Opimizain f Fur-Bun BPM Cnfigurain fr Small-Gap Beam Chamers S. H. Kim Advanced Phn Surce Argnne Nainal Larary 9700 Suh Cass Avenue Argnne, Illinis 60439 USA Asrac. The cnfigurain f fur-un eam psiin mnirs (BPMs) emplyed in small-gap eam chamers is pimized frm 2-D elecrsaic calculain f induced charges n he un elecrdes. The calculain shws ha fr a narrw chamer f widh/heigh (w/h) >> 1, ver 90% f he induced charges are disriued wihin a disance f 2h frm he charged eam psiin in he direcin f he chamer widh. The ms efficien cnfigurain fr a fur-un BPM is have a un diameer f (2 2.5)h wih n un ffse frm he eam. The un sensiiviies in his case are maximized and have gd lineariy wih respec he eam psiins in he hriznal and verical direcins. The un sensiiviies and eam cefficiens are als calculaed fr he 8 mm and 5 mm chamers used in he inserin device sraigh secins f he 7 GeV Advanced Phn Surce. INTRODUCTION Circular un elecrdes are cmmnly used as eam psiin mnirs (BPMs) in a variey f paricle accelerars (1, 2). Fr highly relaivisic filamenary eams f elecrns r psirns, he Lrenz cnracin cmpresses he elecrmagneic field f he charged eam in he 2-D ransverse plane. This resuls in he induced currens n he eam chamer wall having he same lngiudinal inensiy mdulain as he charged eam. When he wavelengh f he eam inensiy mdulain is large cmpared he un diameers, he calculain f he induced charges n he uns may e simplified as a 2-D elecrsaic prlem. In he inserin device (ID) sraigh secins f he 7 GeV psirn srage ring fr he Advanced Phn Surce (APS), eam chamers 8 mm and 5 mm in heigh are used pimize he ID magneic parameers. In his paper he cnfigurain f fur-un BPMs in a small-gap eam chamer is pimized, and BPM sensiiviies and cefficiens are calculaed assuming ha he un elecrdes are flush wih he chamer wall. Wrk suppred y he U.S. Deparmen f Energy, Office f Basic Energy Sciences under Cnrac N. W-31-109-ENG-38.
FIGURE 1. Crss secin f a eam chamer wih a heigh f 2h and widh f 2w. The diameer f he fur un elecrdes fr he BPM sysem is (x 2 x 1 ), and = h + y, a = h y. IMAGE CHARGES Assuming ha he widh f he eam chamer in Figure 1 is much larger han he heigh (w >> h), he induced charges are calculaed y he mehd f image charge. The eam chamer is als assumed have a high elecric cnduciviy and is grunded. Then he verical psiins f he psiive and negaive image charges f a charge λ a (x, y ) are given y +λ a y = 2m(a+) + y = 4mh + y, (m = -, 0, ) and λ a y = 2a + 2m(a + ) + y = 2a + 4mh + y (fr ineger m, <m< ) (1) wih a = h y and = h+y. Fr ease f calculain he verical psiin fr ( λ) is shifed y 2h s ha y'= y 2h= 4mh - y. (In he 3-D gemery he charge λ is a linecharge densiy alng he z direcin.) Then he elecrsaic penial disriuin Φ(x,y) wihin he chamer may e calculaed frm λ z z ( z zm)( z z m) = = = 1 λ z zm m m= ln Φ ( xy, ) ln, (2) 2πε0 z' z' ( z' z' )( z' z' ) 2πε0 z' z' m m m= 1 m= m where ε 0 is he permiiviy in free space, z = x + i y, z' = x + i y', z m = x + i ( 4mh+y ), and z' m = x + i y' = x + i ( 4mh y ). Equain (2) may e simplified a clsed frm
z x iy sin ( ) Φ( xy, ) = π λ Re ln sin ( ' 4hi z x + iy 2πε0 π ) 4hi x x = y y cshπ cshπ λ ln 2h 2h 4πε x x 0 y y cshπ + cshπ 2h 2h (3) where y' is shifed ack y + 2h in he final expressin. The induced charge densiies per x/h in he p and m surfaces f he chamer, σ and σ, are calculaed frm [-ε dφ/dy] y= h σ σ λ cs py =, 4 csh px ( x) sin py λ cs py =. (4) 4 csh px ( x) + sin py Here p = π/2 and y seing h = 1 he crdinae sysem is nrmalized he half heigh f he chamer. By adding up he induced charges in he p and m surfaces in Equain (4), he al induced charge, which shuld e prprinal he sum signal fr a ypical fur-un BPM sysem f Figure 1, is given y x2 s s 2 s 1 x1 Q = Q ( x ) Q ( x ) = ( σ + σ ) dx+ ( σ + σ ) dx x1 x2 = λ 1 2 x2 x1 cs pycsh p( x x) { 2 2 csh px ( x) sin py cs pycsh p( x + x) + }. 