Drift Distortions in Alice TPC Field Cage

Transcription

1 AICE / 97- Iteral Note / TPC Drift Distortios i Alice TPC Field Cage D. Vraic Abstract Calculatios of drift distortios for the TPC field cage with two coaxial cyliders with closed eds are preseted. The calculatio ethod is based o the solutios of aplace s equatio i cylidrical coordiates. It is applied to the Alice TPC to study a variety of possible defects i desig ad costructio of the field cage. Distortios are calculated due to fiite width of the equipotetial strips, errors i resistor chai values, orderig of the resistors, shorted strips, isatch of the field cage cylider groud ed voltage with the pad plae, deforatios of the pad plae, etc. The results show that the precisio i echaical costructio has to be better tha.5. The fiite width of the equipotetial strips is ot a proble ad a proper orderig of resistors i the resistor chai ca sigificatly reduce errors due to the tolerace of the resistace. A shorted strip pair has large distortio ad it would be ecessary to be repaired i order to avoid large correctio.

2 . Itroductio If we assue that the distortio is relatively sall, so that the error i axial electrostatic field copoet (drift directio) ca be igored, the the deviatio i the electro path i the radial directio is equal to the followig itegral (Radial Distortio Itegral, RDI) [],[]: ρ( ρ, ) Eρ ( ρ, ) d E () where ρ, are radial ad aiuthal coordiates of the electro creatio poit. It is assued that E cost so that for evaluatio of the itegral we eed to solve aplace s equatio usig just the error potetial. For the oral case with a agetic field parallel to the axis, the agitude of the distortio is the sae ad its directio is a fuctio of the ωτ (drift gas ixture).. aplace s Equatio i Cylidrical Coordiates I cylidrical coordiates aplace s equatio takes the for [3]: The separatio of variables Φ Φ Φ Φ ρ ρ ρ ρ φ leads to the three ordiary differetial equatios: Φ( ρφ,, ) R( ρ) Q( φ) Z( ) d Q Q dφ (a) d Z d Z k Z or + k Z d d (b) d R dr d R dr + + k R or + k + R d ρ ρ d ρ ρ d ρ ρ d ρ ρ (c) The choice of the sig for the separatio costat k depeds o the boudary coditios for a give proble. The solutios of the first two equatios are eleetary: Q( φ) Ccos( φ) + C si( φ) Z ( ) C3exp( k) + C4 exp( k) or Z () Ccos( k) + Csi( k) (3a) 3 4 (3b) I order that the potetial be sigle valued, ust be iteger. For the radial equatio the solutios are Bessel fuctios or odified Bessel fuctios: R( ρ) C5J ( kρ) + C6 Y ( kρ) or R( ρ) C I ( kρ) + C K ( kρ) For coaxial cyliders with rotatioally syetric boudary coditios : Q( φ) Cφ + C ( C 5 6 (3c) ) (4a)

3 Z ( ) C3exp( k) + C4 exp( k) or Z () Ccos( k) + Csi( k) R( ρ) C5J( kρ) + C6Y ( kρ) or R( ρ) C I ( kρ) + C K ( kρ) For k we have eleetary solutios for all three equatios: 3 4 (4b) 5 6 (4c) Q( φ) Ccos( φ) + C si( φ) or for Q( φ) Cφ + C ( C Z () C+ C 3 4 R( ρ) C5ρ + C6 ρ or for R( ρ) C l( ρ) + C If there is o φ depedece i the proble we have the siple solutio: Φ( ρ, ) ( C + C )( C l( ρ) + C ) ) (5a) (5b) 5 6 (5c) (6) 3. Radial Distortio forula Sice the error potetial vaishes at the ed caps of the field cage, the sie fuctio is the appropriate choice for Fourier expasio with additioal coditio for the separatio costat k : k π (7) is the legth of the field cage, ad is iteger fro to ifiity. Fro equatios (4) follows expasio for the potetial: [ ( ) ( )] ( ) Φ( ρ, ) A I k ρ + B K k ρ si k The coefficiets A ad B ca be evaluated usig the potetials o the boudaries: Φ( ρ, )si ( k ) d A I( kρ) + B K( kρ) (9a) Φ( ρ, )si ( k ) d A I( kρ) + B K( kρ) (9b) where ρ ad ρ are ier ad outer radius of the field cage cyliders. Fro (8) ad usig the derivative relatios: d dx I x I x ( ) ( ) d dx K x K x ( ) ( ) follows the expressio for the radial copoet of the electric field: (8) E ρ [ ( ) ( )] ( ) d ( ρ, ) Φ( ρ, ) k A I kρ B K kρ si k dρ () V Fro RDI (), E, ad the itegral of () we get as fial result for the radial distortios: where: [ ( ) ( )]( ) ρ( ρ, ) a I k ρ b K k ρ cos( k ) () a A [ V D S K k S K k ( ρ) ( ρ )] (a) 3

