HOM calculations of new RF cavities for a super B-factory *

Transcription

1 SAC PUB 79 Octoe 4 HOM calculation of new RF cavitie fo a upe B-factoy * A. ovokhatki Stanfod inea Acceleato Cente, Stanfod Univeity, Stanfod, Califonia 9439 Atact High aveage HOM powe geneated y eam in a vacuum chame of electon-poiton collide can limit achievement of high cuent. It i of geat concen fo futue upe B-factoie of vey high luminoity, otained fom high eam cuent and hot unch length. We can minimize the HOM powe y chooing the ight RF cavity hape. Hee we peent eult of compute pectum analye of diffeent kind of cavitie, which ae aleady ued o can e ued in B-factoie. Talk peented at the Supe B Factoy Wokhop in Hawaii Honolulu, Hawaii, USA Januay 9 Januay, 4 * Wok uppoted y Depatment of Enegy contact DE AC 76SF55.

2 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu HOM calculation of new RF cavitie fo a upe B-factoy A. ovokhatki SAC, Stanfod, CA 945, USA High aveage HOM powe geneated y eam in a vacuum chame of electon-poiton collide can limit achievement of high cuent. It i of geat concen fo futue upe B-factoie of vey high luminoity, otained fom high eam cuent and hot unch length. We can minimize the HOM powe y chooing the ight RF cavity hape of. Hee we peent eult of compute pectum analye of diffeent kind of cavitie, which ae aleady ued o can e ued in B-factoie.. ITRODUCTIO In ecent time, a tong inteet ha developed in exploing the paamete egime ove which electonpoiton factoie may ultimately poduce high 36 cm - ec - luminoity[-]. Electon and poiton eam aveage cuent of ten of ampee ae needed to achieve thi luminoity. Additionally unch length of eam mut e vey hot, of the ode of eveal millimete. To compenate enegy lo due to ynchoton adiation, RF powe of ten megawatt i needed. The aveaged powe of wake field geneated in a vacuum chame can each utantial value, a it cale quadatically with cuent and invee with unch length. A thi powe come fom the eam kinetic enegy it mut e compenated y additional RF powe. It i of geat concen fo futue upe B-factoie of vey high luminoity, otained fom high eam cuent and hot unch length. Main pat of wake field come fom RF cavitie. We can minimize thee field y chooing the ight cavity hape. Hee we peent eult of compute pectum analye of diffeent kind of cavitie, which ae aleady ued o can e ued in the toage ing of electon poiton factoie.. WAKE FIEDS I RF CAVITIES Inteaction of electomagnetic field of a chaged unch with a vacuum chame lead to excitation of additional field. We call thee field wake field. Thee ae two main ouce of wake field. They can e geneated due to finite conductivity and uface oughne of the mateial of a vacuum chame. In pinciple thi pat can e educed y chooing high conductivity mateial, high quality uface tooling o inceaing tanvee dimenion of vacuum chame. Anothe ouce i geometical dicontinuitie of vacuum chame. We can make a vey mooth vacuum chame, ut we can not avoid uing RF cavitie which ecome the main non homogeneity in the mooth ing. Beam geneate electomagnetic field in a cavity. Some pat of thee field tay in a cavity in the fom of the main RF mode and ome High Ode Mode (HOM. Thi HOM ae uually called HOM ellow cut-off fequency. In a time othe pat of the wake field leave a cavity and popagate in the vacuum chame. Thee field ae called HOM aove cut-off fequency. Vey high fequency pat of the wake field tavel togethe with unche fo vey long time. The powe of the excited main mode effectively detemine the powe which eam take out fom the cavity. The HOM powe i diipated in the wall of the cavity o vacuum chame and aoe a Joule loe. Fig. how a naphot of electic foce line of wake field, excited in the PEP-II empty cavity y a ingle unch. At thi moment a unch i coing the ight exit of the cavity. Figue : Wake field electic foce line (geen and lue line in the PEP-II cavity at the time when a hot unch i leaving the cavity. Bunch denity i peented y a ed line. Bunch length i.8 mm... Geen wake function and wake potential The powe of the wake field i natually the ame powe that eam loe. To calculate the eam powe loe we need to find electomagnetic field along the tajectoy of the eam. When we know electomagnetic field, then we can calculate the integal of electic longitudinal component along the unch tajectoy. Thi integal i called wake potential W ( τ : W( τ Ez( t, z z c( t τ dt Vaiale τ define time inteval etween paticle in a eam. Sometime we will alo ue ditance etween paticle cτ, a they tavel with a peed of light c. Wake field and wake potential can e calculated in the time domain y integating Maxwell equation. Wake potential of a point chage i a Geen functiong(τ

