A brief outline of the longitudinal single particle dynamics at transition is presented in terms of phase-space mappings. Simple quantitative prediction about the phase-space dilution is made. More realistic simulation (ESME) of the transition crossing is carried out (including various collective and single particle effects contributing to the longitudinal emittance blow up). The simulation takes into account the longitudinal space-charge force (bunch length oscillation), the transverse space-charge (the Umstaetter effect) and finally the dispersion of the momentum compaction factor (the Johnsen effect). As a result of this simulation one can separate relative strengths of the above mechanisms and study their individual effects on the longitudinal phase-space evolution, especially filamentation of the bunch and formation of a galaxy-like'' pattern. 7 refs., 2 figs

Bogacz, S. A.

A brief outline of the longitudinal single particle dynamics at transition is presented in terms of phase-space mappings. Simple quantitative prediction about the phase-space dilution is made. More realistic simulation (ESME) of the transition crossing is carried out (including various collective and single particle effects contributing to the longitudinal emittance blow up). The simulation takes into account the longitudinal space-charge force (bunch length oscillation), the transverse space-charge (the Umstaetter effect) and finally the dispersion of the momentum compaction factor (the Johnsen effect). As a result of this simulation one can separate relative strengths of the above mechanisms and study their individual effects on the longitudinal phase-space evolution, especially filamentation of the bunch and formation of a ``galaxy-like`` pattern. 7 refs., 2 figs

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Longitudinal beam dynamics at transition crossing