4 b B [ V D S I k S I k ( ρ) ( ρ )] (b) D I ( k ρ ) K ( k ρ ) I ( k ρ ) K ( k ρ ) (3) S S ( k) d V Φ( ρ, )si ( k) d V Φ( ρ, )si (4a) (4b) Sice potetials ad fields are additive, to evaluate RDI itegral () we ca use just the error potetial o the boudary. Fro Eq. (4) we see that the distortio depeds oly o oralied error potetial (error potetial divided by drift voltage V ). The uber of ters required i the su () depeds o the for of the error potetial o the boudary. For each particular case, oe ca use the expressio for the potetial [ ( ) ( )] ( ) Φ( ρ, ) V a I k ρ + b K k ρ si k (5) to check the uber of ters required for a reasoable reproductio of the error potetial. Itegrals i the relatios (4) ca be evaluated usig the fast Fourier trasfor. That reduces sigificatly the coputig tie i the case whe it is ecessary to su few hudreds of ters. Ordiary Bessel fuctios ca be see as distorted sie ad cosie fuctios. Modified Bessel fuctios are just Bessel fuctios of pure iagiary arguet ad ca be see as distorted expoetial fuctios [4]. For practical purposes it is useful to itroduce scaled odified Bessel fuctios: SI ( x) I ( x) e x SI ( x) I ( x) e x SK ( x) K ( x) e x (6a) SK ( x) K ( x) e x (6b) Except for sall x, they are slow varyig fuctios (see Fig.). For large x the oly depedece is i the expoetial part. The liitig fors for large x are: SI ( x ) π x SK ( x ) π x (7) Usig scaled odified fuctios we get for the coefficiets (): a b kρ kρ [ SSK( kρ) e SSK( kρ) e ] D kρ kρ [ SSI( kρ) e SSI( kρ) e ] D k D SI ( k ρ ) SK ( k ρ ) e SI ( k ρ ) SK ( k ρ ) e ( ρ ρ) k( ρ ρ) 4

5 k( ρ ρ) k( ρ ρ) ai ( kρ) [ SSK( kρ) SI( kρ) e SSK( kρ) SI( kρ) e ] D k( ρ ρ) k( ρ ρ ) bk ( kρ) [ SSI( kρ) SK( kρ) e SSI( kρ) SK( kρ) e ] D (8a) (8b) I I K K I, I, K, K x Figure Scaled odified Bessel fuctios 4. Distortios fro Fiite Width Equipotetial Strips For the calculatio of this distortio we assue that the gap betwee strips is ifiitely sall. There are two good reasos for that approxiatio. First, distortio for fiite gap sie will be saller, ad secod reaso is that i that case we ca use exact aalytical expressios for field lies (assuig large radii of the field cage). The error potetial due to fiite width of the strips is equal ero at the ceter of the gap ad at the ceter of the strip (see Fig ). It icreases liearly with the distace fro the gap. The Fourier trasfors (6) for this saw teeth for ca be evaluated aalytically, S S π where N s, 4N s,... ( N s is uber of strips over full legth of the field cage ) I this case we have a error potetial with high spatial frequecy ad large arguets for the odified Bessel fuctios so that we ca use the asyptotic expressios (7). The relatio () gets the for 5

