Particle Accelerators, 1971, -Vol. 2, pp. 31-37 © GORDO~, AND BREACJI, SCIENCE PUBLISHERS LTD. Printed in Glasgow, Scotland.

LONGITUDINj\L EFFECTS, OF COJ.JLIDING BEAM SPACE CHARGEFORCES IN ELECTRON-POSITRON STOR~GE RINGS

WITH CROSSING ANGLEStJOHN R. REES

Stanford Linear Accelerator Center, Stanford University, Stanford, California, U.S.A.

In electron-positron storage rings in which the orbits of the colliding beams cross at a substantial angle aphase-dependen~ longitudinal force is exerted on the particles of each beam by the bunches of the oth~r.~he e~ects of thI~ force on both incoherent and cohere~t lO"ngitudinal particle motion are investigated. Thesee.ffe~t~ l?1pose a lImit on the number of particles which can be stored in a bunch in such storage. rings and thehmI~ IS Independent of the crossing angle (provided the crossing angle is sufficiently large) and off the rf accel­eratIng system parameters. The limit is imposed by the requirement for stability of coherent particle motions.

1. INTRODUCTION-

In electron-positron storage rings the electro­magnetic forces exerted by the colliding beams oneach other are very strong by comparison with theself-forces of each beam, and these colliding-beamforces give rise to a variety of effects which limitthe performance of the storage rings. The forcesare highly nonlinear as functions of the particlecoordinates, and characteristically they lead toavoidance of one beam by the other when either orboth beams are too strong, with consequent re­duction of the storage ring luminosity. (Theluminosity is defined as the interaction rate perunit reaction cross section.) Generally speakingthe. particles congregate ~n traveling bunches,each one of which is spreaq. around a position ofsynchronous phase on the radiofrequency accelera­ting wave. When the beams are not colliding sothat the strong colliding-beam forces are absent,the unperturbed beams assume Gaussian densitydistributions in both transverse and longitudinalcoordinates under the influence of radiationfluctuations and damping. When the beams collidethese equilibrium distributions may be disturbed,and asa result, there may occur an optimum beamcurrent at which tnaximum luminosity- occurs.In this paper, we study these effects as they applyto the longitudinal or phase coordinate. As weshall see, such effects arise in the longitudinal co­orq.inate only if the 'colliding beams cross at anangle, and in that case there is a limit on the numberof electrons or positrons which can be stored in abunch.

tWork supported by the U.S. Atomic Energy Commission.

In the past in storage rings using head-on collisionsboth incoherent and coherent transverse beamins~abilities have been observed and studied exten­sively because or their importance to achievablereaction rates. In the incoherent transverse in-.stability, the weaker beam is di'srupted by thestronger beam so that the central density of theweaker beam is diminished and the luminosityaccordingly reduced. This effect was first describedby Amman and Ritson who characterized the onsetof serious disruption by the shift in the verticalbetatron frequency for small amplitude oscillationsof the weak beam particles due to the strong beamforces.(l) Both experience and computation haveshown that, when this shift LlQy exceeds a/valuein the range 0.05 to 0.025, ~isruption ensues.(2)

These transverse disruptions of the weak beamhave been interpreted as being due to the stronglynonlinear form of the, (largely transverse) electro­magnetic field of the Gaussian' distribution.<;~herent transverse instabilities caused by the col­lISIons between the beams have also been observedand have been explained in terms of the mutualelectromagnetic forces between t~o strong beams.(3)

In the case of head-on collisions, no net .longi­tudinal impulse is imparted to a particle by thecounter-rotating bunch regardless of its longi­tudinal coordinate. Recently, however, electron­positron storage rings have been designed employ­Ing rather large crossing angles.(4) In these thetrajectories of the colliding bunches 'cross 'eachother at an angle, so that the collisions are nothead-on. Such designs have been evolved inefforts to achieve very high luminosities by storingmany bunches in each beam. With many bunches

32 J. R..REES

(1)

(8)

(6)

where eV is the peak energy gain per revolutionavailable from the rf system and <Xm is the momen­tum compaction coefficient. The superscript dotsdenote differentiation with respect to the dimen­sionless variable ft where f is the orbital frequencyand t the time. This variable counts the number ofrevolutions of the particle. D is the damping timein units of this variable~ These equations treat thediscrete impulses given to· the particle by the rfsystem as being spread out uniformly around theorbit. I This is a good approximation for thecharacteristically low frequencies of the phaseoscillations. We consider a storage ring systemwith the orbits of the colliding beams separated sothat the bunches collide at only one point, and eachparticle of one beam encounters only one bunch ofthe other beam on each revolution, a situationreadily realized in practice. If we then considerthe collision impulse [Eq. (1)] as similarly spreadout around the orbit, we can write the linearized,small-amplitude equation of motion for energyoscillations as follows.

