h e c o n s i d e r a t i o n s g i v e n a b o v e a r e v a l i d o n l y i f t h e s u p e r c o n d u c t i n g c a v i t y i s i n

a t r u e M e i s s n e r s t a t e . On f i r s t s i g h t t h i s c a n o n l y b e t h e c a s e i f t h e maximum m a g n e t i c r f

s u r f a c e f i e l d H ^ " 1 3 * i s s m a l l e r t h a n H c o r H^j i n a t y p e I o r a t y p e I I s u p e r c o n d u c t o r r e s ­

p e c t i v e l y . T h i s s t a t e m e n t h o w e v e r may h o l d o n l y i n e q u i l i b r i u m c o n d i t i o n a n d may t h e r e f o r e

n o t a p p l y t o m i c r o w a v e c a v i t i e s . T h e t r a n s i t i o n f r o m t h e s u p e r c o n d u c t i n g t o t h e n o r m a l

c o n d u c t i n g s t a t e i s a p h a s e t r a n s i t i o n . S u c h a t r a n s i t i o n n e e d s n u c l e a t i o n c e n t e r s a n d i t

i s t h e r e f o r e p o s s i b l e t h a t t h e r e may be a m e t . a s t a b l e o r s u p e r h e a t e d s t a t e b e f o r e t h e s u p e r ­

c o n d u c t o r r e t u r n s t o i t s n o r m a l c o n d u c t i n g s t a t e . T h e maximum f i e l d u p t o w h i c h t h i s

t r a n s i t i o n s t a t e may p e r s i s t i s c a l l e d t h " c r i t i c a l s u p e r h e a t i n g f i e l d H ^ . i n t y p e I

s u p e r c o n d u c t o r s l i k e l e a d f o r e x a m p l e , i s h i g h e r t h a n H^_. F o r t y p e I I s u p e r c o n d u c t o r s

[ N b ^ S n f o r e x a m p l e ) t h e s u p e r c o n d u c t i n g s t a t e p e r s i s t s b e y o n d H^j b u t a t . n y s b e l o w t h e

t h e r m o d y n a m i c a l c r i t i c a l f i e l d H . H a t r i c o n a n d J a m e s h a v e c a l c u l a t e d t h e d e p e n d e n c e

o f H h o n K = by s o l v i n g t h e G i n z b u r g L a n d a u e q u a t i o n s w h i c h a r e b a s e d o n a p h e n o ­

mena i^g l e a l U n - , i ¡ u l s u p i - i C U I L U U I . t i v i L y . r h e i i l e s u i t s h a v e t h e l i m i t i n g i u i m

H s h s n o w s a s m o c J t ; n behaviour as K passes through the interesting value of which sepa­rates type I from type II -uperconductors.

The persistence of the Meiss.-ier state nay be very stable in rt fields. This is expec­

ted because the nucleaticn tine of flux lines is around 1C ' s compared to the 10 " s

typical for the rf period of microwave cavities. Experimentally the rf critica: íi"ld has

been studied for type I superconductors lik>- In, Sn and Pb near their critical temperature

T . Th" results t-f these experiments are ir agreement with theory i ¿ \ For a typical type C 3 1) II superconductor like Nb^Sn the lower critical field H^^ has also been surpassed

One therefore presently assumes that the fundamental limit for is given by thp critical superheating field. Table 1 gives some mati;ial parameters for Pb, Hb and Wb^Sn.

; sie 1

Transition temperature and critical fields : the most frequently used materials i r. rf su-percor.ductiv L ty :

Material T c

UJ !üe]

K H (o) sh

loe;

H 8 X P

s at T^2K

[Oe ]

E m a X

at T=2K lMV/m î

Pb 7.2 BOA loso 9oo J f i 22

Hb 9.2 2000 . 0 24CO 34! i c-w SO

Nb^Sn Iii.2 5100 4OO0 • o r ,o 3 3 ' 88

The maximum magnetic surface field in a cavity excited in the T n ^ ^ - m o d e is close to its

equator and a good rule of thumb is H s / F ^ - -5 Oe/MV/m. If is the ratio which is used :n

Table 1 to compute the maximum accelerating fii d E^ a X frcm HS n ( " l . comparisor. between

and H ge X ^ (compare E G . 116)) shows that evr the latest experiirenta 1 results do not yet

attain the t h e o r y . c a l expectations. This howev- r, from experimental evidence, is due to

the anomalous and point like losses described ir section 5. Today we do not know of any

fundamental limitation which prevents us from re :hing the limiting fields given in Table 1.

The high values for the accelerating field promu 'd especially for niobium and Nb^Sn cavi­

ties make it worthwhile to continue the experimen - >TI efforts.

4. CAVITY DESIGN

The main design critérium for a normal conduct q accelerator cavity is the minimi­

zation of the rf power necessary to maintain a givei accelerating field. The surface

resistance of such a cavity is fixed by the choice of the most suitable material, high con­

ductivity copper. Therefore r/Q^ has to be optimized. In superrnnducting cavities P can be

reduced by orders of magni t ude by c h a n m ng the opterai inq tempera ture and can be made .1 Icost

arbitrarily small for practical purposes if une succeeds in controlling the residual resist

ance. Therefore r / Q 0 is an almost free design parameter and other important design crite­

ria can be considered.

A.1 Electron multipacting and the spheri cal cavity shape

The resonant multiplication of free electron currents (called electron multipacting)

was a very annoying field limitation in practically ¿11 superconducting cavities before

ll!7y. This phenomenon was analysed and virtually eliminated by work done at Stanford ^ ' , 13) 14' 14) Genoa and Wuppertal ' in 1Q77 to 1979 Cavity shapes and, in special cases,

grooving of the cavity surface were proposed which later proved to suppress multipac­

ting up to the highest fields reached so far. Tt is because of this that today all s.c.

accelerator structures are of the spherical or elliptical design as can be seen from

the examples shown in Figs. 4 and 1.

Fig. '1 ibO KHz niobium cavity foreseen fcr the energy upgrade of I.EP

The whole unit has a length of 2.4 n.

showing the rf pewt : coupler (1), frequency tuning system with motor driver,

coarse tuner Í2) ar.d piezcelectrica1ly driven fine Luner (3)

- 748 -

Fig. 6 Cross section of a u-mode accelerator cavity of a design typical before 1'379.

The circle indicates an erea where one-point multipacting preferably takes place.

In the magnified view of this area multipacting trajectories of first, second and

third-order are displayed.

IE the local configuration of the time dependent electromagnetic field is such that

the electron returns approximately to its starting point after one rf period, it can pro­

duce other electrons by secondary emission. If the secondary emission coefficient is larger

than one, this process leads to a resonant multiplication and an avalanche develops. This

avalanche absorbs all the excess r£ power delivered to a cavity in order to increase the

accelerating field. Therefore the field is limited at a sr. called multipacting threshold.

If, in a very simplified picture, one assumes that the electron moves on "cyclotron

orbits" in the magnetic surface field B £ , thon its round-trip freauency u> would be e B s / m .

As the rf period has to be a multiple of the "cyclotron period" one would find the reso­

nance condition 3^ = (1/ni <m:,:/e) (with n = 1,2 . . . ) . One den. tes n as the order of the

multipacting trajeetory.

Although electron multipacting appears not to limit the pcrfonsano- oí superconducting

cavities any longer, a short account of this avalanche phenomenon v . l . he given. "Or" pcirt

niul tipacting", which is the m u l t i p a c t m g variety which has plagued s._. Lav.tivs of the

old design, comes about by the following process:

An electron of a few eV may be released from the cavity wall mo the rf field for

example as a knock-on electron from a cosmic ray event. It is acceler-Jitd by the local

electrjc field E which is perpendicular to the cavity surface and bent backwards to its

crigin by the tangential magnetic surface field H , as shown in Fig. t.

I n an accelerator cavity the time dependent lor electromagnetic s u r f a c f i e l j can

only be calculated with computer codes like SUPERFI: or URMEL '*<~'1 . fin analysis -A

multipacting trajectories and their resonance condit ns can therefore o n l y bf achieved b y

numerical integration. This has been done successful; a t Stanford i r . l ^T? 1 ' ' . Later, a

s im i 1 ar computer code was developed a t Wupper ta 1 . Foi f. he r r'.onance condi 11 on one found in

Stanford approximately :

-, • K ï -

wi th e j = O. d-i - O. ûÉ

Because cf (24) it was firmly believed before 1979 that, despite the -dependence

cf tie surface resistance, high frequencies should bt ..referred for s.c. cavities ir. order

to achieve high accelerating fields. The resonance condition 12A) however is not sufficient

for a multipacting barrier. The impact energy T of the "returning electron" has to be high

enough for the secondary emission coefficient 6 to be larger tiia.i 1 (see Fig. 7 ) .

3 -

a f t e r w e t t r e a t m e n t

b a k e d o u t a t 3 0 0 ° C

g a s d i s c h a r g e d e c r i e d w i t h A r

2- •

T(eV)

5 0 0 1 0 0 0 1 5 0 0

Fig. 7 Secondary electron em i ssion coef f j ci ent 6 of a n iob i um surface after dif ferent

surface treatments as a function at the energy T cf the impacting electrons

Under normal conditions the threshold energy for ¿ to become larger tha:: \ is about

30 to V ) eV. By argon discharge cleaning this threshold can be increased tt: about 1 SO eV.

This explains why it is Irequently possible to pass through a rnultipactinc barrier.

If an electron released from the cavity wall is accelerated by thp electric surface

field, £_,_ sin-jt, away from the cavity wall for half an rf period, then turned around (by

the magnetic field during this half period) and accelerated back to the wall it gains an

energy T of e 2 E j _ 2 / 2 m u i 2 . The Stanford computer calculations shew that this simple estimation

leads in fact to the right order of magnitude and the correct resu.t is:

2 F . J

with E 2 = 4 i 1

Si::ce T has t- be only about 4 0 eV to get á > I, which is fulfilled quite easily f c r typi­

cal E , ( ¿ 4 ) is a much more stringent condition for the occurrence of multipacting.

Experiments in Genoa were performed in 1 9 7 6 and surprisingly high fields were ob-14)

rained, well above the threshold placed by ( 2 4 ] , u, Klein and D. Proch noticed this

peculiarity ar-1 found that, in cavities of "spherical shape" like the ones shown in Figs.

4 and 5, stable multipacting trajectories were not to be found. This is attributed to the

fact that in a spherical cavity E_¡_ is quite large everywhere away from its zero crossing at

the equator. A multipacting trajectory therefore drifts after only two or three impacts to

the equator where E x = O and the rauitipacting electrons cannot gain energy.

Many experiments with spherical or elliptical cavities have been performed since 1979

but r.o one-point muitipacting threshold could be identified. As these thresholds scale with

frequency, the 1 0 . 8 MV/m reached in a 3 5 0 MHz cavity at CERN i s , so far, the best evi­

dence foe the fact that spherical cavities are virtually free of multipacting,

A very special variety of a two-point multipacting in a very close vicinity of the 4 2 )

cavity equator was found and analysed by W. Weingarten . This type of multipacting can

only take place on a very contaminated surface.

4 . 2 Higher-order modes and cavity design

An accelerator cavity is always operated by two power sources. One is the rf generator

which supplies harmonic power to the cavity in order to make up for the power dissipated in

the walls and absorbed by the accelerated beam. The other power source is the bunched

particle beam. The latter is by no means harmonic. It has a discrete but sometimes very

dense Fourier spectrum. Each cavity of an accelerator nodule has, apart fröre its fundamental

™ 0 1 0 a c c e l e r a L i n 9 rcode, an infinite number of eigenmodes at higher frequencies, the so-

called higher-order m o d e s . One of the Fourier lines of the beam may coincide with one of

the HOH's of the module. Then a high field is built up which may lead to a beam instability

but certainly to an unwanted joule heating of the cavity wall or to an excitation of the

cavity to its critical rf field. To avoid such circumstances the external Q of dangerous

higher-order modes have to be reduced to low values. This is done by special antennas, the

so called HOH couplers. The fundamental mode and higher-order mode couplet should be located

al the beam tubes of an accelerator moduli- !as in F w s . -i anj ') in ;rder not to - ¡ i s t u r b

the spherical geometry of the cells. A sufficient loadirg c-f .iOM's Ly ar propr :ai f couplinq

antennas then requires a good cell-to-cell coupling fo- as many modes i possible. A large

iris opening can be helpful but may also be dangerous. Each applicatior 'f a f,.c, c-iwty

needs its special optimization which can be performed today with computer ¡Tr/:rans !:ke 4 3 )

URMEL and TBCI . Such an optimization w.t; carried out for the superconauct, ng cavity fr.r

LEP and resulted in the design shown .n Fig. 4. The almost free choice fcr r,Q_ ••'•ry

much supports such optimization as already mentioned.

A bunch of charged particles which enters a cavity will produce a wakf field which can

react with the bunch itself and, for more than a certain threshold charge, may lead 10 a

disruption of the bunch. Such threshold currents can be increased significantly wher. the

metallic wall which surrounds the beam, and thereby its mirror charge, is at a lar 30

distance. Large iris openings (resulting in low r/Q ) and cavities of low frequencies are

therefore preferred. Superconducting cavities are well suited to fulfill these requirements.

5. ANOMALOUS LOSSES

5.1 Temperature mapping and microscopic defects

The origin of field limitations well below H g ^ and the causes of the residual resist­

ance are the main areas if interest for the research on superconducting cavities. Several

diagnostic techniques have beer developed to study these questions. In the framework of

this seminar only o n e , namely thu "temperature mapping in subcoolcd helium" ''' ', will

be described.

As each energy loss mechanism will finally lead to an increase of the temperature of the cavity wall, temperature measurements are of prime importance to identify causes for field and Q-1 imitations. C. Lyneis at Stanford was in 1972 the first to use a chain of ro­tating carbon resistors mounted a few millimeters from a cavity wall to detect the location

4 6 ) of a thermal instability

This method has since been used by many groups working in this field. The carbon

thermometers (see Fig. 8) used are 5G or IOO ÍÍ (1/8 W or 1/4 H) Allen Bradley resistors,

the bakélite insulation of which is often ground off to increase their sensitivity. Diffe­

rent electric schemes have been used to read the resistance value of the many thermometers

generally used on onv cavity, either an oscilloscope display or an automatic data acquisi­

tion system. During a quench all the energy stored in a cavity is set free and a substan­

tial heat flux develops which leads to film boiling and a marked increase of the tempera­

ture of the helium film close to the quench area. This can be detected easily even in su-

perfluid helium and if the resistor is not in contact with the cavity wall. The detection

of quench areas is certainly a most useful diagn^^tic procedure. A temperature map of the

surface of a cavity well below the breakdown field however, will reveal even more infor­

mation about the nature of high-loss areas. Temperature m.ipping can only be done for bath

- 7 5 2 -

E = copper beryllium spring

Al'i

The first set '-p usud fcr the temperature mapping of a 500 .'lliz spherical cavity is

shown i r. the photographs of Fig. :J.

temperatures above the X-temperature. The main obstacle for a temperature mapping f x p e n -

ment is the fact that only the temperature of the outside of the cavity wall can bc-

meascred wh ich is very e f f ectivel y cooled by the surround i nq iqu i d he liun. In an '•xfri-44 >

ment performed at CERN in 1976 it was shown that temperature mapping car. be carried

out quite well in a subcooled helium bath [favourable subcooled condition: bath temperature

slightly above T s and bath pressure - 1OOO mb) . I n a subcooled bath, bubb 1 »s art- flhsi-nt and

Lh(.-fL-fcre the mi croconvect i r,n produced by bubbles rising from the hr-ated s u r f a c L - is avoidtl.

This reduces the cooling capability of liquid helium substantially and increases the heat

transfer resistance between the niobium surface and the helium.

- "53 -

{ATlmK]

Q Cl Q O O i n o i n Q

— — cu

Fig. 10 E-irly température map of a CLRN 500 MHz cavity at E ¿ = 3.2 MV/m with line like regions of increased temperature due to the impact oE electrons field emitted

4 1 ) by point sources

After these 1'irst measurements the technique of temperature mapping was refined con-48,49)

siderably . The temperature increase ot the intermediate helium layer at che outside cavity surface was calibrated against the heat flux density and the dependence of this calibration on the batK temperature was experimentally determined. The relation between the measured temperature increase and the heat Elux density is very dependent on the "hydro­dynamics" of the flow of the local convection stream in the jubcooled helium bath. All these effects have to be considered carefully. The necessaiy calibration experiments can be carried out at higher temperatures (for example at a frequency of 3 GHz at 3 K) where

Thirty nine carbon thermometers (lOO U, 1/4 W Allen Bradley) slide under spring tension on the cavity wall and can be rotated around the cavity. The resistor voltages ar.d the:r angu­lar position are read by a computer controlled data acquisition system. Figure 10 shows one of the first 3-dtmensional temperature maps of a superconducting, 500 MHz, niobium cavity

41)

operated at an effective accelerating field of 3.2 HV/ra . This measurement was done in a subcooled helium bath at a temperature of 2.3 K. On the x-axis the distance along one circle cf constant latitude around the spherical cavity is plotted. The y-axis shows the number of carbon thermometers (with resistor 1 corresponding to the top of the cavity and resistor 39 to the bottom of the resonator). The vertical axis displays the temperature ncrease AT detected by the carbon resistor. The residual resistance of this cavity was

rather noor (R = 330 ní¡). It can be attributed to the very ncn-uniform high-loss area res 1 ^ at the top cf the cavity. In this early experiment at CERN the clear.-rocra handling was not as well developed as today and already at an accelerating field of 3.2 HV/m one observes strong non-resonant electron loading.

- ?£4 -

Fig. II Spatial distribution of the heat flux density on a 20-cell superconducting

accelerator module for the Darmstadt 130 MeV Recyclotron at E = 4.8 HV/rr.

In a few very important experiments at CERN , such defects were detected in 3 GHz

single-cell cavities by temperature mapping, then cut out of the cavity and analysed with

a scanninq electron microscope. Four of the photographs obtained are displayed in

Figs. 12 to 15.

the well known BC3 losses of the s.c. surface determine solely the heat flux through the

cavity wall.

The spatial distribution of the heat flux density on a 20-ceIl superconducting accele­

rator module for the Darmstadt Recyclotron is shown in Fig. 11. This map is typical for the

present day diagnostic technique and for the specific losses ohserved in a s.c. cavity.

Spikes in the heat flux density are seen on the flat background of uniform losses which are

expected from the BCS part of the surface resistance. Similar spikes have already been o b ­

served in the very first temperature maps at CERN, They are produced by high loss areas on

the rf surface which must be smaller than a Few millimeters in diameter. They are in fact

found tc bo microscopically small and in most cases invisible to the naked eye.

- 755 -

Fig. 12 Tungsten inclusion on a cavity Fig. 13 Nb sphere (presumably originating weld, probably embeded during from a welding bead). TIG-welding. Quench field Diameter: tin urn, E = 4.5 MV/m- Quench field - 6.8 MV/m-

Fig. 14 Microscopic hole in a weld of a 3 GHz cavity, causing a thermal instability close to E = 8 MV/m

Fig. 15 Chemical residue (drying mark). Diameter : 40O um Quench field E =3.4 MV/m.

Foreign material inclusions, beads from electron beam welding, holes in welds and chemical residues were found. During the mounting of the cavity to the vacuum system, to rf couplers or other parts, or during rapid pump downs particles can fall onto the s.c. surfaces. They can heat up in the cavity field to very high temperatures, emit light, cause thermal electron emission so leading to an excessive heating af their environment and thereby induce quenching. If a quench location is detected during temperature mapping a later inspection of the cavity often shows a dark spot composed of a central region and a halo as in Fig. 16. One can assume that this halo is produced by material from the "dust particle" evaporated during high field operation. In cavities mounted horizontally such particles would fall onto the equatorial surface, where the electric field is small. They would not give rise to electron loading and would initiate quenches only if they were

- 7 5 6 -

All the above observations lead to the adoption of very careful cleaning an<3 m o u n t ; n g

procedures for superconducting accelerator cavities. Chemical treatment of the cavities

with clean chemicals, the final rinsing procedures carried out with domineralizec? and dustr"

filtered water, and the mounting of the cavities to the test facility in clean rooms have

improved the reliability with which low residual resistances and high accelerating fields

can be presently obtained.

'i, 2 Thermal i nstabi lit íes and th& virtues of high purity niobium

Defect induced thermal instabilities and electron field emission from point sources

(see section 'ï.JJ are the nain mechanisms which linit the performance of s.c. c a v i U o s . A

defect cr, cavity surface like the c:-rs shown in Figs. \7 tc 1 =• is hinted in the rf field

and the Jissipated energy is transferred to the hcliurn bath. The temperature gradient pro­

duced -jcross the cavity wall may lift Un> temperature of the defect's environment above

the critical temperature of the niobium and a sudden dissipation of the energy stored in

the cavity will result. The threshold field of such a thermal instability can be uicreased

if. the thermal conductivity of ni-obiuir. can be improved . Xn standard, commercial!

reactor-grade niobium the interstitial impurities 0, C and N determine the poor thermal

conductivity . These impurities can be controlled to ,i large extent during electron

beam me 1 ti ng of the raw n iobiun and the consecut ive manu Tac tu ring stcp^> L¡f the shue t

materiai. The residual resistivity ratiu (RRR) of niobiun is proportional to its electronic

thermal conductivity. Typical RRR values of standard niobium range between 20 and -10. Due

to a refinement, in production techniques, niobium of RRR values between 00 and lfio is

commercially available since the end of 1903. This advance was ach.' cved mainly by improving

the vacuum condition and the procedure during the multiple electron beam melting of the

ni obium i ngots. The progress i Ii cavi ty per formancc? compared to the s ta tus of 1983 can be

attributed mainly to this improvement of the thermal conductivity NC Ït only the

obtainable fields have increased, but also the reliability with which the present design

fields of b MV/m can be reached.

An effective procedure to clean niobium or the most critical impurity, oxyqen, is the

evaporation of yttrium onto the niobium surface developed at CORNELL . During this

process the surfaces of a niobium cavity are brought into the proximity of an yttrium foil -5 o

at a pressure of about 10 Torr at 1250 C for several hours. A vapor deposited film of

several urn thickness traps the oxygen which diffuses rapidly from the bulk tc the surface.

The oxygen enriched surface layer of yttrium is then dissolved chemically. Starting from

standard material {RRR = 30) the RR.R value and thereby the thermal conductivity can be

improved by about a factor of three (depending on the initial oxygen c o n t e n t ) . Starting

from hiyh purity commercial niobium, RRR values of up to about boo were obtained at

CORNELL. The same technique has been tried experimentally at CORNELL and KEK using

much cheaper titanium foils at slightly higher temperatures, with similar success. The KEK

results on a single-cell, 5O0 MHz cavity in Table 2 were obtained that way.

Figure ¡7 shows the measured temperature dependence of the thermal conductivity \

of niobium samples of different purity. Curve a) represents the status until 1983 and

curve b) shows the quality of niobium which is now commercially available. Curve c) gives

the thermal conductivity of a niobium sample which was yttrium treated at CORNELL.

10 u 2 ¿ 6 10 Fig. 17 The temperature dependence of thn tlmrma1 conduct i vi I y

of niobium samples of different purity characterized by

its residual resistivity ratio IRRR) , U ) .

a) RRR ^ lu, b) RRR = 1 :?, c) RRR = 1UO.

- 758 -

A v e r y i n s t r u c t i v e d i s p l a y i s s h o w n i n F i g . 1 8 . T h e r e t h e p e r f o r m a n c e o f c a v i t i e s f a b ­

r i c a t e d f r o m n i o b i u m o f d i f f e r e n t p u r i t y i s c o m p e r e d . T h e m e a s u r e m e n t s w e r e c a r r i e d o u t

w i t h s i n g l e - c e i l , 3 G H z c a v i t i e s o f s p h e r i c a l s h a p e e x c i t e d i n t h e ™ 0 j 0 m o d e . E a c h m e a s u r e ­

m e n t i s made a f t e r a new c h e m i c a l t r e a t m e n t w h i c h d i s s o l v e s m o r e t h e n 20 gm o f i h e c a v i t y

s u r f a c e a n d t h e r e f o r e c r e a t e s a c o m p l e t e l y new s u r f a c e a s f a r a s t h e s h a l l o w p e n e t r a t i o n

d e p t h o f t h e r f f i e l d i s c o n c e r n e d . M e a s u r e m e n t s o f t w o l a b o r a t o r i e s (CERN a n d W u p p e r t a l )

a r e c o n t a i n e d i n t h e d a t a . T h e d e p e n d e n c e o f t h e c a v i t y p e r f o r m a n c e or. t h e p u r i t y o f t h e

n i o b i u m o r i t s RRR i s c l e a r l y s e e n .

2D 22 2¿ E (MV/m¡

P e r f o r m a n c e o f s . c . 3 G H z :

«. 5 3 , 5 9 )

i n g l e - c e l l c a v i t i e s f a b r i c a t e d f r o m n i o b i u m o f d i f f e -

T h e Q q v e r s u s E ^ d e p e n d e n c e o f t h e v e r y h i g h p u r i t y n i o b i u m c a v i t y , y t t r i u m t r e a t e d a t

C O R N E L L a n d b u i l t a n d t e s t e d a t W u p p e r t a l i s s h o w n i n F i g . ) H . T h i s c a v i t y i s n o t l i m i t e d

a n y m o r e b y d e f e c t i n d u c e d t h e r m a l i n s t a b i l i t y b u t i t s f i e l d i s l i m i t e d b y e l e c t r o n f i e l d

e m i s s i o n .