2 2 csh px ( + x) sin py dx (5) The induced charges prprinal he signals fr he verical and hriznal psiins f he charged eam, and, may e calculaed frm Equain (4): x2 Q = Q ( x ) Q ( x ) = ( σ σ ) dx+ ( σ σ ) dx, (6) y y 2 y 1 x1 x1 x2 x2 Q = Q ( x ) Q ( x ) = ( σ + σ ) dx ( σ + σ ) dx. (7) x x 2 x 1 x1 x1 x2 Here and are he differences in he induced charges eween he p and m, and righ and lef uns, respecively. As ne expecs frm eam psiin measuremens, is an dd funcin in y and even in x, and is he ppsie. Afer Taylr expansins up he hird rder in he charged eam psiin (x, y ), indefinie inegrals f Equains (5 7) are given y
3 Qs ( x) 1 psinh px p px px an [sinh px] ( y x ) y x ( sinh 1 2 2 2 2 6sinh = + + ), 2 2 4 λ p 2csh px 4 csh px csh px (8) Qy ( x) = y [anh px+ x p λ sinh px ] + y [ p 3 csh px 2 2 3 2 sinh px 3 3csh px 2 4 2sinh px 2sinh px + x p ( )], 3 5 3csh px csh px (9) 2 Qx ( x) 2 2 1 3 3 p 3 = x[ sec h( px) + y p { sec h( px) sec h ( px)}] + x [ { sec h ( px) λ 2 6 2 2 4 5 5 3 1 sec h( px)} + y p { 2sec h ( px) sec h ( px) + sec h( px)}]. 3 12 (10) T he firs rder in y /h and x /h, calculains f he induced charges frm x 1 = 0 x 2 = in Equains (8 10) give = -λy /h, = -λx /h, and he al induced charge Q s = -λ as expeced. The derivaives f Q s (x), (x), and (x) wih respec x/h may e called he effecive induced charge densiies fr he sum, verical, and hriznal signals. The firs erms f he charge densiies and Equains (8 10) are pled in Figure 2. 1.6 1.4 1.2 1.0 sum ver hrz 0.8 0.6 0.4 0.2 0.0 0 0.5 1 1.5 2 2.5 3 (a) charge densiies x/h 1.0 0.8 0.6 0.4 0.2 sum ver hrz 0.0 0 0.5 1 1.5 2 2.5 3 () in duced charges x/h FIGURE 2. (a) Induced charge densiies and () induced charges inegraed frm 0 x/h. The induced charges and densiies crrespnding Q s,, and in Equain (8) are dened as sum, ver, and hrz in he legends, respecively, wih unis f -λ, -λy /h, and -λx /h.
FIGURE 3. 3-D pls f he induced charges fr Q s,, and as funcins f nrmalized un ffse x 1 /h and un diameer d/h n he lef side, and heir cnur pls n he righ side. The respecive unis fr Q s,, and are -λ, -λy /h, and -λx /h.
Fr small uns (e.g., x/h < 0.5), when he eam is lcaed near he rigin, he hriznal eam displacemen is n as sensiive changes in he disances eween he eam and he uns as he verical eam displacemen. This makes he densiy disriuin fr rad wih he peak near x/h = 0.6. The densiy fr, n he her hand, has is peak a x/h = 0. This implies ha, when he measuremens f verical displacemens are criical fr a eam chamer f small heigh, he lcain f he uns shuld include he range f small x/h. Fr uns lcaed in he range f x = 0-2h wih un diameer f 2h, fr example, ver 94%, 99%, and 91% f he availale sensiiviies fr sum, verical, and hriznal can e regisered n he uns. Therefre, any uns lcaed mre han 2h (ne chamer heigh) frm he eam psiin in he hriznal direcin wuld e very inefficien. Shwn in Figure 3 are 3-D pls and heir cnurs fr Q s,, and. The negaive psiin f x 1 is pssile y raing he fur-un sysem wih respec he verically symmerical axis. Fr x 1 = 0 and a un diameer d larger han 2h, i is seen ha Q s,, and d saurae as already expeced frm Figure 2. When he uns are exended h sides f he x-axis y raing he fur-un sysem and he diameer is larger han 4h, he values f Q s and increase y a facr f 2 ecause ms pars f he uns are sill lcaed wihin x/h < 2 where he charge densiies are high. On he her hand, decreases ecause he charge densiy fr is asymmeric wih respec x. Therefre, a fur-un sysem wih a un diameer f apprximaely (2~2.5)h and a un ffse f x 1 = 0 wuld cllec nearly all he induced charges and e he ms efficien. BUTTON SENSITIVITIES Wih x 1 = 0 and d = 2h, where he un diameer d is (x 2 x 1 )h, Q s,,, and heir nrmalized values Q s are calculaed frm Equains (5 7). The resuls give an pimized BPM cnfigurain and are pled in Figure 4 as