.. ~. [2 _(47Tk2

<XmreN) ] - 0 (5)8 + D 8 + wO. R 2 8 - ,yac/>

where wij = - (27Tk<XmeV cos 1Js/E) is the approxi­mate angular frequency of the oscillations in unitsofIt in the absence of collisions (wo is dimension­less), and y = Elmc2• The effect of the collisions isof course to shift the longitudinal oscillationfrequency and, in the case of electrons hitting posi­trons, the shift is downward. We may define anew frequency

2 _ 2 _ (47Tk2<xmreN)

W - Wo R· 2 •yac/>

A convenient beam-strength parameter is defined

L1 = 1 _ w2

= 47Tk2

<xm reN (7)wE Rya~wE'

and we may eliminate some of the parameters· ofEg. (1) in favor of L1 and Wo.

D8 = - L1 ( EwE ) (21/ 2 ac/» fY et2 - y2dt.27Tk<Xm 0

The equations of longitudinal motion of aparticle und~r the influence of the radiofrequency(rf) system and the radiation damping, neglectingfor the moment radiation fluctuations, are

e= eVsin(~+1Js)-(2jD)8~(eVcos1Js)·~

- (2jD) 8 (3)

4>= C7T~Clm) e, (4)

(2)

We describe the longitudinal motion of a particlein the crossing angle storage ring system in terms ofits en~rgy deviation 8 from the synchronous energyE and its phase deviation ~ ffom the synchronousphase 1Js. We ignore transverse motion and con­sider each particle to move on its equilibrium orbit.The energy increment D8 given a particle due to theimpulse imparted by one passage through a counter­rotating Gaussi~n b?nch of N particles is

23/2r mc2Nk fYD8 = - e et2 - y2dt

Ra¢ 0 '

2. FORM OF THE LONGITUDINALIMPULSE

where r e is the classical radius and mc2 the restenergy of the electron, k is the harmonic order ofthe radiofrequency accelerating system, R is thegross radius of the orbit, a¢ is the. standard devia­tion of the Gaussian longitudinal density distribu­tion of the bunch, and y = ~/(21/2 a¢) . (5) Them.inus sign holds for electron-poSitron collisions;it is reversed for, electron-electron collisions. InEq. (1) it is assumed that the product of the bunchlength and the crossing angle is large compared tothe bunch height in the case of a. vertical crossing,or compared to the beam width in the case of ahorizontal crossing. Under these assumptions theimpulse is independent of the crossing angle. Forsmall ~

it is necessary to avoid interactions between countermoving bunches I at places other than the interaction"regions where the detection equipment is located,and this is done by storing the beams in separatestorage rings which intersect at an angle.

With the· introduction of a crossing angle, theimpul~es imparted to the particles of one beam bythe bunches of the opposite/beam are no longerpurely transverse relative to the equilibrium orbitsof the particles and- no longer independent of theirlongitudinal (phase) coordinates. The electric fieldof an·· oppositely moving bunch exerts on a particlea longitudinal force which depends on the phase ofthe particle. Thus Augustin has called attention toconsequent potential longitudinal beam insta­bilities in crossing angle storage rings which areanalogous to the transverse instabilities describedabove. (5) The present paper describes an investi­gation of these effects.

33

a8 = 2-1/ 2 energy unit, acb = 2-1/ 2, (10)

which, together with the choice of energy unit,results in a simplified form for Eq. (8) to speedcomputation. Th~' program was tested to verifythat Eq. (10) held for the weak beam distributionsgenerated in the absence of a strong beam (Ll = 0)over a range of oscillator frequencies and dampingtimes covering the region of interest.

For simulation of the influence of the strongbeam, the results of a p.umerical integration ofEq. (8) were tabulated in memory as a function of 1>at intervals of 0.1 unit. During the simulation, theimpulse as a function of the coordinate 1> wasobtained by interpolation in the table, a very fastprocedure. The speed of the code was faster than20 fLsec per interaction.

