Space Charge Resonances and Instabilities in Rings

Space Charge Resonances and Instabilities in Rings I. Hofmann, G. Franchetti* and A. Fedotov1 *GS7, 64291 Darmstadt, Germany

, Upton NY 11973, USA

Abstract. We show that the influence of space charge on resonances of coasting beams in circular machines can be discussed in terms of a coherent frequency shift leading to a "coherent advantage", and of space charge induced resonant instabilities absent in the zero space charge limit. These mechanisms are exemplified for the SNS ring as well as the SIS 18 of GSI.

INTRODUCTION While resonances in circular machines driven by systematic or imperfection lattice terms have been widely explored since the early times of accelerators, the role of space charge has found comparable attention only in recent years [1, 2, 3, 4, 5, 6, 7]. The most obvious effect of space charge is the incoherent tune shift and spread, which leads to an extended foot print of single particle tunes in the tune diagram. Less obvious is the observation that the response on a resonance may be modified by the coherent motion of all or a large fraction of particles. This leads to an additional oscillating force, which must be added to the external forces and may cause a coherent shift of the resonance condition; furthermore a different response is obtained, depending on whether the resonance is crossed from above or below, which was first studied in Ref. [1]. The main point of this paper is to show that a systematic study of space charge effects needs to consider the existence of both, resonances and instabilities. Such structure instabilities have been studied earlier in the context of linear transport systems [8], but these findings can be applied also to circular accelerators. Our model is that of a coasting beam or, equivalently, a constant line current beam in an ideal barrier bucket. In usual bunched beams in an rf bucket the combined effect of synchrotron motion and line current variation leads to additional effects due to resonance crossing discussed elsewhere in this conference [9].

COHERENT SHIFT OF RESONANCE The analytical basis for calculating coherent frequency shifts is the calculation of eigenfrequencies for charge density oscillations with arbitrary tune and emittance ratios, hence fully anisotropic beams, by using the dispersion relations derived in Ref. [3] for KV beams and in smooth approximation. These eigenmodes can be excited resonantly in a circular accelerator by appropriate external field perturbations (for round isotropic beams see also Ref. [5]); furthermore, they can be excited by space

charge itself, or a combination of both. Hence, the coherent space charge contribution leads to a shifted condition of resonance of the coherent frequency co with a field perturbation at harmonic n according to

co = mvx + / vy + Aco = n.

(1)

Here we express the "mode tune" co in units of the revolution frequency similar to single particle tunes, and introduce Aco as coherent shift due to the resonant density oscillations. The importance of such a shift was first pointed out by Smith [10] for second order, and extended to any order by Sacherer [11]. Numerically, it was demonstrated in Ref. [1] - using particle-in-cell simulation - for an example of fourth order resonance crossing. In order to more easily quantify the effect of the coherent shift, a coherent mode coefficient Cmk defined by the equation n = m(v0 —C m ^Av), thus including the incoherent and coherent space charge effect, was proposed in Ref. [4] for isotropic beams and uncoupled modes, based on the dispersion equations of Ref. [3]. Here m is denoting an azimuthal, and k a radial mode number, while Av = V0 — v. The radial mode number reflects the order of the perturbed space charge potential; the azimuthal one gives the multiple of V0 for vanishing space charge. Note that for non-KV beams v and Av must be understood as rms values. Cmk can thus be interpreted as coefficient of attenuation of the single particle space charge shift Av by the coherent motion, provided that C < 1. The intensity can thus be increased by the factor 1/C compared with the single particle based resonance condition, which expresses the "coherent advantage". For the symmetric breathing mode the result is C = 1/2, which reflects the strong density perturbation by the cross sectional breathing. This allows increasing Av by a factor 2 compared with the corresponding single particle resonance condition n = 2 v; for the quadrupolar mode the coefficient is only 3/4, hence an allowed intensity increase by the factor 4/3. The general case for split horizontal and vertical tunes (also unequal emittances) leads to a coefficient C close to the arithmetic mean, e.g. 5/8, for both modes, hence a "coherent advantage" by the fac-

CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy © 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00 248

tor 1.6. In fourth order we find that C ≈ 0.87 is good tor 1.6. 1.6. In In fourth fourth order order we we find find that that C C≈ w 0.87 0.87 is is aaa good good tor approximation for a broad variation of tune splitting and approximation for a broad variation of tune splitting and emittance ratios. This implies a weakening of the “coheremittance ratios. This implies a weakening of the "coheremittance ratios. This implies a weakening of the “coherent advantage” if fourth order resonances are expected to ent advantage” advantage" if if fourth fourth order order resonances resonances are are expected expected to to ent play role. We note here that modes with C > may play aaa role. role. We We note note here here that that modes modes with with C C> > 111 may may play appear, in principle, as solutions of the analytical KVappear, in in principle, principle, as as solutions solutions of of the the analytical analytical KVKVappear, based dispersion relations. Following the discussion in based dispersion relations. Following the discussion in Ref. [12] they should, however, be discarded as artifact [12] they should, should, however, however, be be discarded discarded as as artifact artifact Ref. [12] of KV-distributions KV-distributions associated associated with “negative energy”. associated with with “negative "negative energy”. energy". of

SECOND ORDER ORDER RESONANCES RESONANCES SECOND APPLIED TO TO SNS SNS APPLIED We first appearance of an imperfection driven We first first discuss discuss the the appearance appearance of of an an imperfection imperfection driven driven second order order resonance resonance for for the fictitious fictitious un-split working working for the fictitious un-split working second point (((v , νV0y )= (4.6, 4.6) driven driven by harmonic = = (4.6, (4.6,4.6) driven by by aaa harmonic harmonic nnn = = 999 0x 0jc, ν 0 )) = point 4.6) νν0x 0y gradient error. The effect of the error resonance depends depends gradient error. The effect of the error resonance depends primarily on on the the tune tune and and not not the the details details of of the the focusing focusing primarily lattice, hence hence we we can can assume assume aaa constant constant focusing focusing for for for lattice, simplicity. The expected resonance condition is condition is is simplicity. The expected resonance condition (2) (2) (2)

An anti-symmetric anti-symmetric error error in in xx and and (as would be proand yy (as (as would would be be proproAn duced by a single quadrupole) then drives a quadrupolar single quadrupole) then drives drives aa quadrupolar quadrupolar duced by a single mode resonance, resonance, whereas whereas aaa symmetric symmetric error is needed to symmetric error error is is needed needed to to mode drive the breathing mode resonance. Note that for sufresonance. Note Note that that for for sufsufdrive the breathing mode resonance. ficiently split split tunes aaa single single quadrupole error will drive split tunes single quadrupole quadrupole error error will will drive drive ficiently both We assume equal emittances in and y, and both modes. modes. We We assume assume equal equal emittances emittances in in xxx and and y, y,and and solve symmetric solve the the envelope envelope equations equations with with aaa symmetric symmetric relative relative gradient error error Fourier Fourier harmonic harmonic of of 0.001 at = as well of 0.001 0.001 at at nnn = = 999 as as well well gradient as aaa small small initial envelope mismatch of 2%. If the sinsmall initial initial envelope envelope mismatch mismatch of of 2%. 2%. If If the the sinsinas gle particle particle tune tune is is exactly exactly on on resonance there is only on resonance resonance there there is is only only aaa gle small beating beating of of the the envelope envelope due due to to proximity proximity of of the the cocosmall herent resonance (Fig. 1 top). The resonance of the in(Fig. 11 top). The The resonance of of the inherent resonance (Fig. phase mode should be expected at ν = 4.4 based on x,y should be be expected at at νvx,y = 4.4 based on on phase mode should xy = the the calculated calculated value value for for C. C. At At this this value, value, in in fact, fact, the the enenvelope significant though not yet maximum velope response response is is significant significant though though not not yet yet maximum maximum (Fig. 111 bottom). bottom). The The complete complete picture picture is is shown in Fig. 222 is shown shown in in Fig. (Fig. for both, both, symmetric symmetric and and anti-symmetric anti-symmetric gradient gradient errors. gradient errors. errors. for Due to to the the nonlinear nonlinear nature nature of of the the envelope envelope response response the response the Due maximum is is reached reached for for even slightly slightly stronger tune dedefor even slightly stronger stronger tune maximum pression, with with aa sharply sharply dropping dropping response response beyond. beyond. As As pression, expected, the zero-current case has infinite response at at expected, the zero-current case has infinite response at ν = 4.5. For the actual simulation the particle-in-cell x,y = 4.5. For For the the actual actual simulation simulation the the particle-in-cell particle-in-cell νVj^ x,y = ORBIT was ORBIT code code was was applied applied to to the the SNS SNS lattice lattice with with aaa workwork, ) = (6.45, 4.6). The response to an erν ν ing ing point point (((v , V ) = (6.45,4.6). The response to an ererν0x 0x 0y 0jc ν0y 0y = (6.45, 4.6). The response to an ror in in aa single single quadrupole quadrupole leads leads to to aaa similar similar behavior of of similar behavior of aaa ror shifted curve curve and and steeply dropping maximum envelope and steeply steeply dropping dropping maximum maximum envelope envelope shifted slightly beyond beyond the the envelope envelope resonance resonance condition. condition. Recondition. ReReslightly sults for for the the out-of-phase out-of-phase mode mode excited excited by by crossing crossing the the sults resonance condition in the y-plane are shown in Fig. 3, j-plane are are shown shown in Fig. 3, 3, resonance condition in the y-plane where intensity is plotted in terms of a normalized tune in terms terms of of aa normalized normalized tune tune where intensity is plotted in shift defined defined as as actual space charge tune shift relative to as actual actual space space charge charge tune tune shift shift relative relative to to shift the distance of the bare tune from the half-integer, i.e. from the half-integer, i.e. i.e. the distance of the bare tune from