3 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu which can e ued fo calculation of the field of unche of any chage line ditiution ρ( τ τ ρ( ω ρ( τexp( iωτ dτ W( τ ρ( τ G( τ τ dτ ρ( τ τ G( τ dτ Fo a ingle unch with the Gauian chage denity and m unch length σ and total chage Q the Becaue of the difficultie of Geen function calculation, Fouie tanfom i the wake potential of a hot unch i ued a an σ appoximation fo the Geen function. We ue code ( ω c ρω ( Qe OVO [3] which give the et eult [4] fo the wake field, geneated y a vey hot ulta elativitic Fequency pectum of the wake potential of 4 mm unch in an axially ymmetic, pefectly conducting, unch in the PEP-II cavity i hown in Fig. 3. Thi vacuum chame pipe. Wake potential of 4 mm unch in pectum i calculated fom the wake potential of 5 the PEP-II cavity i hown in Fig.. Bunch ha the mm length. Gauian chage ditiution. Figue : Wake potential of 4 mm unch in PEP-II cavity. Beam enegy lo k i calculated y k W( τ ρτ ( dτ Fo a ingle unch lo facto i uually nomalized to a unch chage and meaued in V/pC... ongitudinal impedance and fequency pectum of wake potential The Fouie tanfom of Geen function give longitudinal coupling impedance Z( ω Z( ω G( τexp( iωτ dτ The Fouie tanfom of the wake potential W( ω W( τexp( iωτ dτ ρτ ( τ G( τ dτ exp( iωτ dτ ρ( ω G( τ exp( iωτ dτ ρ( ω Z( ω give the poduct of impedance Z( ω and Fouie tanfom of unch chage denity ρ ( ω Figue 3: Real (lue line and imaginay (pink line pat of wake field potential of 4 mm unch in the PEP-II cavity. Dotted line how cut-off fequency. A can e een the pectum ha hap peak in low fequency ange and ha oad-and peak in high fequency ange. Thi ehavio eflect the fact that low fequency mode ae tapped y the cavity and ocillate in the cavity. In eality the width of thee mode ae detemined y the loaded quality facto of a cavity Q due to wall conductivity and coupling with a tanmiion line and HOM load. In ou calculation we aume pefect conductivity of the cavity wall, o the width of thee mode i detemined y the time inteval of the wake potential. In the high fequency ange the mode afte eveal ocillation leak out of the cavity and popagate in the eam pipe. In pinciple the pectum fom a continuum in high fequency ange; howeve tapped mode can e thee too. We can aume that the fequency which epaate diffeent pectum ehavio i the cut-off fequency..3. Cut-off fequency Cut-off fequency i the maximum fequency of a captued mode in a cavity. It i detemined y the ize of a eam pipe. Thee can e eveal cut-off fequencie fo TM and TE mode. We conide the cae of longitudinal electic field o TM mode. Fo a ound pipe of adiu a the minimum cut-off fequency i fo TM mode