F i g . 19

v e r s u s E ^ d e p e n d e n c e o f a

RRR = 3CO s i n g l e - c e l l c a v i t y

w h i c h w a s o b t a i n e d b y t h e

y t t r i f i c a t i o n o f a n RRR = 8 0

c a v i t y . T h e c a v i t y w a s b u i l t a n d

t e s t e d a t W u p p e r t a l a n d y t t r i -59)

f i e d a t C O R N E L L

20 E n [MV/m]

- 759 -

5.3 Progresa in field emission studies

Resonant electron loading has been overcome in superconducting cavities as is des­

cribed in section 4.1. Also, the improving ability to avoid lossy defects on niobium sur­

faces, and the progress in thermaL stability of s.c. cavities, have allowed surface elec­

tric fields of more than 25 HU/m at all frequencies suitable for accelerating structures.

At such surface tields, field-emission induced electron loading is observed and consti­

tutes ^n important field limitation. Already one of the very first temperature ^aps ob­

tained at CERN in 39SO and displayed in Fig. 11 gave evidence for the existence of point­

like electron sources which emit at anomalously low electric surface fields. The measured

emission currents from the point-like sources seen in s.c. resonators do net correspond 74)

to predictions by the Fowler-Nordheim theory applied to an ideal niobium surface. The

origin oí this anomaly is still unknown but it can be assumed that the field emission in

rf cavities is related to the dc field emission from broad area cathodes. At the Driverait

of Geneva, experiments are underway to study the field emission properties cf niobium

samples prepared ; n a similar way to cavity s~r^r~es . The measurements are carried out

in a commercial Vacuum Generators "ESCALAB" UHV System including a scanning electron gun

producing a beam of 0.5 um ir. diameter, a 157° spherical sector electron analyser, a sec-n

dary emission detector and an argon gun. Niobium samples of 1.4 cm diameter can be fixed

to a purpose built manipulator which permits the cathode x-y-z-movemer.t necessary for the

field emission scans. The anode holder can accommodate several units, for example a 1 mx.

dLaneter flat anode and a pointed tunasten anode which has been electrolytically etched to

a micron size tip radius (Fig. 2 0 ) .

Tungsten tip ni the pointed .mode

of the "field emission scanning

microscope" set up at the Univer­

sity of Genev.i together with an

emitting particle on a niobium

Using this anode a high electric field can be produced on a very sma'l area of the

niobium sample. Peak surface fields of SOO MV/m have been me-isuied locally. By movinq the

cathode the anode is scanned automatically across the sample with a 1 urn Getting precision

Figure 21 shows a scanning image of 1 cm" of a niobium surface it different scanning field

and after different treatments of the sample. The scan along each line of one image is

carried out at a constant field. When a field emission sice is encountered, the electric

field is electronically reduced to hold the emission current below a fixed limiL. These

field reductions result in vertical deflections on thu plotted lines. A'ter localizing an

s-Its oí 0. Fischer's group át the University of r.eneva ore as fi

i general one can say that broad area c it nodes seeni to show t • sa.Tif- k i r,d

f]J emission as observed i ri j.c. cavities. In detail the foil w i n : s*. itn

a J i J : Tht- tiw ssions are most certainly no! comi ng 1" r _>m motâl 1 proir-jsi

'•1 ectr i c field tr.haficeir.er, t. The em i sy. ior. sites are usually ass iatt-d w;

rticlc-s, some of tr.em sitting probably rather loosely or. the s'. -ace. Th

l;o;, of these ;..¡jrt:cles is not unique. In a minority of casts r., -rticl

ù ri-si lutio;, or 0.5 »im. The emission from micron-size particles nJ.-rli

nee of the clear, room tc-chnigues applied to fie final csvity tn-a* .r.-nts

y. Another result of the Geneva group is shown in Fig. 21.

Chemically Pol i shed Nc Bakc-cut

Heating &O0°C 3o roir.s

Ik-atinq

• vJYh : : r. Vv, /• r fi

7»

|/ ft

Fif*3d emitting sites en a niobium surface and their son

Upper row: Scan-.ing field 50 MV/ai.

liCwc-r row: Scanning î ¿'-d toOMV/m. From left to right: No bakcout, fiOO°C, \A0C"C {3o mins

All scans are performed with an anode of 1 mm diameter.

ivity to bakeouts.

In the ESCALAS System the samples can be moved under UHV conditions to a station where they can be bak"d out at temperatures up to 2O0Ü C. Figure 21 shows c series of field emission scans oí une and the same sample which prior to each scan was baked out for JC minutes at a given temperature. The number of emitting sites is reduces", considerably after takeout at a temperature of more than SJOQ°C. This interesting observation certainly asks. í;¡r more- studies but it may already be seen as a hint, to apply l,:ih temperature filing und.-t UIIV aiiL clean room conditions for s.c. cavities lo surmount the field emission

¡peetabular result

-ÎOO f ar.d o treated at temperatures - t •.* leas

field Lave been teduieJ - . 5 i¿h «; i

number cf emitters reapf cat . "¡11 =

cations which dissolve in the niob

Lt-mpei ature specific for 'he impur

impurities are cf much greater importance than presently assumed.

shows that, if a sar.¡ .• vr.i

which the emitters a* a T J V V : .

q. ' j; heat ;rca-'-d aqain at Hoc"c, Î •*-•

. L. t;.- Geneva grcu; ; - . • v . ^ -

at high temperatures and whicr. s<_-jregat<

This observation nay indicate that m i c r ^

i,„ CAVITIES COVERED WITH SUPERCONDUCTING THIN Fi LWS

Because of the very small penetration depth cf an electromagnetic field into a super­

conductor, it is very resonable tc deposit thin superconducting films cf special properties

onto a cavity built from an appropriate supporting material. Superconducting lead-r^ated

copper resci.ators were among the first cavities tested for accelerator applications "'.

The Stor.y Brook post accelerator for heavy ions 7 -'* is successfully based on this techno­

logy. I T . mere recent times one has started a programme at CERN tc deposite niobium once a

ropper cavity. The experiments at the University cf Wuppertal on Nb^Sn covered niobium

resonators constitute another example of rf work with super conduct i r.ç thir. films.

(.! Copper cavities sputter-coated with a niobium film

It would be very desirable to have a reliable technique by which a film of pure

niobium of a few '„m could be deposited onto a copper cavity. This would not only improve

the thermal stability at high fields but also give the possibility to produce a niobium

layer virtually free of foreign material inclusions. A feasibility study towards this goal

was starLed at CERN in *9ßO. A method was developed tc coat a 50C MHz cavity made of OFHC

copper with a thin niobium film by dc bias sputtering . Figure 22 shews a schematic view

of the sputterir.fi arrangement. Three properly shaped niobiuin cathc3es are rotated inside -2

the copper cavity at a potential of - 1400 V and ¿n argon pressure of 5-10 Torr. In order

to confine the sputtering discharge to the front of the cathodes, the latter are surrounded

at the back and the sides by a shield at 4 mm distance. This shield is biased at + ño V

with refeience to the cavity wall which is at ground potential. During 24 h a sputtered

film between 1.1 um (equator) and 3.7 urn (iris) is grown. The results obtained in first

experiments with SOO MHz cavities in 1984 were quite encouraging. A maximum accelerating

field of 8.6 MV/m was reached wruch is comparable to the best results from niobium cavities.

The observed reduction of the cavity Q wit!, increasing field however required a furtner

improvement of the experimentol procedure to produce a niobium layer free of defects. In

very recent experiments an accelerating field of more than lo MV/m (Table 2] was achieved 9 9 in one case and, in another case, a low field Q of J.7«lo which reduced tc 2 ' \ 0 at the

5 MV/m design field of the superconducting LEP cavities. Experiments with sputter-coated

copper cavities are continuing at CERN to further investigate the field dependence of the

cavity Q and the technology of the niobium sputter deposition on a four-cell 350 MHz

c a v i t y .

- 762 -

F i g . 22 S c h e m a t i c v i e w o f t h e a r r a n g e m e n t t o s p u t t e r

n i o b i u m o n t o a 5 0 0 MHz c a v i t y a t CERN

6 • 2 Nb Sn c o a t e d a c c e l e r a t o r c a v i t i e s

T h e and t h e r e b y t h e s h u n t i m p e d a n c e o f a s . c . a c c e l e r a t i n g s t r u c t u r e i n c r e a s e s

e x p o n e n t i a l l y w i t h t h e c r i t i c a l t e m p e r a t u r e T^ o f t h e s u p e r c o n d u c t i n g m a t e r i a l ( s e e E q . ( 2 ) ) .

T h e r e f o r e n i o b i u m , t h e e l e m e n t w i t h t h e h i g h e s t T ^ , i s t h e m a t e r i a l most f r e q u e n t l y used

f o r s . c . c a v i t L e s . Among t h e A l 5 - m a t e r i a l s , c h a r a c t e r i z e d by h i g h c r i t i c a l t e m p e r a t u r e s and

c r i t i c a l t h e r m o d y n a m i c m a g n e t i c f i e l d s i'd^) , Nb^Sn g a i n e d e a r l y a t t e n t i o n . I t s T__ o f 1 6 . 2 K,

a o f 2 , 2 and o f J I O O Oe make i t a p r o m i s i n g m a t e r i a l f o r s u p e r c o n d u c t i n g c a v i t i e s . T h e

b r i t t l e n e s s o f t h i s compound i s o f no d i s a d v a n t a g e i n t h i s a p p l i c a t i o n . A Nb Sn l a y e r o f

t y p i c a l l y S um i s f o r m e d on a n i o b i u m c a v i t y by t h e v a p o r d i f f u s i o n p r o c e s s . T h e c a v i ­

t y i s p r o c e s s e d i n a vacuum f u r n a c e a t a r o u n d I lOO C i n a t i n a t m o s p h e r e w i t h a p a r t i a l

p r e s s u r e o f a few 10 ^ T o r r . R e c e n t l y , work w i t h Nb Sn r e s o n a t o r s has been resumed a t 64 )

W u p p e r t a l . A s i n g l e - c e l l and a f i v e - c e l l c a v i t y (3 GHz) h a v e b e e n c o v e r e d w i t h a

N b j S n l a y e r . F o r t h e f i r s t t i m e a d e p t h p r o f i l e o f t h e Nb^sn l a y e r was m e a s u r e d o n a

n i o b i u m s a m p l e w h i c h was t r e a t e d b y t h e v a p o r d i f f u s i o n p r o c e s s t o g e t h e r w i t h a f i v e - c e l l

c a v i t y ( F i g . 2 3 ) . T h i s m e a s u r e m e n t was c a r r i e d o u t u s i n g d i s p e r s i v e X - r a y a n a l y s i s i n a

s c a n n i n g e l e c t r o n m i c r o s c o p e o f 0 . 2 urn r e s o l u t i o n a t CERN. The t i n c o n t e n t n e a r t h e s u r f a c e

s l i g h t l y e x c e e d s t h a t o f s t o i c h i o m e t r i c N b ( S n b u t i s s t i l l b e l o w t h e upper l i m i t o f t h e

s t a b l e Nb-jSn p h a s e I t i s o b s e r v e d t h a t r e m o v i n g t h e f i r s t 0 . 5 o f t h e Nb^Sn s u r f a c e

by o x i p c l i s h i n g s i g n i f i c a n t l y r e d u c e s t h e r e s i ' . u a l r e s i s t a n c e o f a Kb Sn l a y e r .

- 763 -

F i g . 23

Depth profile of the Nb^Sn layer on a

niobium sample which was treated by the

vapor-diffusion process together with a

five cell cavity

0 1 2 3 U depthlum] 6

E 0 |MV/m|

.I L _ I ' • • ' • " i l , ID 1 2 L 6 10° 2 i b 10

Fig. 24

Dependence of the cavity on the

accelerating field and on the cool down

procedure.

a) after fast cool down

b) a f ter s low cool down.

Therefore, all cavities were oxipolished by thus amount, rinsed with deminera.îzed and

filtered water and dust-free methanol before they were mounted in the test sysvem.

To learn more about the seemingly high residual resistance of Nb^Sn and its signifi­

cant field dependence, and about field limitations specific to Nb^Sn, the temperature

mapping technique was applied to single and inulticell cavities. One component of the resi­

dual resistance was found to be dependent on the cool down cycle. The Q versus E curve

(Fig. 24) clearly shows the significant difference between the residual losses after a fast

and a slow cool down of the cavity. A careful study of the temperature maps taken in both

cases indicates that even the residual losses after a slow cool down a r e , at least in part,

caused by the same mechanism. The origin cf these losses is unclear. At present it is

assumed that frozen-in magnetic flux produced by thermoelectric currents and excited at

the Nb.Sn-Nb-inter face is r e s p o n s i b l e

He as u céments at 20 G H z , H GHz .MUÍ i Gil? [-fr fcrrv-d -Jt Wui.r.-r • .J 1 sr-,w 'i.-.r t [,.- r:- ir

t'-sni'jjJ r.'sisiar.cv inapusi-s with f r-•• ¡urr < y. The |i.wc-;i r<"iidu.i¡ i-sist-i- < f- ¡MÍ -L< ; ; 1 was

measured in a five-cell 3 GHz structure and was 2*/ n!.. Scaling this with 1 T O 1 . '. GHz

would result in a cavity Q of about l . 5 - l o ' ° at 4.2 K. Ir. fact 1.5 GHz is the design

frequency of the superconducting 4 GoV electron accelerator presently considered for con­

struction at CEBAF in Newport News, Virginia. This accelerator will be cenp-^sed of niobium

cavities of the CORNELL design and has to be operated at ~¿ K. A latf-r conversicr. to

Nb^Sr. covered resonators appears feasible and makes a further invest igat i orí of |ib,Sn cavi­

ties worthwhile. At 4.2 K, a theoretical of about 0 - : o ' ° is expected for Nb^Sn accele­

rating resonators at this frequency.

The accelerating fields obtained in Nb^Sn cavities are comparable to results from cavi­

ties fabricated from low purity niobium. Temperature maps taken on a five-cell Nb^Sn 3 GHz

cavity at different field levels shown in Fig. 25, show the existence of microscopic

regions of weak superconductivity. Already at low surface fields (lo mT) th-'sc regions

switch to a high loss state and lead to thermal instabilities. At present one can only

speculate about the nature of these switching defects. Impurity inclusions in the r.iobium

base material which disturb the uniform Mb^Sn layer and which become weak superconductors

by the proximity effect are one explanation. The use of the new high purity niobium for the

production <,>i Nb.Sji cavity is therefore a next experimental step.

Fiij. 25 Spatial distribution of the rf lasses in a Nb^Sn coated five-cell cavity

taken .it 2.1' K in subcoolod He at E = 2.ri5 MV/m ( a ) , ?.(• MV/m (b) and

i.'i MV/m tc). With increasing field a few presumably microscopic regions

switch into high loss areas ÍQ in .irbi I r.iry u n i t s ) .

CURRENT ACCELERATOR PROJECTS AND ACHIËVEMEIJTS

As a conclusion of this seminar a brief summary of experiments shall be giv^n wt-.uh

are directed towards the application of superconducting cavities to high energy accelera­

tors. Table 2 displays important parameters and accomplishments of these projects. Fiv-

projects span the frequency range from 350 MHz to 3 G H z . This is nicely documented ny th--

photograph in Fig. 26 where a 3 GHz single-cell cavity is compared to 3 50 MHz sir.gle—ceil

Fiq. 2i> The CERN 350 MHz s.o. cavity in coTp^risen with

a 3 GHz single cell. The cavity is xcur.ti'd hori­

zontally on its vacuum syst.'n. The rj tar in-.t arm

with <i0 gliding cvirboi: thermometers for tenpe-

raturr mapping can be seen.

The taust ambitious prc.gramino is underway at CERN. There, has started the prototype

W'.>rk for the product i en of einht. four-cell cavities of 35? MHz te- be installe.i ;n the LEP

siurj'ji! ring it: order to assess their performance under real eruditions ' '. rf ti.is proves

- 76l> -

Table 2

Best performances of cavities frota present high 'nergy accelerator projects

L A B O R A T O R Y

K A T L P I A L

TLMI'Í RATL'HF. (K)

CERN K E K CORNELL D A R M S T A D T / W I T F H PTA L.

I 3 0 M E V P E C Y C L Q T P O N

7.6

0.6

5.5

0.5

9-CELLS

5.5

5-CELLS

12. i*

# ) C*-vitits fabricated from high thermal conductivity niohiura } y t t r i f i e d ruobii

tc be successful then the energy of LEP will be upgraded to about lOO GeV per beam using a

s.c. accelerating system. The first prototype of such a four-cell unit is shown in Fig. 2 7 .

Fig. 2 7 Prototype of the four-cell 3 5 2 MHz superconducting cavities foreseen for the

energy upgrade of LEP

It had its first test in 1 9 8 5 . The design field of 5 MV/m and the desired of 3 > 1 0 were

obtained at the first attempt . Meanwhile a long term test has been carried out with this

cavity mounted in a horizontal cryostat. The cavitv was operated without problems for 9

2 4 0 0 hours at a fioLd of 7 . 2 MV/m and the design Q q of 5 « 1 0 . The research and development

work at CER:i concentrates at present on the development of the sputter-coating technique

as described in section 6 . 1 , and on the assembly and testing of a complete prototype super­

conducting accelerator module for LEP.

In 1984, a 500 MHz three-cell cavity was tested at KEK in Japan in the TRISTAN Accumu­

lation Ring. This was to demonstrate the feasibility of installing several tens of s.c. ca­

vities to the TRISTAN Main Ring. With an accelerating field of 4.3 MV/m an electron current

of 1mA was successfully accelerated to 5 GeV .

In March 1985, the most recent test of a s.c. system was carried out at DESY. A nine-

cell 1 GHz cavity, operated at 4.5 K with an accelerating field of 2.5 MV/m, stored an

electron b"*"- of 2mA at 7 GeV in PETRA This experiment is still going on and will try

to answer the important question of the long term behaviour of a s.c. resonator in a

storage ring. In addition, a development programme has been smarted recently to design,

build and test a 500 MHz superconducting rf module for HERA. This is in preparation for a

possible energy upgrade of the electron ring.

At CORNELL, a 1.5 GHz s.c. cavity system was tested in CESR in the last months of 1984.

Th e r e , the accelerating fields in the multicell structures were the highest achieved so far.

The 15.3 MV/m of the laboratory test of a fully equipped five-cell cavity show the

virtues cf the new high-thermal-conductivity niobium and they come close to the parameters

Fig. 28 Present status of the superconducting 130 MeV Recycjotron For electrons at the

Technische Hochschule Darmstadt. The small quadrupole and bending magnets of

the recirculation system are seen in the foreground, behind which the first

cryostats of the superconducting li.nac and injector are assembled. The injector

cryostat (background) contains one five-cell and two twenty-cell superconducting

cavities {see Table 2) which are expected to have their first system test in the

first halt of l')8(..

necessary for superconducting colliders. Thf research and development programme at CORNfr.I.L

is aiming at this next step.

fit SLAC tests are conducted with superconducting lead, niobium and Nt (Sii cavities at

2.yrj GHz, and which are driven by very short t- 1 i.si rf pulses. The results obtained indi­

cate that in this mode of operation fields close to the critical valu.es -as. be crnsistently

reached. Th advantages and disadvantages of the pulsed not hod nave be. r; .-ceci, t 1 y discussed

ir. an excellent review .

At CE OAF in Newport News, Virginia, o.-.e is presently planni/ig to convert thi- norma J

conducting design of a 4 GeV c.w. electron accelerator tu a superconducting facility.

Approval and funding of this project is expected and will have a very positive influence on

the further development of rf superconductivity.

At the Technische Hochschule Darmstadt, a I30 MeV superconducting Recyclotron for

electrons is under construction by a Darrastadt-Wuppertal Collaboration. The : ofrigera11cn

system and the cryostat for the linear accelerator, consisting of a 10 MeV ir.]ector and

40 MeV accelerating section, is presently set up and shown in Fig. 2 8 . Five accelerating

structures for this accelerator have been tested so far. Two of them reached inore than

a MV/n, the design Q of 3 * l 0 J being obtained in all cases. They were fabricated in 1084

from stock niobium with low thermal conductivity and a purity typical of niobium sheet

material produced before 1984. The first test of the 10 MeV in]ector is expected in the

first half of 1986. Parallel to this programme, work in Wuppertal is presently concentrated

onto the improvement of cavity performance with regard to the limiting fields, the develop­

ment of Nb Sn resonators and to metrology.

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- 7ö9 -

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51) H. Padamsee, ibid. ref. 1^, p. M 5 .

52) K.K. Schulze, Journal cf Metals ¿ 3 , 33 (1981).

53) H . Lengeier, W. Weingarten, G. Müller and H. Piel, IF.EE Trans. MAG-21 , 1014 (198^1.

T4) H. Padamsee, ibid. ref. 5 3 , p. 149.

55) H. Padamsee, ibid. ref. 53, p. 1007.

5(>) P. Kneisel, Cornell University, SRF 840702 : -.'jy4).

57) y. Kojitna, KEK 119851 private communication.

58) .z. Meuser, Diplomarbeit, Univers i ty of Wuppertal, WUD-B5-19 [ 1 985) .

59) n. Elias, University of Wuppertal (1985), prívate communication.

f-o) Ph . N lederir.iinr., N . Sankarrarnan , R..1 . Ncor , 0. Fischer , ibid . ref . I'.1, n -c)£3 .

01) 0. Fischer, University of Geneva, to be published ii: J. App;. phys. 5-J (19t~íí) .

62} C. Benvenuti, N. Circelli, M. Hauer and W. Weingarten, ibid. ref. 53, p. 1 •', 3.

63) G. Ar no Ids-Mayer , ibid . re F . 9 , p. (>43 .

64) M. Peiniger and H. Piel, TEEE T i g r i s NS-32_, 3fj\¿ (1905)

65) J.P. Charlesworth, I. Macphail, P.E. Madsen, J. Met. Science ^, 580 (1970).

6f>) B. Hillenbrand, H. Martens, H. Pfi.ster, K. Schnit.zke, G. /.i «>g1 t-r, ll-KK Trnnr. . M A G - 1 , 420 ( 1975) .

67) p. KntíÍ5el, J. Amato, J. Ki rchqessnor, K. Nakaj ima, H, Padamsee, H.L. Phillips, C. Reecr, R. Sundel in and M. Tignor, ibid. ref. r>3, p. lOOO.

- 771 -

68) Ph. Bernard, L. Lengeier, E. Picasso, CERN/EF/RF 85-1 (1985) and LEP Noce 524 (1985). 69) G. Ar no Ids-Maye J. , Ph. Bernard, D. Bloess, G. Cava] Lar i, E. Chi aver i, F. Haebcl,

H. Lengeler, R. Stierlin, J. Tückmantol, W. Weingarten and H. Piel, ibil. rf. 6 4 , p. 3587.

70) T. Furuya, K. Hará, K. Hosoyama, Y. Kojima, S. Hitsunobu, S. Noguchi, T. Nakazato, K. Saito, Proc. of the 5th Symp. on Acc. Science and Technology, KEK (1984).

71) B. Dwersteg, W. Ebeling, W.D. Möller, D. Proch, D. Renken, J. Susta and H. V o g e l , ibid. réf. 6 4 , p. 3596.

72) I.E. Campisi, ibid. ref, 53, p. 134.

73¡ Ph. Bernard, G. Cavallari, E. Chiaveri, E. Haebel, H. Lengeler, H. Padamsee, V. Picciarelli, D. Proch, A. Schwettman, J. Tückmantel, W. Weingarten and H. Piel, Nucl. Instr. Meth. 206, 47 (1983).

74) R.H. Fowler, L. Nordheim, Proc. Roy. Soc. Lond. A - 1 1 9 , 173 (1928).

75) J.K. Brennan, R. Coughlin, J. Hasstedt, J.W. Noê, P. Paul, R. PilLay, A. Scholldorf, J. Sikora and C D . Sprouse, IEEE Trans. Nucl. Sei., NS-32, 3122 (1905).

- 772 -

Laboratory Knergy (GeV 1 Acce 1er -it livi length |m, over a\ 1 length \m\

LAL/ORSAY . i Jlc .îi.O

KEK/TSUKÜltA 2.r. VJO .11)11

S LAC/STANFORD _M Utr>ii H M P

Table 1 Larne h n . k s in the world

The Jccelerat ing gradient lies in the range of ii to 1-1 MeV/m. Recei.tiy *l;e ;:i.AC lir.ao

lu\& Lrt-'en up-;r aiJi'U to ï i C.i-V and socn is ox;h'cuhI to r-'ach '-0 • :<_'V «i t h iicv k1yptr ons fol lowyd

by a pulse compression systtm''"'. In the last node of operation the acte : erat i ni gradient wi 11

HI G11 • :- IELD ELECTRON LINACS

J. Le Duff

Labora Loire de l'Accélérateur Linéaire, Orsay, France

ABSTRACT

High-field electron linacs are considered as potential candidates to provide very high energies beyond LEP. Fur almost twenty years Iii 11. iiTinruvnnrnf. h,j-. imcn I'Milr: on l i m r technologies as Lhey have been mostly kept at low and medium energies to Le used as injectors for storage rings. Today, both their efficiency and their performances ¿re being reconsidered, and for instance the pylse compression scheme developed at SLAC and introduced to upgrade the energy cf that lir.ac is a first step toward!*a new generation, of linear accelerators. However tin-, is not enough in terms of power consumption and r:ore oevi'lcpnent is needed tc improve both the efficiency of accelerating structures and the performances of RF power sources.

