SRF OPTIMIZATION OF THE END CELLS IN SRF CAVITIES

Transcription

1 SRF588-7 OPTIMIZATION OF THE END CELLS IN SRF CAVITIES Jonahan W. Luk Univrsiy of California, San Digo, La Jolla, CA 9293 Valry Shmlin Laboraory for Elmnary-Paricl Physics, Cornll Univrsiy, Ihaca, NY Dmiry Myakishv Silvaco Daa Sysms, Sana Clara, CA 9554 INTRODUCTION Th work on incrasing h acclraing gradin E acc in RF suprconducing niobium rsonaors has no rminad. An acclraing gradin of 46 MV/m (CW) has bn achivd a Cornll in 24 [1] 1. Hr, E acc is dfind as E acc L = L, which is h avrag lcric fild in a cll wih lngh L, whr E = Ez cos ω = Ez ( z)cos 2πz λ aks ino accoun h ransi im facor. On of h possibiliis o incras E acc is h opimizaion of h caviy shap [2]. Th limi of h acclraing gradin is imposd by H pk Eacc, a raio ha is fixd by gomry, bcaus H pk is boundd abov by h criical magnic fild H cri,rf, abov which suprconduciviy braks down [1, 3]. E pk Eacc may also impos a limi bcaus E pk bing oo high would caus h dangr of fild mission [3]. Howvr, sinc h limi of E pk can b raisd by br clanlinss and high powr procssing whil H pk is a hard limi, i is jusifiabl o rduc H pk by sacrificing E pk up o a crain bound. Prvious opimizaion has bn don for h innr clls in SRF caviis o rduc H pk Eacc for givn valus of E pk Eacc [2, 4]. Howvr, hr is y o b a sysmaic dscripion of h opimizaion of h nd clls in SRF caviis. Unlik h opimizaion of h innr clls, i is mor rasonabl o considr maximizing V acc wih h opimizaion of h nd clls. This is bcaus h lcric fild, in hory, xnds o infiniy in h ub (bu wih convrging ingral) and h dfiniion of E acc bcoms dpndn on h ub lngh. Th physically significan figur is h valu of acclraion is h valu ha w will maximiz. V acc = Edx, and his In ordr o valua h improvmn by h opimizaion, w compar h raios H pk V acc and E pk V acc wih ha of TESLA. For innr TESLA clls, E pk Eacc= 2. and H pk Eacc= 42 O/(MV/m) [5]. W hus dfin = E pk L 2Vacc and h= H pkl 42V acc wih L = mm (a quarr of wavlngh, λ /4, for frquncy of 13 Hz) bing h lngh of h innr cll so ha comparison can b mad bwn innr and nd clls. For TESLA clls, = h = 1. Sinc w can mak sacrific for, h opimizaion of h innr clls has bn prviously don by rducing h whil fixing = = 1.2. Suppos h minimal h achivd for h innr clls was h, which is diffrn for diffrn R. (For R = 35 mm, for xampl, h =.8996, i.., 1% lss han h cas of TESLA.) W would hn lik o minimiz max{/1.2, h/h } for h nd clls. Th minima mus b obaind such ha E pk and H pk ar aaind a h innr nd. In his cas, w xpc h filds in h innr half of h nd cll o b idnical o h filds in h innr clls and, hrfor, max{ 1.2, h h } = 1. 2= h h. A prvious sudy has shown ha smallr iris radii would furhr incras h acclraing gradin [6]. This is hrfor an incniv for sudying h ffc of diffrn iris radii on h filds in h nd clls. W also sudid h possibiliy of diffrn bam pip radii bcaus broadr bam pip can hlp o xrac highr ordr mods. THE GEOMETRY FOR OPTIMIZATION For his opimizaion, w mploy h sam consrucion of h profil lin as ha of h innr clls [4]. This is consrucd as wo llipic arcs (Fig. 1). I has bn shown ha his shap achivs a br acclraing gradin han ha of h circular arc sraigh sgmn llipic arc profil lin as in TESLA [3]. 1 Rcns ss (July, 25) of anohr rnran Cornll caviy a KEK (Japan) confirmd his xprimn wih h rsul of 47 MV/m (K. Saio, priva communicaion).