4 f cut off [ GHz] Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 3 c a ν π.474 a Fig. 4 alo how intinic impedance (R/Q of tapped [ m] In the fomula we ued the fit zeo of the Beel functionν.448. Cut-off fequency i hown y a dotted line in Fig. 3. mode y pink cicle, which i calculated ove tep K of lo integal accoding to the fomula R/ Q k ω.4. Fequency pectum of enegy lo The fequency pectum of enegy lo will give u infomation aout how much enegy a unch loe fo mode ellow o aove cut-off fequency. Uing invee Fouie tanfom we can peent lo facto k in the following fom k W ( ω ρ( ω π ρω ( Z( ω π o facto i a pue eal function ecaue of the popetie of impedance function, whoe eal pat i ymmetic with fequency and imaginay pat i antiymmetic. et u intoduce fequency lo integal K( ω a π ρ ( ω Re{ Z( ω} ω ( K( ω Re{ W( ω ρ( ω } π Index mean wake potential of a ingle unch. In thi definition total integation give lo facto: K ( k. Fequency lo integal fo the PEP-II cavity i hown in Fig. 4 y a ed line. Tapped mode make tep in lo integal. o facto fo 4 mm unch i.85 V/pC. In fequency inteval up to 6 GHz enegy lo achieve 75% of it value. Enegy lo aove cut-off fequency i.389 V/pC o 48%. Figue 4: Spectum, lo integal and intinic impedance fo the PEP-II cavity..5. Powe lo of a tain of unche Analytical eult peented in thi and following chapte ae in ageement with eult aleady achieved y othe autho [5]. Becaue of upepoition of electomagnetic field, the total field i a linea um of field of each chage. If we have a tain of equal unche paced in time yt, then wake potential fo the -th unch i um of wake potential of peviou unche W ( τ W ( mt + τ m The wake potential um fo a tain with unch pacing equal to RF wavelength i hown in Fig. 5. Figue 5: Wake potential fo a tain of unche, paced y RF wavelength. It can e een that wake field i gowing with nume of unche, taking the hape of the main RF cavity mode with ome high fequency ocillation. We can ue equation ( to deive fomula fo the eam powe lo. Enegy lo of the th unche i Powe lo π Fo equal k ρ ( ω Re{ Z( ω} π P + can e defined a P + T ρ ( ω ρ ( ω T k k Re{ Z( ω} unche line chage denity i ρ ( t ρ ( t mt m

5 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 4 It Fouie tanfom i.7. Powe lo aove cut-off fequency ρ( ω ρ( ω exp( imωt m The pectum hee i a moe o le mooth function of fequency and we can change the eie peentation exp( iωt ρ( ω ack to the integal exp( iωt Then powe lo i P ρ ( ωm Re{ Z( ωm} m T in ωt ( + ρ( ω P Re{ Z( ω} ρ( ωm Re{ Z( ωm} ωt in πt πt ωcut off Finally, the eam powe lo aove cut-off fequency Fo a long tain of unche, left function unde integal take the fom goe to an infinite um of δ -function P Ieam T ( k K( ωcut off Ieam Rlo in ωt ( + et u call lim δ ( ω ωm ωt Rlo T ( k K ( ωcut off π in T m a lo impedance. π ω m m T and powe i P ( ( Re{ ( } m Z d T m δ ω ω ρ ω ω ω Afte integation we get well-known eult P ( Re{ Z ( } m m m T ρ ω ω We can analyze thi fomula fo fequency egion elow cut-off fequency and aove..6. Powe lo elow cut-off fequency Fo tapped mode impedance can e decied a y naow-and eonance of fequencyω, loaded quality facto Q ω and intinic impedance ω Z R / Q ω ω Q Z( ω Z ω ω ω + iq then powe lo fo the eam cuent Z I Q ω Q eam i T 3. CAVIES FOR SUPER B-FACTORY Fo upe B-factoy we need cavitie with vey mall lo impedance. We can ty to change the hape of exiting cavitie o the RF fequency too, in ode to deceae impedance. Seveal hape of a cavity ae peented elow. 3.. PEP-II type cavitie The hape of the PEP-II cavity i hown in Fig.. Fit modification can e lage apetue of the eam pipe fo the ame RF fequency. The hape of a cavity with a doule ize eam pipe i hown in Fig. 6 and it pectum i hown in Fig. 7. Figue 6: PEP cavity with doule eam pipe. ( ω m c eam ( m ω ω ω ( m ω Thee i only one HOM mode elow cut-off fequency. + Q Thi mode can e effectively damped y an appopiate ω load. The lo impedance i two time malle than PEP- Q y II cavity, R / Q of the main mode i alo le, ut only P I e σ ω ω The way to diminih powe lo i to deceae applying HOM load and pope detuning of tapped mode. y 35%.