1. INTRiT-l'CTION

After i nt ro.hicing briefly the needs for higher '¡radien! eleclron luie-ir accelerators

by showing that simple ext r.ipolat ion of proüelit techno loi] i es would ui I ryim; to reach

much higher energies, we shalj.review different way:; of improving i ht-ne conventional tech­

niques and their related problems. Since Inili gradient means also high RF power we iili.ill

also present anil discims a new type of RF power source based on the double aim of reaching

i •h higher peak power with short pulses; and having much higher ef f i<: i curies

The present report will .iot. deal with lot-ally new accelerating techniques such as

laser-plasma and wake-field accelerators, since they are taken up by other lecturers.

2, EXTRAPOLATION OF PRESENT TECHNOLOGIES

Tlier<- are three large electron/positron lir.acs operating in the world (Table 1) as

injectors for storage r:»:]^ (although LAL and SLAC were initially built as high-energy

physics facilities!.

be as much as 17 MeV/m. Two bunches, electrons and positrons, will be simultaneously acce­

lerated, then transferred in the two arms of a circular transport system in ^ - . h a v a y that

they will collide once at a giver location. This will be the first linear colLider {SIC)

coming into operation in the world, at an energy level comparable with LEP stage 1. Tt will

serve as a test bed for future linear colliders as well as fo studying the intermediate

boson Z . o

In order to reach many hundred GeV or a few TeV in the center of ir.ass with electrons

and positrons, it appears that linacs are better suited than storage rings since circular

machines would lead to enormous power being radiated in the bends; remember that in L E ? , ope

rating at 100 GeV per beam, each particle will loose 2.6 % of its energy per turn. Clearly

the 5LC scheme, with a single linac plus a circular transport systen, will be a!io avoided

f ' r higher energies^ and future linear colliders will consist of two liriacs firing against

each other.

Consider now a first step in the linac energy, by rougnly one order of magnitude,

using present SLAC technology for the accelerating sections. Table 2 gives the resulting

constraints aiiJ three possible schemes have been considered, knowing that :

E - [P, I * acc input

Pa . c . • f r o p 'V \ P K

eleratina gradient, F the input RF power at the structure, input r

the linac repetition frequency, N^,, P^, being respectively the t-tai nu:iLer jr .-iL

the peak power and the efficiency of each klystron.

SLAC Super SLAC Super SLAC Supei SLAC (" today) (1) <?) 13)

E iGeV)

L [Km I

\ P K iMWj E [MV/m] acc f \Hz\ rep sections/klystron

pulse length (us]

P IMW]

2.5

13.7

2400

30

2.5

137

240

3800

100

180

4

2.5

I 370

950

100

1G0

1

2.5

1370

Table 2 nxtrapolation of SLAC up to 300 GeV

strc

the

Clearly a higher gradient keeps the linac length to a reasonable value but introduces

.., constraints on the power. For instance new power sources need to be developed at

level of 1 CW peak. The state of the art in klystrons is 50 Mw peak with 5 i.s RF pulse

GeV Power/1 irjc Coiner: t s

200 conventions 1

need ior new power

neeJ for new pow sources and new structures

new accelerating method.-;

Table i Possible s Lages for a linear collider alor.t a LKP diameter

has b'.'c.-i r.;r.:nizcii t.y ..-.itehiini the accélérati raj structures to the pu Ire compressor <sec

.,eiit sc-Ctic-.".; , but even uo i: :ti t,::: c-..:!,¡ c I c WI tî. lin.' I.Hi' st.cr.j::i_- at t':.L-

same enei'jy i second case uses an accelerating gradient cf 12b MeV/n which has been

already reached ui; u-xpermental structures. However it can only work if o m - uses 1 GW, s":ior'_

pulse H¡' p'iwer ¡v.'irL'i-s wliU'h ii, i.ot «.11: t y > • I . in L-jt.li i;asiü; a h i u i rej t i i un. rute oí

I'JUij lit '',.iü -•..'i.iii-iered.

Improvements on the power consumption may come from improvements in the efficiency of

accelerating struct ires and also from some tricks such as for instance the use of pulse

current trains that can lowur the repetition rate for .he same luminosity^'.

3. RF COMPRESSION SCHEME

3.1 - Present, situation

Although this scheme can be considered as an already exist

wi.i 1 hwhiltí tu r e e l l the p;inciples since one can expect to i-.:.r.

system in the near future.

The pulse compression is schematically represented on F I E .

technic!-e

first part of the

1 Phase shifter f\

-0- Klysto>r> jj~3db Coupler

Pulse compression scheme

long pulse from the klystron is stored in a couple of low loss cavities. At a liven t iir.e t^

the input signal LO the klystron is rapidly TT shifted so that the energy is now reflected

at the entrance of the storage cavities and directly goes to the structure, in addition the

stored energy flows out of the cavities and also goes to the structure making the peak ener­

gy during the time interval I^j'^' much higher than it would have been fron: a direct, feed

of the structure.

The method can Le either used to increase the energy of an existing linac, or t«i save

on the total number of power sources for a given output energy :

?) SLED (SLAC_Energj/ Doubler^

Storage cavities are placed between the klystrons and the accelerating structures. The

RF pulse length is 5 j.s and since the filling time of the structure is .H ,¡s one can adjust

the switching time such that t^ - t^ - .8 ,is. However the compression scheme lias a poor effi­

ciency and the maximum improvement factor on the peak power that, one can expect is about 3

leading to an improvement factor ^3 on the accelerating gradient.

LIPS_(LEP Injector_Power S a v e r ) " 1

The scheme is used to reduce the number of klystrons by a factor 2 as shown on Fin- 't.

Instead of feeding two structures from a single klystron, one can i >ed 4 structures with the

- 77ö -

same total beam energy if the system is adjusted to increase the effective p ak power by a

factor ¿. This improvement factor is lower than "he previous one and can be obtained freo

a "i.'j ,,s klystron pulse length, the filling time of the LEP injector Linac (LID structures

U-::. : I-.'-s. J T will Le seen later T H A T the unprovcnerit factor : ; .AIR.ly D-'p.:.-: -j-,, L R . ( - : : ' : -i

klystron pulse length when all other parameters, such as for instance the filling time of

the accelerating structures, have been optimized.

± 1 1 1 1 1 i i i i i.J

L? PA Y J

LP P/2 PA [P/2

1 1

i i i i i j

1 1 1 1 1 1

P/2 P/2 1 IP/Í ..... • i i i 1 1 I

1 1 1 1 1 1

Fi p. 3 The LIPS scheme

Up to here the- maximum improvement factor one can reach with a conventional electron

linac, Ly adding a compression system, is of the order of 1.7 and from there existing linacs

nicht bo ,'jble to operate with an .uceli-r-n i n i -iraiient u: \i.<: order 'if MV/:¡., tulun ¡ into

.ï(..vJ,jni ,il:,o thfr up-to-lotr. klystrons and as;¡ui:iirn Uiat < UJII klystrt;r. I cfl: ,i •; i n ; 1 structure.

As will be seen in the next section, ,m add i I. i -. IM 1 improvement fat: tur •:nu i» < <l.t n i n. ...J by

optimizing the parameters of the accelerating structures to match properly the compression

system.

J . - Optimization of TW accelerating structures for SLED operatic.: 7)

The RF" pulse shape due to the compression

scheme, worked out by Z.D. Farkas et al.', ir.'

shown on Viy,. 4. There are two regions of

interest : region 1 which corresponds to a

continuous increase of the stored energy in

the cavities and region 2 which, after a n

phase shift at the klystron input occuring at

time t j , corresponds to a relatively fast

decay of the stored energy in these cavities

to the benefit of the accelerating field. Pulse shape due to compression

For a unit rectangular klystron pulse the combined field entering the accelerating

section is :

-L/t E ] ( t ) = (a - I ) for 0 * t 1

2Qc/^it-ií) is the filling time of the storage cavities

is the unloaded quality factor of the storage cavities

is the coupling coefficient of trie storage cavities

2p/(l + 6)

a l 2 - e * C ) .

For a given peak power P fron the klystron (rectangular pulse) the accelerating field at

the input of the section would be :

where r , v , Q are respectively the shunt impedance per unit length, the group velocity o go and the quality factor of the first accelerating cell.

In a constant impedance structure all the cells are identical, and hence r, v . Q wil

remain constant along the structure.

Due to power dissipation in the cells the amplitude of the propagating field will

decrease exponentially. At a given azimuth z the field beco:..es :

-lu/2v fi)z E(z,t) = E It- At(z) ] e 9

where index í,2 refers to the two different time intervals as previously defined. Here aga

the expression needs to be multiplied by E q for Û given peak power P from the klystron.

¿t(z) is the wave propagation time from the origin up to z :

Ltízi =

It looks interesting to use the normalized variable z' = ? where L is the length of

the structure. Then :

At = T z' a

with L

a v go

Depending whether the time t- At appears to be below or above t j , the field Ej or E., should

be used. That tells us that a field discontinuity will appear at some location zj in the

structure such that :

If zj ' 1 the energy gain along the structure is the contribution of two field inte­

grals :

E_( t - ¿t U ' ) ]e E It - /itízM le

where now L represents the time at which the particle traverses the structure (the transit

time of the particle is negligible compared to the filling time of the structure),

Let cali V, a:¡d V the integrals relative to C, and t,. One lets ;

V = V, + V_

V, t z M = - ( o - 1) —

V (z¡) = (u- 1)

T . i 1 T , 1

with : — = —

1 I _ -x 2 • r c - ¿q •

It is intercstin-j to look at the behaviour -JÍ the function Viz,') in the interval

û «" zj < 1. It can Le shown numerically that for each value of ;:j th'ire is a value of it

which maximizes the energy gain. This has been taken into account in the plots of Vic.. 5,

where it appears that the maximum energy 'jam corresponds to fcj = 1 , which means that the

bean should enter the structure at time t = t^ = tj -* : an*; fiat the width of the t-oir;pres-

sed pulse must be equal to the filling time of the structure.

Yig. 6 Multiplication factor versus the filling time of it constant impedance structure

- 780 -

This leads to

SI-"-However this is not the exact energy multiplication factor since for a un iL pul se ente

mstant impedance structure the energy gain over a unit length is :

Hence the real multiplication factor is the ratio VM / V

0 - F o r each value of ^ there is

a value of i, hence a value of T , which maximizes this multiplication factor as seen on c r

F i « . 6 .

ft similar treatment for the case of a constant gradient structure would show that the

efii iency of this type of structure is either the same, at low filling time, or slightly

less, at ,igh filling time, than the efficiency of a constant impedance structure. Hence we

shail proceed with constant impedance structures in what follows.

It has been seen that for a given structure length there was an ensemble of optimum

values for 5, T _ and which realize the correct matching between the SLED pulse and t;ie

accelerating structure. It is interesting to look in more detail at the performances of

these structures vrrsus different parameters, like the parameter setting of the storage

cavities ÍQ , &) , the length and the aperture of the accelerating structures, the width of

the direct peak power pulses from the klystrons.

For a constant impedance structure, fed by a klystron peak power pulse P, t^> through

a couple of storage cavities with a IT phase shift at time t - t - T , the energy gain is ;

where R = rL i3 the total shunt impedance of the structure = 1,/v its filling time.

The fact that for a given length there is an optimum value for means that there is

an optimum value for v , hence for the iris aperture 2a of the structure. To illustrate g G) this point let us consider the cell characteristics of the LEP injector linacs (LIL) which

operate at 3 G H ? in the 2i/3 mode ;

= l'.2ü0

r = öf) - 3 - i> (2a) 2

v /c = <2a) '••2i/mi •1

where 2a, the iris diameter, is expressed in cm the shunt impedance r is in Mii/m.

- 7S1 -

Ir" a s i n g l e s t r u c t u r e i s f e d b y o n e k l y s t r o n t h e a v e r a g e a c c e l e r a t i n g g r a d i e n t b e c o m e s

52 MeV/'in. A s m a l L e r v a l u e f o r (2a) w o u l d l e a d t o a U i c j U e r g r a d i e n t , f a r i n s t a n c e 2a = 1 . 6 5 c m

g i v e s 75 MV/m a n d t h e c o r r e s p o n d i n g s t r u c t u r e l e n g t h i s 1 . 3 m e t e r s .

F i n a l l y w i t h s h o r t c o n s t a n t i m p e d a n c e s t r u c t u r e s o p t i m i z e d t o n a t c h t h e S L E D c o n d i ­

t i o n s , a n d c o m m e r c i a l l y a v a i l a b l e k l y s t r o n s o n e c a n g e t c l o s e t o I'.Ki neV/in i n a s l i ^ r t t e r m

f u t u r e .

F i g u r e 7 s h o w s t h e e v o l u t i o n o f t h e RF p e r f o r m a n c e s v e r s u s t h e i r i s d i a r . e t e r , f o r d i f ­

f e r e n t s t r u c t u r e l e . i g t h s . A s t h e leii-:t:i i e c r e a s u s t h e i r i s d i a : . - . t e r o s o i-LrvJstî , i

t o g e t t h e maximum g a i n c o r r e s p o n d i n g t o t h e r i g h t m a t c h i n g v a l u e f o r . . I n a l l c a s e s ¿

a n d : h a v e b e e n o p t i m i z e d , c

T h e maximum e n e r g y g a i n s o b t a i n e d f o r e a c h s t r u c t u r e l e n g t h a r e p l o t t e d o n F i g . fl a s

w e l l a s t h e c o r r e s p o n d i n g v a l u e s o f a n d : ^ w l i i c h c l e a r l y r e i r . a i n c o n s t a n t .

A s y s t e m a t i c s t u d y o f t h e e n e r g y g a i n a s a f u n c t i o n o f t h e o t h e r p a r a m e t e r s , l i k e r ^ ,

Qc a n d Q | ^ = c t e ) l e a d s t o t h e f o l l o w i n g c o n c l u s i o n s ;

- n e i t h e r Q n o r h a v e i n f l u e n c e o n t h e o p t i m u m v a l u e o f i ^ . B o t h g i v e a l i t t l e e f f e c t

o n t h e o p t i m u m e n e r g y g a i n . T h e o p t i m u m v a l u e o f T _ c h a n g e s w i t h

- t h e o p t i m u m v a l u e o f c h a n g e s w i t h t h e w i d t h t ^ o f t h e d i r e c t k l y s t r o n w a v e . F o r

l o n g p u l s e s o n e c a n h o l d a I c n . j e r f i l l m . j t i n e , but t h a t r . e a n s a s m a l l e r a r w : j n - f o r a f i x e d

s t r u c t u r e l e n g t h . An i m p o r t a n t i n c r e a s e o f t h e e n e r g y g a i n f o l l o w s a n i n c r e a s e o f t 7

- o n e o f t h e m o s t i m p o r t a n t f e a t u r e s , - o n s : JeriM: t h e r e s u l t s p l o t t e d -r. F i t - . . 3, i s

t h a t t h e t o t a l e n e r g y g a i n f r o m o n e k l y s t r o n s o u r c e w i l l b e h i g h e r i f t h e p o w e r i s s h a r e d

b e t w e e n s m a l l e r s t r u c t u r e s , f o r t h e same t o t a l l e n g t h . T h i s f a c t i s i l l u s t r a t e d o n F i g . 9 ,

a s s u m i n g n o p o w e r l o s s e s i n t h e RF n e t w o r k s , a n d k n o w i n g t h a t t h e e n e r g y g a i n f o l l o w s t h e

s q u a r e r o o t o f t h e i n p u t p o w e r . O f c „ " r s e s m a l l e r s t r u c t u r e s , w h e n o p t i m i z e d , w i l l h a v e

s m a l l e r a p e r t u r e s a n d t h e i n t e r e s t i n g r e s u l t i s t h a t t h e minimum s t r u c t u r e l e n g t h w i l l d i r e c ­

t l y d e p e n d o n t h e beam a p e r t u r e r e q u i r a m o n t . F o r i n s t a n c e a minimum a p e r t u t o o f 1 . 3 cm w o u l d

l e a d t o a d e s i g n l e n g t h o f I .8 m f o r L I ! t y p e c e l l s , a c c o r d i n g t o F i g , 1 0 ,

I n o r d e r t o d e s i g n a n o p t i m u m l i n a c s t r u c t u r e u n d e r S L E D o p e r a t i o n , it i s u s e f u l t o

d r a w d e s i q n c u r v e s h a v i r . o t h e m a i n d e s i g n p a r a m e t e r s , P, , > Q, O a n d t „ . S u c h a d e s i o n ^ k l y s t r o n K Kc ¿

e x a m p l e i s shown on F i g . 1 1 . I f o n e i n t r o d u c e s a d e s i g n c o n s t r a i n t s u c h a s ' ¿ a ' n i ¿ n= í'.Ocn:

o n e g e t s d i r e c t l y t h e r e m a i n i n g d e s i g n p a r a m e t e r s w h i c h i n t h e p r e s e n t c a s e a r e :

; = 2 . 1 2 y s c

L = 2 . 5 m T = , P >>s a

li = 8 t = 4 . 2 u .

F i g . 8 Maximum e n e r g y g a i n a s a f u n c t i o n o f t h e s t r u c t u r e l e n g t h F i ^ . 10 Opt imum a p e r t u r e o f a n i r i s l o a d e d s t r u c t u r e v e r s u s t h e s t r u c t u r e l e n g t h

Fig, U Design example : P = 40 MW, Q = 15200, Q c = 180Û00, t2 = 5 us

4. ULTIMATE ACCELERATING GRADIENTS IN CONVENTIONAL STRUCTURES

A question can be raised now ; can we reach in practice the gradient previously men-

tionned and can we go even further ?

The answer to the first part of the question is mainly related to breakdown I Í K . J L B in

warm structures and will be treated in this section. If no limitation occurs one way tc .:O

further consists of improving both the efficiency of accelerating structures and the peak

power of RF sources [their efficiency too) and this will be treated in the next two sections.

¿.1 - The Kjlpatrick criterion

Breakdown phenomena may occur at high field level on the walls of accelerating struc­tures a/iJ they art' nut very Well understood ai. ui.rowvr- freguejicieii. The study Jor.e by

9)

Kilpatrick is one of the few investigations of breakdown phenomena and was in the past

very often referred to by accelerator designers. He empirically derived a relationship

between frequency and maximum electric field :

f = 1.643 E 2 exp(- B.S/E I max raax

where f is the RF frequency in MH and E is the maximum electric field in MV/m. At. ^ z max

f = 3000 MH this relation predicts E = 46.8 MV/m. z max

The corresponding maximum accelerating field now depends on the type oE structure. For ins­

tance disk loaded waveguides have a ratio " w a [ ^ i ;a c c

o i c i i e o r d e r of 2, hence the maximum

expected gradient would be 23 MV/m. This could be one reason why accelerating gradients ¡.ave

been kept below this value for a long time, but certainly another good reason was that the

klystron peak power was still low and long accelerating structures w.^re x.aking a botter use

of this power in terms of maximum beam energy per klystron [no SLED) . The overall iir.ac

length was not a big worry at that time.

- 784 -

Accelerating structures with higher shunt impedances would lead to lower j:ia>:i::,uj:. ac;_i —

leraiu.- gradients since, as a matter of fact, the ratio Em & y / K

a c a increases when tïi'.- shunt

impedance increases.

Since recently the need for higher gradients became more and more obvious and new

checks of the Kiipatrick criterion Locain.- of real coiicer:. -J: ¿ lew ¡..__CL-S. A Ï ; J r__..iK ¡ L

I S now believed that the Kiipatrick criterion la pessimistic, at louyt umJ-vr pul.si-1 RJ

conditions.

10)

U ,2 - The experiment at VARÍAN

The experimental set up is shown on Fi(î- 12, where a single nose cone cell is fed by

a magnetron Q . 6 M/i, 4.4 usl -

VÎ;;. 1."' .*• cross-sectional vi-w of 'he cavity test system

In this experiment the repetition rate could be varied between 70 and .300 pp» while

the. output peak pow&r could be varied from Ü.2 to 2.6 MW.

The type of cavity which is u_ed h a high shunt impedance, as much as 130 Mii/m, at

3 G'r ar.u the CORRASPONÚÍNN E ,,/£ raf ) ij of the order of S. THE OBSERVED BREAKDOWN _, wal1 acc J._i"Ut corresponded to an accelerating rield of 30 -"v/m and a naximun field of 240 MV/n on

the inner surface of the ceil. With different geometries corresponding to different E w a 11"^acc C A t i a s t i l e !caxim*L*rr field was roughly the sane.

It was also observed that above a certain level of wall polishing there was no effect

on the breakdown limit. The limit also was found to be independent of the repetition fre­

quency in the ranqe previously -ntioiu^l.

- 785 -

From t h i s ^ x p e r i n e n t o n e c a n c o n c l u r a t h a t t ' maximum s u r f a c e e l e c t r i c f i e l d c a n te­

a t l e a s t a s h i g h a s t i v « t i m ^ c t n e l i m i t p r e d i c t ' - b y K i l p a t r i c k . E x t r a p o l a t i n g t h i s r e s u l t

t o d i s k l o a d e d c a v i t i e s o n e c a n e x p e c t a t l e a s t a c c e l e r a t i n g g r a d i e n t s c l o s e t c 120 MV/m.

4 . U - E x p e r i m e n t s a t SLAc'1''^1

T h e f i r s t h i g h g r a d i e n t t e s t a t S L A C w a s

d o n e o n t h e n o m a l S L A C a c c e l e r a t i n g s t r u c ­

t u r e s . I n o r d e r t c i n c r e a s e t h e g r a d i e n t two

k l y s t r o n s o p e r a t i n g i n t h e S L E D mode w e r e

c o m b i n e d , s o t h a t e a c h o f t h e f o u r s e c - i o n s

n o r m a l l y f e d b y o n e k l y s t r o n c o u l d r e c e i v e a n

i n p u t p e a k p o w e r a s h i g h a s 87 MW ( F i g . 1 3 ) .

T h e c o r r e s p o n d i n g S L E D f i e l d : n t h e s e - -

t i o n s w a s t h e n u p t c 32 MV/m o n t h e a x i s a

Co MV/.T. o n t h e w a l l s . A t t h i s l ^ v e l no b r e a k ­

down o c c u r e d i n t h e s e c t i o n s .

R e s o n a n t s t r u c t u r e u s e d f o r t h e

s e c o n d e x p e r i m e n t . T e s t p o i n t s i

c a t e l o c a t i o n s o f t h e r m o c o u p l e s 13 T h e f i r s t e x p e r i m e n t a l c o n f i g u r a t i o n

I n o r d e r t o i n c r e a s e t h e g r a d i e n t a s e c o n d e x p e r i m e n t w a s s e t u p i n w h i c h a s h o r t d i s k -

l o a d e d s t r u c t u r e w a s d e s i g n e d t o o p e r a t e i n t h e 2TT/3 S . W . mode ( F i g . 1 4 ) .

I'he c a v i t y f e d by 30 MW RF p e a k p o w e r d i d n o t show b r e a k d o w n p r o b l e m s a f t e r a s h o r t p r o ­

c e s s i n g . T h e maximum e q u i v a l e n t t r a v e l l i n g w u v e a c c e l e r a t i n g a n d s u r f a c e f i e l d s i n t h e . s e

c o n d i t i o n s w e r e r e s p e c t i v e l y 133 MV/m a n d 2 5 9 MV/m.

H o w e v e r i t s h o u l d b e n o t i c e d t h a t i n t h i s e x p e r i m e n t c o n s i d e r a b l e X - r a y r a d i a t i o n w a s

d e t e c t e d a r o u n d t h e s e c t i o n c o r r e s p o n d i n g t o a s t r o n g f i e l d e m i s s i o n .

5 . A SURVEY O F A C C E L E R A T I N G S T R U C T U R E S

P r e v i o u s e x p e r i m e n t s t e l l u s t h a t a c c e l e r a t i n g g r a d i e n t s o f t h e o r d e r o f 100 MV/m c a n

b e a c h i e v e d w i t h c o n v e n t i o n a l d i s k l o a d e d s t r u c t u r e s , b u t t h i s w i l l n e e d v e r y h i g h p e a k

- 78o -

power and correspondingly high average power to fit the luminosity requirements in a linear

collider.

Efforts have already been made to improve the efficiency of accelerating structures

and at least four types of accelerating structures, either operating in l.-band or in S-band,

have been developed for the acceleration of electrons (Fig. I S ) . One can ;:.aku the following

remarks :

- the disk loaded structure is very well known since it has been used for a long time

in linac design. It has a relatively low shunt impedance but a very üood ra_io E / E

i runnr innr i r JUUUUUUUL

a) Disk loaded

r Í • X-

? 0 e

O

O Á 1 1*.