2 LENGTH OF BEAM PIPE Fig. 1: Gomry for calculaion. W only opimiz h our half of h nd cll bcaus for h ohr half w choos a gomry idnical o ha of h innr clls, which has alrady bn opimizd. For h gomry in Fig. 1, w hav hr indpndn variabls o opimiz, namly, A, B and a. Th ohr axis of h llips, b, is fixd by gomry sinc h wo llipss mus hav a common angn a h conac poin. R is fixd by h opimizaion of h innr q cll. L is chosn by uning o h corrc frquncy and for h nd cll will b dsignad as L, Fig. 2. (This is diffrn from h opimizaion of h innr clls whr L= L i was fixd o b λ /4 and h frquncy is und by changing R.) Th frquncy ha is usd is 13 MHz, q bu h opimizaion is valid for any frquncy: on nds only o scal all h dimnsions. THE CODE FOR OPTIMIZATION Th SLANS cod [7] is usd for his sudy. SLANS cod is dsignd for numrical calculaions of monopol mods of axisymmric caviis. I uss a fini lmn mhod of calculaion wih a msh of quadrilaral biquadraic lmns [8] and provids high accuracy of h fild calculaion. In a sudy comparing diffrn cods for calculaing h filds in a sphrical caviy, SLANS was shown o hav h bs prformanc [9]. For h opimizaion of h nd clls, a spcial cod TundCllEnd was wrin by h auhor of SLANS. This cod is basd on TundCll cod which was usd arlir for opimizaion of innr clls [6] and is som kind of wrappr cod ovr SLANS cod. Th TundCllEnd cod auomas h uning h frquncy of h nd cll by changing is lngh L. I also maks i possibl o calcula a s of clls wih diffrn paramrs: half-axs A, B and a (Fig. 2). Th ingraion V = acc Edx as dscribd abov can b approximad by numrical calculaions wih an appropria choic of h uppr ingraion limi. I has o b a sufficinly larg y fini numbr so ha h rror is sufficinly small. I is also imporan ha h ingraion limi is no oo larg for ohrwis rror will aris from h larg msh siz. W aim a an accuracy of four dcimal placs in and h. Sinc and h varis from.8 o 1.3, hy can b akn, for simpliciy, o b 1 and hus h absolu rror can b akn as h rlaiv rror. This, in urn, by h formula of and h, is givn by h rlaiv rror of V acc, i.., V acc V acc. Th lcric fild of h fundamnal mod in h bam pip follows an xponnial dcay αz, whr 2 2 α = 2π f c f c, wih f c, f and c bing h cuoff frquncy, frquncy of h wav and h spd of ligh rspcivly. Th calculaion of f c is dscribd in h scion of Highr Ordr Mods. Supposing h caviy xnds from o L c and h pip xnds from L c o L c +L (Fig. 2), hn h rlaiv rror of V acc is givn by R V V acc acc = Lc Lc+ L L c L + L c c Lc L + L αl To carry ou h approximaion, w us h cas of = 35 mm. Th firs fracion is calculad numrically o b.54. Prvious calculaions wr don wih L = 3. Wih his choic of L, h rlaiv rror was R In ordr o rduc h rror, w incrasd L o 4R and h rlaiv rror bcam 1-4. Thrfor, h rror in and h was also 1-4, as dsird. RESULTS Th opimizaion, wih h mhod mniond abov, has bn don for nd clls wih R a = R = 3 mm, 32.5 mm, 35 mm (Fig. 2). Th opimizd gomric paramrs of h our half of h nd clls ar prsnd in Tabl 1. As mniond abov, h paramrs of h innr half of h nd clls ar akn as ha of h opimizd innr clls. In Tabl 1, h acclraing volags of h opimizd nd clls ar also compard o ha of h opimizd innr clls. I is shown ha h nd clls, afr opimizaion, hav largr acclraion han h innr clls. Th diffrnc is mor significan for smallr R (Fig. 3)..

3 Fig. 2: Dimnsions of h nd cll wih R a = R. Tabl 1: Dimnsions (in mm) of h opimizd nd clls and comparison of V in nd clls and innr clls. R A B a b R q L V nd /V innr V nd V innr =.9944 V nd V innr = 1.41 Fig. 4: Filds along profil lin of h nd cll bfor (lf) and afr (righ) opimizaion. Vnd/Vinnr In gnral, afr h opimizaion, h cornr bwn h nd cll and h bam pip bcoms sharpr. Afr h opimizaion, E pk is aaind a boh nds of h nd clls, i.., h wo local paks ar qual. Fig. 4 illusras his by showing a comparison bwn h lcric fild along h profil lin bfor and afr opimizaion for R = 35 mm. RESULTS FOR SINGLE-CELL CAVITY R Fig. 3: Th dpndnc of V nd /V innr on R. Th opimizd rsuls for h nds clls can also b usd o consruc singl-cll caviy (Fig. 5) o carry ou an xprimn on h acclraing gradin. Th goal is o consruc a singl-cll caviy ha has an acclraing gradin highr han ha of h caviy which achivd a world rcord acclraing gradin [1].