249

FIGURE 1. In-phase mode envelope evolution at the single FIGURE 1. 1. In-phase In-phase mode mode envelope envelope evolution evolution at at the the single single FIGURE particle resonance condition =4.5 (top), and at the point of x,y particle resonance resonance condition condition ννvx,y =4.5 (top), (top), and and at at the the point point of of particle JCJ=4.5 ν =4.4 (bottom) for ν =4.6. coherent resonance, x,y coherent resonance, v y=4A (bottom) for v =4.6. 0,x,y coherent resonance, νx,y X =4.4 (bottom) for ν0,x,y Qxy =4.6. 1.8 1.8 normalized normalizedenvelope envelope

+∆ ∆ω ω= = 9. 9. 22ννxx +

1.6 1.6

out-of-phase out-of-phase in-phase in-phase zero-current zero-current

1.4 1.4 1.2 1.2 1.0 1.0 1.0 4.3 4.3

4.4 4.4

4.5 4.5

4.6 4.6

tune tune

FIGURE 2. Maximum envelope response (normalized to iniFIGURE 2. 2. Maximum Maximum envelope envelope response response (normalized (normalized to toiniiniFIGURE =4.6 as as tial envelopes) for both eigenmodes and bare tune tial envelopes) envelopes) for for both both eigenmodes eigenmodes and and bare bare tune tune ννV0,x,y as =4.6 tial 0,x,y 0 ^=4.6 function of depressed incoherent tune. function of of depressed depressed incoherent incoherent tune. tune. function

∆ . The solid vertical line indicates the coher/∆ sc Av /Av The solid solid vertical vertical line line indicates indicates the the cohercoher∆ ννsc ννinc /∆ inc 5C /nc.. The ent resonance condition, where the normalized tune shift ent resonance resonance condition, condition, where where the the normalized normalized tune tune shift shift ent equals 1/C, the long-dashed the r.m.s. incoherent resoequals 1/C, 1/C, the the long-dashed long-dashed the the r.m.s. r.m.s. incoherent incoherent resoresoequals nance condition, and the short-dashed line the incoherent nance condition, condition, and andthe the short-dashed short-dashed line line the the incoherent incoherent nance limit due to the small-amplitude particles in waterbag limit due due to to the the small-amplitude small-amplitude particles particles in in aaa waterbag waterbag limit distribution. As with the simple envelope model above, distribution. As As with with the the simple simple envelope envelope model model above, above, distribution. we find again that the resonance happens at significantly we find find again again that that the the resonance resonance happens happens at at significantly significantly we

higher particles in in good goodagreeagreehigherintensity intensitythan higher intensity than for for single single particles ment with the theoretical expectations. In this context the ment withintensity the theoretical theoretical In this the ment with the expectations. higher than for single particles in context good agree-