Fermi National Accelerator Laboratory Longitudinal Beam Dynamics at Transition Crossing S.A. Bogacz Fermi National Accelerator Laboratory P.O. Box 500, Batavia, Zllinois 60510 November 1991 * Presented at the 5th ZCFA Beam Dynamics Workshop, Corpus Christi, Texas, October 3-8, 1991. 4s Operated by Universities Research Assoclatkx~ Inc. under Contract No. DE-ACOZ-76CHO3000 with the United States Depanmenf of Energy llhdaimer This report was prepared as an account of work sponsored by an agency of the United States Gouernment. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefullness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof: Longitudinal Beam Dynamics at Transition Crossing S.A. Bogacz Accelerator Physics Department, Fermi National Accelerator Laboratory*, P.O. Box 500, Batavia, IL 60510, USA Abstract A brief outline of the longitudinal single particle dynamics at transition is pre- sented in terms of phase-space mappings. Simple quantitative prediction about the phase-space dilution is made. More realistic simulation (ESME) of the tran- sition crossing is carried oat (including various collective and single particle ef- fects contributing to the longitudinal emittance blow up). ‘Ibe simulation takes into account the longitudinal space-charge force (bunch length oscillation), the transverse space-charge (the Urnstitter effect) and finally the dispersion of the momentum compaction factor (the Johnsen effect). As a result of thii simulation one can separate relative strengths of the above mechanisms and study their indi- vidual effects on the longitudinal phase-space evolution, especially fdamentation of the bunch and formation of a “galaxy-like” pattern. 1.0 Introduction We start with a simple description of the longitudinal beam dynamics near transition. A pair of equations of motion is integrated analytically and expressed in terms of time evolution matrices and longitudinal phase-space trajectories. Mapping of equal density contours across transition yields quanti- tative predictions about the longitudinal emittance blow up. Our simple model do-es not include any nonlinearities nor intensity dependent forces. To illustrate more realistic phase-space dilution effects, we employ a sacking code ESMEl (to simulate transition crossing in the Main Injector). One of the advantages of the simulation compared to existing analytic formalisms, e.g. baaed on the Vlasov equa- tier?, is that it allows us to consider the collective effecta in a self-consistent manner with respect to the changing accelerating conditions. Furthermore, this scheme enables us to model nonlineatities of the longitudinal beam dynamics, which are usually not tractable analyticaily3. 2.0 Single Particle Dynamics at Transition The longitudinal motion, energy-phase oscillations (E-Q), near transition is governed by the fol- lowing set of equations * Operated by the Universities Research Association under contact with the U.S. Department of Ene~’ dt eVwo - = 7 (sin+ - sin@,) , dt where the frequency slip factor, q, depends now on energy offset, E, and also on time. It is given by the following expression q=a,+(a,+2a l-ao2)+++e"s';pt, EP -/t where I& is the synchronous phase. The momentum compaction factor,a , is defined by the following expansion g= a, 6+ a, Sz +...., CO s=P, PO where C, is the nominal closed orbit path length, and AC is the increase in path length for an off-mo- mentum particle. The coefficients a, and a, are geometrical properties of the lattice, given by where angle brackets ( . . . ) denote averaging weighted by bend angle. The quantities beiig averaged are component dispersions in a momentum expansion of the total dispersion function,;l ,given below x = x0 6 +x1 s= + . . (5) One can notice that the first and third terms in Eq.(2) cancel identically. To simplify these equations even more one introduces new set of canonical variables, 13-p. and dimensionless time in units of the nonadiabatic time, T = 0, defined by T= J F ($)’ “$ Sin@rllcOS@,l The set of equations of motion in reduced variables simplifies as follows ml * _ sin($, + 0) - sin@, dr- 1 co& (6) Q where the oxfticient a= &(at+iG)Icot@,l 0 (8) is proportional to the so-called Johnsen time, defined as the rms average of ban&ion crossing time de- lay (with respect to the synchronous particle) taken over the entire bunch. The strength of this effect7 is directly related to the formation of long tails and longitudinal emitlance blow up. However, to reduce OUT model even more - to a minimum, but still nontrivial case of transition crossing - we turn this nonlinearity off for the time being, by setting a = 0. Applying linear approximation to the RF fo- cussing in Eqs.