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Electron Density Enhancement in a Quasi lsochronous Storage RingC.Pellegrini and D. Robin

University of California at Los Angeles, Department of Physics405 Hilgard Avenue, Los Angeles, California 90024AbstractWe establish the condition for stable single particlemotion in a storage ring with very small momentumcompaction, and very short bunch length, the Quasilsochronous ring. We discuss how this condition can beachieved and its applications to colliders and synchro-tron radiation sources.

IntroductionThe six dimensional phase-space density of an electronbeam in a storage ring is determined by the emission ofsynchrotron radiation, and by the transverse and lon-gitudinal focusing forces determining the particle tra-jectories In the simplest case of uncoupled horizontal,vertical and longitudinal motion, the phase spacevolume occupied by the beam can be characterized bythe product of its three projections on the single degreeof freedom planes, the horizontal, vertical, and longitu-dinal emittances.To minimize the beam phase space volume we canminimize the transverse and longitudinal emittances. Inthe case of the transverse emittances this problem isvery important for synchrotron radiation sources, andhas been studied by several authors [l]. The resultshave been used to build high brightness synchrotronradiation sources like the ALS at Berkeley and the APSat Argonne.A method to minimize the longitudinal emittance , andproduce electron bunches with a short pulse length,small enemy spread and large peak current has beenproposed and discussed recently [2] by C. Pellegriniand D. Robin. Such a beam would find applications insynchrotron radiation sources for the production ofpicosecond long high brightness radiation pulses, andin lepton colliders, like the meson factories, where it canlead to larger luminosity for the same beam current,since a shorter bunch allows the use of strongerfocusing and smaller transverse area at the interactionpoint.This method uses a ring in which the revolution periodis weakly dependent on the particle energy, Quasilsochronous Ring (QIR), in other words a ring with amomentum compaction nearly zero. In this paper we WIIIextend the previous analysrs of the conditions for stable

We wish to thank M. Cornacchla and E. Forest faruseful dlscusslons. Work supported by DOE contractDE-FGO3--9OEF.40565 398 0-7803-01358/91$01.00 @IEEE

single panicle motion in such a ring, and give simplecriteria for the estimate of the energy spread and phaseacceptance of a QIR.Single Particle MotionWe consider only the uncoupled longitudinal motion,using as variables the phase @respect to the RF system,and the relative energy change respect to the referenceparticle, b = ( E - E r ) / Fr. In our approximation weneglect the betatron oscillation amplitude, and thetransverse displacement of a particle is a function of 6only, and can be written as

(1)where 11 is the linear dispersion function and 11describes the first non linear correction.The change in phase with energy respect to the refer-ence panicle is determined by the change in revolutiontime, i.e. the change in velocity, and the change in thelength of the trajectory.The change in trajectory length is-Il.= If ((1 +.\./p)?+ ,.ytls-[. (2)d RTwhere the integration is done on the reference trajectory(RT) of length L, and a prime indicates a derivativerespect to the RT arc length, s. Assuming x, x to be small,expanding (2) to second order in band using (1) wehave~~=w,,.~~r=w,;P(tx,b+a,,b')

wherea = ;j,,f(i,-i (3; y;I r :1 11 j 3 Ia,=- I I ,I);+- :

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the square of the derivative of the linear dispersion, isalways positive, while the second term, which can becontrolled with sextupoles, can be either positive ornegative, and allows us to control the value o f thelongitudinal chromatic&y and make it zero.Using these notations, the equations for the longrtudinalmotion (synchrotron oscillations) can be written asI$ = h(a,6+ a,62) (6)

e I/ 0b=-- ~sin(~+b,)-sln(O,)}-~+~ (7)2IlE,where a prime now indicates a derivative respect to cu 0 1,h = w RF cuO is the harmonic number, VO is the peakRF voftage,r the damping time,+ the fluctuation term,and C$ the phase of the reference particle.We study initialfy equations (6) and (7) neglectingdamping and fluctuations, to determine the phase spacearea for stable single particle motion (closed, boundedphase space tra jectories). In this case the system canbe described by the Hamiltonian

H=H, + iha2b3 (8)

points is smaller than the energy acceptance given by(11) the separatrix through this unstable point willdetermine the corrected acceptance, which is now givenapproximately by(12)

If the oppos ite is true, we will have two separatrices Ofthe usual fo rm, as in fig. 1, separated by an increasingdistance as we make the ratio of the as larger.