4 Th opimizd valus of boh h innr clls of mulicll caviis and singl-cll caviis ar prsnd in Tabl 2. Th dimnsions usd for h singl-cll caviy wr h sam as ha in Tabl 1. Th filds in ach cll ar plod (Fig. 6) for R = 3 mm. Filds for R = 32.5 mm and 35 mm hav h sam parn. I should b nod ha h filds in h nd clls consis of wo pars: h lf par is idnical o h lf half of h graph (a), and h righ par is idnical o h righ half of h graph (b). Tabl 3 shows h valus of and h for h singlcll caviy consrucd wih h gomric paramrs of h opimizd nd clls. I is xpcd ha h singl-cll caviy of R = 3 mm, 32.5 mm, 35 mm would show improvmns of.63%,.55% and.41% rspcivly from h opimizd nd clls as in Tabl 1, i.., improvmns of 1.3%, 1.1% and.82% rspcivly from h innr clls. This is bcaus h improvmn obaind by rplacing an innr half cll by an nd half cll should b h sam. This xpcaion is rachd wihin rror. Fig 5: Shap of singl-cll caviy. Tabl 2: Th dimnsions (in mm) of muli-cll and singl-cll caviis. Innr clls of Muli-cll Caviy Singl-cll (using nd half of nd clls) R A B a b L A B a b L R q Tabl 3: and h valus for h singl-cll caviis. R h h / h/h V singl /V innr (a) Innr cll of muli-cll caviy, R = 3 mm (b) Singl-cll caviy, R = 3 mm Fig. 6: Filds along h profil lins of innr clls of h muli-cll caviy and of h singl-cll caviy. HIGHER ORDER MODES I is anohr goal of h opimizaion of h shap of h nd cll o xrac highr ordr mods (HOM) in h caviis. Th HOMs includ h dipol and quadrupol mods and hy boh hav a ransvrs componn of lcric fild, which will disor h bam. Ths HOMs can b go rid of if h bam pip radius (R ) is chosn approprialy so ha h frquncis of h HOMs ar highr han h cuoff

5 frquncy of h bam pip and consqunly h HOMs will propaga ou of h caviis. Th cuoff frquncis of h TE 11 (dipol) mod in h bam pip wih various R ar shown in Tabl 4. Thy ar calculad by f = ' 11 / 2πR µ ε, whr ' 11 = is h firs roo of drivaiv of h Bssl funcion J 1. Th frquncis of h fundamnal and highr ordr mods in h caviy ar shown in h disprsion curvs (Fig. 7). Th disprsion curvs show h phas dpndnc of h frquncis of ach mod in an R = 35 mm caviy. Thy ar found by calculaing h ignmods insid a 9-cll caviy wih h opimizd cll, i.., h (diffrn) opimizd valus of h paramrs of h innr and nd clls ar usd. Frquncy Disprsion Curvs Phas Monopol 1s Diplo 2nd Dipol Quadrupol Fig. 7: Disprsion curvs for R =35 mm. In Tabl 4, w also calcula h rquird R (=55.82 mm), abov which h cuoff frquncy will b lowr han h frquncis of h HOMs. I should b nod ha vn so h problm of HOMs is no oally liminad bcaus som HOMs can ransform so ha hy ar no propagad ou of h ub. Morovr, by incrasing R w hav o mak wo sacrifics. Firs, h anuaion of h fundamnal mod of h caviy will b slowr. Scond, as w will show in h nx scion, h acclraion will b lowr. Tabl 4: Cuoff frquncis of TE 11 mod for diffrn R. R /mm f cuoff /MHz OPTIMIZATION WITH LARGER BEAM PIPE RADIUS In ordr o rduc HOMs, w amp o opimiz wo gomris wih largr R. Th firs gomry is shown in Fig. 8. R is changd o b diffrn from R a and w invsiga is impac on V acc. Fig. 8: Th shap wih R a < R. Th rsuls for Ra = 3 mm and Ra = 35 mm ar shown in Fig. 9. I is shown ha as w incras R o comba HOMs, w sacrific V acc. This rad-off is mor significan for Ra = 3 mm, whr V nd /V innr dcrass mor rapidly. In fac, for h R a = 3 mm cas, i is impossibl for E pk o b aaind a h innr half for R > 4 mm. V/Vi R Ra=3 Ra=35 Fig. 9: V nd /V innr for various R a and R (unis in mm). Anohr gomry ha has bn opimizd is shown in Fig. 1. I should b nod ha h half-cll of h innr caviy is shown for clariy bu was no usd in h gomry for calculaions. A i, B i and a i ar chosn as h opimizd valus of h innr clls and A, B and a ar o b opimizd as abov. Bfor h opimizaion, a, b, and c ar chosn arbirarily. R was chosn o b largr han R a so ha h HOMs can b allowd o propaga ou wihou much sacrific in V acc. Th sam chniqu was usd wih h on-cll KEK caviy and h wo-cll caviy of h ERL injcor ha is undr dvlopmn a Cornll Univrsiy. W hop o gnraliz his chniqu o muli-cll caviis.