(7) further reduces them to the second order Airy equation given below dz sp*rp=O aal (9) e*&p=o. Here the upper sign refers to ‘above’ and lower ‘below’ transition situations. A convenient way of illus- trating solutions of this system of equation of motion is to introduce propagators for phase-space tra- jectories. A trajectory is defmed in p-I3 space by the following column vector PO) X(T) = [ 1 em . (10) To propagate the above trajectory across transition one has to construct a time evolution matrix, M, for E4.(9), defined according to x@(r) = WT,-T) x(-r) , (11) where z = 0 was chosen at transition. Explicit form of the matrix M can be written in the following foml WM=[ :, -)T) [; ;]A-‘(%) , (12) where the matrix A is given below in terms of two orthogonal Airy functions Ai and Bi and their detivatives Ai Bi(-z) A(T) = 1 A?(-T) Bi’(-r) ’ The Airy functions are in turn expressed in terms of the Bessel functions of the fmt kind AX-T) = $ V,,(5) + J.. (5) 1 Ii-3 ’ (13) Bi(<) = 1 . (14) The mapping defined by M allows one to propagate any equal density contour across transition. Figure 1 illustrates evolution of initially elliptical contour. Since M represents a linear mapping the contour remains elliptical, but its area gets compressed approaching transition. After transition the area enclosed by the contour expands and the final contour represents the same longitudinal emittance as the initial one. The final ellipse is stretched and rotated with respect to the initial one. We can conclude qualita- tively that when the kinematic nonlinearities are present (a f 0) shearing will filament the contour and it will gradually fill up the smallest matched (straight) ellipse, which includes the final contour in Figure 1. To get a qualitative result for a realistic situation, where both kinematic nonlinearities and in- tensity dependent forces are present, one resorts to numerical solution of the original set of equations of motion, where Johnsen effect and the space charge forces are built in explicitly. This will be carried out in the next section. P Figure 1: Equal density contours: below transition (T = -1). at transition (‘c = 0) and above transition (7= 1) 3.0 Longitudinal Phase Space Tracking with the Space Charge The tracking procedure used in ESME consists of turn-by-turn iteration of a pair of Hamilton-like difference equations describing synchrotron oscillation in +E phase-space (0 5 $5 2n for the whole ring and E = E - E,, where E, is the synchronous particle energy). Knowing the particle distribution in the azimuthal direction, p(o), and the revolution frequency, coo, after each hnn, one can conshwt the longihdii wake field induced by the coherent space charge force4 when Vi(~) = eo, Cp” Z,+(“~,)ei”$. “=--o (15) Z,.,(“Q) = nZ, m* 1 1. 1+21*; Here, a and b are the radii of the beam and the smooth vac”“m pipe, respectively. The above force is defocusing below and focusing above the transition. Therefore it corra‘ts the equilibrium bunch length to be longer below and shorter above the transition (compared to the case without any space charge). This yields bunch length oscillation above the transition set off by nonlin- ear bunch length oversho&. 4.0 Implementation of the Urnstitter and Johnsen Effects As a result of the transverse space charge forces each particle suffers a horizontal bet&o” tune shift, which is proportional to the particle density, p($), at the given longitudinal position $I. This tune shift translates directly into the change of yt, Close to the transition, when q goes through zero, even very small corrections to ‘(t play dominant role and they govern the longitudinal beam dynamics. One of the features of ESME code is that each particle has its own yt, which allows for straightforward implementation of the Umstiuer effect (described above). Similarly, to account for the dispersion of the momentllm compaction factor (Johnsen effect). different parts of the bunch (particles with different momentum offset) are allowed to cross transition at different times. Both contributions to the yt shift are summarized below6 1 2 A U, = 2hrPRB{7a* 0 -p(e) - (2al + a, - ao2) 5Q P 5.0 ESME Simulation - Main Injector As a starting point for OUT simulation a single bucket of the Fermilab Main Injector in +-E phase- space is populated with 5000 macro-particles according to a b&Gaussian distribution matched to the bucket so that 95% of the beam is confmed within the