6 ρ π ρ π π ρ( ρ, ) w exp ( ρ ρ ) exp ( ρ ρ) cos( ) π ρ w ρ w w (9) where ρ, ρ are ier ad outer cylider radii ad w is strip width w ( is the legth of the field cage) N s Fro Eq. (9) it follows that the distortio decreases expoetially with the distace fro the cage walls axial distortio occurs for the electros produced at w (strip ceter) iial distortio occurs for the electros produced at w (gap ceter) the distortio is such that the electros are pushed away fro the cage walls Fro Eq. (9) follows siple expressio for distortio close to the cage wall (we take oly the first ter i suatio): w distortio π exp( π w d ) () where d is distace fro the wall. For dw distortio. -3 w, ad for a typical strip width of c we have distortio of oly 4µ! I order to test this results we ca use the exact for of the field for the ero gap ad ifiite radius derived fro coforal appig u+ iv l x+ iy si c () This is a very well kow appig used for the calculatio of the field ad equipotetial lies for the wire grid of the MWPC s. I our case we have the copleetary proble so that u ad v have iterchaged their roles ad for the field lies we get the relatio (w): u e y acosh + cos π x () π 4 Figure Field ad equipotetial lies for field cage strips (drift i horiotal directio) 6

7 STRIP Figure 3 Field lies aroud the gap STRIP Fig. shows the part the field cage field lies for a ifiite sall gap (width w ad height w). For a gap of the fiite sie circular field lies aroud the gap are defored ito ellipses as show i Fig. 3. If we assue that driftig electros follow the field lies, tha fro relatio () it follows that axial distortio at the distace of strip width equals to distortio w i full agreeet with previous results. Fig. 4. shows axial distortio as a fuctio of the distace fro the cage wall. As oe ca see, cotributios of higher ters i the relatio (9) are egligible for distaces greater tha w/3. I Fig.. we ca see that for distaces less tha w/3 the field lies start to ed up i the strips which eas that the electros produced i that regio are collected o the strips. Relatio () gives.8 w for that turig poit. 5 TERMS FIRST TERM SECOND TERM EXACT DISTORTION - DISTORTION / W - Fig 4 Maxial distortio as a fuctio of the distace fro the cage wall DISTANCE / W Because the driftig electros follow the field lies, oe ca expect distorsios i coordiate close to the field cage walls. If we assue that the drift velocity does ot chage (field itesity costat), the the distortio is equal to the differece betwee path legth for electros wigglig alog the field lies ad the coordiate of the productio poit. Electros driftig over the full drift legth will have axial distortio. Fig. 5 shows results of the caculatios usig first ter i equatio (9) ad exact field lies (). The path legth differece is less tha µ at the distace fro the cage wall equal to the strip width. 7

8 CACUATED EXACT DISTORTION (c) - - Figure 5. Drift legth distortio (i c) as a fuctio of the distace fro the field cage wall for total drift legth of 5c ad strip width of.c DISTANCE / W 5. Radial Distortio fro Resistor Errors i the Resistor Chai For this calculatio the resistors have the uifor rado values betwee.99 ad. egaohs (.5% tolerace). The diesios of the filed cage are take to be close to the values for the Alice TPC: 5 c legth of the field cage cylider (Max drift legth) ρ 5 c outer cylider radius ρ 8 c ier cylider radius N s 8 uber of strips (strip width w. c ) Results fro the previous chapter show that the local distortios of the drift field decrease expoetially as a fuctio of the distace fro the field cage wall. Therefore, we ca expect that the tolerace i resistor values has uch less ifluece i the agitude of the distortio tha sortig of the resistors. The error potetial was calculated usig a liear iterpolatio of the discrete values give by φ k k Ri i N s i R i k N s where R i are radoly produced resistor values ad R. Figures 6. to 9. show error potetials (dots) ad calculated error potetials (solid lies) for four differet resistors orderigs. 8

