Optimum geometry for strip cathodes or grids in MWPC for avalanche localization along the anode wires

NUCLEAR

INSTRUMENTS

AND

METHODS

163 ( 1 9 7 9 )

83-92;

Q

NORTH-HOLLAND

PUBLISHING

CO.

OPTIMUM GEOMETRY FOR STRIP CATHODES OR GRIDS IN MWPC FOR AVALANCHE LOCALIZATION ALONG THE ANODE WIRES* E. GATTI, A. LONGON1

Istituto di Fisica del Pofitecnico, Milan, Italy H. OKUNO t

Brookhaven National Laboratory, Upton, New York, U.S.A. and P. SEMENZA

Politecnico, Milan, Italy

Optimum geometry for strip cathodes or grids in MWPC for avalanche localization along the anode wires through measurement of induced pulses is calculated. Resolution limits due to electronic noise are established as a function of the required time resolution. It is found that the optimum ratio of sensing electrodes width to cathode plane-anode plane distance is about 1. Maximum resolution requires, if simple centroid methods are used, to estimate the avalanche position with a small number of cathode signals: the arising non-linearity problems are discussed.

1, Introduction The purpose of this paper is to find the theoretical limits due to amplifiers noise and geometry choice in determining the position of an avalanche in a MWPC along the anode wires by the method of the induced pulses on an array of perpendicular adjacent cathode stfips~-3), or sets of sensing grids 4) connected to charge sensitive amplifiers. The considered geometry is shown in fig. I: the lower array of electrodes could be the array of segmented cathodes or the groups of sensing grids 4) suitably connected. In this figure the relevant parameters a, D and s of the geometry are defined.

2. Induced pulses and electronic noise In order to solve the problem we have first to determine the amplitude of the induced pulses with the given geometry and the associated electronic noise with a given time resolution and avalanche size. Let us consider a) the effective radial electric field near an anode Eel(r) = Eeer(R) R/r, (1) Research supported in part by U.S. Department of Energy Contract no. EY-76-C-02-0016 and in part by Istituto Nazionale di Fisica Nucleare (Italy). On leave from Institute for Nuclear Study, Univ. of Tokyo, Tokyo, Japan.

f

Fig. 1, MWPC with readout cathode strips.

where R is the anode wire radius, and b) the radial electric field which would exist near an anode wire at the normalized coordinate 2 = x/D of the avalanche position, due to the unit voltage applied to the considered strip while all other electrodes would be kept at zero voltage

Ew(r, 2) = Ew(n, 2) R/r.

(2)

Following the Wilkinson treatment 5) we obtain, for the voltage pulse on the considered strip, the expression:

V(z) = ~CCREw(R' 2) In (1 +v),

(3)

q being the avalanche size, C the capacitance of the readout electrode and r a normalized time

84

E. GATTI et al. (4)

z = t/[R/211Eeer(R)] = t/To,

where/1 is the ion mobility and To is the time for the ions to travel in a uniform field Eofr(R) the path R/2. Shaping the waveform of eq. (3) by a rectangular bipolar filter of total width Am (as shown in fig. 2), we get the output waveforms shown in the same figure as well as the peak values M for different ;!.,1. The output peak voltage can be written

account azimuthal asymmetry of the effective fields and of the weighting functions, can be exploited for localizing the avalanche in the direction z orthogonal to the anodesT): this problem is not dealt with in this paper. Let us calculate the factors appearing in eq. (5). In appendix A RE,~(R, 2) is calculated: the result can be expressed by the semiempirical expression REw

Correspondingly, considering only series white noise introduced by the input amplifiers we get the noise contribution 2

~2 =

en

-

K 1 1 22K4 K

3 ×

x (arctg {x/K3 tanh[K2 (2 + D)I} - arctg {x/K3 tanh IK2 (2 - D)I} ) ,

(7)

(6)

22m To' where e~ is the unilateral noise voltage density VV Hz. e 2 takes into account not only the considered amplifier noise but also the contribution of the noise injected by the contiguous electrodes connected to amplifiers in the charge amplification configuration. The correlation between signals at different amplifier outputs, due to this noise contribution, is neglected. We note explicitly that, for the geometry considered, any azimuthal asymmetry of the avalanche 4,6) does not produce changes in the ratio of pulse amplitudes induced at the cathodes. Azimuthal asymmetry of the avalanche, taking into 18

( R, a1)' 2)

(5)

e = ~cc REwM.

