SL-FL 9-5 The ffec of he Meal Oiaion on he Vacuum Chambe Impeance nani Tsaanian ambug Univesiy Main Dohlus, Igo agoonov Deusches leconen-sinchoon(dsy bsac The oiaion of he meallic vacuum chambe inenal suface is an accompanying pocess of he chambe fabicaion an suface eamen. This cicumsance changes he eleco ynamical popeies of he wall maeial an as consequence he impeance of he vacuum chambe. In his pape he esuls of longiuinal impeance fo oiise meallic vacuum chambe is pesene. The suface impeance maching echnique is use o calculae he vacuum chambe impeances. The loss faco is given fo vaious oies an oiaion hicness. The numeical esuls fo he unulao vacuum chambe of uopean XFL poec ae pesene. Inoucion The nowlege of he vacuum chambe impeance in acceleaos is an impoan issue o povie he sable opeaion of he faciliy fom he machine pefomance an beam physics poin of view []. The impeances of meallic ype vacuum chambes have been suie, fo eample, in [-], incluing he analyical pesenaion of he longiuinal an ansvese impeances fo laminae walls of iffeen maeials [8-] an finie elaivisic faco of he paicle [,]. The meal-ielecic vacuum chambe impeances have been suie in ef [-4]. In [4] he longiuinal an ansvese impeances fo he uopean XFL [5] ice vacuum chambe ae calculae base on he fiel ansfomaion mai echnique []. geneal appoach o evaluae he impeances of he muliplaye vacuum chambe is he fiel maching echnique. In his pape, he longiuinal impeance of meallic vacuum chambe wih inenal suface oiaion is suie fo ulaelaivisic beam case. The eplici analyical soluion is obaine fo vaious hicnesses of he meallic laye an oiaion eph. Base on he obaine esuls he impeance of uopean XFL unulao aluminium vacuum chambe is calculae fo vaious oiaion ephs. The eplici analyical soluion fo longiuinal impeance of wo-laye ube is obaine. The eac fomula fo ula elaivisic poin lie chage moving on ais is inouce in ems of suface impeance. I is shown ha in small oie eph asympoic limi he suface impeance on bounay wih vacuum can be esimae as a sum of oie an meal suface impeances. Base on his soluion he monopole impeances fo aluminium beam pipe wih iffeen oie laye hicness ae evaluae.
. eflecion of elecomagneic fiels on bounay of wo maeials Plane waves of any polaiaion can be escibe as a supeposiion of waves wih pepenicula (-moe an paallel (-moe polaiaions o he plane of incien [6, 7]. Le us invesigae he eflecion of hese waves wih pepenicula an paallel polaiaions (fig. fo he case when maeial has finie hicness. The case wih infinie wall hicness can be foun as a limi. Fo monopole beam impeance, which will be iscusse in ne he chape, only -moe is elevan. So in his chape we will mainly focus on -moe. -Moe y -Moe y -Moe y -Moe -Moe y -Moe y -Moe y Fig. eflecion of plane waves wih pepenicula (lef an paallel (igh polaiaions fom maeial suface which has finie hicness. The -componen of magneic fiel fo -moe an he same componen of elecical fiel fo -moe ae ( ( ( ( ef in ef in (, (, ep ep ep ep + wih cos sin, m ef in Whee he uppe ine escibe he maeial numbe, own ine escibe he polaiaion an poecion of fiel. Fom Mawell s equaion we ge he ansvese componen of elecic an magneic fiels fo boh ins of polaiaions coesponenly SL-FL 9-5
SL-FL 9-5 (, (, y (, y (, cos cos in ef ( ep( + ep( in ef ( ep( ep( By using he view of fiel componens we can easily fin he suface impeance ( ( (, (, (, (, (, (, (, y cos + (, y (, y + (, y cos Whee an ae he eflecion coefficiens. Taing o accoun he wave veco view fo suface impeance we ge + ( ±,, (, [cos] (, The eface wave fiel componens fo moe ae (, (, a y, in ef ep( b ep( in ef [ a ep( + b ep( ] wih in, ef, m, y n fo moe (, (, a y, in ef ( b ep( in ef [ a ep( + b ep( ] ep Fo suface impeance we ge ( (a, (, y y, (, ( (, y + (a (,, (, (, (, y (, y + y, whee, b, / a, ( whee he uppe ine (a shows he fis bounay of secon maeial.
