INSTABILITY DYNAMICS WITH ELECTRON CLOUD BUILDUP IN LONG BUNCHES

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1 J C NSTABLTY YNAMCS WTH ELECTRON CLOU BULUP N LONG BUNCHES M. Blaskiwicz, BNL 9B, Upto, NY 97-, USA Abstract For log buchs, trailig dg multi-pactig causs th lctro cloud dsity at th trailig dg of th buch to b sigificatly largr tha th cloud dsity durig th arly ad ctral portios of th buch. Th ffctiv wakfild is modld ad compard with had stimats. stabilty simulatios, which iclud th ffct of oliar spac charg, ar dscribd. NTROUCTON Fast lctro cloud istabilitis hav b obvsrvd i th Los Alamos PSR [,,,,, 6] ad may b problmatic for th Oak Ridg SNS [, 6]. For both ths machis th trasvrs lctro oscillatio priod is much lss tha th buch lgth. Th lctro motio is arly adiabatic ad th lctro dsity ca strogly icras as th tail of th buch passs. This is illustratd i Fig., which shows CSEC [6] simulatio rsults for th PSR. Th scodary missio paramtrs of th stailss stl wall agr wll with th modl i [7]. t is likly that sigificat wall coditioig would b dd bfor th PSR could rach a buch itsity of, but th author kows of o dtaild data udr such coditios ad it smd mor rasoabl to adopt a wll dfid stat. ay cas, th purpos of th prst papr is to dscrib computatioal tchiqus. d C/m, J ma/cm PSR, 8 uc, Stailss, pc/cm /tur tim s d p / d d i bam J wall Figur : CSEC simulatios for a high itsity PSR with a uscrubbd stailss stl bam pip. Work supportd by U.S..O.E. udr cotracts E-AC- OR7 ad E-AC-98CH886. blaskiwicz@bl.gov ELECTRON CLOU WAKEFELS Cosidr a proto buch with li dsity ad radius, itractig with a lctro cloud with li dsity ad radius. Assum both clouds hav uiform dsity ad ar ctrd horizotally with. Lt ad b th vrtical offsts of th lctro ad proto ctroids, rspctivly. Th, th t forc pr uit lgth o th protos is! #"$ '& +, '&* ' -. & +, whr -. is th li charg dsity of lctros withi th bam ad is th t forc pr uit lgth o th lctros. Th lctros hav small logitudial vlocitis, so cosidr a sigl logitudial slic. fi th tim dpdt lctro bouc frqucy /& * '& whr 6 87:9* is th istataous proto currt through th slic. Th quatio of motio for th lctro ctroid is ; & < ; & /& >=, Th avrag lctric fild o th protos du to th lctros is E 'F -. & G, A Equatio will b rfrrd to as th full modl of th lctro forc. Th ctral modl of th forc corrspods to rplacig -. with th th valu of H-. i th ctr of th buch. Th ctral modl glcts th flash o th tail. Thr ar svral approimatios ivolvd i th drivatio of quatio ad it is prudt to chck its accuracy. Toward this d CSEC was modifid to allow for a arly cylidrically symmtric lctro cloud NCSEC. Th proto bam was tak to b roud, but with a tim dpdt vrtical offst,,. Withi th roud bam pip of radius th lctric fild du to th protos is KML NG O & < & & & QP & ' &R A 6

2 L } L _ w J A Z ; ; Th cod allows for uiform proto dsity profils with S"T$U & & P &, ad smooth proto profils with & P & P &. t is assumd that is small ough so that th lctro dsity ca b approimatd at ach tim stp as V V W P VYX W ] Z*[\ 7 W whr & & P & ] ad ^ for a particl o th positiv _a` ais. Th particls wr bid oto a radial grid with E msh poits. fi th triagl fuctio b dcf g hg g igj lk g hg A 8 Th first stp was to loop ovr th m lctro macroparticls itrpolatig oto th radial grid. This producd th arrays ad q po r b rts Uu X q po r rts Hu r b W r r is th charg pr uit lgth of th v th lctro is th sam as for X whr u macro particl. Th forc du to CSEC. For th pottial du to th lctros is giv by w qb W whr X zy #"$!W & W & W r &O{ 9. Th wak forc o th protos was obtaid by avragig ovr a disk of