The results of several typical runs are shown inFig. 1. In each run the particle is started at4> = 0, 8 = 0 and run for 1000 damping times. Thecoordinates 1> and B are sampled at intervals of adamping time and. sorted into the histogramsshown in Fig. 1. The sampled coordinates are alsoused to compute their rms values and the overlap

grams which, by appeal to the ergodic hypothesis,are considered to be the weak beam bunch distribu­tion functions in 1> and 8. Also certain statisticalproperties of the distributions are computed.

The rf accelerating system is treated according toEq. (3) and Eq. (4) as a linear restoring force andaverage radiated power is considered to varylinearly with energy, so the particle dynamicsbetween impulses are those of a damped harmonicoscillator.. Radiation fluctuations are simulated byintroducing instantaneous increments in the variable8 distributed randomly and uniformly in the intervalb.etween - Wj2 and + Wj2. Such a spectrum hasrms value !Brms = WI121/ 2• It was found unneces­sary to randomize the phase at which these fluctua­tions were applied in the range of oscillatorfrequencies of interest. The impulse, simulatingthe collision with the strong beam, Eq. (8), isapplied once each turn.

For purposes of computation, we measure energyin units of (Ewo/2n:kcxm) so that for .4 = 0, a e = acb

where aB is the rms energy deviation of the testparticle. Also we expect

ae = !Brms n¥d2/ 2 = W(nfd/48)1/2, (9)

wherenfdis the number offluctuations occurring in adamping time. The width W of the rectangularrandom-increment spectrum is adjusted to give,for LJ = 0,

t Strictly speaking, the notion of a potential well is notapplicable to the system in a real storage ring; because boththe rf accelerating system and the collision impulses' act atcertain discrete, highly localized places around the orbit.However, the impulses are weak enough that the charac­teristic fr(1quencies (e.g., wo) are sufficiently low that theconcept of an approximate potential is useful.

AS

COLLIDING BEAM SPACE CHARGE FORCES

3. INCOHERENT INSTABILITY

The lon~itudinal incoherent instability can be'most clearly understood by studying the behaviorof a weak beam under the influence of a strongone. Under these conditions we can assumethat the strong-beam distribution i& undisturbed bythe weak beam and the impulse is given correctly byEq. (8). Because of the typically low frequencies ofphase oscillations in electron storage rings, we canvisual~ze the influence of the impulse in terms of its(averaged) effect on the potential well in which aparticle of the 'weak beam moves.t .The rf systemcreates a potential well which, in the region ofphase which would be occupied by a \damped,unperturbed weak beam, is adequately approxi­~ated by a parabola whose curvature is propor­tIonal to the unperturbed frequency w. Theintroduction of the strong beam impulse alters thesha~e of the potential w~ll, adding a potential pro­portl~nal to an integral over 1> of Eq. (8) andreducmg the curvature at 1> = o. At strong beamstrength Ll = 1, the curvature at 1> = 0 has beenreduced to zero, ~nd so correspondingly has thesmall amplitude frequency. At beam strengthsgreater than one (Ll > 1), a hump appears in thecenter of the potential well, producing two separ­a~ed .poten~ial ~inima. The weak beam always<;lIstrIbutes Itself In the well so as to maintain a fixedrms energy spread, determined by the radiation con­stants and independent of the shape of the well andtherefore of the strong beam strength. Thus weanticipate that the weak beam should be broadenedor dispersed in phase when the strong beam distortsthe potential well and that, beyond unity strongbeam strength as defined above, the weak beamshould begin to separate into two parts.

To test this picture we coded a simulation pro­gram to run on the SLAC IBM 360/91 computer.We chose computer simulation to study the prob­lem primarily because we have been engagedrecently in similar work, so the tools and techniqueswere readily at hand. In the program the weakbeam distribution is developed by following, the~otion of a single test particle for many dampingtImes and sampling its state of motion once eachdamping time. The samples are sorted into histo-

J. R. REES

(c)

pp,p""p,pppp 'PP'PP'PPPPPP'"PPP"PPPP''''·'''''''''PP'P'P'PPPPPPPPPPPPPPPPfl',PPPPPPPPPP'PPPP9P, ......,fIIPp"PPOPPPPP"PPPppp,Pp,PPPP

P P'PPPPfIIPPPPPPPPPPPPPPPPP,PPPPPPPPPPppp,p,ppp,ppp,ppppppppppppp,pp"ppppp It

pp,ppppppppp,ppppppppp,p",p,pp""ppppp,pp

En R=I.288

(f)

(b)

R-3.207

(d)

