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 ?
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 )
- 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 ,
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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
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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
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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
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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.
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- 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.
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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.
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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
L . .
13 fci..i. kl.G l<i. 15.*.
5 S.9
. Jt' '
2D iS.S 6 . Ë 5 . "
J LIJITX. LA Li .LL 1
-
3 S .3
. i ? ;
I« fil.J l.B.2 5.S 5.6
L. 5.9
J LIJITX. LA Li .LL 1
-
3 S .3
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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;
<1«T
1961
. Sweden ï coolei. Bl
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).
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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 :
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 ) .
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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 ( Œ ) 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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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.
- 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
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
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