llk = d.rk/(qsdd), ate selected by considering the subset of lattice points ( T k, &) = ( ( k, l / P ) / n t, (k,

Transcription

1 997 P a r t i c l e A c c e l e r a t r Cnf., Cariada, May 2-6, 997. Vancuver, B.C., BNL EFFECTS F NCHERENT SPACE CHARGE FRCE EEVED N THE FAST HEAD-TALNSTABLTY 3u) 2 5 # M. Blaskiewicz,AGS Dept. BNL,Uptn NY 973-5USA Abstract The effects f incherent space charge frces n the fast head-tail instability are studied numerically. t is fund that incherent space charge frces can dramatically inaease the threshld current fr a fixed wall impedance. PHYSCAL MDEL AND SMULATN ALGRTHM Fr the purpses f beam dynamics ne transverse and the lngitudinal degrees f freedm are cnsidered. Let B dente the machine azimuth, which increases by 2n each turn and will be used as the time-like variable. The time is dented by t and w is the angular revlutin frequency f a synchrnus particle. Cnsider a single particle and let r ( 6 ) = t - B/w dente the time delay between this particle reaching B and the synchrnus particle reaching 6. The lngitudinal equatin f mtin fr a single particle is apprximaedas (4) n equatin 4 the new functins X(r) and W ( T )are in- trduced t smth ut the particle-particle frces. f ne takes the limit N 3 and then takes the limits f X(r) ging t adeltafunctinand W ( r ) W ( T )equatin, (2) is recvered. The number f mam-particles is cntrlled using the parameter nt. The initial lngitudinal variables, n,and llk = d.rk/(sdd), ate selected by cnsidering the subset f lattice pints ( T k, &) = ( ( k, l / P ) / n t, (k, l/z)/nt), with k, and k, integers, which are inside the unit Circle. The initial lngitudinal crdinates f a macr particle are derived via (??cy uk) = L(Tk,&) ( (- Bz)l (lp)) /2 /R&, where the parameter p determines the bunch shape and RE = V;. The smthed density in lngitudinal phase space is prprtinal t (L2- r2- v2)j, and results in a line density a (L2 - G)p/2.During the simulatin the lngitudinal variables were updated nce per turn using a rtatin with angle 2rs. Figure illustrates the selectin f ln@tudinalcrdinates. The initial values f the transverse variables Z k and dr W(r r )p(6, r ) < z(,r ) >. (2) P k = d x k / d d were btainedusinga randm number generatr. The transverse dynamics cnsists f tw parts, single -T particle dynamics and multiparficiedynamics. The scheme n equatin (2)z is the bare betatrn, C,,2 character- invlves a single particle update fllwed by a multiparticle izes the peak strengthf the incherent space charge frce, update and is repeated M times per turn. The applicatin f and 4 6, is ) the line density f the particles which vancllective frces nce per turn, as is usually dne in leptn ishes fr frf > T,the half length f the bunch. The transmachines [l],is nt sufficient since space charge tune shifts verse center f the beam as a functin f azimuth and de- are large. The single particle update is given by a translay is < z(6, r) >, and the causal cherent frces due t fer matrix with a bare betatrn phase advance 2n,/M. wall impedances are characterized by the wake ptential The multiparticle update cnsists f a kick frm the space W (r). Extending the mdel t include chrmaticity and charge and wake frces. The space charge kick is given by lng range wake frces is straightfrward, but will nt be cnsidered here. N Equatins() and (2) are slved by particle tracking. The (5) bunch is mdeled as N interacting macr-particles. The equatins f mtin fr kth macr-particle are taken t be where Csc= 2nCSc/NM. The kick due t the wake ptential is (3) where sis the synchrtrn tune. The single particle equatin f mtin fr the transverse degree f freedm 2 is apprximated as - Wrk supprted by United States Departmenr f Energy 8TRlBUTN F THS DCUMEW S UNLM- Jk