radius. NCSEC rsults for a uiform dsity bam ar show i Figurs through. Corrspodig figurs for a smooth distributio ar show i figurs through 7. d C/m. d p d KV a p = cm, p = mm. tim s Figur : proto ad lctro li dsity withi a uiform bam for 9 turs of PSR. Th proto bam had a cm radius ad uiform trasvrs dsity. E V/cm, d C/m - - KV a p = cm, p = mm d p d E full E ctr tim s Figur : NCSEC simulatios for PSR. Th proto buch was offst by mm for.s startig at s. Valus of '~ ad -. for 9 turs ar show. Also show ar th avrag trasvrs fild from th simulatios ad th two aalytic modls. E = V/cm KV a p = cm, p = mm E ctr E full tim s Figur : NCSEC simulatios for PSR. Th proto buch was offst by mm for.s startig at s. First cosidr th uiform travrs profil. Th dsitis for 9 cosctutiv turs ar ovrplottd i Figur. This givs a idicatio of th statistical accuracy. Figur shows th li dsitis as wll as th simulatd wakfilds ad th two modls. Th lctro oscillatio was startd by offsttig th proto bam by mm for. aoscods startig at s. Th simulatio rsults li somwhr btw th full ad ctral modls. Figur shows rsults wh th offst bgis at s. Nt cosidr Figurs through 7. Th valu of was chos so that th pottial diffrc btw th ctr of th pip ad th pip wall wr th sam for th two cass. Th lctro dsitis show i figur ar clos to thos i Figur, but th wakfilds ar quit diffrt. Figur 6 shows oly th ctral modl sic v it ovrstimats th wakfild. Both th modls ar show i Figur 7 but th ctral modl is o wors tha th full

3 } A ; ] d C/m. d p d smooth a p =.cm, p = mm. tim s E = V/cm smooth a p =.cm, p = mm E ctr E full tim s Figur : proto ad lctro li dsity withi th bam for 9 turs of PSR. Th trasvrs proto bam dsity scald as P E = V/cm W Z & tƒh. smooth a p =.cm, p = mm E ctr tim s Figur 6: NCSEC simulatios for PSR corrspodig to Figur. Th proto buch was offst by mm for.s startig at s. Th avrag trasvrs fild from th simulatios ad th ctral aalytic modl ar show. modl. th actual PSR th bam has a smooth trasvrs profil so Figurs through 7 show th appropriat rgim. Th simulatios show i Figurs through 7 took about miuts pr tur for a sigl lctro cloud slic. For a istability simulatio o ds to apply lctro cloud kicks about tims pr btatro priod to accuratly modl th dtuig with btatro amplitud, ad at last priods d to b modld to s th istability. t is ot clar whthr th ois lvls i Figurs ad 7 ar suffictly low to rmov spurious growth. With ths thigs i mid th author chos to adopt a simpl wakfild modl for istability simulatios. Th ctral modl appars to fit th PSR bst, ad this modl will b usd for th rst of th ot. ] Th timlik variabl for th protos is azimuth T, whr is th logitudial Srrt-Frt coordiat ad is th avrag radius of th acclrator. Th tim- Figur 7: NCSEC simulatios for PSR corrspodig to Figur. Th proto buch was offst by mm for.s startig at s. Th avrag trasvrs fild from th simulatios ad th two aalytic modls ar show. lik coordiat for th lctros is tim as masurd i th lab fram. Th lctro cloud will b approimatd by slics, ad th ] coordiats of th slics ar uiformly spacd aroud th rig. Th lattic fuctios for th proto bam, ad hc th bams trasvrs dimsios, vary with azimuth. May slics would b rquird to faithfully modl th variatio i lctro bouc frqucy du to th bam siz variatios. stad, a dampig factor will b itroducd ito th lctro quatio of motio. ; & ; & /& *= G<? For th PSR ad SNS o fids ˆ^Š. SPACE CHARGE AN OTHER YNAMCAL CONCERNS / ; A With quatio ad assumig uiform focusig th vrtical quatio of motio for th protos is ; &, ; ] & &Œ, P & +, P i MŽ - Œ whr Œ is th vrtical tu chromaticity is asy to iclud, & -. /&, '& masurs th strgth of th lctro cloud, ad MŽ - Œ is th vrtical compot of th spac charg forc. For horizotal motio th trm proportioal to & is abst. Prvious work [6, 8] assumd that th spac charg forc was liar i th trasvrs offst. Th PSR has a smooth trasvrs profil so som variatio i spac charg tu shift with btatro amplitud is prst. O optio is to do a full solutio of th two dimsioal Poisso quatio [9, ]