LongiTud

b) Juínili; Gyn or cru-;s bar

cl Dis* and •. ... • d) Side coupled

Fíj;. 1' lièrent type nf accelei i-, stn:-tures

- The junqle jyir. structure has been first developed at low frequency. Since it has no

revolution sytnmetry it. i, hard tu stu-jy this struct ui e with .-ompu ter c-'J.les, .jnd t,.-[u;e it

rujeds prototype worn. However it i expected to jet from this structure an improved

shunt impedance with a high group velocity.

- The disk and washer structure is an open structure, as the previous one, which makes

the wall losses smaller and correspondingly leads to a higher shunt impedance. It has ulso

a higher Q but not a higher r/Q.

- The side coupled structure has a very high shunt impedance but a very bad ratio

E / E -max acc

- 787 -

T h e l a s t two s t r u c t u r e s a r e q u i t e c o ^ p l i c a t e d t o b u i l d , a n d u p t o now t h e y h a v e b e e n

m o s c l y c o n s i d e r e d i n t h e S . W . mode a c c o r d i n g t o t h e i r h i g h s h u n t i m p é d a n c e .

F r o n t h e p o w e r c o n s u m p t i o n p o i n t o f v i e w i t i s w e l l r e c o g n i z e d t h a t f o r a ç i v e n t y p e

o f s t r u c t u r e , o p e r a t i o n i n t h e S . w . mode i s l e s s e f f i c l e n t [ a l t b o u c i n o t obv l o u s when c o n s i d e ­

r i n g s m a l l l i n e a r a c c e l e r a t o r s ) t h a n o p e r a t i o n i n t h e T . .Tio-Je i f c o r r e c t c a t c h i n g o f c n e

s o u r c e i s made i n b o t h c a s e s 1 " " . H e n c e i t i s s t i l l p r e f e r a b l e t o c o n s i d e r T . w . s t r u c t u r e s

f o r v e r y h i g h e n e r g y l i n a c s and i n t h a t c a s e t h e p a r a m e t e r s o f r e a l i m p o r t a n c e a r e r / Q ,

E / E a n d v Jc. F o r t h e s e r e a s o n s i t i s b e l i e v e d t h a t t h e j u n a If* gym s t r u c t u r e -nay b e - . o -max a c c g

me a g o o d p o s s i b i l i t y b . . t s t i l l n e e d s n o r e d e v e l o p m e n t . I n t , ; e n e a n t i m e t h e o l d d i s k l o a ­

d e d s t r u c t u r e w i l l r e m a i n a g o o d c a n d i d a t e .

A n o t h e r a d v a n t a g e o f t h e T . w . a c c e l e r a t i n g s t r u c t u r e c o m e s f r o m t h e f a c t t h a t i t c a n

b e u s e d i n t h e S L E D m o d e . M o r e o v e r , i f t h e g r o u p v e l o c i t y i s h i g h t h e k l y s t r o n p u l s e c a n b e

made v e r y s h o r t a n d c o r r e s p o n d i n g l y t h a p e a k p o w e r c a n b e i n c r e a s e d w h i c h i s t h e r : i h t J i r e c -

t i o n t o f o l l o w i n t h e n o n S L E D c a s e .

T a b l e 4 c o m p a r e s t h e p e r f o r m a n c e s o f s e v e r a l s t r u c t u r e s i n t h e T . W . ir.ode, a t d i f f e r e n t

14) f r e q u e n c i e s . T h e d i s k a n d w a s h e r i s a l s o shown f o r c o m p a r i s o n .

D i s k - L o a d e d

( a = 1 . 1 6 cm} 56 1 3 , 3 0 0

D i s k - L o a d e d

( a = 1 . 5 0 cm) 4 6 1 3 , 0 0 0

D i s k a n d W a s h e r 76 3 2 , 0 0 0

J u n g l e Gym ( IT/2) 51 9 , 0 0 0

J u n g l e Gym ( u / 3 ) 60 9 , 0 0 0

J u n g l e Gym ( n / 2 ) 61 7 , 5 0 0

J u n g l e Gym IT/2) 71 7 , 5 0 0

5712 MHc

J u n g l e '-¡ym (TT/2) 72 6 , 5 0 0 . 2 0 6

J u n g l e Gym (rc/j) 85 6 , 5 0 0 - 1 0 6

T a b l e 4 c o m p a r i s o n o f s t r u c t u r e f o r a c o l l i d e r

I t i s s t i l l w c . r t r . v h i l e d e v e l o p i n g s h o r t d i s k l o a d e d s t r u c t u r e s i n t h e f r a . n e o f i m r - o v e d

p o w e r s o u r c e s .

6. RF POWER S O U R C E : THE LASERTRON

Up t o now p u l s e d k l y s t r o n s h a v e b e e n u s e d t o p r o v i d e h i g h RF p e a k p o w e r t o e l e c t r o n

a c c e l e r a t i n g s t r u c t u r e s . P e a k p o w e r s up t o SO ¡-TW w i t h i n 5 v-s p u l s e l e n g t h h a v e a l r e a d y b e e n

a c h i e v e d b u t t h e e f f i c i e n c y o f t h e s e d e v i c e s is s t i l l b e l o w 50 %. A h i g h e r p e a k power k l y s ­

t r o n , a b o u t 150 MW, a t a s h o r t e r p u l s e l e n g t h , a b o u t f u s , i s u n d e r d e v e l o p m e n t a t S L A C ^ ' ,

- 7 S H -

and it seems very difficult to go much higher, As a matter of fact high ;

accelerating voltage which would » educe the bundling efficiency, hence tl

contained in the fundamental and the extraction efficiency.

ither

;ii r

To overcome these difficulties, altnough

not proven to be fundamental limitations, a

new microwave RF power tube has been recently

p r o p o s e d * ^ ' i n which a photocathode, illu­

minated by a modulated laser, emits short,

dense current pulses which, after being acce­

lerated, traverse an output cavity where the

RF beam modulation is extracted {Fig. 1 6 ) .

Here, a high accelerating voltage is

necessary to compensate for the space charge

forces which otherwise would distort the

emitted short bunches and reduce the extrac­

tion efficiency. Since in principle the laser

can provide a train of such bunches with a

given repetition rate, the accelerating vol­

tage car; bo u.c. Schematic of a ph<

microwave powei tocathodc source

Considering the fast pulsed photo-emission it is believed that the maximum cha

can be extracted per pulse from the photocathode is equal to the superficial charge

2 -

where is the accelerating field at the photocathode, S the useful area of the photocar.ho

de, d the distance between the cathode and the anode, C the gun capacitance and V the acce

lerating voltage.

This maximum charge is twice the space charge limit, showing that the limitation of

such a tube ±3 very different from rhat of a klystron. As a matter of fact, if f ^ 13 the

repetition frequency of the laser pulses, the average curren! per laser burst is :

I RF CV

and the beam power : p , = £ C V 2

b RF

while in a klystron the maximum current is related to the voltage through a parameter k

called perveance ;

k - I ZV« 5

A 2-D simulation of the lasertron has been already performed which, for a given accele­

rating voltage, shows an increase of the energy spread and an increase of the bunch length

above a certain average beam current, or beam power (Pig. 1 7 ) .

Momenlum Spieaa UP''R%I and

Pulse lenglfr (Degrees) IDI

V : IOhV and _t<i., 60/

10 00 -Dû Aveiage beam power IMWI

?S9 -

2.0

1.9

1.6

1.7

1.6

I,/lo for 500 kV

1.0 10 Average beam p I M W I

F i ¿ . 17 Pulse length and ¡noinentum spread Fig. i8 Amplitude of the fundamental versus beam power versus >m power

A corresponding decrease of the ratio 1 j / ' I0 ' w n e r e I

1 i s tne amplitude of the firs

harmonic of the current modulation, is shown on Fig. Í0, leading to a decrease of the

efficiency and to a saturation of the extracted RF power. This is shown ci: Kip. 19 for 19)

the case of the prototype under consideration at SLAC . Improvement of t; saturation

level would follow an increase of the accelerating voltage. 20)

The main parameters of the SLAC prototype are given in Table 5. The -/er level

comparable to the peak power of the best klystron, and this is a first step i check the

lasertron principle before envisaging much higher peak power.

V s 4 0 0 KV

50 100

Average beam power ( M W l

Fig. 19 efficiency and RF power versus beam power

Peak RF Output Power 35 MW Beam power 50 MW Efficiency 7'] s, Voicage 400 kV dc Peak Pulse Current 735 A Cathode Diameter 3 cm Average Pulse Current {- Peak/6! 126 A Optical Pulse Length 60 ps FWHM Optical Pulse Separation 35Ü ps for a 2356 MH¿ Rate Microwave Pulse Length or 1 '„s Optical Pulse Train (Comb) Length Average Power (Power i Supply Limited! < 4 kW Peak Electric Field in Gun Reqion < 15 MV/m Electric Field on Planar Cathode 10 MV/m Maximum Magnetic Focusing Field r-.2 T

parameters of the SLAC prototype lasertron

Prototypes are also under consideration in Japan and in France.

Among the difficulties encountered in designing a lasertron it is worthwile mentioning

the high current photocathode. Rememtierin¡; the poor efficiency of lasers, the photocathcde

must have a very good quantum efficiency. Unfortunately it happens that efficient cathodes,

like AsGa for instance, show poor lifetime. On the contrary, metallic cathodes are robust

but with a poor quantum efficiency.

Modulated lasers at s-band or C-band frequencies, with long pulse trains and hin) repe­

tition rate have to be developed also, with optical frequencies either in the visible (green)

or in the VUV.

Figure taken from reference 20) .jjves a good idea of t.'io lawc-rtron i/ecrxctry as well

as the technologies involved.

Fig. 20 Geometry of the SLAC prototype lase..tron

REFERENCES

1) SLAC Linear Collider (SLC) : Conceptual design report, S L A C - R c ; t (I'JIiO), 2) 2.D. Farkas et al., SLED : A method of doubling SLAC's energy. Proceedings of the

9th International Conference on High Energy Accelerators, Stanford, 1974, (SLAC, Stanford, 1 9 7 4 ) ,

3) G.T. Konrad, High power RF klystrons for linear accelerators. Proceedings of the l'-*54 Linear Accelerator Conference, Seeheirr. (R.F.A.) 1^84.

4) J. Le Duff, LAL/R-V85-03, Orsay (1985).

5) F. Bulos et al.. Physics with linear colliders in the TeV CM enenjy region. SLAC-PCB-3002. Also, contribution to the Proceedings of the Summer Study cn Elementary Particle Physics and Future Facilities, 1982, Snowmass, CO.

6) The LEP Injector Study Group, LEP design report, Vol. 1 : the LEP injector chain, CERN-LEP/TH/83-29 ; LAL/RT/Q3-09 (1933).

7) J. Le Duff, Optimization of TW accelerating structures for SLED type :nodos of operation LAL/RT/84-01, LAL-Orsay (1984).

8) R. Belbeoch et al.. Rapport d'études sur le projet des linacs injecteurs de Lt-.F (LILI ; LAL/RT/82-01, Orsay (1982).

9) W.D. Kilpatrick, Criterion for vacuum sparking designed to include both RF and ;x:. UCRL-2321, (1953).

10) E. Tanabe, Voltage breakdown in S band linear accelerator cavities. Proceedings of 1983 Particle Accelerator Conference, SANTA FE, 1983, IEEE Trans. Nucl. Sei., NS-30 (1983).

11) H.A. Hogg et al.. Experiments with very high power RF pulsea at SLAC. IEEE Trans. Nucl. Sei., NS-30, (1983).

12) J.W. Wang and G.A. Loew, Measurements of ultimate accelerating gradients in the PLAC disk-loaded structure. Proceedings of the 1985 Particle Accelerator Coiifwrei.ee, Vancouver, 1985,

IEEE Trans. Nucl. Sei., NS-32, (1985).

13) P.B. Wilson. IEEE Trans. Nucl. Sei., NS-26, 3255 (1979).

14) P.B. Wilson, IEEE Trans. Nucl. Sei., NS-28, 2742 (1981).

15) M.T. Wilson, P.J. Tallerico, US patent n°4 , 3 P , 072- 1 /26/1 981.

16) M. Yoshioka et al., Laser-Triggered RF sources for Linacs in TeV r t.jion. Proceed inns of the 19B4 Linear Accelerator Conference, Seeheim (R.F.A.) 1984.

17) H. Nishimura, Particle simulation code for non relativistic electron bunch in laser-

tron. Proceedings of the 1984 Linac Conference, Seeheim [R.F . A .) 1984.

18) W.B. Herrmannsfeldt, Computer simulation of the Lasertron, S^AC/AP-21.

19) P.B. Wilson, Private communication. ¿0) E.L. Garwin et al.. An experimental program to build up a multïnegawa11 La sert ron f e r

super linear colliders. Proceedings IEEE, NS-32, (19S5) .

F R E E E L E C T R O N L A S E R S : A S H O R T R E V I E W O F T H E T H E O R Y A N D E X P E R I M E N T S

G . D a t t o l i , A - R e n i e r i a n d A . T o r r e *

D I P . T I B , D i v i s i o n e F i s i c a A p p l i c a t a , C e n t r o R i c e r c h e E n e r g í a F r a s c a t i ,

C . P . 6 5 - 0 0 0 4 4 F r a s c a t i , R o m e ( I t a l y ) .

A B S T R A C T

T h i s n o t e i s d e v o t e d t o a s h o r t r e v i e w o f t h e t h e o r e t ­

i c a l a n d e x p e r i m e n t a l a s p e c t s o f F r e e E l e c t r o n L a s e r s

( F E L ) . We w i l l d i s c u s s b o t h r e c i r c u l a t e d a n d s i n g l e -

p a s s a g e F E L s a n d t h e i r r e l e v a n t d e s i g n p r o b l e m s .

1 . I N T R O D U C T I O N

S i n c e l a s e r s o u r c e s h a v e b e e n e x p e r i m e n t a l l y d e m o n s t r a t e d t h e c o n ­

c e p t o f a " u n i v e r s a l " o r " r a d i o - l i k e " c o h e r e n t l i g h t s o u r c e h a s b e e n r e ­

c o g n i z e d a s a p o w e r f u l t o o l f o r a l a r g e n u m b e r o f p o t e n t i a l a p p l i c a t i o n s .

T h e c o n c e p t o f f u l l y t u n a b l e l a s e r s i s t h e r e f o r e a s o l d a s l a s e r

P h y s i c s .

T h e r e s e a r c h a c t i v i t y i n t h i s f i e l d , d e v e l o p e d t h r o u g h t h e y e a r s , i s

s u m m a r i z e d i n F i g . 1 w h e r e w e h a v e p l o t t e d t h e p o w e r a g a i n s t t h e w a v e l e n g t h

( a n d w a v e l e n g t h r a n g e ) o f t h e c o m m o n l y c o n s i d e r e d t u n a b l e c o n v e n t i o n a l

s o u r c e s . M a n y o f t h e l i g h t s o u r c e s o f F i g . 1 a r e f a r f r o m b e i n g r e a l t u n a ­

b l e l a s e r s . I t i s s e l f - e v i d e n t t h a t t h e u l t i m a t e t u n a b l e l a s e r h a s n o t b e e n

d e v e l o p e d , b u t i t s d e s i r e d p e r f o r m a n c e c a n b e e a s i l y o u t l i n e d :

a ) s t a b i l i t y

b ) l o n g l i f e

c ) e a s i l y m a n a g e a b l e

d ) h i g h p o w e r

e ) e a s i l y t u n a b l e v i a e x t e r n a l s e t t i n g s t o a n y s e l e c t e d f r e q u e n c y .

T h e s e a n d m a n y o t h e r " s c i e n c e - f i c t i o n " p e r f o r m a n c e s w i l l b e t h e

c h a r a c t e r i s t i c s o f a r e a l t u n a b l e l a s e r . T h e a r e a s o f a p p l i c a t i o n o f t h e s e

k i n d s o f s o u r c e s a r e a s w i d e a s t h e i r v e r s a t i 1 i t y a n d i n c l u d e s u c h d i f f o r e n t

f i e l d s a s s p e c t r o s c o p y , r e m o t e d e t e c t i o n , p h o t o c h e m i s t r y e t c - 1 ' .

T o g i v e a n e x a m p l e , c o n v e n t i o n a l t u n a b l e l a s e r s l i k e e x c i m e r s , d y e s

a n d h a r m o n i c g e n e r a t i o n h a v e b e e n u s e d f o r U V s p e c t r o s c o p y , w h i l e c o l o r

c e n t e r l a s e r s h a v e b e e n e x p l o i t e d i n t h e m i d d l e I R s p e c t r o s c o p y a n d h a v e

p r o v i d e d s u b s t a n t i a l a d v a n c e s . T h e u s e o f r e a l t u n a b l e s o u r c e s i n c o n n e c -

* ) D e p t . o f P h y s i c s a n d A s t r o n o m y , D a r t m o u t h C o l l e g e , H a n o v e r 0 3 7 5 5 N . H .

( U S A )

- 795 -

CENTER LASER

10"' 10° 10 ' 1 0 2 10- 1

F i g . 1 C o m p a r a t i v e c h a r t b e t w e e n F E L a n d c o n v e n t i o n a l c o h e r e n t s o u r c e s .

A v e r a g e p o w e r v s k . C u r v e s F E L ( S P ) ; s i n g l e p a s s a g e F E L a v e r a g e

p o w e r v s A 1 s t a n d 3 r d h a r m o n i c r e s p e c t i v e l y , m a x i m u m e l e c t r o n

b e a m < e . b . ) p o w e r 2 0 MW, K = l , \ u = 5 c m , N = 5 0 , L c = 6 m , T h = 1 2 U S ( 6 = 5 % .

S t r a i g h t c u r v e s : F E L s t o r a g e r i n g a v e r a g e p o w e r v s K w i t h o u t ( c o n ­

t i n u o u s ) a n d w i t h ( d a s h e d ) T o u s c h e k e f f e c t r e s p e c t i v e l y . Î = 1 0 0 r.

a n d o p e r a t i n g p a r a m e t e r o f L E D A - F , s e e R e f . 2 . i ^ c = c a v i t y l e n g t h ,

Ö E d u t y c y c l e l .

t i o n w i t h p h o t o a c o u s t i c s p e c t r o s c o p y a n d p h o t o t h e r m a l r a d i ó m e t r y w i l 1 r e ­

s u l t i n m o r e u s e f u l a n d f l e x i b l e d e t e c t i o n t e c h n i q u e s 1 * . I n t h e f i e l d o f

p h o t o c h e m i s t r y t h e t u n a b l e s o u r c e s w i l l a l l o w m o r e r e l i a b l e a n a l y t i c t e c h ­

n i q u e s a n d l a s e r - b a e e d c h e m i c a l p r o c e s s i n g r a n g i n g f r o m c o n t r o l l e d t h e r m a l

c h e m i s t r y t o l a s e r i n i t i a t e d r a d i c a l r e a c t i o n s .

T h e a b o v e l i s t o f p o t e n t i a l a p p l i c a t i o n s o f t u n a b l e s o u r c e s m a y b e

c o m p l e m e n t e d w i t h m o r e t e c h n o l o g i c a l s u b j e c t s s u c h a s m e c h a n i c a l p r o c e s s ­

i n g , i n f o r m a t i o n t r a n s f e r a n d c o m m u n i c a t i o n . T h e n o t i c e a b l e i n t e r e s t i n t h e

f i e l d i s t h e r e f o r e f u l l y j u s t i f i e d . T h e s t a t e o f t h e a r t o f t h e t u n a b l e

s o u r c e s h a s b e e n d i s c u s s e d i n R e f . 2 w h e r e a c o m p r e h e n s i v e r e v i e w o f t h e

e x p e r i m e n t a l r e s u l t s a n d o f t h e l i t e r a t u r e h a s b e e n g i v e n . A s a c o m m e n t t o

F i g . 1 w e n o t i c e t h a t a m o n g t h e s o l i d s t a t e l a s e r s t h e m o s t r e l i a b l e i s t h e

A l e x a n d r i t e , w i t h a t u n i n g b a n d w i d t h o f a b o u t 1 0 0 0 Ä , o p e r a t i n g a t r o o m

t e m p e r a t u r e . T h e c o l o r c e n t e r l a s e r s h a v e a m o r e s i g n i f i c a n t t u n i n g r a n g e ,

b u t r e q u i r e i m p r o v e m e n t s i n t e r m s o f s t a b i l i t y a n d l i f e - t i m e . T h e m o s t

l i m i t i n g f a c t o r o f d y e l a s e r s i s r e p r e s e n t e d b y t h e s t a b i l i t y , e v e n t h o u g h

t h e i r t u n a b i l i t y o v e r t h e y e a r s h a s b e e n i m p r o v e d , t o c o v e r t h e r a n g e b e ­

t w e e n t h e v i s i b l e a n d t h e n e a r I R . H o w e v e r , t h e p o w e r l i m i t a t i o n b e y o n d

1 0 urn i s e v i d e n t . I n t h i s r e g i o n a n d i n t h e s h o r t w a v e l e n g t h r e g i o n ( V U V ,

X - r a y s ) t u n a b l e n o n - c o n v e n t i o n a l s o u r c e s a r e p r o v i d e d b y t h e F r e e E l e c t r o n

L a s e r s ( F E L ) 3 * .

- 794 -

In Fig. 2 the tunability curve of FELsis shown, the continuous line ranges from VUV to the microwaves. Although the basic mechanism of FEL allows a wide tunable range, this device, as it stands, does not provide the universal laser we are talking about. A fully tunable FEL requires, indeed, a "universal" accelerating electron machine able to provide an electron beam with continuous varying energies from MeV to GeV region. Even this kind of machine does not exist, but its characteristics for FEL ap­plication can be easily listed:

a) easy energy tunability b) modest size c) high beam power (average and peak) d) good beam qualities (small energy spread and emittances) -

An overview of the design characteristics of an accelerating electron device dedicated to FEL has been presented in Ref. 4. In that paper the performances of storage rings, diode machines, induction linacs, elec­trostatic devices and R.F. accelerating machines have been discussed within the framework, of their relevance to FEL. In Fig. 3 we have sum­marized the relative range of energy and current of accelerating devices. These are impressive, from MeV and kA to GeV and hundreds of mA.

10 : »3

El MeV)

|01 ' i l l . L

1 0 ' 3 1 0 " 2 10"' 10° 10' 1 0 ; 10-,3 1 0 4

Á (/-ml

*-I.BI.-[.I.NL ( U V E R H Ö R E ) V-IICSB (SINGLE STAGE) l~ UC.SB (TWO STAGE?) » - AT&T-RELl,

FRASCATI (ENEA) a - C K PROJECT + - TRW STANFORD X - I.ANI. (LOS ALAMOS) ® - M S N W - B A C

<D - S ÏANFORl) j_ - NOVOSIBIRSK txl-ORSAV ® -FRASCATI (1NFN) 0 - BROOKILH'EN D - MK H I - S T A N F O R D 0 - YEREVAN T - SOU-RING

Fig. 2 FEL- scenario

795 -

I DIODE

1Û3

104

10J

10a

10;

10°

IQ"'

10° 1Û1 10' i o : i

ElMeVl

Fig. 3 Current vs. Energy for existing accelerators

Such a flexible accelerator is very far from the present technological capabilities. Nevertheless, exploiting such different tools as relative energy tunability, higher-harmonics emission, undulator gap variations etc. an FEL can provide a range of tunability much larger than that of the con­ventional sources.

In the next sections we will briefly summarize the main features of the FEL theory with particular emphasis on the design criteria of both recirculated and linear devices. The final section is devoted to concluding remarks where we complete the comparison with the conventional lasers.

2. FEL: THEORY AND DESIGN CRITERIA

In an FEL a beam of ultrarelativistic electrons interacts with an undulator magnet (UM) where it undergoes transverse oscillations and emits radiation at a fixed wavelength; the radiation is stored in an optical cavity, it reinteracts with the copropagating e-beam and is amplified.

An undulator magnet is a spatial array of magnets arranged as in

- 796 -

Fig. 4, with spatial period A and it was originally proposed as a tool to U 5 J enhance the brightness of the synchrotron light

We will give a simple heuristic explanation of the central emission frequency in an undulator. Madey 6 * has shown that for ultrarelativistic e-beam the undulator field can be treated as a radiation field with wavelength

and density number of pseudophotons 7 *

( 2 )

where a is the fine structure constant, r is the classical electron __o

radius, k is the undulator parameter and B (= B Q for helical undulator, B o / V 2 for linear undulator) is the average on axis field <B Q is the on axis field).