2 e T calculate the sumsne starts with $ =, uses q e e 8 Figure : nitial lngitudinal crdinates fr nt = and p =. There are 36 w mi macr-pahcles. n the surface, equatins (5) and (6) appear t require (N2) t btain the -Phdes. This wuldmakesimulatinswitblargen untenable. Fr apprpriate chices f X(r) and W ( T )the peratin cunt drps t ( N lg(n)).the trick is t generalize the phasr technique which is usually emplyed t retain the cumulative effects f multiple passages thrugh a resnant structure[2]. The smthing functins tin (5)is given by &/csc =XkSlk - s2, Where j= = and Nte that these recurrence relatins are stable and that the kicks fr all N particles require ( N ) calculatins after the particles have been srted in arrival time. The srting prcedure is dne when the rk s are updated, nce per tum. Next,cnsider the kick due t the transverse wake field. A smthed versin f the step functin wake given by W ( 7 )= W f r r > a n d W ( t ) = f r t < is i- -c where Cw is a cnsfant and X ( t ),= exp( -al,-l), as befreadjusting the clstant s as c w(r) yields Fk = -[s; 27rW NM where s, XkSlkf - s2:, (7) - s2,/2 S2$/2], () k =c x j. 2 SMULATlNRESXTS The algrithm described in the previus sectin has been implemented in frtran cde. Results with the smthing functin A2).( and the step functin wake will be presented. Values f nt, p, s,re,2,m, W, and the peak value f the incherent space charge tune shiftascwere chsen. Fr the results presented here M = 24, z = 2.9, and s=. -.. The simulatins were allwed t cntinue until the betatrn scillatin shwed a clear expnential grwth rate. The grwth rate f the expnential was identified as the grwth rate f the mst unstable mde. The values f the

3 . ) 9 x _ -E i g l tau_e/r grwth rate were insensitive t the seed value f the ran- dm number generatr and were the same in bth single and duble precisin. Figure 2 shws the grwth rate f the mst unstable mde as a functin f re/t fr a system which wuld be highly unstable in the absence f space charge frces. Frm the figure the best values f re were.5,.2 and. fr nt = 25,5, and, respectively. Figure 3 shws m(,/,) versus AJqs. While there is a nticeable rise in grwth rate after the initial decay m(,/,) stays belw l%f its value in the absence f space charge. A threshld value f Asc/, = 2 is inferred frm this plt Figure 4 is the main result. The threshld value f the wake ptential increases with space charge tune shift. Results fr smth (p = ) and bxcar ( p = -/2) line densities are similar. f real beams behave in this way the fast head tail instabilitywill rarely, if ever, be seen in lw energy hadrn machines. r;. 7. : le45 t Figure 2 m(2/s) versus re/t with p = and Asc/S= 2 fr varius values f nt: nt = 25, slid line; nt = 5, lng dash; nt =, shrt dash. The value f the step functin wake was five times larger than the threshld value with ASc=.. le dscls Figure 3: m(z/s)versus A,,/, with p = fr varius values f nl.: nt = 25, slid line; nt = 5, lng dash, nt =, shrt dash. The value f the step functin wake was 2 5 times larger than the threshld value with AS, =. P3 3 REFERENCES [] D.Bmdt, GL.Sabbi B. Zaer EPAC94pg 66 (994). [2] A. W. Cha, bf Physics f Cllective Beam nstabilities in High Energy Acceierata. Wiley,pg 2 4, (993). p] see eg. W.H. Press, B E h e y, SA. Teuklsky, W.TVetterhg Numerical Recipes,Cambridge University Press, (986). 2 3 dsc/s Figure 4: Threshld wake in units f the threshld wake fr ASc= versus A&,,/&, fr different line densities: p =,slid line; p = -/'2, lng dash.

4 DSCLAMER This reprt was prepared as an amunt f wrk spnsred by an agency f the United States Gvernment. Neither the United States Gvernment nr any agency theref, nr any f their emplcyecs, makes any warranty, express r implied, r assumes any legal liability r respnsibility fr the accuracy, cmpleteness, r usefulness f any infrmatin, apparatus, prduct, r prcess disclsed, r represents that its use wuld nt infringe privately wned rights. Reference herein t any specific cmmercial prduct, prcess, r service by trade name, trademark, manufacturer, r therwise des nt necessarily cnstitute r imply its endrsement, rccmmendatin, r favring by the United States Gvernment r any agency theref. The views and pinins f authrs expressed herein d nt necessarily state r reflect thse f the United States Gvernment r any agency theref.

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