4 ª k W W ˆ Š µ µ A with may logitudial slics alog th buch. This appars to b computatioally difficult ad th author is ot awar of ay, publishd, buchd bam simulatios usig this approach. A simplr modl will b adoptd. Lt th particls cross th rfrc azimuth with offsts r ad r at tims r. Th dirct spac charg kick at coordiats is approimatd as MŽ ' Œ qb G whr th two particl Gr s fuctio is 8 Z>[<\ v Z*[\ Ev & A Th paramtr v cotrols th oliarity of th spac charg forc ad cotrols th siz ad logitudial smoothig lgth of th spac charg forc. Th first stp i th algorithm is to grat a uiform msh i. Th msh spacig, š, is aroud t tims smallr tha th smoothig applid latr. Nt grat th four arrays Ž Ž Ž MŽ ž qœ q qb qœ Z*[\ Z*[\ \@ Ÿ \@ Ÿ Z*[\ Y\ž Ÿ Z>[<\ \ž Ÿ Each of ths arrays is smoothd usig two applicatios of a covolutio with $ : Ug g,, whr is th quivalt lgth of th fial smoothig fuctio. This is do usig a algorithm whos opratio cout scals liarly with th umbr of msh poits. Fially, th sum ruls for siusoids is usd to apply th forc i a algorithm that scals liarly with th umbr of macro particls. Th trasvrs sigl particl dyamics is hadld usig matricis whr th phas advac ca b a fuctio of th btatro amplitud octupolar dtuig ad th momtum offst chromaticity. Th logitudial motio is updatd usig a liarizd RF forc. Noliar RF has b icludd but o rus producd ay qualitativ ffcts for th PSR paramtr rgim. To obtai th iitial coordiats o dfis a logitudial dimsio ml. Nt crat th st of ordr pairs of half itgrs P P P & P P & m & A This is a uiformly spacd st of lattic poits o th pla which li withi a circl of radius m. A ordrig is imposd actually assigd durig cratio ad th ormal- izd coordiats ª«P P A smooth logitudial profil is obtaid by th mappig! / ^ g ªgž v v b b b b Tš Eš Eš Eš A tmf ar gratd. g ªg & Xž * ²±³ Xž so that th li dsity is proportioal to + &, & ±³ X@ & A Whil this assigmt limits th possibl logitudial profils, is gratly supprsss startup fluctuatios. To grat trasvrs coordiats start by fidig four uiform dviats withi th uit -sphr, ª W X W µ &. Normalizd btatro coordiats ar obtaid usig with ad rplac by. trms of th two trasvrs actios ad Œ, with Sj8, Sj Œ, ad P Œ j ; th Vlasov dsity is Œ º¹» ¼ Œ P Œ» ¼ Œ & ¼ ¼ Œ Th two dimsioal charg dsity is smooth ad fairly wll approimatd by &. probability dsity 6 ½ & ¾ &, with & P & j & PRELMNARY RESULTS rho,y = rho a p - -y ka p =. Gaussia Q-Q y, /Q sc Figur 8: Normalizd tu distributios from th simulatios ad for a roud Gaussia bam. Th Gaussia rsults wr obtaid usig first ordr prtubatio thory with th itgrals computd umrically. Figur 8 shows th ormalizd spac charg tu shift distributio with v ¾ A for a rctagular buch À, ad compars it with th rsults for a roud Gaussia bam with th act Coulomb forc. Wh viwd i this way th distributios ar fairly similar ad th valu v½ A ' will b adoptd. Figur 9 shows a scattr plot of vrtical tu vrsus logitudial positio for th PSR. Th pak tu shift is somwhat largr tha has b rportd prviously [], but is ot absurd. Figur shows a scattr plot of th vrtical positio alog th buch for th PSR just byod thrshold, aftr turs. Figur shows th volutio of th cohrt actio for th sam paramtrs. Also show ar simulatios usig liar spac charg with th sam avrag tu shift, ad rsults with dirct spac charg turd off. Spac charg dstabilizs