SL-FL 9-5 Taing ino accoun he bounay coniions which leas o have coninuous angenial componen of wave veco,,, sin Using ne elaion of wave vecos beween wo meias ( We can easily fin ne esul (a (a (, (, sin + sin + α + α + Whee α sin Fo he secon bounay he suface impeance eas ( y, (b,, (, y ±, e, (, α ( (4 y,,, (, y +, e Fo hi maeial he fiel componens fo moe ae (, g (, n fo moe we have (, (, g g ep ep ( (, y g ( ep( y, y, ep ( n fo suface impeance we ge ne esul coesponenly whee,, y 4
SL-FL 9-5 ( ( (, (, (, (, (, (, (, y y, (, y (, y (, y y, sin sin whee is hicness of maeial. Using ne noaion an equaion ( we ge α sin (5 ( ±, (, α In he case when,, >> he α, appoimaion is vali. Now by maching ( (b he impeances (, wih (, coesponenly,, we fin he unnown coefficiens an,, y, m ( e whee, α (, (6 +, (a ( By maching (, wih (, we fin he eflecion coefficiens fo boh, polaie fiels inepenenly,, u, (, wih u +, u u ( ( / cos a (, a (, cos / ( In he case when he hi maeial is pefec conuco (, fo suface impeance fomula is simplifie (a (a y, (, anh( (, anh( y, y, y, y, an( y, an( y, y, (7 Whee he y-componen of wave veco in secon maeial has he ne view 5
SL-FL 9-5 y, sin In hin oie eph (<< limi suface impeances will eas (a (a (, (, sin Fo non-magneic maeials his fomula is simplifie (a (a (, (,, sin, Whee is elaive ielecic pemeabiliy. Fo incien angle π / we ge (a (a (, π / (, π /,, Le ewie his equaions in ne fom (a (a (, ( L L ( L L, sin, (8 Since he unis of paamees L, ae he same as fo inucance, he hin ielecic coaing impac on beam impeance can be consiee as influence of inuco impeance. In he case when he hi maeial is no pefec conuco in mos pacical cases he >> is fulfille an he suface impeance fo boh in of moes ae equal an eas as ( ( ( ( ( ( κ( κ Whee κ( is conuciviy of conuco wih τ - elaaion ime. + τ 6
SL-FL 9-5 Taing ino accoun noaion in eq.(6 afe some manipulaions fo suface impeance on bounay beween vacuum an fis maeial we ge (a (a y, (, (, y, an( an( y, y, y, an + + y, ( y, + an( y, an ( y, + + + an( y, (9 The coefficiens, we igh in ne fom y, y, ( ( ( ( Fo hin ielecic (<< he asympoic fomula will eas (a (,, sin, ( ( ( (a ( (, + sin + Finally aing ino accoun noaions in eq.(7 an smallness of ielecic laye hicness we ge (a L κ, (, ( ( + +, (, + ( ( Whee meaning of he inees L an κ ae inuco an conuco coesponenly.. Beam Impeance Consie he elaivisic poin chage Q moving wih spee of ligh along he ais of unifom, cicula-cylinical wo-laye ube of inne aius (Fig.. The chage isibuion is hen given by Q(,, Qδ ( δ ( δ ( v. The secon laye has infinie hicness while he hicness of fis one is. The coss secion of he ube is ivie ino hee concenic egions: (vacuum, + (fis laye, + < (secon laye. 7
SL-FL 9-5 I b φ Fig. Geomey of he poblem In geneal, when he chage has non-eo offse, ue o cuen aial asymmey he fiels aiae in he ube have all si componens,,,, an, while in paicula case when i is on ais only hee componens ae ecie,, an hey ae inepenen on aimuhal cooinae ef [4]. The Mawell s equaions in fequency ( c omain (, ~ e fo moe will eas (. ( J (. (. Whee c 77Ω is he impeance of fee space. c Fom equaion (. an (. follows ha he longiuinal componen of elecic fiel is consan cons ( Subsiuing o he secon equaion we ge ( Le us ewie above equaion in ne fom J + 8
9 ( J + ( Taing ino accoun he elaion beween cuen an cuen ensiy J S J I S The inegaion of equaion ( gives π π I + n fo aimuhal componen of magneic fiel we ge I + π Finally all componens of M fiel will ea + + c c c e e I e I π π (4 The beam impeance will be b I The unnown coefficien shoul be foun fom bounay coniion. s The suface impeance on fis bounay coul be foun using maching echnique. In ems of suface impeance fo his coefficien i is easy o ge following epession SL-FL 9-5
SL-FL 9-5 s I π + Finally fo beam impeance we ge c s b s π + c s Whee suface impeance can be foun using he fomulas eive in pevious chape whee shoul be aen ino accoun ha he elecomagneic fiel ecie by ulaelaivisic chage has only ansvese componens which leas of π / incien angle. L s ( L L L κ s( s ( + s ( whee κ wih s ( κ κ( κ( + τ This fomula coincies wih hin coaing asympoic limi of eac soluion fo beam impeance of wo laye ube eive in ef [,].. Numeical esuls In his secion we pesen esuls of influence of hin oie laye on loss faco an enegy spea. Wae poenial of a Gaussian bunch wih σ 5 m ms lengh was calculae b 7 5 fo aluminium ( κ.7 Ω, τ 7. s beam pipe wih 5mm aius. To evaluae loss faco an enegy spea ne fomulas have been use Loss faco < W > negy spea W(s λ(s s < (W < W > > (W(s < W > λ(s s Whee W (s is a wae poenial an isibuion funcion. λ s ( s ep σ is a nomalie Gaussian πσ b b
SL-FL 9-5 a Loss faco [ V/pC/m ] 5 b Oie hicness [ nm ] negy spea [V/pC/m] 5 Oie hicness [nm] Fig. Loss faco an enegy spea vesus oie hicness fo iffeen in of oies. In figue ae ploe loss faco an enegy spea vesus oie laye hicness. The calculaions ae one fo hee ype of oies wih elaive ielecic pemiiviy, 5 an.