~J ~ R=2.008HE e (8) -0423

-4 -2 0 2 4

~

fj. -S.OO~ =2.268CTE =0.702

PP'PP''''PP''''P

P ppp"pPPPPPP,P'PPPPPPP

fj. -tOO~=0.911

CTE =0.701

fj. -3.00~ =1.420·CTE =0.682

• PO' P

"'''''IIPPP''PPPP

","",..,p,'PPPpppo.",...."p"""

PPPP".PPP~PPpppppppp

II"'""""""""",PP,pp"""""""",'PPH'""""",'P"""'p,p"",",p""""""ppp,p""p,pp""p,p,p

",PltPP"""",PP" .."",pppppppp,ppppppppppp,ppppp,,.ppp,,,,,,,,,,,,,P""""

PPPPPppppppppppppppppppppppppppPtl......P9P..' ..P9...... 'PJJpppppp"' ....

P 9Pppppp~pppPPPPPPPPPPPPPPP"O,P••"

-4 -2 0 2 4ep

,PPPPp,PP'p,••""PPPP"PP"'PP'PPPPPPo

20

20

80

60

20

40

60

60

80

80

40

40

100

o100

o100E

Ee

EE

g R=1.609

(e)

ee R=2.548~E~~ (8)-0173

(0)

; R=1.018

-4 -2 0 2 4

~

fj. =2.00~=1.I38

CTt =0.693

~=O.O

~ =0.720CTE =0714

ppp~pppp~DoIlPPPP

P"9'P"'"''''''''PPPPPPPPPStpppppppp

ppp,pp"""."""pppp,p"""p""",,P,PPPP9PPP'-'PPPPPPPPPpppppp,ppppo"ppppppp,,'

PttPPPPPPPPPPPt'pppppppppp..,,.,pp,p,p,,,,,,,P,",,,.,,,,P "",IIt",",p"","""",,,,,,,,

Itlt. pppp, pp~pppp~p~

'P.P~'PPPp pppppppp",",pp""p,pp,pppp,p"

,ppp"pppppp,pp,p,pp,pp~ppp,ppppppp,pppppppp"

~"PP'PPP,pppppp"p,pp,.pp

ppppppp,p"poppppppppp",pppppppppppp,ppppopppp"pp,,"pP,",,,,pp,,,pppp,,,,,PPPPp,

p,.ppppppppppppp,ppppppp,ppppppppppppp,pppppp"pppp,ppppp

."",'PPPPPPPP'PPPPPPPPPPPPP'fJ,PltltPPPP"PP,P,P,PP"P'P"PPPPPp,p,pp,ppppppp","",pppppPPP""

,pppp,PPPPPPPPPPPPPPPPPPPP"PPPPPIt, "Pltp,.",,..P.. P.PP",'P"PPPP"'PPPP,PP,, ,

fj. =5.0004> =1.802CTE =0.700

"p",pppp~p,

PPPP'PPPPPPPP"P"pppOPPPPPPPDf)Opppp PP,PPPPPPII'Pl'PPl'l'

P"ltOlPC-OP'PItPOPPPPPP pppoppp"ppp,pp,pp., .. "pOPPPPPPPPPPppppppppppppppDOpppppp, ....pp"P,p

-4 -2 0 2 4ep

34

100

80

60

40

20

a100

80>-u

60zw::::>0

40wa:::I.L

20

0100

80

60

40

20

a

FIG. 1. Equilibrium distributions in phase and energy at different values of the beam strength parameter Lt.For all distributions WO/21T = 5 X 10-3

, the damping time is 2000 revolutions and there are 1000 samples in thehistogram. On each histogram is shown the corresponding values of Lt, the strong beam strength parameter,Clef>, the rms spread in phase, Cl€, the rms spread in energy, R, the ratio of Clef> to its unperturbed value of 0.707and, e, the overlap integral.

COLLIDING BEAM SPACE CHARGE FORCES 35

0.2

and those of the beam-rf-system interaction(6)

Cl..j 0.6-0:::w6 0.4

(12)

Ol.-----.L----~-----'---------'

0.1 0.3 1.0 3.0 10.0

BEAM STRENGTH PARAM ETER, b.

FIG. 2. The overlap integral e of the weak beamand strong beam distributions as a function of thestrong beam strength parameter LJ.The integral isnormalized to one for complete overlap whichoccurs for LJ = O. Error flags indicate estimatedrms statistical spread.