6 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 5 Figue 7: Spectum of a cavity peented in Fig.6. Anothe poiility i to inceae RF fequency two time keeping the ame eam pipe ize. The hape of uch a cavity and it pectum i hown in Fig.8 and Fig.9. Figue : Optimized hape of the cavity. 3.. Single-mode cavity If we have only one mode (main elow cut-off and thi mode i an acceleating mode with intinic impedance R/Q then powe lo i ω R P Ieam T ( k (4 Q Supeconducting cavitie of Conell CESR-III and KEKB ae ingle-mode cavitie. Cavity hape and wake field of a hot unch in the CESR-III type cavity [8-9] i hown in Fig.. Figue 8: 95 MHz cavity. Figue : Electic foce line of a hot unch in CESR- III type cavity. Figue 9: Spectum of a cavity which i hown in Fig.8. o impedance of thi cavity i 3.4 time malle than impedance of PEP-II cavity fo a unch of.8mm length. R/Q of the cavity i 66 Ohm. Optimization of the cavity hape (Fig. impove R/Q a little. Thi cavity ha R/Q78 Ohm and 8% le lo impedance. Figue : Wake potential of 4 mm unch in the Conell CESR-III-type cavity.

7 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 6 Wake potential of a unch of 4 mm length i hown in Fig.. o integal and intinic impendence fo thi cavity ae hown in Fig. 3. Figue 3: Spectum, lo integal and intinic impedance fo Conell CESR III type cavity. Bunch length i 4 mm. KEKB cavity ha a little it diffeent hape []. o integal fo thi cavity i hown in Fig. 4. In thi example the unch length i.8 mm. Figue 6: Spectum fo the KEKB upeconductive cavity with tape. Bunch length i.8 mm. One moe example of a ingle-mode cavity and it pectum ae hown in Fig7 and Fig.8. Thi cavity ha R/Q3 Ohm and the mallet lo impedance of Ohm fo.8 mm unch. Figue 7: Single-mode cavity. Figue 4: Spectum, lo integal and intinic impedance fo KEKB type cavity. Bunch length i.8 mm. Cavitie with lage eam pipe uually have tape. Unfotunately, thee tape change wake potential damatically. Fig. 5 how pictue of electic foce line of wake field jut at the moment when unch i in the ight tape. Spectum fo thi cae i hown in Fig. 6 Figue 8: Spectum of a ingle-mode cavity. Bunch length i.8 mm. Figue 5: Electic foce line of the wake field in KEKB upeconductive cavity. At thi moment unch i in the cone of the ight tape. An additional advantage of the ingle mode cavity ytem which ha an automatically low R/Q, i that it i tonge againt the coupled-unch intaility [8]: it need le fequency detuning f I R/ Q inφ f V RF c