SL-FL 9-5 a 5 nm Loss faco [ V/pC/m ] 4 nm nm nm nm b Oie 5 nm negy spea [V/pC/m] 4 nm nm nm nm Oie Fig 4. Loss faco an enegy spea vesus oie ielecic pemiiviy fo iffeen hicness. In figue 4 ae ploe loss faco an enegy spea vesus oie elaive ielecic pemiiviy. s we see fom boh figues (, 4 he influence of oie laye in a wos case when aleay a 5 nm hicness inceases he loss facos moe han 6%. Since i is unnown he popeies an hicness of oie laye which is appeaing uing manufacuing of acceleao pas in ne figue we mae invesigaion fo he wos oie. In figue 5 is ploe he loss facos vesus oie hicness fo iffeen beam pipe aiuses.
SL-FL 9-5 a mm Loss faco [ V/pC/m ] 4mm 6mm 8mm mm Oie hicness [ nm ] b mm negy spea [V/pC/m] 4mm 6mm 8mm mm Oie hicness [nm] Fig 5. Loss faco an enegy spea vesus oie hicness fo iffeen beam pipe aiuses. In uopean XFL poec he unulao secions ae esigne wih aluminium vacuum chambes wih ellipical coss secions. The ellipse iamees in boh planes ae 5mm an 8.8mm coesponenly. Losses in ha secion can be esimae by oun beam pipe moel wih aius 4.4mm. s we can see fom figue 5 aleay a 6nm oie hicness he losses inceases wo imes wih espec o ieal case (no oie.
SL-FL 9-5 efeences. B.W.oe an S..Kheife, Impeances an Waes in igh-negy Paicle cceleaos (Wol Scienific, Singapoe, 997... ene an O. Napoly, in Poceeings of he Secon uopean Paicle cceleao Confeence, Nice, Fance, 99 (iions Fonies, Gif-su-Yvee, 99, pp. 46 48... Piwinsi, I Tans. Nucl. Sci. 4, 64 (977. 4.. Piwinsi, epo No. DSY-94-68, 994, p.. 5..W. Chao, Technical epo No. 946, SLC-PUB, 98; see also.w. Chao, Physics of Collecive Beam Insabiliies in igh negy cceleaos (Wiley, New Yo, 99. 6. B. oe, Pa. ccel., (97. 7. D. Jacson, SSCL epo No. SSC-N-, 986. 8. M. Ivanyan an V. Tsaanov, Phys. ev. ST ccel. Beams 7, 44 (4. 9..M. l-haeeb,.w. asse, O. Boine-Fanenheim, W.M. Daga an I. ofmann, Phys. ev. ST ccel. Beams,, 644 (7.. M. Ivanyan an V. Tsaanian, Phys. ev. ST ccel. Beams, 9, 444 (6.. N. Wang an Q. Qin, Phys. ev. ST ccel. Beams,, (7.. M. Ivanyan e al, Phys.ev.ST ccel.beams, 84 (8... Buov an. Novohasii, INP-Novosibis, epo No. 9-8, 99. 4.. Tsaanian, M. Ivanyan, J. ossbach, PC8-TUPP76, Jun 4, 8. 5. M.laelli,.Binmann e al (eios, XFL. The uopean X-ay Fee- lecon Lase, DSY 6-97, July 6. 6... Collin, Founaion fo Micowave ngineeing, McGaw-ill, 966. 7. M. Bon,. Wolf, Pinciples of Opics,(986 4