44 ν4

νyy y νy

ment with the theoretical expectations. In this context the 33

1

2.75 2.75 3

3.5 3.5H 3.5 3.5

normalized envelope

normalized envelope normalized envelope

2.75 2.5 2.5 2.5

2.25 2.25

2.25

22

3 3 I •••th••••••••• th th 4order order 3 44 thorder

1.75 2

1.75

order 24nd order nd

1.75 1.5

2 order

1.5

2nd order

1.5 1.25

2.5 2.5 2.5 2.53.5 3.5 3.5 3.5

1.25 1.25

1.25 1.5 1.75 2 1 normalized 1.25 1.5tune1.75 shift 2 0.75 1 normalized 1.25 1.5tune 1.75 2 shift normalized tune 1.75 shift 0.75 1 1.25 1.5 2 normalized tune shift

2.25

2.5

2.25

2.5

2.25

2.5

FIGURE 3. 3. Maximum Maximum envelope envelope response response for SNS lattice as FIGURE for SNS lattice asas FIGURE 3. normalized envelope response for SNS lattice function of tune shift for ,νfor )=(6.45,4.6). FIGURE 3.Maximum Maximum envelope SNS lattice as 0x,v 0y function of normalized tune shift and andresponse for ((vνQx )=(6.45,4.6). 0y function of normalized tune shift function of normalized tune shiftand andfor for(ν (ν0x ,ν,ν0y )=(6.45,4.6). )=(6.45,4.6). 0x

0y

question arises, arises, if if fourth fourth order order resonances resonances might question might play play aa question arises, if33 fourth order resonances might play aa question arises, if fourth order resonances might role over the 10 turns required for filling the ring. role over the 103 turns required for filling the ring. For For 3 turns role over thevalue 1010of turns required for For role over the required forfilling filling the the ring. ring. For the typical C = 0.87 the corresponding normalthe typical value of C = 0.87 the corresponding normalthe typical value of C = 0.87 thecorresponding corresponding normalthe typical value of C = 0.87 the normalized tune shift would only be 1.15, hence nearly cancelized tune shift would only be 1.15, hence nearly cancelized shift would onlybebe1.15, 1.15,hence hencenearly nearly cancelcanceling thetune second order “coherent advantage”. ized tune shift would only ing the second order "coherent advantage". ing the second order “coherent advantage”. ing the second order “coherent advantage”.

RESONANCE AND AND INSTABILITY INSTABILITY FOR RESONANCE RESONANCEAND ANDSIS18 INSTABILITYFOR FOR RESONANCE INSTABILITY FOR THE THE SIS18 THESIS18 SIS18 THE