6 Fig. 1: Shap wih diffrn R a and R. W opimizd h caviy wih R = 35 mm. W s a = 9.28 mm, b = 12 mm and c = 3 mm (and hus R = 5 mm). This iniial choic of a and b is basd on h curvaur of llips in h opimizd innr cll. Wih his choic of a, b and c, w obaind an opimizd valu of V =. 9875, which is narly 8% br han h V i ai valu obaind for h R = 5 mm cas for h abov gomry. Th dpndnc of h opimizaion on a, b and c is also sudid. W vary hs paramrs whil fixing ohr paramrs as abov, including h prviously opimizd paramrs A, B and a for h nd cll. b and c ar changd oghr in ordr o kp R consan. I was shown ha max{/1.2, h/h } incrass wih incrasing b (Fig. 11). max{/1.2,h/h} Fig. 11: Th dpndnc of max{/1.2, h/h } on b in mm. In ordr o minimiz and h, w choos b = 7 as w sudy a. a is varid from 4.53 mm o 6 mm and i is shown ha max{/1.2, h/h } dcrass (Fig. 12), i.., V acc incrass in h nd cll. Th dcras lvls off a b a = 5 mm. W hn opimiz his gomry wih a = 5 mm, b = 7 mm and c= 6 mm. For his gomry, nd V innr V is opimizd o b 1.26, a valu ha is largr han ha of a = mm bu smallr han ha in Tabl 1. I is apparn ha as a, his gomry nds o h on in Fig. 2 and V nd V innr incrass asympoically o 1.41, h valu in Tabl 1. max{/1.2,h/h} Fig. 12: Th dpndnc of max{/1.2, h/h } on a in mm. CONCLUSION Th nd clls can b opimizd o obain V acc br han ha of h innr clls. Alhough his improvmn is small (abou.5%), his sudy nvrhlss provids a sysmaic discussion of h possibl improvmn ha can b givn by h opimizaion of h shaps of h nd clls. Th possibiliy of combaing Highr Ordr Mods by incrasing h bam pip radius is also sudid. Th final shap has o dpnd on h rad-off bwn h rducion of Highr Ordr Mods and h incras in a

7 acclraing gradin. This, in urn, will dpnd on furhr sudy on Highr Ordr Mods as h bam pip radius is incrasd. REFERENCES [1] R. L. Gng, H. Padams, A. Saman, V. D. Shmlin. World Rcord Acclraing Gradin Achivd in a Suprconducing Niobium RF Caviy. PAC 25, Knoxvill, TN, May 25. [2] V. Shmlin, H. Padams. Th Opimal Shap of Clls of a Suprconducing Acclraing Scion. Cornll Univrsiy LNS Rpor SRF ; 22; TESLA Rpor [3] H. Padams, J. Knobloch, T. Hays. RF Suprconduciviy for Acclraors. John Wily & Sons, Inc., [4] V. Shmlin, H. Padams, R. L. Gng. Opimal Clls for TESLA Acclraing Srucur. Nucl. Insr. and Mh. A496 (23) 1-7. [5] D. A. Edwards (d.). TESLA Ts Faciliy Linac- Dsign Rpor. DESY Prin, March 1995, TESLA [6] V. Shmlin. Opimizd Shap of Caviy Clls for Aprurs Smallr han in TESLA. PAC 25, Knoxvill, TN, May 25. [7] D. G. Myakishv, V. P. Yakovlv. Th Nw Possibiliis of SuprLANS Cod for Evaluaion of Axisymmric Caviis, 1995 Paricl Acclraor Confrnc and Inrnaional Confrnc on High-Enrgy Acclraors. May 1-5, 1995, Txas, pp [8] R. Paryl. SUPERLANS Companion for PC. Cornll Univrsiy SRF/D [9] S. Blomsnykh. Sphrical Caviy: Analyical Formulas. Comparison of Compur Cods. Cornll Univrsiy LNS Rpor SRF , 1994.