1.4 r-----,----~------r---J

OVERLAP INTEGRAL@ 1.2 vs~ STRONG-BEAM STRENGTH~ 1.0 -r---__

<.9W

~ 0.8

and similarly for K2• These are coupled, damped­oscillator equations. The free phase-oscillationfrequencies for the uncoupled oscillators (K1 =

K2= 0) are (wi - exi)1/2 and (w~ - (X~)1/2, respec­tively. The characteristic equation for the normal

integral of the strong beam,and weak beam distri­butions. The overlap 'integrale, which is theintegral over ep of the product of the weak beamdistribution and the (unperturbed) strong beamdistribution, is computed by summing contribu­tions of the form exp[ - (ep2/2a~)] where ep is thesampled coordinate and a~ =0.5 as in Eq. (10).At the end of the simulation computation the sumis normalized so that it is equal to one for completeoverlap, which occurs for L1 = O. As a measure ofthe statistical fluctuation in the results of thesimulation, we find an rms spread of the order of1000-1/ 2 in the values of the rms value of B fordifferent runs with the same parameters, indicatingthat samples taken at intervals of a damping timeare statistically independent.

The distributions of 'Pig. 1 show that ·there islittle disruption or dilution of the weak beam up toL1 = 1. At 4 = 2 serious broadening in ~ has begun,and at L1 = 3, the 'weak beam has separated intotwo lobes. The overlap integral is plotted in Pig. 2as a function of L1. The luminosity is proportionalto the beam strength and to the' overlap integraland we define their product as the relative lumino­sity which is plotted against;L1 in Fig. 3. It is clearthat, from the. point of view of incohet;ent disrup­tion of the weak beam by the strong beam, beamstrengths well above L1 = 1 (the value for which thesmall amp1i~ude. frequency w goes to zero) arepermissible.

4. COHERENT INSTABILITY

5

RELATIVE LUMINOSITYvs

STRONG-BEAM STRENGTH'

I 2 3 4

BEAM STRENGTH PARAMETER, D:.

FIG. 3. Relative luminosity as a function of thestrong beam strength parameter L1. The straight,dashed line shows the luminosity which would beachieved if there were no disruption due to thestrong beam. Error flags indicate estimated rmsstatistical spread.

1.8 .-------r------...----.----.-I ------,------,

II RELATIVE LUMINOSITY FOR1.6 ;----- UNIT .OVERLAP INTEGRAL

II

II

II

II

II

II

"Ii

>- 1.4r-eno 1.2z3 1.0-.-l

w 0.8>~ 0.6-.-l

~ 0.4

0.2

(fi1 + 2ex1¢1 +Wrepl = K2(~1 -~2)

¢2 +2tx2¢2 +w~ ep2 = K 1(ep2 -ep]), Cl1)

where !Xl and (X2 are both positive damping ratesand include both the effects of radiation. damping

Since the integrity of the colliding bunches ispreserved to rather large values of the beamstrength parameter, it is reasonable to study thecoherent longitudinal motion in the approximationof rigid bunches in the range L1 < 1. This is astrong-beam-strong-beam case. For this purpose,we assume that each bunch of each beam is stableunder the influence of its own rf system in theabsence of the other beam and that each bunch ofone beam collides with only one bunch of theopposing beam. Again spreading out the inter­actions and, in a~dition, linearizing the impulse asin Eq. (2), we write for- the motion of bunch 1 andbunch 2,

36 J. R. REES

(17)

(16)

(19)

The coherent longitudinal interaction betweenbunc4es in an electron-positron storage ring systememploying- a crossing angle places a limit on thecharge which may be stored in each bunch. At thislimit, there will be no substantial incoherentdisruption of either bunch by the other. Thelimit is such th~t the frequency of small-amplitudeincoher,ent phase oscillations of individual particlesis reduced by the factor 2-1/ 2 from its unperturbedvalue. The limit is conservatively estimated fromthe small-amplitude analysis to be

5. 'CONCLUSIONS

The actual limit in practice may be considerablyhigher owing to the stabilizing influence of theintrinsic nonlinearities. As an example, for theintersecting· storage rings designed at SLAC, thecrossing angle of 10° satisfies the assumptionsmade in Eq. (1), R = 35m, OCm = 0.03 and atE = 2 GeV, a e = 0.96 MeV. For these' parametersN~0.5 x 1012

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge the valuableprogr'amming help of Mrs. Nancy Spencer. Thanksare also due to Dr. Phil Morton, Dr. Martin Lee,Dr. Burton, Richter and Dr. Matthew Sands forvaluable discussions during the course of this workand to Dr. J. E. Augustin for sending us hisunpublished report.