5 Á Â Æ Ç Q y icohrt uC. PSR, Q y, =.9 logcohrt actio uc, kv, C/m oliar SC liar SC o SC.7 tim s Figur 9: Scattr plot of btatro tu with logitudial positio from th simulatios. y mm uc, kv, C/m - tim s Figur : Scattr plot of vrtical vrsus logitudial positio for a PSR bam just byod thrshold currt ad volvd ovr turs. th bam ad oliar spac charg is lss dstabilizig tha liar spac charg. Simulatios for various paramtrs ar show i Figur. Notic that that th rd trac X corrspods µ to Š à X µ protos for th sam protos ad th pik trac to Ä Ã buch lgth. Th lctro X µ dsity is oly 'Å highr for th cas with Š¾Ã. Clarly, th valu of th lctro dsity is critical for stimatig th thrshold. Figur shows th corrspodig situatio for half th umbr of protos ad half th RF voltag with th sam lctro dsitis. Notic that Figur shows half th turs of Figur. Halvig th umbr of protos ad th RF voltag whil kpig th umbr of lctros costat, dstabilizs th bam. Figur shows thrshold data for th PSR. Ulss thr is a srious rror i th simulatios, th liar scalig of thrshold voltag with bam itsity is a coicidc. That is to say, th lctro cloud dsity must vary appropriatly with itsity to obtai a liar scalig. Th wak dpdc o buch lgth also appars to dpd o th dtails of th lctro cloud. 6 8 tim turs Figur : Evolutio of th cohrt actio for a bam just byod thrshold with oliar spac charg rd. A simulatio with liar spac charg ad th sam avrag tu shift violt shows a strog istability, whil glctig spac charg blu yilds a stabl bam. logcohrt actio ,.,7,.,7,.,,.8,7,.,.8 uc, kv, C/m. 6 tim turs, ŸO\È. All th simulatios assumd a Figur : Simulatios for various valus of m H Ÿ ad É<Ê RF voltag. CONCLUSONS Th ffct of th lctro burst from sigl pass multipactig o th lctro cloud wakfild has b studid. For bams with a uiform trasvrs dsity th actual wakfild was btw th prdictios of two simpl modls. For bams with a smooth trasvrs dsity th ctral modl, which assumd a costat lctro li dsity, producd th bst approimatio. Th PSR corrspods to th scod cas. A simpl simulatio tchiqu to iclud th dpdc of spac charg tu shift o btatro amplitud was itroducd. Th fr paramtr v was chos so that th tu distributio for th modl agrd fairly wll with th tu distributio for a roud Gaussia bam. stability simulatios for PSR showd that oliar spac charg was lss dstabilizig tha liar spac charg. Nglctig spac charg tirly producd th most stabl simulatios. X µ

6 Ì Ë logcohrt actio N/, d C/m, T b s, kv.,.,7.,.8,7 tim turs Figur : Simulatios for various valus of ml, Ÿ ad ŸO\. Both th simulatios assumd a É'Ê RF voltag. X µ [] H. Qi, E. Startsv, R. C. avidso Phys. Rv. ST Accl. Bams 6. [] S. Cousiau, J. A. Holms, J. Galambos, A. Fdotov, J. Wi, R. Mack Phys. Rv. ST Accl. Bams 6 7. thrshold voltag kv stabl bam historical 9 6 ustabl bam bam charg microcoulombs Figur : Rproducd from [6]Courtsy R.J. Mack. Thrshold RF voltag vrsus bam itsity. Th thrshold RF voltag is th smallst RF voltag for which th bam is stabl. Th historical curv rprsts th situatio bfor th dirct Í ƒ upgrad ad th tdd ru of. Thrshold curvs ar th d of th ru for ijctd buch lgths of, 6, ad 9 s ar show. REFERENCES []. Nuffr t. al. NM A p 99. [] M. Plum, PSR Tch Not PSR [] M. A. Plum t. al PAC97 p 6. [] R. Mack, AP Cof. 8, p6, 998 [] V. ailov t.al PAC99 TUA. [6] M. Blaskiwicz, M. A. Furma, M. Pivi, R. J. Mack Phys. Rv. ST Accl. Bams 6,. [7] M. A. Furma, M. T. F. Pivi Phys. Rv. ST Accl. Bams. [8] M. Blaskiwicz, Phys Rv S.T Accl Bams 998. [9] H. Qi, R. C. avidso, E. Startsv, W. W. L, PAC, 69,