The relation (1) can be understood as follows. The vector potential of the undulator field can be written as

( 3 )

As a consequer.ee the magnetic field is given by

B = V x A = Je {-iBQ exp(iz/\^] it

{x, y, ¿ = unit vectors)

From (4) we get also

V x B = V x V x A = -V 2A = 1 / A Z A

<4)

UNDULATOR MAGNET

MAGNET AXIS

LIGHT PULSE

ELECTRON OBSERVER TRAJECTORY

Fig. 4 Undulator magnet geometry

- 79 7 -

Fig. b CompLon scatteiing di agi din

w h i c h c l e a r l y c o n t r a d i c t s t h e M a x w e l l e q u a t i o n . T h e l a s t t e r m o n t n e r i g h t

h a n d s i d e o f ( S ) c a n h o w e v e r b e r e i n t e r p r e t e d , a c c o r d i n g t o R e f . 3 , a s a

k i n d o f " p h o t o n t r a s s " . T h i s i s i n d e e d t h e p r i c e t o b e p a i d w h e n o n e t r e a t s

f r o m t h e v e r y b e g i n n i n g t h e u n d u l a t o r v e c t o r p o t e n t i a l i n t h e t r a n s v e r s e

g a u g e . F u r t h e r m o r e , t r a n s f o r m i n g t h e v e c t o r p o t e n t i a l t o t h a t s e e n b y a

r e l a t i v i s t i c e l e c t r o n m o v i n g t h r o u g h t h e UM we h a v e

A - = . i ? e | B Q \u e x p | i Y / * U ( Z * + ( J e t ' ] } y

( 6 )

(ß = v z / c , y 2 = (1 -ß2)'1}

t h u s y i e l d i n g f o r t h e m a g n e t i c a n d e l e c t r i c f i e l d t h e e x p r e s s i o n s

B ' = rfe í - i v B 0 e x p | i Y / * U ( z ' + ß c t ' ] } x

E'-Ke i - i Y ß B o e x p / i y / A u < z ' + f i c t ' ] ¡ y

a n d f i n a l l y o b t a i n i n g t h e " M a x w e l l e q u a t i o n "

i i

- ' X - C dtJ + ^

From t h e f i r s t o f e q u a t i o n s ( 8 } i t f o l l o w s t h a t t h e " m a s s t e r m " i s n e g l e c t ­

e d o n c e y >> 1 . F u r t h e r m o r e . s i n c e ß ~ 1 t h e f i e l d s ( 7 ) a p p r o x i m a t e c l o s e l y

t o t h o s e o f a r a d i a t i o n f i e l d . S i n c e a 1 i g h t p u l s e mus t b e a t a d i a t i o n

f i e l d i n a l l t h e r e f e r e n c e f r a m e s , t h e f i e I d d e f i n e d by ( 7 ) m u s t r e m a i u a

" r a d i a t i o n " f i e l d i n t h e l a b o r a t o r y f r a m e t o o . T r a n s f o r m i n g b a c k i t s w a v e ­

l e n g t h t o t h a t f r a m e o n e f i n d s

A* = ( 1 * P ) \ . ( 9 )

t h u s g e t t i n g f o r ß ~ 1 t h e r e l a t i o n ( 1 } . F i n a l 1 y t h e e q u a t i o n ( 2 ) i ¡; a

s t r a i g h t f o r w a r d c o n s e q u e n c e o f t h e d e f i n i t i o n o f t h e f i e l d e n e r g y d e n s i t y .

S i n c e t h e u n d u l a t o r f i e l d c a n b e t r e a t e d a s a n o i d i n a i y e l e c t i o m a g n e t -

i c w a v e , i t s i n t e r a c t i o n w i t h t h e e l e c t r o n c a n b e v i e w e d a s a s e a t t e r i n g .

A c c o r d i n g t h e r e f o r e t o t h e w e l l known f o r m u l a o f t h e Comp t o n s c a t t e j n i g ,

t h e w a v e l e n g t h o f t h e s c a t t e r e d l i g h t a t a n a n g l e u i s g i v e n by ( s e e K i g . b )

K = Z\ 1 - ß c o s U 1 + ß

E x p a n d i n g f o r s m a l l a n g l e s a n d u l t r a r e l a t i v i s t i c e n e r g i e s o n e f i n d s

A

A = ( 1 + k 2 * Y

2 6 2 ) ( I I I

T h e c o r r e c t i v e t e r m k 2 i s d u e t o t h e t r a n s v e r s e e l e c t r o n m o t i o n a n d i t i s

a n a l o g o u s t o t h e e f f e c t s u g g e s t e d b y B r o w n a n d K i b b l e i n t h e a n a l y s i s o f

t h e e l e c t r o n m o t i o n i n an i n t e n s e l a s e r w a v e , vrliere a n i n t e n s i t y d e p e n d e n t

c o n t r i b u t i o n t o t h e C o m p t o n w a v e l e n g t h s h i f t w a s f o u n d ( s e e R e f . 3 f o r

f u r t h e r c o m m e n t s ) .

One o f t h e m o s t p e c u l i a r c h a r a c t e r i s t i c s o f t h e l i g h t emi t t e d b y a

c h a r g e d p a r t i c l e r u n n i n g i n a UM i s t h e b a n d w i d t h . T h i s q u a n t i t y c a n b e

e a s i l y e v a l u a t e d a c c o r d i i . y t o t h e f o l l o w i n g s i m p l e a r g u m e n t :

1 ) T h e d u r a t i o n o f t h e e m i t t e d l i g h t p u l s e i s l i n k e d t o t h e d i f f e r e n c e

b e t w e e n e l e c t r o n a n d p h o t o n f l i g h t t i m e s ( s e e F i g . 4 >

N\

A t = ( 1 - p ) z ( 1 2 )

( N i s t h e n u m b e r o f u n d u l a t o r p e r i o d s ) .

2 ) A c c o r d i n g t o t h e i n d é t e r m i n a t i o n p r i n c i p l e t h e b a n d w i d t h c a n b e

e a s i l y e v a l u a t e d f r o m

3 ) C o m b i n i n g b o t h ( 1 1 ) a n d ( 1 3 ) we g e t t h e r e l a t i v e h o m o g e n e o u s b a n d ­

w i d t h

The Q t j . n i n a t i o n " h o m o g e n e o u s " a n d i n h o m o g e n e o u s d e r i v e s ( w i t h t h e s a m e

m e a n i n g ) d i r e c t l y f r o m t h e s t a n d a r d t h e o r y o f t h e p h o t o n e m i s s i o n b y a t o m s

o r m o l e c u l e s . A s i s w e l l k n o w n , t h e s p e c t r u m o f an N - p e r i o d l i g h t p u l s e

h a s a w i d t h g i v e n b y ( 1 4 ) , w h i l e i t s s h a p e i s 3 *

s i n v / 2 w - w f ( u > > " ( „ „ ) 2 , v = 2 « N - 2 - <iu = - ( 1 5 )

T h e s p e c t r u m g i v e n b y E q . | 1 5 ) p l a y s a f u n d a m e n t a l r o l e i n t h e FEL t h e o r y a n d

i t h a s b e e n p l o t t e d i n F i g . 6 t o g e t h e r w i t h t w o e x p e r i m e n t a l s p e c t r a

s p o n t a n e o u s e m i s s i o n . F o r a r e c e n t a n d d e t a i l e d a n a l y s i s t h e r e a d e r i s

r e f e r r e d t o R e f . 4 . We a r e now i n t e r e s t e d i n s t i m u l a t e d e m i s s i o n a n d g a i n -

By t h e f o r m e r we m e a n e m i s s i o n i n t h e p r e s e n c e o f o t h e r e . m. m o d ^ s a n d

v a r i a t i o n o f t h e i n t e n s i t i e s o f t h e t h o s e m o d e s . A r i g o r o u s a n a l y s i g o f t h e

g a i n c a n b e f o u n d i n R e f . 3 , b u t h e r e we w i l l e v a l u a t e t h e g a i n f u n c t i o n

i n a r a t h e r d i r e c t w a y . T h e g i i i n m e c h a n i s m c a n b e u n d e r s t o o d a s a b a l a n c e

b e t w e e n a n a b s o r p t i o n a n d a n e m i s s i o n p h o t o n p r o c e s s . The g a i n f u n c t i o n

w i l l b e t h e r e f o r e g i v e n b y t h e d i f f e r e n c e b e t w e e n t h e p r o b a b i l i t i e s o f e m i t ­

t i n g a n d a b s o r b i n g a p h o t o n . T h e f u n c t i o n a l f o r m o f e m i t t i n g o r a b s o r b i n g a

p h o t o n i s t h e s a m e a s E q . (15), t h e o n l y d i s t i n g u i s h i n g f e a t u r e i s t h e

( S t a n f o r d 8 } a n d O r s a y 9 ) ) . The a b o v e b r i e f c o m m e n t s a r e r e l e v a n t t o t h e

-10 - 6 6 10

10.78 it

J w 1 0) 2?

cl

^600 4100 4200 4000 3800 0

F i g . 6 U n d u l a t o r m a g n e t f o r w a r d e m i s s i o n s p e c t r u m : ( a ) T h e o r e t i c a l s p e c ­

t r u m ; ( b ) E x p e r i m e n t a l s p e c t r u m ( S t a n f o r d . R e f . 8 ) ; ( c ) E x p e r i ­

m e n t a l s p e c t r u m ( O r s a y , R e f . 9 ) .

- «ou -

électron recoil, so that

g(w )« f(u - £ ) - f <w f r. )

-V - [-

4 n z AL ! k 2 Au _ 2 d sine/2 2

' E o

if 2,, < I T

e-beam peak current and I Q = e c / r0 ( - l ^ x l C A ) is the Alfvén current. Sxnce

emission at higher harmonics occurs in undulator magnets on and off axis, an analogous gain formula for higher harmonics can be derived ( see Ref. 10

for a general formulation). In particular, linear undulators alluw odd-harmonic emission on axis

and the relevant gain can be written as

. sin nv /2 9„<"> •= < * 5 7 " I I ' - " = 1.2....

y n ï ¿ I n n l t ' l i u ' o

n-th harmonic: filling factor

(J (-) n-th cylindrical Bessel function)

- 301 -

The analysis we have developed so far is relevant to a sma11 - signal, single - mode, homogeneously - broadened FEL operation. The strong signal and multimode behaviour will be treated below. By homogeneous broadening we mean an FEL operating with an electron beam whose energy spread and emit-tances produce negl igibly srnal 1 effects .

It is well known that those beam qualities produce both a broadening of the emission line and a reduction of the gain 3 * . The value of the in-homogeneous linewidth in terms of the beam emittances and energy spread is 3 1

The u-coefficients are the ratio between the inhomogeneous and homogeneous widths and have played a crucial role in the design and optimisation of an FEL device 1 1 *.

In particular,

u = 4No , o. = r.m.s. energy spread

where e , are the radial and vertical emittances, o the transverse e-beam dimensions, h^ ^ are coefficients depending on the undulator geome­try, namely h x = h y = 1 for helical undulators and h x =-ft, h y =2+ 6 ( 6 < < l ) for the linear case, with polarization along the y-axis. Physical 1 y fi is the magnitude of the sextupolar term along the x-direct.ion 3 ' .

It is worth noting that the inhomogeneous broadening due to the emit-tances consists of two distinct contributions; the first due to the angular divergence, the second to the finite beam size which explores regions of different magnetic field strength.

The expression Eq. [21} suggests that one can choose an optimum a ^

* , y , V 2

Therefore one finally gets

is the beta amplitude func­

tion, Eq. (22]amounts to = [1/2/h^ ||'^* <yA u)/kn.

- soz -

" x . y • 2 N ^ | h x . y ' î T i V ^ • < " >

We must r e m a r k , h o w e v e r , t h a t w h i l e t h e c h o i c e o f E q . (22) m i n i m i z e s t h e e f ­

f e c t o f t h e inhomogeneous b r o a d e n i n g i t may, a t t h e same t i m e , c r e a t e

d i f f i c u l t y w i t h t h e f i l l i n g f a c t o r . T h e r e f o r e , i n s e r t i n g t h e a u x i l i a r y c o n ­

d i t i o n t h a t t h e e . b . c r o s s s e c t i o n i s o f t h e o r d e r o f t h e l a s e r mode w a i s t ,

we g e t

E s J2~\h T n ( - ) N A . ( 2 4 ) x , y v 1 x , y 1 * v '

(25)

C o m b i n i n g E q . (23} and \24,\ one c a n a l s o g e t a c o n d i t i o n on t h e s e x t u p o l a r

t e r m s I h x . y l

when a v e r y s m a l l beam s e c t i o n i s r e q u i r e d we can n e g l e c t i n E q . ( 2 l J

t h e inhomogeneoue b r o a d e n i n g i n d u c e d by t h e u n d u l a t o r f i e l d i n h o m o g e n e i -

t i e s . W i t h t h e r e q u i r e m e n t t h a t t h e beam c r o s s s e c t i o n i s o f t h e o r d e r o f

t h e l a s e r w a i s t and t h a t ( 2 3 ) be l e s s t h a n u n i t y we f i n d t h e c o n d i t i o n s 1 1 ^

t % \ , ß ~ NX . ( 2 6 ) x , y K x , y u

To g i v e an i d e a o f t h e u p a r a m e t e r s on t h e spontaneous e m i s s i o n and t h e

g a i n , we h a v e p l o t t e d i n F i g . 7 t h o s e f u n c t i o n s a g a i n s t v f o r d i f f e r e n t

v a l u e s o f u . I t i s e v i d e n t t h a t w i t h i n c r e a s i n g v a l u e s o f t h e inhomoge­

neous p a r a m e t e r s t h e c u r v e s a r e b o t h w i d e n e d and r e d u c e d .

L o n g i t u d i n a l mode l o c k i n g a r i s e s f o r FEL o p e r a t i o n w i t h bunched

e - b e a m s . I t has i n d e e d been shown t h a t i n t h i s s i t u a t i o n a n a t u r a l p h a s e

l o c k i n g i s i n d u c e d b y t h e FEL i n t e r a c t i o n , and t h e s t r e n g t h o f t h e c o u p l i n g

b e t w e e n t h e modes i s g i v e n by the f u r t h e r p a r a m e t e r

where az i s t h e e l e c t r o n bunch r . m . s . l o n g i t u d i n a l l e n g t h . The l a r g e r i s

H c - t h e g r e a t e r i s t h e number o f c o u p l e d modes. The bunched e-beam s t r u c ­

t u r e i s a l s o r e s p o n s i b l e fo t t h e so c a l l e d FEL l e t h a r g i c b e h a v i o u r , i . e . t h e

s l o w down o f t h e l i g h t p u l s e due t o t h e i n t e r a c t i o n and t u e n e c e s s i t y t o

s h o r t e n t h e c a v i t y l e n g t h w i t h r e s p e c t t o t h e n o m i n a l r o u n d - t r i p p e r i o d t o

keep t h e s y n c h r o n i z a t i o n be tween l i g h t and e l e c t r o n bunches The i m -

- S05 -

/ m .1

'' J/ V \ - r * e \ A , 7 i , < \ T - f = -

/ ' -b)

• S P i - y i t — , i 1 " V ^ M - 8 \ - 4 A 1 8 v B »

J- c) d)

t

—"^4 B V

Fig. 7 Inhomogeneous b r o a d e n e d g a i n : ( a ) M £ = M J Î

= H y = 0 ' ' ( b ) ^ £

- 1 ' ^ x ' ^ y - 0

( c ) p e = û , u x = l ( M y = Û ; ( d ) M E = M x = M y = l .

p o r t a n c e o f t h i s e f f e c t f o r s h o r t - p u l s e o p e r a t i n g FEL d e v i c e s w i l l be d i ­

s c u s s e d b e l o w .

The above n o t i o n s a r e t h e m i n i m a l t h e o r e t i c a l b a c k g r o u n d t o u n d e r s t a n d

t h e FEL o p e r a t i o n ; i n t h e n e x t two s u b s e c t i o n s we w i 1 1 d e s c r i b e i n some

d e t a i l FEL d e v i c e s i n s t o r a g e r i n g s and s i n g l e - p a s s a g e d e v i c e s .

3 . FEL STORAGE RING OPERATION

We h a v e s e e n t h a t t h e p r a c t i c a l r e a l i z a t i o n o f à B'EL r e q u i r e s an

e - b e a m w i t h good q u a l i t i e s , n a m e l y l a r g e p e a k c u r r e n t and r e l a t i v e l y low

e n e r g y s p r e a d and e m i t t a n c e . A s t o r a g e r i n g (SR) p r o v i d e s a v e r y good

e -beam f o r t h e FEL o p e r a t i o n .

I n t h e s e d e v i c e s t h e e -beam i s c o n t i n u o u s l y r e c i r c u l a t e d t h r o u g h t h e

i n t e r a c t i o n r e g i o n and as a consequence t h e e n e r g y s p r e a d and t h e e m i t -

t a n c e s i n c r e a s e . T h e r e f o r e , a c c o r d i n g t o t h e a rguments p r e s e n t e d so f a r ,

t h e i n c r e a s e o f t h e .••nhomogeneous b r o a d e n i n g r e d u c e s t h e FEL a m p l i f i c a t i o n .

T h i s d y n a m i c a l b e h a v i o u r i s p e c u l i a r t o s t o r a g e r i n g FELs. A c o r r e c t

d e s c r i p t i o n o f t h e s t o r a g e r i n g F£L o p e r a t i o n r e q u i r e s i n d e e d t h e s e l f -

c o n s i s t e n t a n a l y s i s , t u r n by t u r n , o f b o t h t h e l a s e r and t h e e l e c t r o n

beams. S t o i a g e r i n g FELs have b e e n s u g g e s t e d f o r l a s e r o p e r a t i o n i n t h e

s h o r t w a v e l e n g t h r e g i o n f i x m v i s i b l e down t o VUV and x - r a y ( s e e F i g . 2 ) .

J u s t t c s t a r t t h e s e i n t r o d u c t o r y r e m a r k s we show i n F i g . 8 t h e l a y o u t o f an 1 2 1

SR d e s i g n e d f o r FEL o p e r a t i o n ' . The mach ine has a t w o f o l d symmetry

( p r o b l e m s a r i s e f o r n o n - s y m m e t r i c a l s t r u c t u r e s ) . The symmetry i s p r o v i d e d

b y t h e i n s e r t i o n o f t w o , l o n g , l o w - f i e l d u n d u l a t o r s ( f o r t h e FEL o p e r a t i o n )

ar.d t w o , s h o r t , h i g h - f i e l d u n d u l a t o r s t o enhance t h e s y n c h r o t r o n r a d i a t i o n

- S04 -

F i g . 8 S t o r a g e r i n g l a y o u t ( d e s i g i . s t u d y L E D A - F 2 ; B = b e n d i n g m a g n e t ;

F , D = f o c u s i n g a n d d e f o c u s i n g ( H o r i z o n t a l ) q u a d r u p o l e m a g n e t ;

S = s e x t u p o l e m a g n e t ; WM ( F E L ) = u n d u l a t o r m a g n e t f o r F E L o p e r a ­

t i o n . - WM = h i g h - f i e l d u n d u l a t o r m a g n e t f o r e n h a n c i n g s y n c h r o t o n

r a d i a t i o n e m i s s i o n .

e m i s s i o n ( t h i s f a c t w i l l b e c l a r i f i e d b e l o w ) .

T h e f o c a l i z a t i o n i s p r o v i d e d b y a n a l t e r n a t e d i s t r i b u t i o n o f h o r i ­

z o n t a l f o c u s i n g a n d d e f o c u s i n g q u a d r u p o l e m a g n e t s ( F a n d D i n F i g . 8 ) . A l s o

i n s e r t e d i n e a c h q u a r t e r o f t h e m a c h i n e a r e t w o b e n d i n g m a g n e t s a n d t w o

s e x t t i p o l e s t o min iJT i i i e t)iL- d e p e n d e n c e o f t h e t r a n s v e r s e o s c i l l a t i o n f r e q u e n c y

1 2 )

o n t h e p a r t i c l e e n e r g y . T h e f r e e s p a c e b e t w e e n t h e q u a d r u p o l e s o f t h e

l o n g s t r a i g h t s e c t i o n i s u t i 1 i z e d f o r t h e i n j e c t i o n o f t h e e l e c t r o n s i n t o

t h e m a c h i n e a n d t o i n s e r t t h e R F a c c e l e r a t i n g s y s t e m w h i c h :kv<.'loniU's t h e

e l e c t r o n t o h i g h e r e n e r g i e s t h a n t h e i n j e c t i o n o n e a n d s u p p l i e s t h e e n e r g y

l o s t b y s y n c h r o t r o n e m i s s i o n i n t h e b e n d i n g m a g n e t s .

S i n c e t h e p a r t i c l e s e m i t s y n c h r o t r o n r a d i a t i o n t h e o f f - e n e r g y p a r ­

t i c l e s t e n d t o r e d u c e , t u r n b y t u r n , t h e e n e r g y s h i f t f r o m t h e s y n c h r o n o u s

o n e s w i t h a d a m p i n g t i m e

— * U = V 4

w h e r e E q i s the machine n o m i n a l e n e r g y , T t h e r e v o l u t i o n p e r i o d , U q t h e e -n e r g y r a d i a t e d p e r t u r n and p t h e b e n d i n g magnet r a d i u s (assumed i d e n ­t i c a l f o r a l l t h e m a g n e t s ) .

The betatron motion too is damped, with damping times

The above expressions are only approximate. The correct ones involve the so-called damping partition numbers , namely

T U o

According to the Robinson Theorem ^ ' the J numbers obey the following identity

In any case Eg. (29) and |30) are good approximations for a typical plain machine (J g ~ 2, J

x - 1 • J y = 1 " T h e e x a c t expression should contain a small correction to take into account the (eventual) radial gradient in the bending magnets).

After these few remarks on SR physics let us briefly discuss what are the achievable e-bearo qualities.

3.1. Emittances

The smallest emittances in an s.R. are achieved with the magnetic 14 )

lattice suggested by Chasman, Green and Rowe . Such a magnetic structure consists of M achromatic bends and, according to Krinsky 1 5 ^ , sommer

c x = < 7 - 7 x 1 0 m.rad) v V J M-1 (32)

which i s very accurate for M > 4 ^ ' . In actual storage ring design it is di f f i cult to achieve the win i murt va lue given bv ' 3 2 ) a n d a o r e rr»-!. ' stic estimate is

3.2 Energy spread

According to the explanations given so far, it may be thought that, due to damping, the e-beam becomes point like. This is not the case. The synchrotron radiation is, indeea, emitted m quanta of discrete eneroy

w h i c h g e n e r a t e a k i n d o f n o i s e . A s a c o n s e q u e n c e t h e e l e c t r o n s u n d e r g o a

d i f f u s i o n m e c h a n i s m , c o u n t e r a c t i n g t h e d a m p i n g , t h e r e s u l t i n g e n e r g y B p r e a d

1

9 J R P

C q = 3575 5^5 5 3 . 8 4 x l 0 - ' 3 m . ( 3 5 )

B e s i d e t h e q u a n t u m e x i c i t a t i o n t w o o t h e r e f f e c t s m a y c a u s e b e a m h e a t i n g ,

n a m e l y t h e " T o u s c h e k e f f e c t " a n d t h e " a n o m a l o u s b u n c h l e n g t h e n i n g " . We

w i l l d i s c u s s t h e m w i t h i n t h e f r a m e w o r k o f t h e c u r r e n t - l i m i t i n g f a c t o r s .

3 . 3 e - b e a j n c u r r e n t

C u r r e n t l i m i t a t i o n i n S R ' s i s d u e t o s u c h r e a s o n s a s b e a m - g a s i n t e r a c ­

t i o n , i o n t r a p p i n g , i n t r a - b e a m s c a t t e r i n g , a n o m a l o u s b u n c h l e n g t h e n i n g e t c .

F o r a m o r e c o m p l e t e d e s c r i p t i o n o f t h e s e e f f e c t s t h e i n t e r e s t e d r e a d e r m a y

l i k e t o Eee t h e p a p e r b y L e D u f f i n R e f . 1 9 . I n t h i s n o t e w e w i l l b r i e f l y

d i s c u s s t h e T o u s c h e k a n d b u n c h l e n g t h e n i n g e f f e c t s .

W h e n t w o p a r t i c l e s p e r f o r m i n g t r a n s v e r s e o s c i l l a t i o n s c o l l i d e , a p a r t

o f t h e i r t r a n s v e r s e m o m e n t u m i s t r a n s f o r m e d i n t o a l o n g i t u d i n a l m o m e n t u m

c h a n g e . A s a c o n s e q u e n c e , o n e p a r t i c l e g a i n s a n d t h e o t h e r l o o s e s m o m e n t u m .

I f t h e m o m e n t u m v a r i a t i o n i s l a r g e r t h a n t h e m o m e n t u m a c c e p t a n c e o f t h e SR

b o t h p a r t i c l e s a r e l o s t ' . T h i s i s t h e T o u s c h e k e f f e c t a n d i t s c o n s e ­

q u e n c e i s a r e d u c t i o n o f t h e b e a m l i f e - t i m e . T h e p r o b a b i l i t y o f i n t r a b e a m

s c a t t e r i n g i n c r e a s e s w i t h t h e b e a m d e n s i t y . T h e r e f o r e , t h e e f f e c t i s a l s o

a l i m i t i n g f a c t o r o f b o t h e m i t t a n c e a n d c u r r e n t d e n s i t y . C a l c u l a t i o n s o f

t h e m a x i m u m a c h i e v a b l e c u r r e n t a n d b e a m l i f e - t i m e h a v e b e e n g i v e n a n d c a n

b e f o u n d i n R e f s . 2 0 .