M=1.5924

2_~_cREw

14

,-

I~- / A ~ '......~)-RESPONSE, 1.1~~/ \! ~

where K1, K2, K3, K4 are given in table 1 and are calculated for the fixed value of the parameter: D/R = 666 (for instance D = 1 cm, R = 15/~m). The total impulse induced on all cathodes can be evaluated by calculating REw(2)in the limit for a/D-+oo and is, as expected, independent of the position of the avalanche. We get REw~ =

K2 K1 x/K3 arctg x/K3,

(8)

and recalling eq. (5), we have:

q ~, v o = v o ~ = 2 c ,=_o

(9)

RE,,~ M.

REw~ is plotted against b = s / D in fig. 3. The decrease of induced charge on the cathodes or grids, as anodes are closer, is in agreement with results in ref. 4. REw(2) is plotted as given by eq. (7) for a-,O (that is for a cathode strip of width 2da) and for several values of anode wire spacing b = s/D in fig. 4. (For details of calculation see appendix A.) It is worth noticing that the spatial distribution of induced charge gets narrower as the anode grid spacing is reduced.

8 TABLE

2

0

1

Values of K1,

6

,

i

i

~

i

,

i

~

s

T-

t

a 40 eo t eo ~60 2ao Fig. 2. Waveforms at the output of filters, whose response is shown in the figure, when the input pulse is In (1 +t/To).

K2

and

K3.

b = s/D

K1

K2

0 0.2 0.5 1.0

0.002355 0.03537 0.05650 0.06806 0.07413

½n 1.299783 1.0807 0.94096 I~

K3

0 0.168587 0.414901 0.687805 1.0

AVALANCHE

I

--

I

LOCALIZATION

85

I

REw,~

pF

6,11]z

100

Do

4-11]z

60

g

/

o

o

~

2 • 102

2C

I

Oi

0

L

i.

.5

1

1

b: ! D

distance). 1

I-

I

- I

-

.o

\

\

,

0

\ 2

1

1

I

.l,

.2

.6

15

Fig. 3. P ~ w ~ (ratio of the charge induced on all cathodes to the avalanche charge when r = 6.389 T o) as a function ofs/D: anode wire spacing s normalized to D (anode plane-- cathode plane

70-~~

l

O.

a_ 0o

I

Fig. 5. Capacitance per unit length of a cathode strip for the geometry shown in the figure.

where if(or, ~0) is the Legendre standard form of the elliptic integral of the first kind. C is plotted in fig. 5. D O is 2D if the anode wires are neglected or D if they are considered as a conducting plane: large or small s/D, respectively. The other conducting plane in the geometry of fig. 5 can simulate the container of the MWPC. For convenience in estimating the electrode capacitance of other geometries, the capacitance of a strip with respect to an indefinite plane separated by a gap g is calculated in appendix B and plotted in fig. 6. 3. Evaluation of avalanche position The output peak voltage V, as given by eq. (5), is now a known function of the geometry and of the avalanche position 2.. The nth cathode strip provides an estimate of ~.

\\

3 D

Fig. 4. REwl(dalD) number proportional to induced charge on a cathode o f infinitesimal width 2 da as a function o f ~. = x/D: avalanche position x normalized to D (anode plane-cathode

L

.B

plane

pF

_ ~,g~_

~a _

T

d~stance).