--.----0.01 i 1 iL --

5

+j--; I 111 -- _--:0.01-1 2

H,=$ha,b- e L o-{cos($+ $,I+ Qsln@,l (9)2nE,When the longitudinal chromaticity is zero (6) and (7)are the usual synchrotron oscillation equations. Thespace phase trajectories, as shown in fig. 1, are char-acterized by one stable and one unstable fixed points.Assuming cos I$, > 0 , the small oscillation frequencyaround the stable fixed point isv, = {ha,e~~,cos~,/7_~E,}* (10)The separatrix through the unstable fixed point definesthe stable oscillation region. The maximum stableenergy displacement is

When we reduce the value of a ,, the energy acceptancegiven by (11) becomes larger. However the longitudinalchromaticky term becomes now important. and thephase-space trajectories are modified, as shown in fig.2. There are now two stable fixed points and twounstable fixed points, located at &I = 0 , 41= n - 2 b, and6 = 0. b = - a , / a 2. These points will define two sepa-ratrices. as shown in fig. 2, passing through the twounstable points and surrounding the stable points. Forcos I$, > 0 the point at (0,O) is stable and the point (0,- a , / a 2) is unstable. If the distance between these two

Figure 1: Longitudinai phase space trajectories. In thiscase a:, = 0.

-0.02 I \ 7- n4 d

Figure 2: Longitudinal phase space trajectories . In thiscase a?jel = 100.399

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Notice that in all cases we have now two stable regionsin phase space. In the usual case they look like in Fig.1, RF buckets, and their separation is so large thatparticles in the second bucket would have a verydifferent energy and could not stay in the ring. In thecase of small momentum compactlon, and an accep-tance determined by (12), the two regions have adifferent shape, Fig. 2. We WIII call them Alpha-buckets.The difference In the central energy of the two Alpha-buckets isthe same as the expression (12) ofthe energyacceptance. These two regions could both containparticles which can survive tn the ring. However one canstill fill only one of them.The phase acceptance can be also easily determinedby finding the intersection of the separatrices with thecurvesb = 0. b = -n, /a?.

ConclusionsWe have established the phase space structure for aQIR, and given a simple expresslon for the ring energyacceptance. We must also notice that these results arebased on a simplified model of the CUR. We areneglecting the effect of betatron oscillations andexpanding the equation only to second order in& A morecomplete description requires a full non linear trackingof the pamcle motion in the ring. These results canhowever be used as a guide to the design of the ringlattice. It is for instance apparent the tmportance in thering design of controlling simultaneously both themomentum compaction and the longitudinal chroma-ticity, so that their ratio will remain large enough toprovide the needed energy and phase acceptance. Thisrequires that we now control and make zero or nearlyzero the two transverse chromaticrties and the longi-tudinal chromaticity. This can be done using at leastthree families of sextupoles. As usual these sextupolesmay reduce the ring dynamic aperture, and theirplacement and number will need to be properly selectedto keep this reduction to a minimum. We are nowstudying the lattice of rings of the QIR type that canproduce bunch length in the millimeter or submillimeterregion.The peak current in such short bunches can be madelarge by using the longitudinal coupling impedancereduction with bunch length, and the synchrotron radi-ation damping, as discussed in [2].References1. See for Instance Proc. Of the ICFA Workshop on LowEmittance Beams, J.B. Murphy and C. Pellegrmi em..Brookhaven Natlonal Laboratory Report (1987).2. C. Pellegnni and D. Robin, Nucl. Instr. and Meth. A301,27-36 (1991).

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