L e t u s n o w d i s c u s s a l i t t l e m o r e q u a n t i t a t i v e l y , t h e s o c a l l e d a n o m ­

a l o u s b u n c h l e n g t h e n i n g . T h e p h e n o m e n o l o g i c a l m o d e l 2 1 ^ p r e d i c t s t h a t w h e n

t h e b u n c h c u r r e n t e x c e e d s a c e r t a i n t h r e s h o l d v a l u e L h e e n e r g y s p r e a d a n d

t h e b u n c h l e n g t h w i l l b o t h i n c r e a s e w i t h t h e s t o r e d c u r r e n t i n t h e b u n c h .

* ) S t r i c t l y s p e a k i n g t h i s i s a s i n g l e - p a r t i c l e T o u s c h e k e f f e c t . H o w e v e r ,

a n o t h e r e f f e c t t a k e s p l a c e n a m e l y t h e m u l t i p l e T o u s c h e k e f f e c t , i n

w h i c h t h e e n e r g y t r a n s f e r b e t w e e n p a r t i c l e s d o e s n o t l e a d t o p a r t i c l e

l o s s e s b u t a p p e a r s a s a n o i s e s o u r c e f o r t h e p a r t i c l e m o t i o n . T h e o b ­

v i o u s c o n s e q u e n c e i s a n i n c r e a s e o f t h e e n e r g y s p r e a d i n t h e b e a m .

- 807 -

It can be shown that, under specific conditions the following two equalities hold 1 5 ' 2 0 > :

a -- = e , R = machine r a d i u B

s (36)

, ev I Z

where o z is the longitudinal bunch length, a the momentum compaction- v ß

the machine tune, T the average current and |2 n/n| the characteristic impedence. The peak current is given to

(37)

e I | ^ | = 2nE ooo2 . (38)

We now have all the moot important parameters to write down the gain for an SR FEL. We mUBt underline that Eq. (38) has a particularly interesting mean­ing. It shows that the energy spread is not only an undesireable featur< in the sense that it reduces the gain but, since larger energy spread allows 1arger peak currents, a suitable balanee between the two competitive ef-fects may give rise to an "optimum" energy spread for FEL operation. The explicit expression of the gain function is ( 2 L - L A )

g(u») = 2 5 M 2 f ( v . M u ) 139)

c |Z n/n| c x c

where f(...) is the inhomogeneous gain function and reduces to the ordinary gain function when all the u's are zero. The presence of the coupling coef­ficient H C

I N E <3- 1 i s due to the bunched beam operation and, therefore, to the longitudinal phase locking. Within this framework it is not an extra independent variable but, according to £qs. (27/ and(36|may be written as

( 4 0 )

Once the emittance is fixed by Eqs. (32-37) and by the further condition of Eq. (26] one can find the optimum a by looking at the maximum of the func­tion p-2 f ( . . . ) against u £ .

Analogous optimization criteria can be found for the current limita-

- SOS -

tion due to the Touschek effect, but we will not discuss this esse since the optimisation procedure closely follows that developed above. Let us now quickly discuss the power achievable with an SR FEL (for a complete analysis the interested reader is referred to Refs. 19-20).

At the beginning of this subsection we have briefly outlined the SR FEL dynamics and saw that the FEL interaction acts as a kind of noise which,

22 ) in the chosen hypothesis , induces a diffusion counteracted only by the damping due to the synchrotron emission in the bending and undulator mag­nets . We can, therefore, expect that the average laser power P L will be related to the synchrotron emission one. The relationship can be stated more quantitatively as follows. The laser process itself degrades the e-beam quai i ties, then the gain decreases and the laser is switched off. We must wait a time of the order of the damping time to have a new laser pulse. The average laser power is therefore approximatively given by 2 2 >

— E <

where AE is the maximum energy variation, N is= the number of particles in the beam and P is the synchrotron radiation power given by (see Eq. 26)

Using the above scaling law and the chine J we have plotted in Fig. 1

wavelength and have also included Touschek effect.

design parameters of the LEDA-F ma-the FEL-SR power levels against the the current limitation due to the

4- S INGLE-PASSAGE FEL OPERATION

The first FEL operation was accomplished with the Stanford supercon­ducting Linac. This electron source vas characterized by extremely good beam qualities which made it an almost unique tool for the first experi­mental attempts.

Even though an ideal machine for FEL operation, the superconducting Linac has long been considered an impracticable solution for FELs owing to its technological complexity and large operational costs 4 * . However, recent progress in superconducting cavity technology made the machine operation less critical, in principle, and reduced considerably the costs. It is therefore desirable that, with its special characteristics, this

- 8 0 9 -

a c c e l e r a t i n g d e v i c e b e c a r e f u l l y r- • > n s i d e r e d f o r F E L o p e r a t i o n . S i n g l e -

p a s s a g e F E L * s u s i n g m o r e c o n v e n t i o n a l s o u r c e s h a v e b e e n p r o p o s e d a n d u p t o

n o w h a v e o p e r a t e d w i t h a n i n d u c t i o n L i n a c 2 3 ' , a n R F L i n a c 2 4 ' , a V a n d e r

G r a a f m a c h i n e 2 5 * a n d a m i c r o t r o n z & ^ .

T h e c h a r t o f e x i s t i n g e x p e r i m e n t s i s s h o w n i n F i g . 2 . A r a t h e r d e -

t a i l e d r e v i e w o f t h e l o w e n e r g y a c c e l e r a t o r s d e d i c a t e d t o F E L o p e r a t i o n h a s

b e e n m a d e i n R e f . 4 . H e r e w e w i l l b r i e f l y d i s c u s s a f e w o f t h e c h a r a c t e r i z ­

i n g f e a t u r e s o f e a c h e - b e a m s o u r c e .

T h e m o s t c o m p r e h e n s i v e r e v i e w o n L i n a c s h a s b e e n g i v e n i n R e f . 2C.

T h e r e a r e e s s e n t i a l l y t w o t y p e s , n a m e l y t h e R F a n d t h e i n d u c t i o n . T h e f i r s t

c a n p r o v i d e c u r r e n t s o f t h e o r d e r o f h u n d r e d s o f mA a n d ( f o r F E L o p e r a t i o n )

a n e n e r g y o f h u n d r e d s o f M e V . I n d u c t i o n L i n a c s c a n f u r n i s h e - b e a m s o f t e n s

o f k A a n d t e n s o f H e V 4 * ; t h e a d v a n c e d t e s t a c c e l e r a t o r i s i n d e e d d e s i g n e d 2"11

t o p r o v i d e a b e a m o f 1 0 k A a t 5 0 M e V

I t i s c l e a r t h a t c o n v e n t i o n a l R F L i n a c s c a n b e d e d i c a t e d t o C o m p t o n

r e g i m e F E L s , w h i l s t i n d u c t i o n L i n a c s F E L o p e r a t e i n t h e s o c a l l e d h i g h g a i n

c o l l e c t i v e r e g i m e 3 ' . R F L i n a c s h a v e b e e n o p e r a t e d a r o u n d t w o f r e q u e n c i e s ,

3 G H z ( S - b a n d ) a n d 1 . 3 G H z ( L - b a n d ) . T h e m a i n l i m i t a t i o n o f t h e s e m a c h i n e s

i s t h e l a r g e e n e r g y s p r e a d w h o s e m a i n s o u r c e s a r e t h e v a r i a t i o n i n R F a c ­

c e l e r a t i n g f i e l d a l o n g t h e l e n g t h o f t h e b u n c h a n d s i n g l e - b u n c h b e a m l o a d ­

i n g 2 0 ^ . A t y p i c a l e n e r g y s p r e a d i s 1 % a t 5 0 M e V w h i c h c a n r e s u l t i n a t o o

l a r g e i n h o m o g e n e o u s b r o a d e n i n g t o e n s u r e l a s e r a c t i o n . A t y p i c a l m e a s u r e t o

o v e r c o m e t h i s l i m i t a t o n i s t o c o m p r e s s t h e e n e r g y b y a n o r d e r o f m a g n i t u d e .

T h e t y p i c a l l o n g i t u d i n a l m i c r o b u n c h l e n g t h v a r i e s f r o m 3 - 4 p s f o r

s u p e r c o n d u c t i n g L i n a c s , t o 6 a n d 1 5 p s f o r 3 a n d L b a n d n o r m a l L i n a c s r e ­

s p e c t i v e l y . A c c o r d i n g t o E q . ( 2 8 1 S - b a n d L i n a c s ( a s w e l l a s s u p e r c o n d u c t i n g

o n e s ) m a y c r e a t e p r o b l e m s f o r F E L o p e r a t i o n w i t h l o n g u n d u l a t o r s a t F I R

w a v e l e n g t h s . T h e u s e o f t h e e n e r g y c o m p r e s s i o n m e c h a n i s m m a y s o l v e t h i s

p r o b l e m t o o , b u t i n e v i t a b l y c r e a t e s d i f f i c u l t i e s w i t h t h e p e a k c u r r e n t . A

c r i t e r i o n t o o p t i m i z e b u n c h l e n g t h a n d p e a k c u r r e n t h a s b e e n d i s c u s s e d i n

R e f . 1 1 . L a r g e p e a k c u r r e n t s c a n , i n p r i n c i p l e , b e o b t a i n e d w i t h a L i n a c ,

b u t l i m i t a t i o n s a r i s e f o r t h e a v e r a g e c u r r e n t . U s i n g a s u b h a r m o n i c b u n c h i n g 2 4 )

t e c h n i q u e 1 0 0 A p e a k c u r r e n t i n 3 0 p s h a s b e e n o b t a i n e d a t L o s A l a m o s ' .

A t O s a k a U n i v e r s i t y 3 0 ^ t h e s a m e m e t h o d i s u s e d t o o b t a i n v e r y h i g h s i n g l e

b u n c h c u r r e n t ( 3 k A i n 1 6 p s ) .

L e t u s n o w b r i e f l y d i s c u s s t h e p r o b l e m o f e m i t t a n c e i n L i n a c s . i n

F i g . 9 w e h a v e p l o t t e d t h e p r o d u c t o f t h e n o r m a l i z e d e m i t t a n c e s a g a i n s t t h e

a v e r a g e c u r r e n t s o f t h e m o s t r e p r e s e n t a t i v e s a m p l e o f e x i s t i n g l o w - e n e r g y

a c c e l e r a t o r s 1 1 T h e s e t w o g u a n t i t i e s a r e r o u g h l y c o r r e l a t e d b y a n e m ­

p i r i c a l r e l a t i o n s h i p

- BIO -

10"

10'

1

J 1 0 3

Í 10-*

io-5

10"

10"

•SLAG LINAC /

• » B S m SLAC INJECTOR (SHB)

*N1NA INJF.CTOR

SLAC INJECTOR / . S L A C G I N (NORMAL) . l I w r w m r

• UCBS

SCA (SUB-HARMONIC BUNCHER) / • ^ C A (PULSED GUN? |

1mA 100mA 10A

10mA 1A 100A l ( A l

Fig. 9 Current V B emittances for existing accelerating devices without radiative damping

Í 4 3 )

w h i c h h a s b e e n e x p l o i t e d t o g e t s c a l i n g r e l a t i o n s h i p s f o r t h e F E L g a i n a n d

p o w e r H o w e v e r , s i n c e n o f u n d a m e n t a l l i m i t o t h e r t h a n t h e c a t h o d e e m i s ­

s i o n e x i s t s , o n e m a y e x p e c t t h a t w i t h c a r e f u l r e s e a r c h t h e l i m i t o f E q . 4 3

c a n b e i m p r o v e d b y a n o r d e r o f m a g n i t u d e o r m o r e 3 1 K

I n d u c t i o n L i n a c s , a s a l r e a d y n f - t i o n e d , c a n p r o v i d e v e r y h i g h b e a m c u r ­

r e n t s , b u t a r e , a t t h e m o m e n t , 1 . - * zed i n e n e r g y ( t e n s o f M e V ) a n d p u l s e

d u r a t i o n ( t e n s o f n s ) . A s t o t h e . s r g y s p r e a d i t c a n b e d e r i v e d f r o m t h e

f o l l o w i n g r e l a t i o n s h i p 28)

-3 / a ~ 1 0 Vl + y • • ( 4 4 )

T y p i c a l v a l u e s a r e a r o u n d a f e w % a . A s f a r a s t h e e m i t t a n c e i s c o n c e r n e d ,

t h e a b o v e c o n s i d e r a t i o n s r e l e v a n t t o R F L i n a c s a l s o h o l d f o r i n d u c t i o n

o n e s .

* ) T h e e m i t t a n c e s a r e e x p r e s s e d i n c m . r a d .

- 811 -

4 . 2 M i c r o t r o n s

A r e v i e w o f F E L e x p e r i m e n t s i n p r o g r e s e w i t h m i c r o t r o n s a n d t h e i r r e ­

l e v a n t t e c h n o l o g y h a s b e e n m a d e i n R e f . 4 . T h e m i c r o t r o n s o f f e r w i t h

r e s p e c t t o t h e L i n a c s t h e i m p o r t a n t a d v a n t a g e o f a m u c h l o w e r e n e r g y

s p r e a d w h i l e t h e p u l s e l e n g t h i s l o n g e r t h a n t h a t o f t h e S - b a n d L i n a c s .

T h e s e t w o e f f e c t s c a n b e e x p l a i n e d b y t h e e n e r g y c o m p r e s s i o n m e c h a n i s m 3 2 )

w h i c h i s a u t o m a t i c a l l y s e t u p b y t h e m i c r o t r o n o p e r a t i n g p r i n c i p l e

T y p i c a l v a l u e s o f e n e r g y s p r e a d a r e

K a ~ 1 . 5 x 1 0 rr^ ( 4 5 )

e ^ o

w h e r e E r i s t h e r e s o n a n t e n e r g y g a i n p e r o r b i t a n d E Q i s t h e n o m i n a l m a ­

c h i n e e n e r g y .

F r o m E g . I l l ) a n d t h e a b o v e r e l a t i o n s h i p , w e f i n d a c o n d i t i o n o n t h e

r e s o n a n t e n e r g y g a i n t o a v o i d e n e r g y s p r e a d p r o b l e m s , n a m e l y

(46)

T a k i n g i n t o a c c o u n t t y p i c a l o p e r a t i n g F E L m i c r o t r o n p a r a m e t e r s a n d t h e

f a c t t h a t E ^ < 1 M e V , t h e c o n d i t i o n o f E q . ( 4 6 ) i s l a r g e l y s a t i s f i e d . T y p i c a l

v a l u e s o f t h e b u n c h l e n g t h a r e a r o u n d 2 0 - 3 0 p s . L i m i t a t i o n s c a n a r i s e f o r * )

F E L o p e r a t i o n a t l o n g w a v e l e n g t h s ( F I R ) o r w i t h l o n g u n d u l a t o r s . F o r

t h e m i c r o t r o n e m i t t a n c e s t h e c o n c l u s i o n s a r r i v e d a t f o r t h e L i n a c s 1 1 '

a l s o h o l d .

T h e m o s t s e r i o u s d i s a d v a n t a g e o f a m i c r o t r o n i s t h e p e a k c u r r e n t w h i c h

c a n r e a c h o n l y a f e w A m p e r e s . T h e i n t r i n s i c l i m i t s a r e t h e a m o u n t o f p o w e r

w h i c h c a n b e p u m p e d i n t o t h e c a v i t y , a n d t h e c a t h o c V g e o m e t r y . T h e r a c e ­

t r a c k raicrotrons h a v i n g a s e p a r a t e i n j e c t o r s e c t i o n c a n b e u s e d t o o v e r ­

c o m e t h i s d i f f i c u l t y a n d e n j o y b o t h t h e a d v a n t a g e s o f l i n a c s a n d c o n -

4 )

v e n t i o n a l m i c r o t r o n s

4 . 3 V a n d e r G r a a f a c c e l e r a t o r s

F i n a l l y w e m e n t i o n c h e V a n d e r G r a a f a c c e l e r a t o r s . A m a c h i n e o f t h i s

t y p e h a s b e e n a l r e a d y e x p l o i t e d a s a n e - b e a m s o u r c e f o r t h e U C S B F E L e x -2 5 )

e x p e r i m e n t T h e b e a m o f a V a n d e r G r a a f a c c e l e r a t o r i s c h a r a c t e r i z e d * ) We m u s t h o w e v e r u n d e r l i n e t h a t t h e F E L o p t i m i z a t i o n i s a r a t h e r c o m ­

p l i c a t e d p r o c e s s , w h i c h s h o u l d b e c a r r i e d o u t w i t h r e g a r d t o t h e v a r i o u s

e f f e c t s c o n t r i b u t i n g t o t h e g a i n . T a k i n g t h e s e e f f e c t s i n t o a c c o u n t

s e p a r a t e l y m a y l e a d t o m i s l e a d i n g c o n c l u s i o n s .

by e x t r e m e l y good q u a l i t i e s . For e x a m p l e , t h e UCSB a c c e l e r a t o r has

f u r n i s h e d a beam w i t h an e m i t t a n c e ( n o r m a l i z e d ) a t 2 . 5 MeV o f a b o u t

7 . 5 x l 0 ~ m . r a d . F u r t h e r m o r e t h e beam has a c o n t i n u o u s s t r u c t u r e and can

r e a c h a v e r a g e c u r r e n t s o f a few Amperes w i t h maximum e n e r g y o f t e n s o f

MeV 4 , 3 3 )

4 . 4 G e n e r a l F e a t u r e s o f S i n g l e - P a s s FELs

We have d i s c u s s e d so f a r t h e a c c e l e r a t o r p e r f o r m a n c e s r j t h e r t h a n

t h o s e r e l e v a n t t o t h e l a s e r . We w i l l now b r i e f l y d i s c u s s the main c h a r a c ­

t e r i s t i c s o f t h e s i n g l e - p a s s FEL g a i n and s a t u r a t i o n .

Assuming t h a t t h e m a i n l i m i t a t i o n on t h e beam c u r r e n t i s t h e RF power ,

we can w r i t e FEL s i n g l e passage g a i n as f o l l o w s 3 '

g. = g° ¡fleq [ 8 ; u ; u ; u ,u ] h --- h e l i c a l y h a h ^y 1 M c M x ^ y M c

q„ = QÍ! .üfeq [G ;u ;np , nij , nu | , 2 - l i n e a r ( 4 7 )

where 6 i s t h e machine d u t y c y c l e , P [MW] i s t h e e-beam power i n megawat ts

and 6 and 9 a r e t h e " d e l a y - p a r a m e t e r s " g i v e n by

4N tu Í T ( 4 8 )

£ , n

where ÓT = T c - T , T t h e c a v i t y round t r i p p e r i o d and T t h e b u n c h - b u n c h

t i m e ( s e e F i g . 1 0 ) .

The q u a n t i t y í feq^ r e p r e s e n t s t h e maximum v a l u e c f t h e m u l t i m o d e g a i n

f u n c t i o n and c o n t a i n s a l s o t h e dependence on t h e d i f f e r e n t p a r a m e t e r s

e n t e r i n g t h e p r o c e s s .

The t y p i c a l b e h a v i o u r o f eq a g a i n s t 8 i s shown i n F i g . 1 1 , toge t h e r

w i t h t h e d i m e n s i o n l e s s l a s e r power \ . I t s h o u l d be n o t i c e d t h e ' the

maximum g a i n and t h e maximum o u t p u t l a s e r power do n o t c o r r e s p o n d t o t h e

same v a l u e o f 0 . T h e r e f o r e o p t i m i z a t i o n o f t h e g a i n does n o t r e s u l t i n t h e

maximum o u t p u t l a s e r power .

The a v e r a g e l a s e r power can be e v a l u a t e d a c c o r d i n g t o t h e f o l l o w i n g

f o r m u l a 3 4 *

P L |MW] = P [ M W ) f | H z ) x ( B ) ( T M | u r l - T R [ u s J ) ( 4 9 )

G l JuLnL

F i g . 1 0 e - b e a m s t r u c t u r e f r o m a n R F m a c h i n e : r. m i c i ' o b u n c h t i m e d u r a -

0 0 2 0 4 0.6 0.8 1.0

F i g . 1 1 G a i n f u n c t i o n a n d d i m e n s i o n l e s s l a s e r p o w e r v s tl

w h e r e f i s t h e m a c h i n e r e p e t i t i o n f r e q u e n c y , i s t h e e - b e a m m a c r o p u l s e

d u r a t i o n ( s e e F i g . 1 0 ) a n d i R i s t h e p u l s e r i s e - t i m e l i n k e d t o t h e g a i n b y

t h e f o l l o w i n g e x p r e s s i o n

M m ] B I MS I - 0 . 1 4

w h e r e g i s t h e g a i n a s a f u n c t i o n o f t h e a b o v e p a r a m e t e r s , y l f i s t h e c a v i t y

l o s s a n d L C i s t h e l e n g t h o f t h e c a v í t y . I n F i g . 1 t h e c u i v e s r e p t e s e j i t .

t h e a v e r a g e o u t p u t p o w e r o f a n T E L o p e r a t i n g a t t h e 1 s t f i n d 3 t d h a i m o n J c

r e s p e c t i v e l y , w i t h a n e - b e a m p o w e r o f 2 0 MW. I t i s e v i d e n t t h a t i n t h e r e ­

g i o n 1 0 < A ( u m ) < 1 0 0 t h e F E L , i n p r i n c i p l e , m a y g e n e t < i t e U i g p i p o w e r t h a n

t h e c o n v e n t i o n a l s o u r c e s .

5. CONCLUSIONS

In this note we have presented a review of both the storage rings and single-pass FELs. We have emphasized the prob 1 ems relevant to the ulectron sources and laser light output but no mention hnu bi- ii mnJe of the cavity and undulator technology whicli are dealt with more com­pletely in Ref. 4 . we have also stressed that the future development of FELs as a "workhorse" for tunable applications strongly depends on the reliability of the electron source.

The use of the FEL for industrial applications will depend on its cost being relatively modest. In Kef. 3 4 a comparative cost analysi s of FEL with other lasers was carried out and the results are summarized in Fig. 1 2 . It is clear that the FEL is competitive when it is operating with a iii'j.!; efficiency extraction system (in the figure with an efficiency of 1 0 % )

As a concluding remark we would like to stress that the goal of this paper has been twofold, namely to give a review of the basic ideas and problems underlying the FEL physics, and to indicate how this new laser device must be realistically considered within the framework of tunable sources.

FEL (EFFICIENCY 1%}

5

« 10 1 FEL (EFFICIENCY lO'/J

10 100

Fig. 1 2 FEL cost/watt vs \ with efficiencies of 1 % and 1 0 %

*) The analysis has been limited to single-pass operating FELs where a market analysis for the electron source can be carried out. For the SR the technology is so speci fic that a marko t anal ysi s makes no sense.

R E F E R E N C E S

1 ) B . D . G u e n t h e r a n d R . G . B u s e r , I E E E J . Q u a n t u m E l e c t r . 16 ( 1 9 6 2 ) .

2 ) B . D . G u e n t h e r a n d R . G . B u s e r i n R e f . 1 p . 1 1 7 9 ,

3 ) G . D a L L o i i a n d A . H u n i e r i , L a s e r H a n d b c k , t-d \ y 11. ' ^ ; ; '. -h .i;.fí

M . S . B a s s ( N o r t h - H o l l a n d C o m p a n y , A m s t e r d a m 1 9 B 5 ) , V o l . I V , p 1 .

4 ) U . B i z z a r r i , F . C i o c c i , G . D a t t o l i , ^ . D e A f i g e l i s , E . F l o r e n t i n o , G . P . G a l l e r a n o , T . L e t a r d i , A . M a r i n o , G . M e s s i n a , A . R e m e n , E . S a b i a a n d A . V i g n a t i , t o b e p u b l i s h e d i n " L a R i v i s t a d e l N u o v o C i m e n t o " -

5 ) H . M o t z , J . A p p l . P h y s . 2 2 , 5 2 7 ( 1 9 5 1 ) .

6 ¡ J . M . J . M a d e y , J . A p p l . P h y s . 4 2 , 1 9 0 6 ( 1 9 7 1 ) .

7 ) G . D a t t o l i a n d A . R e n i e r i , F E L H a n d b o o k , E d . b y W . B . C o l s o n , C . P e l l e g r i n i a n d A . R e n i e r i , ( N o r t h - H o l l a n d C o m p a n y , A m s t e r d a m ) , t o b e p u b l í r h e d .

8 ) L . R . E l i a s a n d J . M . J . M a d e y , R e v . S e i . I n s t r ; . i t i . 5 0 , 1 3 3 5 ( 1 9 7 9 ) .

9 ) Y . F a r g e , A p p l . O p t i c s 1 9 , 4 0 2 1 ( 1 9 3 0 ) .

1 0 ) W . B . C o l s o n , G . D a t t o l i a n d F . C i o c c i , P h y s . R e v . 3 1 A , 8 2 8 ( 1 9 8 5 ) .

1 1 ) G . D a t t o l i , T . L e t a r d i , J . M . J . M a d e y a n d A . R e n i e r i , I E E E J Q E - 2 0 , 6 3 7 ( 1 9 8 4 ) .

1 2 ) R . B a r b i n i , G . D a t t o l i , T . L e t a r d i , A . M a r i n o . A . R e n i e r i a n d G - V i g n o l a , I E E E T r a n s . N u c l . S e i . N S - 2 6 , 3 8 3 6 ( 2 9 7 9 ) .