Let us now consider the capacitance C appearing in eq. (5) for the geometry of fig. 5. C is given, in F/m, by (appendix B) C = 4%

K(k)/K'(k),

(10)

w h e r e K is t h e c o m p l e t e elliptic integral a n d k is from the two equations

calculated

fi

iI L

_F ½n, arcsin

-

Do2'

_E(½n, arcsink-~) = (~o + ~-~o)2,

(11)

(12)

01

o

I

.1

I

.2

I

.3

I

.4 __0 a

Fig. 6. Capacitance for a unit length of a cathode strip having a given gap g with respect to an infinite electrode plane as shown

in the figure.

86

E. G A T T I

with a variance

¢z

2

[d V ()L-- 2 n a/D)/d2] 2,

(13)

provided that the total avalanche size is assumed known by an independent measurement of negligible error. Weighting each ;L estimate in the optimum way, i.e. with a weight inversely proportional to the variance, we get an overall estimate, exploiting the information given by all cathodes, having a variance 1 ~2

~=

+~ Z

n=

--o0

f=

~

~,n

+~

[ d V ( 2 - 2 n alD)/d2] 2

Z

et al.

increases; however the advantage is only appearent, as far as the resolution is concerned, because, decreasing the anode spacing, V decreases as already shown in fig. 3. In fig. 8 e~, as given by eq. (14), is plotted taking into account also the factor ~2/V2~, (17) (a part from a multiplicative constant) in order to study the influence of the capacitance change versus a/D. In this case the accuracy in the avalanche localization shows a maximum versus a/D also for an avalanche position in front of the interelectrode gap. The value of a/D which gives the maximum is shifted toward the lower side.

ZI~ --o0

,,e

~2

V2

(14)

to

f~=L----

Vfl~ ~, [dV(2_2na/D)/d2]2 n =

-oo

.8

where

~b:a2

d V ( 2 - 2 n a / D ) _ ?_~M_~ x d2

.6

x l+K3tanh2{KEi2_(2n_l)_~l }

.4

-l+Katanh2{K2[2,(2n+l)Dl} J.

(15)

The ratio

+ ~

°o 1.8

V~z

,

a

•2

¢2/V ~Z _ ,=~-~ [ d V ( 2 - 2 n a/D)/d2] 2 e~z

a

.4

.6

.8

1.8

1.2

1.4

v=~

(16) .8

deduced by eq. (14) has been plotted for different anode spacings corresponding to b = 0.2, 0.5, 1 in figs. 7a, b, c. In each of the figures the three curves shown refer to a position of the avalanche in front of the center of the cathode 0. = 0) or in front of the gap between two adjacent cathodes (2 = a/D) or the intermediate case 2 = ½a/D. An optimum width shows up for localizing the avalanche near 2 = 0 while no maximum shows up for the region around 2 = aiD. The value of the maximum increases when anode wires become closer (small b) because V(2), referred to a given electrode, decreases its width; the form factor [eq. (16)]

D

.8 b:O.5

.4

2

~

,

.2

.4

.6

.8

LO

. 1.2

1.4

AVALANCHE LOCALIZATION

87

./,,z

1.0

.B

A = -a.6

b 1 .4

~.~ =0 .2

6--

0

o

.2

,4

.6

.8

Lo

~.2

t.4

Fig. 7. Figure of merit for avalanche localization as a function of a/D: semielectrode width to anode plane-cathode plane distance ratio. The three curves refer to the avalanche facing one cathode center it = 0 or the gap between two cathodes ). = a/D or the intermediate situation it = ~a/D. Anode spacings s = 0.2D, 0.5D, 1D. Change in capacitance of cathode vs a/D is not taken into account.

5

~/,

.3

.4

.6

.5

.7

.8

.b

i --"~ o

Fig. 8. Resolution, in arbitrary units, for anode spacing s = 0.5 D, stray capacitance of 10pF, for a MWPC with 0.5m long cathodes, as a function of normalized cathode width a/D. The two curves refer to position of avalanche at 2 = 0, or ). = a/D, i.e. facing intercathode gap. and ~2

e~b =

N

N

~

(2.--2b)2.