9 .8.8 ERROR POTENTIA ERROR POTENTIA (c) (c) Fig. 6. Error potetial for radoly sorted resistors Fig. 7. Error potetial for the resistors sorted as copleetary pairs 4 ERROR POTENTIA ERROR POTENTIA (c) (c) Fig. 8. Error potetial for the resistors sorted i Fig. 9. Error potetial for resistors sorted i ascedig order descedig order For Figures 6, 8, ad 9 the calculatio was doe with 5 ters i the expasio, ad for Figure 7 with 45 ters (usig fast Fourier trasfor) i order to reproduce the fiest details of the error potetial. Factor 4 was used for calculatio purposes. Oe ca see that for particularly bad sortig of the resistors i ascedig or descedig order produces a relative error i the potetial of the order of - 3. But, if oe sorts resistors as copleetary pairs (resistor with iial resistace, resistor with axial resistace, etc...) the relative error i the potetial ca be reduced ore tha te ties. Calculatio of the radial distortios caused by differet sortig of the resistors are doe for three radial positios: 45c (5c away fro the outer cage wall), 5c ad 5c. Results are show i Figures to 3. As expected, the orderig of the resistors has a sigificat effect o the distortio. Sortig resistors as copleetary pairs allows to use.5% tolerace resistors resultig i radial distortios of oly less tha 5µ. Next iteratio (orderig copleetary pairs) would iprove that value, but ot sigificatly. 9

10 .4.3 ρ45c ρ5c ρ5c.4.3 ρ45c ρ5c ρ5c.. DISTORTION (c).. -. DISTORTION (c) DRIFT ENGTH (c) DRIFT ENGTH (c) Fig.. Radial distortio as a fuctio of drift distace for radoly sorted resistors Fig.. Radial distortio as a fuctio of drift distace for resistors sorted as copleetary pairs ρ45c ρ5c ρ5c.4 -. DISTORTION (c).3.. DISTORTION (c) ρ45c ρ5c ρ5c DRIFT ENGTH (c) Fig.. Radial distortio as a fuctio of drift distace for resistors sorted i ascedig order DRIFT ENGTH (c) Fig. 3. Radial distortio as a fuctio of drift distace for resistors sorted i descedig order

11 6. Radial Distortio Caused by Shorted Strips For this serious error i the field cage we ca expect large distortios because it would ea a global chage of the drift potetial. Here we cosider a coditio where two stripes of the outer field cage wall are coected together. Igorig the error potetial due to the fiite sie of the strips, we have for the error potetial alog the outer field cage wall: φ( ) φ( ) N s N s N s for < s for > s where s represets the locatio of the shorted strips. Substitutig this expressios i the equatios (4) gives s cos π S π ( N ) s S Resultig distortio for the full drift volue is show i Fig. 4. Fig. 4. Distortio resultig fro the shorted strips o the outside field cage wall at 5c The sig of the distortio is reversed. As oe ca see, this is a rather catastrophic sceario ad oe has to repair the coected strips, or to deal with the huge distortio correctios (up to ore tha c!).

12 7. Distortio Caused by the Edcap Misatch The icorrect atchig of the ed cap potetial with the field cage cylider ca be caused by echaical isatch or by wrog potetial. The followig calculatio was doe for the case whe δv -4 correspodig to V for kv drift voltage, or echaical displaceet of 5µ. The for the error potetial o the ier ad outer cylider we have expressio: ad fro (4) follows φ() δv V S S δ π Resultig distortios for differet radial distaces iside the field cage are give i Fig. 5. ρ85c.4 DISTORTION (c).. -. ρ5c ρ5c ρ45c ρ65c ρ85c ρ5c ρ5c -.4 ρ45c DRIFT ENGTH (c) Fig. 5. Distortios for differet radial distaces resultig fro a voltage isatch of -4 at the groud plae. The effective positio of the groud plae depeds of the gatig grid offset voltage ad it ca be tued to get iial distortio, but if the ed plae is ot parallel to the equipoteital strips withi -4 (correspodig to 5µ over 5c) we ca expect to have a siilar agitude of the distortio.