1 3 ) K . R o b i s o n , P h y s . R e v . I l l , 3 7 3 ( 1 9 5 8 ) .

1 4 ) R . C h a s m a n , G . K . G r e e n a n d E . M . R o w e , I E E E T r ¿ n s . N u c l . S e i N S - 2 2 , 1 7 6 5 ( 1 9 7 5 ) .

1 5 ) S . K r i n s k y , F r e e E l e c t r o n G e n e r a t i o n o f E x t r e m e U l t r a v i o l e t C o ­h e r e n t R a d i a t i o n , E d . b y J . M . J . M a d e y a n d c . P e l l e g r i n i A I P , 1 1 8 , s u b s e r i e s o n O p t i c a l S c i e n c e a n d E n g i n e e r i n g , A P S , 4 4 ( 1 9 8 5 )

1 6 ) M . S o m m e r , D C I I n t e r n a l n o t e 2 0 / 8 1 ( 1 9 8 ¿ ) .

1 7 ) D . T o t a u x , D C I I n t e r n a l n o t e 3 0 / 8 1 ( 1 9 8 1 ) .

1 6 ) S e e e . g . G . V i g n o l a , N u c l . I n s t r u m . a n d M e t h o d s A 2 3 6 , 4 1 4 ( 1 9 6 5 ) .

1 9 ) J . L e D u f f , J . d e P h y s . C l , 4 4 , 2 1 7 ( 1 9 8 3 ) .

2 0 ) H . W i e d e m a n n , J . d e P h y s . C l , 4 4 , 2 0 1 ( 1 9 8 3 ) .

2 1 ) A . R e n i e r i , L N F R e p o r t 7 6 - 1 1 ( 1 9 7 6 ) ; A . W . c h a o a n d J . G a r a y t e , P E P - 2 2 4 ( 1 9 7 6 ) .

2 2 ) A . R e n i e r i , 1 1 N u o v o C i m e n t o , 5 3 B 1 6 0 ( 1 9 7 9 ) . G . D a t t o l i a n d A . R e n i e r i , I l N u o v o C i m e n t o , 5 9 B , I ( I 9 Ö 0 J .

2 3 ) T . T . O r z c h o w s k y , B . A n d e r s o n , W . M . F a w l e y , D . P i o s n i t z , E . T . S h a r l e m a i i n S . Y a r e m E , D . H o k i n s , A . C . P a u l , A . M . S e s s l e r a n d J . W u i L e l e , P h y s . R e v . L e t t 5 4 , 8 8 9 ( 1 9 8 5 ) .

2 4 ) R . W . W a r r e n , W . E . S t e i n , M . T . L y n c h , R . L . S h e f f i e l d j t i d J . S . F r a s e r , N u c l . I n s t r u m . a n d M e t h o d s , A2 3 7 , 1 8 0 ( 1 9 3 5 ) .

- 81(1 -

2 5 ) L . R . E l i a s , J . Hu a n d G . R a m i a n , N u c l . I n s t r u m . a n d M e t h o d s . A 2 3 7 ,

2 0 3 ( 1 9 8 5 ) .

2 6 ) U . B i z z a r r i , F . C i o c c i , G . D a t t o l i , A . D e A n g e l i s , G . P . G a l l e r a n o , 1 . G i a b b a i , G . G i o r d a n o , T . L e t a r d i , G . M e s s i n a , A . M o l a , L . P i c a r d i ,

A . R e n i e r i , E . S a b i a , A . V i g n a t i , E . F l o r e n t i n o a n d A . M a r i n o , P r o c . o f t h e L a k e T a h o e F E L ( 1 9 8 5 ) C o n f e r e n c e ( t o b e p u b l i s h e d ) .

2 7 ) A . F a l t e n s a n d D . K e e f e , P r o c . o f t h e 1 9 P 1 L i n e a r A c c . C o n f e r e n c e

L o s A l a m o s , N a t . L a b . L A - 9 2 3 4 - C , p . 2 0 5 ( 1 9 8 1 ) .

2 8 ) s e e e . g . R . K . C o o p e r , P . L . M o r t o n , P . B . W i l s o n , D . K e e f e a n d A . F a l t e n s , j . d e P h y s . C I , 4 4 , 1 8 5 ( 1 9 8 3 ) .

2 9 ) S e e e . g . G . S a x o n , N u c l . I n s t r u m . a n d M e t h o d s , A 2 3 7 , 3 0 9 ( 1 9 3 5 ) .

3 0 ) S . T a k e d a , 2 n d J a p a n - C h i n a J o i n t S y m p . L a n 2 h o n ( 1 9 8 3 ) ( u n p u b l i s h e d ) .

3 1 ) S e e e . g . W . A . B a r l e t t a , J . K . B o y d , A . C . P a u l a n d D . S . P r o n o , N u c l .

I n s t r u m . a n d M e t h o d s , A 2 3 7 , 3 1 8 ( 1 9 8 5 ) .

3 2 ) s e e e . g . s . R o s a n d e r , J . d e P h y s . C I , 4 4 , 2 3 3 ( 1 9 8 3 ) .

3 3 ) L . R . E l i a s , P h y s i c s a n d T e c h n o l o g y o f F r e e E l e c t r o n L a s e r s , E d . b y

S . M a r t e l l u c c i a n d N . C h e s t e r , ( P l e n u m P r e s s , Nt:w Y o r k , 1 9 8 3 ) .

3 4 ) G . D a t t o l i , T . L e t a r d i , J . M . j . M a d e y a n d A . R e n i e r i N u c l . I n s t r u m .

a n d M e t h o d s , A 2 3 7 , 3 2 6 ( 1 9 8 5 ) .

ISIS, THE ACCELERATOR BASED NKUTRÜN SOURCE A T RAI.

D A Cray and G H Rees

Rutherford Appleton L a b o r a t o r y IIK

1 . INTRODUCTION

During the Oxford Accelerator School a tour and description were arranged of the

Rutherford Appleten Laboratory 1 s new neutron source. Subsequently, on the last day of

the school, a seninar was given on the high- i n tens i t y performance of the sctirce ' s rapid

cycling synchrotron. Details of the talk and seminar are repeated here.

The design specification for the pulsed neutron source called for peak fluxes of iK " 2 - 1

therral and epithermal neutrons > 1 0 l b n cm sec in pulses of duration 10 us at a

repetition frequency of 50 H z . To achieve this goal at RAL the method adopted has been

the construction of a 50 H z , 800 MeV proton synchrotron to provide 2.2? 1 0 1 3 protons per

pulse at a heavily shielded target of depleted uranium 238. The initial reaction in the

target is the production of fast neutrons by spallation and fission- This is followed by

the slowing down of the neutrons to thermal and epithermal energies by associated

moderators.

¿uch a spallation neutron source allows significant advances compared with existing

high flux reactor sources. 'ihe ef fect ive flux is much greater than that aval]ah le f rom

reactors for the higher energies of the neutrón spectrum. This increase in neutron flux

will be a major benefit to a wide range of condensed matter studies, especially for the

case of the epithermal neutrons at energies of several electron volts.

2. LINAC AKT) SYNCHROTRON

The synchrotron injec r is a 70,4 MeV II linac with U Alvarez tanks operating at

202.5 Mftz, The pre-injector is a 665 kV Cockroft-tfalton set with a medium gradient

accelerating column, using glass Insulators. The H ion source Is of the Penning type

and uses a mixture of hydrogen gas and caesium vapour. It is a direct extraction source

with a duty cycle of 2.5% (50 H z , 500 u s ) .

A transfer line tAkes the 70.4 MeV beam line from outside to inside the synchrotron

magnet ring. Here it includes a 202.5 MHa debimcher cavíty to control the input beam

momentum spread; there is also a septum magnet, for Injection from an inside machine

radius. Diagnostics are p r o v d e d for emittance and momentum spread measurements.

The injection straight .ection oE the synchrotron is approximately 5 m in length.

It houses 4 septum-cype dipo' magnets for creating o localised bump of the closed orbit.

The first of the bump magneto ies adjacent to the injection septum magnet and the region

between the 2 central magnet; is used to house the foil which scrips H ions ¿o protons.

- SIB -

The synchrotron lg divided into 10 superperiods with each stiperperiod containing a

pair of doublet quadrupoles» a long straight section, a singlet quadrupole, a combined

function gradient-bending magnet and a medium length straight section. The quadrupoles

and bending magnets are connected in a 50 H z , series resonant circuit together with

associated capacitors and a common choke with 10 secondary windings (ex-NINA). The

superperiod and magnet design? were chosen for compatibility with the stored energy of

the existing choke and capacitors. In principle, the magnet current may be locked in

frequency to a fixed 50 Hz frequency or to the 50 Hz mains; In practice it is necessary

to lock to the fixed frequency for adequate stability.

Vacuum system components are entirely of metal or ceramic apart from the ferrite of

the injection and ejection magneis. The design has aimed for simplicity and reliability

to minimise the maintenance in the high radiation environment. Each superperiod has 3 _l

triode titanium sputter ion pumps of capacity 400 e sec with additional pumps at the _3

ferrite locations. The system is pumped down to 10 Torr via a roughing line which

passes through the ring shielding to external carbon vane, sorption pump units and a J?

turbo-pump. Tne ion pumps reduce the pressure to 5 10 Torr within 10 hours, ultimately _e

reaching 10 T D T X. There are no sector valves in the ring but all-metal valves are

included in the injection and extraction hearn lines.

Eddy currents preclude the use of solid metal chambers within the large aperture,

rapid-cycling magnets; ceramic chambers are used for both the main and the correction

magnets. Sections of chamber are formed, typically 300 mm In length, by isostatically

pressing 97.6X pure alumina powder in a mould, machining the pressed powder, firing at

high temperature and subsequently grinding the external surfaces to a high tolerance.

Final chambers are formed from the individual sec ci ans by glazing and dowel]ing the

mating surfaces and then heating the free-standing assembly to 1100 ° C in a furnace to

allow the 0.25 mm glass layer to fuse into the ceramic, bonding the surfaces. End

flanges are also of ceramic and are sealed to adjacent flanges by re-usable Indium

T~sea]s.

Special radio frequency shields ar-> inserted within the ceramic chambers to reduce

the coupling impedance of thi proton beam to its environment. The shields are made as

rectangular chambers of stainless steel rods > supported in insulating frames, with the

rods lying paral lei to the beam direct Ion. In the bending magnets the side rods are

replaced by 2 mm thick solid, stainless steel plates, standing vertically. Each rod and

side plate is connected to the adjacent straight section by compact, ceramic capacitors

which present a high impedance at 50 Hz but a low one at and ahove 100 kHz.

Acceleration from 70.4 to 800 MeV Is achieved via 6, double-gapped, ferrite-tuned

cavities, operating from 1.347 to 3.09 MHz at harmonic number 2. The net peak

accélérât tng voj cage has to vary smoothly f rom 3 to 160 kV per turn in the J 0 as

acceleration period and the design intensity corresponds to a high level of beam loading.

At present four of the cavities are in operation and are adequate for acceleration to 550

MeV. The final two are to he installed In mid-1986. Each cavity is powered by its own

- 819 -

RF amplifier, Lhe final stage of which contains two 250 kW tetrodes in parallel. For che

present stage of running, only one of the tetrodes is included per ampl 1 fier but the

second must be added to control the maximum beam load i..g level s. The power amp 1Ifiere

have been designed for ease of removal with electrical and coolant connections nade via

quick disconnect terminâtíor.s In a region that may be shielded if It Is found necessary.

Extraction is achieved by three fast kicker magnets and an extract ion septum

magnet. The plane for extraction is vertical and a closed orbit bump at the septum

locatior reduces the kicker requirements. Each kicker is of a push pull design with the

ferrite split at the mid-point by an electrical ground plane. Each half magnet is

powered via a pulse forming network and coaxial thyratron switch. The voltage on the

system Is 40 kV, the peak current 5000 A and the required kick rise-ti^ie 225 ns. The

septum unit lies above the synchrotron and encloses its curved (21°) vacuum chamber which

is joined to the top of the straight section beneath. The septum chamber is non-magnetic

but the adjacent straight section is of mild stepl to reduce the septum leakage field at

the beam. Parameters of che magnet are field level 1 T, bending length 1.8 m and septun

thickness 10 - 15 mm. In the extraction straight are a number of beam loss protection

units; the low energy units are made of copper and graphite and the high energy units of

stainless steel.

There is a long (150 ta) baaraline from the extraction point to the target station,

it consists almost entirely of ex-NIMBOD, large aperture q u a d m p o l e s and dipnles. The

power dissipation in the line Is high, over BOO M i , because of the large bend angles

involved. The downstream end of the line lies In the neutron experimental hall and in

this region it is heavily shielded. In Che future an intermedíate target station Is to

be installed In the line. This is to feed a powerful inuon spin resonance beam line and

experimental station. At the end of the main beamllne the proton beam is focussed to a

70 mm diameter spot at the incident end of the uranium target. Beam profiles along the

line are measured using strip secondary emission detectors.

3. TARGET STATION'

The target cons is :.s of a cooled uranium target, assoc iated moderators, reflectors

and decouplers, a bulk shield and shutter system and a remote handling facility for

dealing with spent and replacement targets.

Target material is depleted nraaium 238 and, because of Its p o o r thermal

conductivity, it is segmented inte a number of plates with intermediate parallel cooling

channels. Heavy water is used for the cooling and the uranium plates are clad with

ZIrcaloy-2 to avoid corrosion and to contain fission products.

The interaction of the 800 Mi?V proton beam with the target material Is a

combination of spallation and nuclear excitation. Fast neutrons and other secondary

particles result directly from spallation and subsequently from fission and evaporation

after nuclear de-excitation. Secondary particles undergo further interactions leading to

- S :Ü -

a p a r t i c l e c a s c a d e . U r a n i u m i s used i n p r e f e r e n c e t o o t h e r h e a v y m e t a l s as t h e r e I s a

f u r t h e r f a c t o r o f 2 i n c r e a s e i n t h e n e u t r o n y i e l d s due t o t h e t i s s i o n s .

The t a r g e t a r r a y I s 3 4 0 mn l o n g w h i c h i s 207. l o n g e r t h a n t h e r a n g e l e n g t h f o r

8 0 0 HeV p r o t o n s . The u r a n i u m d i s c s a r e 90 mn i n d i a m e t e r and t h e y a r e mounted i n

r e c t a n g u l a r p i c t u r e f r a m e s c o n t a i n e d i n a s t a i n l e s s s t e e l v e s s e l . N e u t r o n p r o d u c t i o n I s

a p p r o x i m a t e l y 26 . i e u t r o n s e s c a p i n g t h e t a r g e t p e r I n c i d e n t p i o t o f i ; t h e a v e r a g e n e u t r o n

e n e r g y I s 2 MeV and t h e r e a r e a b o u t 10% o f t h e n e u t r o n s w i t h an e n e r g y g r e a t e r t h a n

15 MeV. Power d i s s i p a t i o n i n t h e t a r g e t i s 2 0 0 kW f o r Î 8 0 uA of i n c i d e n t p r o t o n s . A

c h o i c e o f 90 mm i s made f o r t h e t a r g e t d i a m e t e r a s z c o m p r o m i s e b e t w e e n f a s t n e u t r o n

p r o d u c t i o n and c o u p l i n g t o t h e a s s o c i a t e d m o d e r a t o r s .

T h e r e h a s t o b e a m e t a l l u r g i c a l bond b e t w e e n t h e u r a n i u m p l a t e s and t h e c l a d d i n g o f

Z i r c a l o y - 2 a n d t h i s i s a c c o m p l i s h e d î y h o t . ' . s o s t a t i c p r e s s i n g . The c o o l i n g c h a n n e l

b e t w e e n t h e i n d i v i d u a l p l a t e s i s 1 . 7 5 mm and s u b s e q u e n t s w e l l i n g o f t h e u r a n i u m may

r e d u c e t h i s t o 1 mm. Heavy w a t e r i s t h e c o o l a n t r a t h e r t h a n l i g h t w a t e r as t h e p r e s e n c e

o f t h e h e a v y w a t e r i n t h e e x t e n s i v e c o o l i n g m a n i f o l d s a c t s as a n e u t r o n r e f l e c t o r . F o r

s a f e t y t h e r e a t e s e c o n d a r y and t e r t i a r y l i g h t w a t e r c o o l i n g c i r c u i t s c o u p l e d v i a h e a t

e x c h a n g e r s t o t h e p r i m a r y h e a v y w a t e r c i r c u i t .

F a s t n e u t r o n s a r t s l o w e d t o e p i t h e r r a a l and t h e r m a l e n e r g i e s by m o d e r a t o r s . Two a r e

s i t e d j u s t above t h e t a r g e t and two b e l o w , a l l i n v i n g g e o m e t r y . A t y p i c a l s i z e o f

m o d e r a t o r i s 100 x 1 0 0 x 50 m m 3 , s m a l l enough t o r e s t r i c t t b e n e u t r o n p u l s e d u r a t i o n t o

5 - 100 us ( d e p e n d a n t on >.) , a r e q u i r e m e n t f o r t h e t i m e o f f l i g h t e x p e r i m e n t a l s t a t i o n s .

A l l m o d e r a t o r s h a v e e x t e r n a l d e c o u p l e r s on a l l f a c e s e x c e p t t h e e x i t f a c e and t h e w h o l e

a r r a y i s c o n t a i n e d w i t h i n a r e f l e c t o r . The d e c o u p l e r s p r e v e n t n e u t r o n s t h e r m a l i s e d

o u t s i d e t h e m o d e r a t o r f r o m e n t e r i n g i t and d e g r a d i n g t h e n u t p u t p u l s e . The r e f l e c t o r

s c a t t e r s h a c k f a s t n e u t r o n s I n t o t h e m o d e r a t o r , e n h a n c i n g t h e o u t p u t y i e l d by n F a c t o r

o f 3 .

E a c h m o d e r a t o r i s d e s i g n e d t o o p t i m i s e i t s p e r f o r m a n c e o v e r a p a r t i c u l a r r a n g e o f

t h e n e u t r o n e n e r g y s p e c t r u m . Two t y p e s c o n t a i n a m b i e n t - t e m p e r a t u r e l i g h t w a t e r , one t y p e

l i q u i d m e t h a n e and one p a r a - h y d r o g e n a t 20 ° K . The l o w e s t t e m p e r a t u r e m o d e r a t o r p r o v i d e s

t h e l o n g e s t wave l e n g t h n e u t r o n s , 4 - 10 A . E n d o s l n g t h e t a r g e t and m o d e r a t o r s i s t h e

r e f l e c t o r w h i c h conta ins b e r y l I iurn a n d heavy w a t e r . T o t a l p o w e r d e p o s i r e d i n t h e

r e f l e c t o r I s 7 . 2 kW, tha ï : i n t h e m o d e r a t o r s i s 1 kW and t h a t i n t h e d e c o u p l e r s i s 9 kW.

T h e r e i s a l a r g e h u l k s h i e l d s u r r o u n d i n g t h e t a r g e t w h i c h r e d u c e s t h e r a d i a t i o n

l e v e l i n a c c e s s i b l e a r e a s t o < 7 . 5 uSv h r . A d d i t i o n a l s h i e l d i n g i s n s e d a r o u n d t h e 18

n e u t r o n beam p o T t s i n t h e s h i e l d and a r o u n d the. 18 n e u t r o n heam t u b e s a n d d e t e c t o r s . The

o v e r a l l s h i e l d h e i g h t i s 7 m, t h e t h i c k n e s s 4 . 3 m and t h e o u t e r 0.25 m l a y e r i s c o n c r e t e

l o a d e d w i t h 1% b o r o n . I n t h e f o r w a r d d i r e c t i o n , t h e s h i e l d i n g I s e x t e n d e d t o w a r d s t h e

r e m o t e h a n d l i n g c e l l .

W i t h i n t h e s h i e l d i n g i s a t a r g e t v o i d v e s s e l , a s h u t t e r s y s t e m , s h i e l d i n g i n s e r t s

- 821 -

and a plinth and shield door. A cylindrical p r e s s u r e vessel with 18 double aluminitas windows 1B used to contain a helium atmosphere around the target and a closed cooling circuit for the helium removeB about 5 kw Erom the vessel va11B. The shutter s y s t e n

constats of tvo abutter vessels, each containing nine. 22 tonne shutters. These are made of concrete and iron and are used to isolate an Individual neutron beam line. The shielding i n B e r t s are prefabricated steel boxes with recoveable shielding blocks packed around the neutron beam tubes so that each beam tube may be readily ie-designed. The concrete includes two caverns. The final component of the shield Is the 90 tonne, 4.5 m

thick door at the downstream end of the target, ahead of the remote handling cell and the services region. There is a seal between the door and the shield to contain the atmosphere of helium. The target-moderator assembly ls cantlievered fron the shield door.

when it is necessary to obtain access to the target assembly, the shield door is rolled backwards on rails until the target assembly is in the remote handling cell; the door then completes the back wall of the cell. Four master-slave manipulators are used in the cell for removing a spent target, replacing with a new one and for any maintenance on the components of the assembly. To facilitate use of the manipulators, Che cell Is provided with two large, zinc bromide windows. Viewing I B supplemented by TV cameras. Irradiated targets may be stored in any of three storage wells in the floor of the cell and be removed via an acceBs hatch.

4. EXPERIMENTAL FACILITIES

A wide range of instruments are used with the eighteen neutron beam lines ¡ at present eight are in ope - H t i o n and In the future up to cwenty five Day be accommodated. In addition to neutron scattering science, n e u t r i n o physics will be undertaken at ISIS In a large neutrino blockhouse adjacent to the target. A brief description only is given of the present experimental facilities.

A B p e c t r o m e t e r named IRIS is used to measure quasi-elastic scattering in processes such as dlffuslonal motion in liquids and rotational and translational dynamics of molecules. Good time of flight resolution is achieved with a 40 m cold neutron guide and with energy analysis by crystal analysers in back re fleet Ion. The resolution of the instrument ls In the range around 50 uev.

LAD is a l i q u i d s ' and amorphous materials' diffractometer. Tt is used to study the structure factors of non-crystalline materials and also as s medlum-resolntion, hiph intensity diffractometer. This Instrument had been previously tested on Che Harwell pulsed n e u c r o n source.

HRPD ls a high resolution powder difEractometer and Includes a 96 ro thermal neutron guide, It allows a large number o f structural parameters to be determined from powder measurements and i t may also be used to study phase transit ions and line broadening effects.

- H 22 -

TXFA is a tine-focissed crystal analyser; HET Is a high energy transfer inelastic

spectrometer; LGQ is a low-Q diffrac tometer ; EVS is an electron volt spectrometer for

epi thermal neutrons and POLARIS ts for study as a polarisation spectrometer. POLARIS

uses a neutron polarising filter, Sm 149, in the incident neutron hean and it will be

developed for performing inelastic polarisation experiments for investigating electronic

and nuclear magnetism.

Several -nechanical chopper systems have been developed for use with the neutron

instruments. A magnetic bearing has been incorporated in the chopper for the HET

spectrometer; there are three rotors allowing peak transmission at 0.25, 0.5 and 1.0 eV

energies for 1 ps pulses.

The neutrino facility, KAkHEN, has been initiated by the Karlsruhe Laboratory, FRC.

It consists of a 5500 tonne iron shield, housing two detector svstems, and is located

14 m from the ISIS target station. The inside dimensions of the blockhouse are

10 x ¿ s 3 m 3 . One detector is a total energy calorimeter using liquid scintillator and

the other is a high precision tracking device for measuring neutrino-electron scattering.

5. HIGH INTENSITY PERFORMANCE OF THE ISIS SYNCHROTRON

The main features involved In high intensity operation oí the synchrotron are the

performance of the linac injector, the efficiency of the H injection process, the bunch

formation in the ring, the trapping efficiency, the heavy beam loading, the crossing of

betatron resonances, the possibilIty of instab H i t les, the extraction eff iciency and the

activation of machine components.

At the time of the Oxford meet ing, the maximum bean: injected at low re pet it ion

frequency had been lo' protons per pulse. The best performance had been 9 uA on target,

corresponding to 4.5 1 0 1 2 protons accelerated per pulse (for 5.5 1 0 1 2 H ions inject. H

at 12.5 Hz. Si nee chat time the performance has been improved to 40 M A on target

corresponding to 5 lO* 2 protons per pulse at 50 Hz.

A schematic lay-out of ISIS is shown in Fin- 1, the scale of which may be judged

from the 52 m synchrotron dianeter and the 150 m beam line to the target station. The

linac is shown In the Forefront of the figure.

The injector has not yet met its design specification. Typical performance figures

have been output currents of 5 mA fnr pulse lengths up to 20D us at 25 and 50 Hz and

currents of 3.5 mA for pulse lengths up to 450 us at lower repetition frequencies. These

figures are set both by the H ion source and by the face that, as the average current is

raised, there is increased frequency of breakdown of the 665 kV accelerating column. The

mechanism of breakdown Is not understood but the present performance has been achieved

only after improved pumping at the high voltage end of the column and after installing

inter-electrode shields in the column to intercept ions before they reach the glass

insulators. Future plans include a thorough cleaning and check of the column and the

- 824 -

d e v e l o p m e n t o f a new I o n s o u r c e . T h e r e a r e no f u n d s a v a i l a b l e f o r Che d e v e l o p m e n t o f a n

RFQ.