8 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 7 whee V the voltage pe cavity and φ the ynchonou phae. Alo a HOM damping ytem i le complicated fo a ingle mode cavity. Futue plan include moe numeical tudie of the polem, finding optimum hape of the cavity Shot unche Fo hot unche the lo facto can e etimated y the following expeion [6] : Zc g k π a σ whee g i the ize of cavity gap, a i the adiu of the eam pipe, Z π i impedance of fee pace. o facto goe up with deceaing unch length and ecome the main paamete fo the powe lo d g P Ieam T k Ieam Z π a σ Whee we intoduced he d T a the ditanced etween unche Minimum powe lo The minimum ditance etween unche i the RF wavelength. The adiu of the eam pipe i limited λ RF y the main mode and can not e moe than λ RF /. The cavity gap i appoximately half of the wave length, o the minimum powe lo i detemined y the atio of the RF wavelength and unch length λrf P Ieam Z π σ The coeponding minimum lo impedance i λrf Rlo Z π σ Thi minimum impedance i hown in Fig. 9 togethe with calculated impedance fo diffeent cavitie. It i inteeting to note that ingle-mode cavity (PEP-II SC, peented in Fig. 7 ha malle lo impedance, ut thi i ecaue of malle gap ize ( g << λ RF / of the cavity. 4. COCUSIOS High aveage cuent, hot unch length eam needed fo Supe B-factoie can give ie to lage amount of HOM powe, a high a hunded of kw pe cavity. Thi powe mut e damped y the wate cooled aoe. Mot of the powe lot y the eam i in mode elow GHz fo mm unche. Beam in a ingle mode cavity excite le HOM powe. A ingle mode cavity need to e a upeconducting cavity, a it ha mall intinic impedance R/Q. Paamete of all peented cavitie ae compiled in the Tale. Figue 9: Minimum lo impedance (lue line and lo impedance of diffeent cavitie. c ACKOWEDGEMETS The autho wih to thank John Seeman, Michael Sullivan and Stephen Weathey fo the inteet and fuitful dicuion. Wok uppoted y Depatment of Enegy contact DE-AC3-76SF55. REFERECES [] J. Seeman et al., Deign tudie fo a 36 upe factoy, Poceeding of the PAC 3, Potland, p.3 (3. [] K. Ohmi et al., Recent development in deign fo e+e- collide, Poceeding of the PAC 3, Potland, p.345 (3. [3] A. ovokhatki, to e pulihed ometime [4] O. Meincke, A.Wagne, B.Zotte ew Wake Filed and Bunch engthening Code, S-ote-97-7 (AP, CER, Switzeland (997. [5]. Meminga et al., Specifying HOM-powe extaction efficiency in a high aveage cuent, hot unch length SRF envionment, Poceeding of the XX Intenational inac Confeence, Monteey, p.86 (. [6 A. ovokhatki On the etimation of the wake potential fo the ulta elativitic chage in an acceleating tuctue, Pepint IP 88-39, Intitute of uclea Phyic, ovoiik (988. [7] B.Zotte and S.Kheifet Impedance and Wake in High-Enegy Paticle Acceleato, Wold Scientific, Singapoe, 998. [8] A.Chao, M.Tigne, Hand ook of Acceleato Phyic and Engineeing, Wold Scientific, p.537 (998. [9] S. Belometnykh, On the BB Cyomodule o Facto Calculation, Conell note SRF , Ithaca (999. [] KEKB B-factoy deign epot, KEK Repot 95-7 (995.

9 Supe B Factoy Wokhop in Hawaii. Januay 9-, 4, Honolulu 8 Tale : Cavitie paamete HOM Powe o in the Supe B-factoy Cavity f36khz h349 Cavity Fequency Pipe R/Q Bunch Total Aove Beam Bunch Wake HOM type adiu length o cut-off cuent chage Voltage Powe [MHz] [mm] [Ohm] [mm] [V/pC] [V/pC] [A] [nc] [kv] [kw] PEP-II CESR-III PEP-II CESR-III KEKB-SC with tape 75 KEKB-SC-T no tape PEP-II-age PEP-II CESR-III KEKB-SC-T PEP-II-age PEP-II-age ew PEP-II ew PEP-II PEP-Ellip PEP-SC