contour of the longitudiul emiuance of 0.4 eV- sec. Each macro-particle is assigned an effective charge to simulate a bunch intensity of 6~10~~ pro- tons. The fmt set of results, Figure 2, corresponds to the situation when only intensity-dependent coher- ent forces are present (a1 = 0). The simulation is carried out over a symmetric (with respect to the tran- sition) time interval of 2700 turns. Figure 2 represents a sequence of the longitudinal phase-space snap- shots taken every 400 turns. One can clearly see dilution effects leading to extensive filamentation of the beam at transition. It yields the longitudinal emittance blowup (100%) and beam loss (5%) at tran- sition. The second set of simulations incorporates in addition to previously discussed coherent space charge forces also the Johnsen effect. The dispersion of the momentum compaction factor, al, is as- signed a value of 5x1r3 and both the longitudinal and transverse space charge forces described by Eqs. (15) and (16) are used in the simulation. Again, the phase-space snap shots are illustrated in Figure 3. One can see fast development of long tails contributing to the longitudinal emittance blow up (150%) and substantial (30%) beam loss, since the particles from the tails quickly stream to the unstable phase- space region. 6.0 Conclusions One can see from our simulation that the presence of large al has crucial impact on beam degrada- tion at transition. One can look at the Johnsen effect using simple physical picture of instantaneous phase-space configurations. Particles with large positive momentum offset cross transition scxmer than the synchronous particle and they end up “seeing” unstable phase-space region long before the syn- chronous phase is “snapped” (& + n - & at the transition crossing for the synchronous particle). They follow unstable orbits in phase-space and eventually leave the bucket (long tail formation). Similarly, for particles with large negative momentum offset transition crossing is delayed with respect to the synchronous particle. After the synchronous phase “snap” they are still below transition and drifting into unstable region, which contributes to formation of the second tail (see Figures 2 and 3). 7.0 References 1. J.A. MacLachlan, FERMILAB TM-1274 (1984) 2. S. Krinsky and J.M. Wang, Particle Accelerators, 17, 109 (1985) 3. S. Stahl and S.A. Bogacz, Phys. Rev. D, 37, 1300 (1988) 4. J.A. MaLachIan, FERMILAB m-446 (1987) 5. S.A. Bogacz and K-Y Ng, Phys. Rev. D, 36,1538 (1987) 6. A. SQrenssen, Particle Accelerators, 6,141 (1975) 7. S.A. Bogacz. Transition crossing in the Main Injector - ESME simulation, Proceedings of the F III Instability Workshop, Fermilab, July 1990. -~j ~ ..~~i ,_II ~Ir;~~.i;~,In~ ~“. ~^~ I~ _. “. .” ._. ,~~n I I __..,....,~,..., 8<-.0..0~‘1 I_ .I~ 4‘4 f A ,...; -‘ 3 ~~2 2 .. ~“~ ( :j _~ -. . . I L,l, ..eI,,,.el .._..,^” j ,..c~a, SIC ..,;;.., I-~.j~ ,.e”.^. ,.. ..~~.._ ,..“.~j Ilj 3 . ...:..., I-.nj ,..“.~. . ..< ; ‘“~ 3 1 .,O,” ,^,.L,D- ,..^..,.O1 ,..“..,I” .,.. .oo ,...<~O, TIC ..;.., ..?A... ~..‘A.. ;. /:.:- A., 1 .i~l 3 .,L., .&..- ~..,L.. ..,1 2 .:.. ..,, 2 I ~ ,: 2 .,, j ! ~j’ji 3 i -l ~.... 2: ~..~~ B : a .‘l” i \ ~._~ 2 y ‘\ 1 --TV “Tmpm”,n~G 1 ~ .“. i’i: s(,.~~y _, .I.. . . . .~11 ,_....... e,.“., ,,i .~,n. lll,,,“. ,~.i ..,.” .j.R ~nr 1~1:/~1, n ..,,,.. ~I .A,... ..,::.... L ,~,LiI ..f.... .il~ : . ...:.., ,A.... ,..L.. I ..~. 1 “~j i “DD 2 I i 2 ,Ijl j ~ J ~“;; a f!G ~,.~~ 4 ~,_~ 1 ~ I :::I’;,,. j ~._I ..I. . . I -_... “~ ..,. “_, .~. i’i&.“,b.,y~ Figure 3: Phase-space dilution for (at = 5x10-3) illustrated by a sequence of the longitudinal phase- space snapshots taken every 400 turns. I;,” ,^,._.~. II0 I.... ~I CIm..,“e .“.” ,~00 ,j.,c~ol .=< . ..L.. .‘.. 2: . . . . :. ,A- ..:..., 1 ..:A~., .&... ,.x.. I. ._~ .-~ .“~ .“. ~~ ..I~ ~“~ I”~ I. i :. -~~ ~_..)_...“. .I. .-. __..,_ ~..,_., . rs...%6- -. ,... .-. .-. ,.5.., :~: ~ ,,:L~., ‘?l~~;jZ?;YE ..:... / “~.” ,.,. l,i. ;.q”y;F,>m~-;;’ _:~ :=:., z$ 2::: 1. A.... ,A.., “0,” ,^,.~,~, - ~_ ..-:‘“~~s.c,:*y~;’ ..;;... ..‘... . ..‘A.. ;. ,..:.... ..:,.., .I~ . ..L.. .A... ,,,?. . . . . .I. ,“. I... . . ~~ ..~. . . ._. ._~ -. ~_. .,I. . . . . ._. i.i:..,~,b-,i” _. ..~. .-~ ,I. _.... “d,C.“. Figure 2: Phase-space dilution for (at = 0) illustrated by a sequence of the longitudinal phase-space snap-shots taken every 400 turns. 

Longitudinal Beam Dynamics at Transition Crossing

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Longitudinal Beam Dynamics at Transition Crossing

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