The o u t p u t beam e m i t t a n c e s f r o m t h e l i n a c h a v e b e e n f o u n d t o be a s e x p e c t e d w i t h

9 5 1 o f t h e beam w i t h i n t r a n s v e r s e ( u n - n o r m a l i s e d ) e m i t t a n c e s o f 2 0 « u r a d m. A t

c u r r e n t s up t o 1 mA, momentua s p r e a d n e a s u r e r a e n t s i n d i c a t e 9 S 1 o f t h e beam w i t h i n & p / p _3

v a l u e s o f * 1 . 2 10 a n d a d e b u n c h e r c a v i t y i s u s e d r o u t i n e l y t o r e d u c e t h e s p r e a d t o _(«

< i 5 10 .

5 . 1 H c h a r g e e x c h a n g e i p j e ç r l o n ^

Up t o 3 0 0 t u r n s h a v e been i n j e c t e d I n t o t h e s y n c h r o t r o n w i t h h i g h e f f i c i e n c y by

a p p r o p r i a t e f i l l i n g o f h o r i z o n t a l and v e r t i c a l b e t a t r o n phase s p a c e f o l l o w i n g t h e

s t r i p p i n g o f H i o n s t o p r o t o n s . L a r g e a l u m i n i u m o x i d e s t r i p p i n g f o i l s , 120 mm x 4rt mm,

h a v e b e e n d e v e l o p e d w i t h i n t h e l a b o r a t o r y »nd h a v e p r o v e d h i g h l y s a t i s f a c t o r y i n

o p e r a t i o n . T h e y h a v e a t h i c k n e s s o f 0 . 2 5 u .

O v e r 9 8 J o f t h e i n p u t H i o n s a r e s t r i p p e d t o p r o t o n s a n d a b o u t LH% t o H°

p a r t i c l e s . T h e r e i s a s e p a r a t i o n o f t h e H° beam f r o m t h e p r o t o n s a f t e r p a s s a g e t h r o u g h

t h e i n j e c t i o n bump m a g n e t j u s t d o w n s t r e a m o f t h e f o i l ( t h e t h i r d o f t h e f o u r bump m a g n e t s

i n t h e l o n g i n j e c t i o n s t r a i g h t ) . A n o n - d e s t r u c t i v e m o n i t o r o f t h e i n j e c t e d beam t s

o b t a i n e d by u s i n g a n i n t e r n a l s c i n t i l l a t o r and a n e x t e r n a l TV c a m e r a t o v i e w t h e

s e p a r a t e d H ° beam. F l u c t u a t i o n s o f t h e i n j e c t e d beam a r e r e a d i l y s e e n on t h i s m o n i t o r .

A s e c o n d i n t e r n a l s c i n t i l l a t o r h a s b e e n vised t o o b s e r v e t h e i n j e c t e d beam a f t e r o n e

r e v o l u t i o n i n t h e r i n g . T h e two m o n i t o r s h a v e b e e n u s e d t o g e t h e r t o o b t a i n c o r r e c t

v e r t i c a l a l i g n m e n t o f t h e i n j e c t e d b e a m .

I n j e c t i o n o c c u r s o v e r i n t e r v a l s o f up t o A5Û u s , commencing 5 5 0 u s b e f o r e t h e

minimum o f t h e b i a s e d , s i n u s o i d a l g u i d e f i e l d o f t h e m a i n I S I S m a g n e t s . S t a c k i n g i n

h o r i z o n t a l p h a s e a p a c e i s a u t o m a t i c a l l y o b t a i n e d fcy h o l d i n g a l l bump m a g n e t f i e l d s

c o n s t a n t w h i l e t h e m a i n g u i d e f i e l d f a l l s d u r i n g i n j e c t i o n . T h e I n p u t d i s t r i b u t i o n may

be a l t e r e d b y p r o g r a m m i n g o f t h e bump m a g n e t s . F o r v e r t i c a l f i l l i n g , t h e beam i s swept

v e r t i c a l l y i n t h e i n j e c t i o n l i n e w i t h a c o r r e l a t i o n b e t w e e n l a r g e v e r t i c a l b e t a t r o n a n d

s m a l l h o r i z o n t a l b e t a t r o n o s c i l l a t i o n s i n t h e r i n g and v i c e - v e r s a .

Beam l o s s d u r i n g i n j e c t i o n i s o b s e r v e d b y r a d i a t i o n m o n i t o r s a d j a c e n t t o t h e

I n j e c t i o n s t r a i g h t b u t no i n j e c t i o n l o s s i s seen on r a d i a t i o n m o n i t o r s a d j a c e n t t o t h e

o t h e r s t r a i g h t s e c t i o n s . I n F i g . 2 some f e a t u r e s o f t h e i n j e c t i o n l o s s a r e s h o w n . The

t i m e b a s e i s 2 0 0 ps c n and t h e i n j e c t i o n i n t e r v a l l a 90 u s ; t h e u p p e r t r a c e i s f n r t h e

r a d i a t i o n m o n i t o r n e a r t h e i n j e c t i o n s t r a i g h t , t h e c e n t r e t r a c e f o r t h e m o n i t o r n e a r t h e

end o f t h e i n j e c t i o n l i n e and t h e l o w e r t r a c e i s t h e f i r s t t r a c e r e p e a t e d b u t w i t h t h e

bump m a g n e t s s w i t c h e d o f f 1 5 0 us e a r l i e r . L o s s c o n t i n u e s a f t e r I n j e c t i o n , d e c r e a s e s and

t h e n i n c r e a s e s a g a i n e v e n t h o u g h t h e e q u i l i b r i u m o r b i t i n t h e m a c h i n e i s m o v i n g away f r o m

t h e i n j e c t l o n s e p t u m . The l o s s c e a s e s once t h e o r b i t bump f s r e d u c e d . The t o t a l l o s s

c o r r e s p o n d i n g t o t h e u p p e r t r a c e i s o f w h i c h ^2Z i s t h e s t r i p p i n g l o s s and t h e

- 825 -

Ufert n u t

7 t í

j / 1 r - . /

i / V

^ 3

Fig . 2 I n j e c t i o n Beam Loss Fig. 3 Bunch Formation (200 n s/division)

l a t e l o s e i s thought to correspond t o h o r i z o n t a l beam growth of ^ 5 mm in a t ime of

100 u s , for an i n j e c t e d beam of 3 1 0 1 2 pro tona . Further s t u d i e s of the e f f e c t are

needed to i d e n t i f y the growth mechanism.

5 .2 Bunch formation

The RF i s Bwitched on 145 us b e f o r e the guide f i e l d minimum (T = 0) and i s he ld at

c o n s t a n t frequency and c o n s t a n t v o l t s / t u r n u n t i l T - 0 . Subsequent ly , the frequency i s

r a i s e d to keep the beam centred i n the aperture and the v o l t s / t u r n are r a p i d l y i n c r e a s e d

from 3 kV t o 80 kV by T - 1 ras and t o 112 kV by T • 5 ms. This i s the mode of o p e r a t i o n

for a c c e l e r a t i o n t o 550 HeV. When two further c a v i t i e s are added, the v o l t s / t u r n w i l l be

r a i s e d t o 156 kV by m i d - c y c l e and a c c e l e r a t i o n w i l l be to 500 MeV.

The most efficient operation has been with the debuncher cavity powered and a

narrnw momentum spread injected, ûp/p = l 5 10 . Particles undergo a quarter of a

Synchrotron oscillation by T = 0, at which time two smooth bunch shapes have developed.

Later motion is non-adiabatic with filamentation present and the develc-iment of

non-equilibrium bunch distributions.

The shape of both bunches Is double-humped by T E 100 us and periodically returns

to this form, but with more complex forms at intermedíate times. Typical patterns at low

intensity (6 1 0 1 1 protons per hunch) are shown sequentially In Viß. 1 for T = 100, 175,

225 and 300 us. As the intensity is increased, the shapes become smoother due to the

enhanced effect of the long!tudIna1 space charge forces. A one-dlmensio:iaI langi tudinal 2)

space charge tracking code has heen developed to study the bunch development and it

is of interest to see if the code continues to predict the motion at increased intensity.

Puring the ear ly coramiss iontng, the pronounced double-humped bunches J ed to

complications for the dipnle-mode beam control loop. Incorrect ]Imitina led tn spurious

phase detector signals and the formation of narrow, dense bunches osci'lacing through the

main bunch distribution. Filteri . i fc of the bunch signals before U n i t i n g removed the

spurious effects.

The bunches become progressively smnother as acceleration prnieeds and appear

stable at the present maximum levels of accelerated beam [3 1 0 L 1 protons per bunchl. The

trapping efficiencies are typically R5 to 902 but decreasLnj'. at the highest level of beam

loading observed.

Runch areas appear larger than that corresponding to the Injected momentum spread.

Some of the Increase is due to scattering nf the beam as it circulates through the foil

prior to RK switch-on. This introduces a tail in the momentum distribution but it is not

large enough to explain the effect observed,

5.3 Beam loading

Present Intens 11 i es have been oh ta ined only wi th the aid nf f eed-forward bean

loading compensation. Each nf the four cavities has been powered by Its nwn t'îass B

power amplifier and feed-forward signals have been introduced in the first 2 ms of the 10

ms acceleration period. At the start of acceleration the RF voltage is low and then it

is ,-idvani .igeniis ro keep the cavities tuned to resonance and not to detune them for

reactive earn loading compensation.

Fut.re plans include the installation of two further cavities and the addition of a

''lass A i- wer s;t.ige In parallel with each Class R stage to provide greater linearity for

the feed-:nrward compensation. Also, there will he some reduction In the shunt Impedance

or the t,- :.-it ies ami some stabilisation of the gain of the feed-forward signals.

5.4 Betatron resonances

The betatron Q-values have heen measured throughout acceleration and found Co be

i, \% lower than the values predicted fron magnetic measurements. Trim quadrupole

correction magnets have been powered to adjust the tunes, both for chromatic correction

during the injection period when the beam is spi ral 1ing towards the cent re of the

aperture a.id subsequently. The performance is, in general, not sens i tive to the tune

correction apart from in an interval late in the acceleration cycle when a slow vertical

orbit bump is introduced to reduce the kicker magnet requirements for fast extraction.

The only betatron resonance effect ohserved Is associated with the slow orbit bump

for extraction. The effect has been observed on a scintillator which may je inserted at

the input of the extraction septum magnet. With the Q-values uncorrected, a beam spot is

observed on the scinti1 lator with a dense core and four prominent wings, characte ri st ic

of a fourth order resonance. It is thought tn be the coupling resonance 2Q^ - 2Q^_ = 1.

The effect Is removed by adjusting the Q-values away from rj^ = 4.2b, = 3.7b, after

which there is very little loss of beam during fast extraction.

5.5 Instabilities

No instabilities are observed during acceleration for the maximum beam Intensities

achieved, b 1 0 1 2 protons per cycle. There is, however, the growth of the circulation

beam following injection, described in section 5.1.

Each ceramic vacuum chamber in the ring includes a special RF shield, designed for

a low beam coupling Impedance. Quadrupole and bending nagnet chamhers are shown in

F i g . 4 and the experimental set-up fnr comparing the longitudinal inpedance of the

shield with that of a solid chamber in !•' t H- 5. The resist Ive component of Che shle 1 d

is restricted by ensuring good RF contact between the shield and the neighbouring vacuum

chamber components. The reactive component of the longitudinal impedance is lowered b>

arranging for the shield wires to approximately follow the low energy beam profile. Even

with this care, the space charge contribution to the longitudinal coupling Impedance,

Z/n, is - j 700 ii at low energy and - j 170 [i a t high energy.

The Impedance values are such that the design currents are above the threshold

levels for longitudinal and transverse Instabilities. For the coasting beam longitudinal

microwave instability, the predicted initial growth time for the residual 202.5 MHz linac

bunch structure is of order 100 us and It is planned to look tor this effect while

operating the synchrotron as a 70.4 HeV storage ring. It is believed that an initial

tail In ¿p/p may develop which inhibits further growth. Possibly this is contributing to

the horizontal beam growth observed after injection.

The synchrotron operates below transition and with the natural negative values of

the chromaticities. Trans-erse coherent instabilities may arise for higher intensities

at the end of the Injection interval; the most likely frequency is 140 kHz for the lowest

Fig . 5 Impedance Measurements

v e r t i c a l c o h e r e n t mod<: a n d t h e n e x t m o s t l i k e l y f r e q u e n c i e s a r e f o r m o d e s n e a r 1 0 " y.hz.

S p a c e h a s b e e n l e f t i r t h e m a g n e t l a t t i c e f o r t h e i n c l u s i o n o f a s o t o f n c t o p o l e l e n s e s

t o c o m b a t t r a n s v e r s e I n s t a b i l i t i e s , i f n e c e s s a r y .

5 , 6 C o l l e c t i o n o f ] o s t beam

A h o r i z o n t a l 70 - 100 MeV beam l o s s c o l l e c t i o n s y s t e m h a s b e e n I n s t a l l e d w h i c h i s

d e s i g n e d t o l o c a l i s e much o f t h e l o s s i n o n e l o n g , s t r a i g h t s e c t i o n o f t h e s y n c h r o t r o n ' * 1 .

T h e p u r p o s e o f t h e s y s t e m i s t o r e s t r i c t Tadi3tion damnpe and a c t i v a t i o n o f m a c h i n e

c o m p o n e n t s w h i l e o p e r a t i n g a t h i g h i n t e n s i t y . T h e m a j o r l o s s i n e x p e c t e d t o b e t h e beam

l o s s d u r i n g t r a p p i n g .

T h e p r i m a r y i n t e r c e p t i n g u n i t i s p l a c e o a t an i n s i d e m a c h i n e r a d i u s , n e a r t h e

u p s t r e a m end o f a l o n g , s t r a i g h t s e c t i o n . T h i s i s j u s t a f t e r an F q i i a d r u p o l e s o t h a t t h e

d i s p e r s i o n i s h i g h and th¿ r a d i a l m o t i o n o f un t r a p p e d heam n e a r .i n a x i m u m . T i m s r h e

l o c a t i o n g i v e s a g o o d i n t e r c e p t i o n e f f i c i e n c y a n d a l s o a h i g h p r o b a b i l i t y o f c a p t u r i n g

o u t s c a t t e r e d b e a m o n f u r t h e r c o l l e c t o r s l o c a t e d d o w n s t r e a m i n t h e s t r a i g h t s e c t i o n .

T h e c o l l e c t o r s a r e m a i n l y o f g r a p h i t e b u t t h e p r i m a r y c o l l e c t o r h a s n l i p o f c o p p e r

n t t h e d o w n s t r e a m e n d t o e n h a n c e t h e a n g l e o f o n t s c a t t e r . The u s e o f t h e g r a p h i t e i s t o

r e d u c e t h e r e s u l t i n g a c t i v a t i o n l e v e l s .

* * *

R E F E R E N C E S

1 . V 'J K e m p s o n , C W P l a n n e r ,i:id V f I ' n g h , i n J U L - i i on d vn;»ini r s a m ! m i t l t i t u r n c h . : r > ; , '

e x c h a n g e i n j e c t i o n i n t o t l i t f ; i s t e y e I i n j ; s y n c i i r u i r o . - i Tor tin? S.VS*, I I"! . 1 ; I ' . - .uis .

N u c l . S e i . V o l X S - Z 8 . p . 3G85 ( 1 9 K 1 ) .

2 . S K o s e i e l n i ü k , p r i v a t e «"«cumin i c a t i o n .

1 . .1 A H i r s t , Q H R e e s a m i .1 V T r o t i n a n , S N S 7tl - 100 MeV H o r i K o n t . i l lienm i . o s s

C o l l e c t i o n , RAI. I n t e r n a l R e p o r t , K N S / • ; / N 2 / J M , N.iy 1 9 8 1 .

9.1o / - 831 -

LIST OF PARTICIPANTS

P.

ANDERS, K.

ANTON, F.

AUNE, B.

BAARTMAN, R

BARBER, D.

BAR1ALUCCI,

3AUDRENGM '

BAZZANI, A.

BECK, R.A.

BIAGINI, U.E.

BOURAÍ, C.

BRANDT, D.

BRINKMANN, R.

BUYTAERT, J .

CAPPi, R.

CHABERT, A.

CHAN, D.

CHEHAB, R.

CHOHAN, V.

CORNEL IS , K.

CRAWFORD, J . F .

CROHJE, P.

DMNEU.I, A.

DALLIN, L.

DECKER, F - J .

DENIMAL, J .

DJ IUI, K.

FARVACCJUE, L.

FERCH, M.

FISCHER, C.

FONG, K.

FRIESEL, D.L.

GENUREAU, G.

HAEBEL, E.

HAGEL, J.

HARDEKOPF, R.A.

HF IKKINEN, P.

HERRERA, J . C .

IVANÛV, S.

JEANSSON, J.

JOHANSSON, A.

KALLBERG, A.

KARAN1ZÜULIS, E.

Kiel U n i v e r s i t y , Fed. S e p . Germany

Bonn U n i v e r s i t y , Fed. Rep. Germany

CEN-Saclay, G i f - s u r - Y v e t t e , France

TRIUMF, Vancouver, Canada

DESY, Hamburg, Fed. Rep. Germany

INFN-LNF, F r a s c a t i , I t a l y

CERN, Geneva, S w i t z e r l a n d

Padua U n i v e r s i t y , I t a l y

GAÑIL, Caen, France

INFN-LNF, F r a s c a t i , I t a l y

CGR MeV, Buc, France

CERN, Geneva, S w i t z e r l a n d

PESY, Hamburg, Fed. Rep. Germany

U n i v e r s i t y of Gnent, Belgium

CERN, Geneva, Swi tzer land

GAÑIL, Caen, France

Chalk River Nuclear L a b o r a t o r i e s , Chalk R i v e r , Canada

LAL, Orsayt France

CERN, Geneva, S w i t z e r l a n d

CERN, Geneva, S w i t z e r l a n d

SIN, V i l l i g e n , S w i t z e r l a n d

Nat ional A c c e l e r a t o r Centre (CSIR), Faure , South A f r i c a

CERN, Geneva, S w i t z e r l a n d

U n i v e r s i t y of Saskatchewan, Canada

DESY, Hamburg, Fed. Rep. Germany

c / o P . N . I . N . , S t r a s b o u r g , France .

CEN-Saclay, G i f - s u r - Y v e t t e , France

Lab. Nat. Sa turne , CE'Í-Saclay, Gi f-sur -Yvet te , France

I n s t . f. Angewandte P h y s i k , Frankfurt , Fed. Rep. Germany

CERN, Geneva, S w i t z e r l a n d

TRIUMF, Vancouver, Canada

Indiana U n i v e r s i t y , Bloomington, USA

GAÑIL, Caen, France

CERN, Geneva, S w i t z e r l a n d

CERN, Geneva, S w i t z e r l a n d

Los Alanos Nat ional Laboratory, JSA

Research I n s t i t u t e of P h y s i c s , Stockholm, Sweden

Brookhaven Nat ional Laboratory, Upton, NY, USA

FOM, Amsterdam, Nether lands

Research I n s t i t u t e of P h y s i c s , Stockholm, Sweden

Tandem A c c e l e r a t o r Laboratory, Uppsala Sweden

Research I n s t i t u t e of Physics, Stockholm, Sweden

I1ESY, Hamburg, Fed. Rep. Germany

KARLSSON, M. Royal I n s t i t u t e of Technology, Stockholm, Sweden

KOECHLIN, F. CEN-Saclay, V\f-sur-Yvette, France

KOSCIELNI AK, _ S. Rutherford ¿api e t on Laboratory, C h i l t o n , U.K.

KOUTCHûUK, J -P . CERN, Genev , S w i t z e r l a n d

KRÄMER, D. Max-Planr. I n s t i t u t , He ide lberg , Fed. Rep. Germany

KRAUSE, U . GSÎ, Darmsi i t . Fed. Rep. Germany

•ÍO'GLtR, H. CERN, Geneva, Swi tzer land

ONNE, R . A. NIKHEF-H, Amsterdam, Netherlands

KHURSHEEU, A. CERN, Geneve, Swi tzer land

LAGNIEL, J.M. CEfJ-Saclay, G i f - s u r - Y v e t t e , France

LAWRENCE, G.P. Los Alamos 1 a l i o n a ! Laboratory, USA

LEE, i -Y. Oak Ridge 'i ional Laboratory, USA

LELEUX, G. Lab. Nat . S j r n e , CEN-Saclay, G i f - s u r - Y v e t t e , France

LEHAI RE , J-L. C£N-Saclay, > i f - s u r - Y v e t t e , France

LIËUVIN, M. I n s t i t u t de S c i e n c e s N u c l é a i r e s , Grenoble , France

LIMBERG, T. 0£SY, h'arabu i. Fed. Rep. Germany

LUSTFELli, H. KFA, J ü l i c h : e d . Rep. Germany

HAAS, R. NIKHEF-K, A- i t erdam, Netherlands

MAGNE, J-C. CEN-Saclay, 3 i f - s u r - y v e U e , France

MA-VE, S.R. UESY, Hamburg, Fed. Rep. Germany

MARCHAND, P. CERN, G e n e v a Swi tzer land

MARTIN, B. H a h n - M e i t n e r - I n s t i t u t , B e r l i n , Fed. Rep. Germany

MEYER-PRUESSNER.R. GSI, Darmst It , Fed. Rep. Germany

M0LLER, S .P . u n i v e r s i t y Aarhus, Denmark

MORGAN, J.G. UKAEA, Culham Laboratory, Abingdon, U.K.

MOSNIER, A. CEN-Saclay, G i f - s u r - Y v e t t e , France

NGHIEM, P. Centre d'Orsay, France

NOLUEN, F. GSI, Sarmst d t , Fed. Rep. Germany

NUHN, H-D. Bonn Univer Ly, Fed. Rep. Germany

OLSEN, D.K. Oak Ridge N fonal Laboratory, USA PALUMBO, L. "La Sapienz U n i v e r s i t y , Rome, I t a l y

PATTER1, P. 1NFN-LNF, F ^ s c a t i , I t a l y

PILAT, F. CERN, Genev , Swi tzer land

PISENT, A. Padua Unive s i t y , I t a l y

PL0UVIE2, E. LAL , Centre d 'Or France

RAICH, U. CERN, G e n e v a Swi tzer land

RAMSTEIN, G. CEN-Saclay, G i f - s u r - Y v e t t e , France

RASMUSSEN, N. CERN. Gene- i, Swi tzer land

REISTAD, D. Tandem Acc- e r a t o r Laboratory, Uppsala , Sweden

RIEGE, H. CERN, Gene<. , Swi tzer land

RINDLFI, L. CERN, Gene. , Swi tzer land

RIPKEN, G. DESY, Hamb! g, Fed. Rep. Germany

RIUNA'JD, J . P . CERN, Gene •., Swi tzer land

RÖDEL, V. CERN, Gene. •-, Swi tzer land

- s.Vi -

RÖHL IN, S. Instrument Aß S c a n d i t r o n i x , Uppsala, Sweden

RUPERT, A. CEN-Saclay, G i f - s u r - Y v e t t e , France

SAYS, L-P. Clermond-Ferrand U n i v e r s i t y , Aubière , France

SCHEMPP, A. I n s t . f. Angewandte Phys ik , Frankfurt , Fed. Rep. Germany

SCHMIDT, F. Hamburg U n i v e r s i t y , Fed. Rep. Germany

SCHMUSER, P DESY, Hamburg, Fed. Rep. Germany

SCHNEIDER, G. CERN, Geneva, S w i t z e r l a n d

SCHNURIGER, J . C . CERN, Geneva, Swi tzer land

SCHUTTE, U. DESY, Hamburg, Fed. Rep. Germany

SELlGMANN, B. Kernforchungszentrum K a r l s r u h e , Fed. Rep. Germany

SERAFINI, L National I n s t i t u t e Nuclear P h y s i c s , Milan, I t a l y

SETTY, A. CGR MeV, Buc, France

SHERWOOD, T CERN, Geneva, S w i t z e r l a n d

SIMPSON, M. L. MRC Cyclotron U n i t , Hammersmith H o s p i t a l , London, U.K.

SIMROCK, S. I n s t i t u t für Kernphysik, Darmstadt, Fed. Rep. Germany

SPADTKE, P. GSI, Darmstadt, Fed. Rep. Germany

STINSON, G. U n i v e r s i t y of A l b e r t a , Edmonton, Canada

THIELHEIM, K-0. Kiel U n i v e r s i t y , Fed. Rep. Germany

TKATCHENKO, A. CEN-Saclay, G i f - s u r - Y v e t t e , France

TOMPKINS, P .A. Texas A S M U n i v e r s i t y , C o l l e g e S t a t i o n , Texas , ;JSA

TRANQUILLE, A. CERN, Geneva, S w i t z e r l a n d

von KEMPIS, A. KFA, J ü l i c h , Fed. Rep. Germany

VOS, L. CERN, Geneva, Swi tzer land

WEIS, T. Frankfurt U n i v e r s i t y , Fed. Rep. Germany

WEISS, M. CERN, Geneva, S w i t z e r l a n d

WI1K, B. DESY, Hamburg, Fed. Rep. Germany

WUCHERER, P KFA, J ü l i c h , Fed. Rep. Germany