We next consider the working diagram of the SIS18 latWe next consider thetheworking diagram ofofthe 18 latWe next consider working diagram theSIS SIS18 tice with 12 superthe periods and diagram triplet focusing. In Fig.lat4 We next consider working of the SIS18 lattice with 12 super periods and triplet focusing. In Fig. 44 with 12the super periods andorder tripletsystematic focusing. In Fig. wetice indicate usual fourth singletice with 12 super periods and triplet focusing. In Fig. wewe indicate the usual fourth order systematic singleindicate the line usual systematic single-4 particle resonance 4νyfourth = 12, order its coherent counterpart we indicate the usual fourth order systematic singleparticle resonance line 4v = 12, its coherent counterpart y particle νy = 12,which its coherent ν + ∆ω =line 12 4(dotted), nearlycounterpart coincides given by 4resonance given by 4v4yyν+y + Aco which nearly coincides particle resonance 412 ν(dotted), = 12, its coherent counterpart y(dotted), ∆line ω= =12 which nearly coincides given with thebylower harmonic of the fourth order coherent with thethe lower harmonic ofofthe fourth order coherent + ∆ ν ω = 12 (dotted), which nearly coincides given by 4 with lower harmonic the fourth order coherent y resonance 2νy + ∆ω = 12/2; furthermore the coherent resonance 2v2yν+yharmonic Aco = =12/2; furthermore the coherent with the lower of the fourth order coherent resonance ω 12/2; furthermore the + ∆ second order resonance condition 2νy + ∆ω = 6,coherent which second order condition 2v2yνy++Aco ==6,coherent which second order condition 6,gradiwhich ∆ωthe resonance 2νyresonance ω a=6-th 12/2; furthermore +resonance ∆by could be excited harmonic imperfection could be excited by a 6-th harmonic imperfection gradicould be excited by a 6-th harmonic imperfection gradisecond order resonance condition 2 ν ω = 6, which + ∆ y shifts are calcuent error (not studied here). The coherent ent error (not studied here). The coherent shifts are calcuent error (not studied here). The coherent shifts are calcucould be excited by a 6-th harmonic imperfection gradilated for an incoherent space charge tune shift ∆νy = 0.4 lated forfor anpractically incoherent space charge shift lated an incoherent space charge shiftAv-y ∆ νy=calcu=0.4 0.4 ent error (not studied here). The coherent shifts are νtune and found independent oftune 0x . and found practically independent of V . ν . and found practically independent of 0jc0x shift latedInfor an incoherent space charge tune ∆νy =co0.4 Table 1 we summarize analytically calculated In Table 1 1coefficients wewesummarize analytically calculated In Table summarize analytically calculatedcocoherent mode relevant for resonances in . and found practically independent of νthe 0x modecoefficients coefficients relevant forthe theresonances resonances herent relevant for inin Fig. 4.mode They are practically independent of both tunes Inherent Table 1 we summarize analytically calculated coFig. 4. They are practically independent of both tunes Fig. 4. They are practically independent of both tunes provided thatcoefficients the incoherent shift isfor notthe comparable within herent mode relevant resonances that the incoherent shift notxcomparable comparable with provided that the incoherent shift isisnot with orprovided larger than the tune splitting between and y,both in which Fig. 4. They are practically independent of tunes or larger than the tunesplitting splittingbetween betweenx xand andy,y,ininwhich which or larger than the tune case corrections are needed. shift is not comparable with provided that the are incoherent case corrections areneeded. needed. case corrections TABLE 1. Some coherent mode coefficients for or larger than the tune splitting between x cients and y,CCin which TABLE 1. Some coherentmode modecoeffi coefficients Cfor for TABLE Some ratios. coherent different 1. emittance case corrections are needed. different emittance ratios. different emittance ratios.

4.0 4.0 4.04.0

4.5 5.0 4.5 ν 5.0 v 4.54.5νvxxνx5.0 5.0 x

FIGURE 4. and coherent shifted resonance lines FIGURE 4. 4.Incoherent Incoherent andand coherent shifted resonance lines FIGURE Incoherent coherent shifted resonance ν =3.4 variable ν . of SIS18 for fixed FIGURE 4. Incoherent and coherent shifted resonance lines lines 0y0y=3.4 and variable V 0x0jc. of SIS18 for fi xed V variable ofSIS18 SIS18for forfixed fixed ν0yν=3.4 andand variable ν0x .ν0x . of 0y =3.4

conducting conducting boundary boundaryconditions conditions[13]. [13].We Wehave haveused usedrms rms conducting boundary conditions [13]. We have used conducting boundary conditions [13]. We have used rms matched waterbag distributions generated by filling aarms matched waterbag distributions generated by filling matched waterbag distributions generated byit filling a matched waterbag distributions generated by filling 4D hyper-ellipsoid uniformly and deforming accord4D hyper-ellipsoid uniformly and deforming it accord- a 4D hyper-ellipsoid uniformly andand deforming it accord4D hyper-ellipsoid uniformly deforming it according to the rms matching. Carrying out the self-consistent ing to the rms matching. Carrying out the self-consistent ing Carrying out the the self-consistent simulation with νmatching. isisrelatively arbitrary ingtotothe therms rmsmatching. Carrying out self-consistent simulation with V0x0jc == 4.9 4.9 (which (which relatively arbitrary simulation with ν = 4.9 (which is relatively arbitrary = 3.4 we find in Fig. that ν in this context) and simulation with ν = 4.9 (which is relatively 0x 0y in this context) and0xV0 = 3.4 we find in Fig.55arbitrary that 3.43.4 we we findfind iny Fig. 5 that ν0yν = = inin this context) and there is a significant emittance growth in in the reFig. 5 that this context) and there is a significant emittance growth in y in the re0y there isis< a asignificant emittance growth in two yininydistinct the region 2.7 ν < 2.95. In fact, we expect y there significant emittance growth in the gion < vy < 2.95. In fact, we we expect expect two two distinct distinct region 2.7 In and fact, y play mechanisms a role cause broadtwo a stopgion2.7 2.7