funcins f he nrmalized eam psiin (y /h, x /h). The un sensiiviies and cefficiens fr y /h and x /h fr he pimized cnfigurain are calculaed frm Equains (8 10). Opimized cnfigurain: Q s = 0.945[1.0 + 0.07143 {(y /h) 2 - (x /h) 2 }+ 0.0842 (x /h) 2 (y /h) 2 ], = 0.9963[{1.0-0.01836 (x /h) 2 }(y /h) + {0.00612 + 0.0295 (x /h) 2 }(y /h) 3, = 0.9137[{1.0 + 1.4649 (y /h) 2 }(x /h) - {0.4883 + 2.7354 (y /h) 2 }(x /h) 3 ], = 1.0542[{1 + 0.0531 (x /h) 2 }(y /h) - {0.0653 + 0.0631 (x /h) 2 }(y /h) 3 ], = 0.9669[{1 + 1.3935 (y /h) 2 }(x /h) - {0.4169 + 2.6159 (y /h) 2 }(x /h) 3 ]. (11)
1.0 0.5 0.0-0.5 sum ver ver/sum (a) x = 0-1.0-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 y /h 1.0 0.5 0.0-0.5 sum hrz hrz/sum () y =0-1.0-1.2-0.8-0.4 0 0.4 0.8 1.2 x /h FIGURE 4. Fr he pimized BPM cnfigurain, nrmalized un psiins f x 1 /h = 0 and x 2 /h = 2 (nrmalized diameer d/h = 2), variains (a) Q s,, and are pled as a funcin f nrmalized verical eam psiin y /h fr x = 0, and () Q s,, and as a funcin f nrmalized hriznal eam psiin x /h fr y = 0. Here Q s,, and are dened as sum, verical, and hriznal and heir respecive unis are -λ, -λy /h, and -λx /h. Figure 4(a) and Equain (11) shw ha he verical signals and wihin 0.7y /h have excellen lineariy in y /h and x /h. This is paricularly impran since verical measuremens are generally criical in a small chamer heigh. The hriznal signals and, n he her hand, are less linear cmpared hse fr he verical as seen frm Figure 4() and he cefficiens f y /h and x /h in Equain (11). In he APS srage ring, eam chamers wih relaively small chamer heighs are used fr he IDs in he sraigh secins (3). Several fur-un BPMs wih un diameers f 4 mm and un-cener separains f 9.65 mm have een insalled fr chamer heighs f 8 mm (h = 4 mm, x 1 = 0.7075h, x 2 = 1.7075h, diameer = 1.0h) and 5 mm (h = 2.5 mm, x 1 = 1.132h, x 2 = 2.732h, diameer = 1.6h). One can see frm Figure 2 ha hese uns are lcaed a relaively inefficien psiins cmpared he pimized case f x 1 = 0 and x 2 = 2h. The un sensiiviies and y and x cefficiens fr he w chamers are:
APS ID chamer (2h = 8 mm): Q s = 0.3178[1.0 + 0.0529 {x 2 - y 2 }+ 0.00778 x 2 y 2 ], = 0.0465[{1.0 + 0.2199 x 2 } y + {-0.0733 + 0.00228 x 2 }y 3, = 0.1144[{1.0-0.00738 y 2 } x + {0.00246 + 0.00827 y 2 }x 3 ], = 0.1464 [{1 + 0.1669 x 2 } y + {- 0.2033 + 0.00502 x 2 } y 3 ], = 0.360 [{1 + 0.0456 y 2 } x + {-0.5049 + 0.00229 y 2 } x 3 ]. (12) The smalles aperure APS chamer (2h = 5 mm): Q s = 0.1957[1.0 + 0.1817 {x 2 - y 2 } - 0.0104 x 2 y 2 ], = 0.02205[{1.0 + 0.7246 x 2 } y - {0.2415 + 0.1285 x 2 }y 3, = 0.1205[{1.0-0.1509 y 2 } x + {0.0503 + 0.0136 y 2 }x 3 ], = 0.1127 [{1 + 0.5429 x 2 } y - {0.0598 + 0.00858 x 2 } y 3 ], = 0.6156 [{1 + 0.0308 y 2 } x - {0.1314 + 0.0146 y 2 } x 3 ]. (13) As seen frm Equains (12) and (13), he ms criical signals fr 8 mm and 5 mm chamers are nly 0.046 and 0.022 f he uni -λy /h. Cmpared, he hriznal signals are ver 0.11 f he uni -λx /h fr h chamers. Even if he nrmalized signals are n small (ecause f he small values f Q s ), ne shuld expec ha he nise/signal rais fr and in Equains (12) and (13) are relaively large cmpared hse in Equain (11). ACKNOWLEDGMENTS The auhr wuld like hank Glenn Decker fr his numerus suggesins and useful discussins cncerning his wrk. REFERENCES [1] Shafer, R. E., Beam Psiin Mniring, presened a he Firs Accelerar Insrumenain Wrkshp, Upn, NY, Oc. 23 26, 1989, AIP Cnference Prceedings 212, 26-58 (1989). [2] Barry, W. C., Brad-Band Characerisics f Circular Bun Pickups, Prceedings f he Furh Accelerar Insrumenain Wrkshp, Berkley, CA, Oc. 27-30, 1992, AIP Cnference Prceedings, 281, 175 184 (1993). [3] Lumpkin, A. H., Cmmissining Resuls f he APS Srage Ring Diagnsics Sysem, Prceedings f he Sevenh Accelerar Insrumenain Wrkshp, Argnne, IL, May 6-9, 1996, AIP Cnference Prceedings, 390, 152 172 (1997).