REFERENCES

1. F. Amluan and D. Ritson, Proc. Intern. ConI. on HighEnergy Accelerators, Brookhaven, 1961, p. 471.

2. See for example E. D. Courant, IEEE Trans. Nucl. Sci.NS-12, No.3, 550 (1965); B. Gittelman, ibid., p.1033; W. C. Barber et al., Proc. V Intern. ConI. onHigh Energy Accelerators, Frascati, 1965 (CNEN­Rome 1966), p. 330; V. L. Auslander et al., ibid.,p.335; R. A. Beck and G. Gendreau, ibid., p. 371;M. Bassetti, ibid., p. 708; Orsay Storage RingGroup,Proc. Intern. Symp. on Electron and PositronStorage Rings, Saclay, 1966, p. 11-4-1;, F. Ammanet al., Letters Nuovo Cimetlto I, 729 (1969).

3. C. Pellegrini and A. M. Sessler, Storage Ring SummerStudy, 1965, on Instabilities in Stored Particle Beams,a Summary Report, Report No. SLAC-49, StanfordLinear Accelerator Center, Stanford University,Stanford, California (1965), p. 61.

4. Double storage-ring systems utilizing large crossing anglesare under construction at the Deutsches Elektronen­Synchrotron, Hamburg, and under consideration as

(15)

from which we can write

L1 = r e(mc2)2yN1TRocm a2

which is obviously independent of rf parameters. (7)

The luminosity is proportional to (Ll l Ll 2/

(at + a~)1/2) where al and a2 are the rms phasespreads, and Eq. (14) can be rewritten as

2 A '2 Aa2L1 1 a1L1 2 1-2-+-2-~ .

01 02

modes exp(f3ft) of Eq. (11) is

f34 + f33[2(oc] + OC2)] + f32[4oci <X2 + (wi - K2) + (~~ - K])]

+f3[2oc](w~ - K1) + 2OC2(wt - K2)]

+(wi-K2)(w~-K1)-K]K2 = O. (13)

The conditions of the Routh-Hurwitz criterionthat there be no unstable normal modes reduce tothe single inequality

(wi - K2)(W~ - K1) - K1 K2=wiw~ - wi Ll 1

- W~Ll2 ~ O. (14)

At this point, it is important to' note that thebeam strength parameterLl is independent of rfsystem parameters (although that fact is notexplicitly evident from its definition, Eq. (7)). Forth~ Gaussian' equilibrium distributions we areconsidering,

It follows that a1and a2 should be made as small aspossible -by increasing the accelerating voltage asmuch as possible. This, of course, results in thehighest possible synchrotron oscillation frequencies.Then assuming the accelerating voltages to beequal we obtain the restriction

Ll ~t (18)

for each bunch. This restriction required toretain coherent stability is obviously more .stringentthan that imposed by the incoherent instabilitytreated in the previous section as can be seen withreference to Fig. 3.

Physically this limit represents the beam strengthat which one of the normal mode frequencies ~as

been pushed to zero. Beyond that point, that modewould be characterized by exponentially growingseparation of the bunches. For large separationsthe linearized analysis breaks down and non­linearities play an important role. It is probabletherefore, that Eq. (18) gives a conservativeestimate of the effective limit.

COLLIDING BEAM SPACE CHARGE FORCES 37

a future improvement to the single ring under con­struction at the Stanford Linear- Accelerator Ceriter,Stanford, California. It is also intended to operatethe Frascati, Italy,' storage ring Adone with a moder­ate crossing angle.

5. J. E. Augustin, Groupe 'Anneaux de Collisions', NoteInterne 35-69, Orsay, France, Nov. 14, 1969. In thispaper a convenient form of the impulse integral isgiVen from which Eq~ (1) is obtained.

6. K. W. Robinson, Cambridge Electron Accelerator ReportCEAL-1010, Cambridge, Mass. (Feb. 1964).

7. The rms energy deviation ae can be obtained by requiringequilibrium between radiation-induced random walkand radiation damping, both processes being in­dependent of rf parameters. Thus, ae is independentof rf parameters. The equilibrium distribution inq)-e-space is a function of (ip2 +W5qJ2), an approximateinvariant of the motion. Then Eq. (15) follows fromEq. (4).

Received 30 July 1970