13 8. Distortio Caused by the Edcap Deforatio The exact solutio of aplace equatio oe ca get oly for few siple boudary coditios. Oe of the is for two coaxial cyliders with liear icrease of the potetial alog the directio ad oe ed side plae (perpedicular to the axis) o a costat potetial (as it is the case of the TPC field cage). The secod ed ca be with differet legth for ier ad outer cylider. Fro the equatio (6) oe gets Φ( ρ, ) D l( ρ) + D where D V V l ρ ρ D V V l( ρ ) l( ρ ) ρ l ρ V ad V are voltages applied o the secod eds of the cyliders ad, are legths of the ier ad outer cyliders respectively. For V V ad we have D ad potetial equal, as oe would expect, Φ( ρ, ) V idepedet of ρ. For V V or (or both) we have radial ad axial copoets of the field equal to: Φ E D ρ ( ρ, ) E ρ ρ ( ρ, ) Φ D D l( ρ ) Isertig this expressios ito RDI () it is possible to calculate radial distortios i the TPC field cage due to defored ed cap, but the deforatio has to have a exactly defied shape i order to satisfy the boudary coditios. As a exaple, let V V ad 5c, 5.5c. The for the deforatio of the ed cap (which is at the sae tie equipotetial surface) we get the shape preseted i Fig DEFORMATION() RADIA DISTANCE (c) Fig 6. Deforatio of the ed cap for 5µ differece i the sie of the ier (ρ8c) ad outer (ρ5c) cylider Resultig radial distortio for several radial distaces is give i Fig 7. 3

14 .4 ρ85c.3 DISTORTION (c).. ρ5c ρ5c ρ45c ρ65c ρ85c ρ5c ρ5c ρ45c DRIFT ENGTH (c) Fig 7 The radial distortio is show for defored ed cap (HV plae) as a fuctio of the drift distace The aplitude of the distortio is as expected fro the results of the previous chapter. 9. Coclusio Siple geoetry of the cylidrical TPC field cage allows to calculate electro drift distortios due to iperfectios usig exact solutios of the aplace equatio, or usig Fourier expasios for approxiate solutios. Where it was possible to copare, both ethods give very siilar results. The results of the calculatio show that axial distortios due to echaical iperfectios are rughly factor two bigger tha the errors i the costructio of the field cage. The local deforatios i the drift field decrease expoetially with the distace (see Eq. ()) ad are therefore egligible. Special care has to be take for the aliget of the ed caps (groud plae ad HV plae) because they itroduce global deforatio of the drift field. The tolerace o resistor precisio ca itroduce sigifficat distortios ad special care has to be take to order resistors i the right way. If resistors are ordered as copleetary pairs i respect to their values, the distortio ca be reduced by factor to coparig to the worst cases of orderig i descedig or ascedig order. With proper orderig of the resistors, the distortio ca be reduced to less tha µ for resistors with.5% tolerace. Refereces [] H. Wiea, Calculatio of Drift Distortio Due to Iperfectios i a TPC Cylidrical Field Cage, STAR Note #53, Aug. 96. [] W. Blu ad.roladi, Particle detectio with Drift Chabers, Spriger-Verlag, 993. [3] J. D. Jackso, Cassical Electrodyaics, Joh Wiley & Sos, Ic. 96,975. [4] M. Abraowit, I. A. Stegu, Hadbook of Matheatical Fuctios, Dower Publicatios, Ic., 97. 4