(21)

( -N+I

N o w we have the m i n i m u m variance achievable with a given geometry. Let us c o m p a r e it with s o m e practical i m p l e m e n t a t i o n s o f the measurement. 4.

Practical

Eq. (18), for 2 b = 0, gives a2

s~b-

N

~

(Z V.f

while eq. (21), for '~-b = a/D (avalanche at the interc a t h o d e gap) gives

+N

Z

--

(22)

-N

implementations

T h e variance o f the estimator

~'b

(2n) z, -N

aS

(18)

- N +N

e~b-

(

Ev.

02

N

v./

( 2 n - 1 ) 2.

(23)

-N+I

-N+I

-N

o f the position o f the avalanche t h r o u g h the centroid o f the c a t h o d e pulses at 2 N + 1 c a t h o d e s is z /~-b

~2 --

+N

+N

E ( 2 , - 2 0 2,

(19)

(Z V.f-N -N

w h e r e A, = 2 na/D is the center coordinate o f the cathode. For an e v e n n u m b e r 2N o f electrodes exploited, we have N

2b =

-s

E

+i N

Z v. -N+I

(20)

But we need the variance o f 2 and not o f the estimator 2b, therefore to obtain e l , we have to multiply eqs. (21) and (22) by (dgt/d,~02. This factor can be calculated f r o m eqs. (18) or (20) where the expressions o f V, as a function o f ;t are inserted. Figs. 9a and 9b s h o w the resolution e~/(~2/ V=:) w h e n ;t = 0 and ;t = a/D respectively as a function o f the n u m b e r o f cathodes considered. T h e centroid calculated with a suitably low n u m b e r o f cathodes achieves a resolution very near the maxim u m one, provided that an odd or e v e n n u m b e r o f c a t h o d e s is c h o s e n according to the avalanche position ;~. Fig. 10 s h o w s for a/D = 0.6 the ideal resolution versus 2 c o m p a r e d with the one achievable with a c o n v e n i e n t choice o f two or three electrodes.

88

E. G A T T I

et al.

, v~

T

O P T I M ~

.f,

J-~O b--~15

~ =06 b=0.5

.4

.2

.3

.I

8

.2

.4

.6

.8

1.0

1,2

A

1.4 8

~*

.8

,2

.4

.6

.8

I.e

D

Fig. 10. Resolution of avalanche position estimation by centroid of signals of two or three cathodes compared with ideal resolution as a function of avalanche position 2/(a/D).

.6

,S

/

b=0.5 aD :0.6 b ;05

OPTIMUM .4

b

3

\

.3 .2 .I

4

2

/5 e e

.2

.4

.6

.8

Le

' L2

a ' ~1.4

Fig. 9. Resolution of avalanche position estimation by centroid of the number of cathode signals shown in the curves as parameter. T h e o p t i m u m resolution is shown for comparison. Position of the avalanche at 2 = 0 in fig. 9a and 2 = a / D in fig. 9b.

Fig. 11 shows the ratio of the signal amplitudes at the cathode having the larger signal and the amplitudes at the contiguous ones. Measuring this ratio 6, one can select for 6 <2.5 the centroid measurement based on two cathodes centroid and for 6>2.5 the one based on three cathodes centroid. However, the centroid calculated with a small number of cathodes introduces systematic errors. Figs. 12a, b,c show these errors on a logaritmic

I

J

.2

.4

16

.e

1.1~

Fig. 11. Ratio (5 of the signal amplitudes for two neighbouring electrodes as a function of the avalanche position A/(a/D).

scale versus a/D for different numbers of electrodes used and different positions of the avalanche: for 2 = 0.25 a/D, 0.5 a/D and 0.75 a/D, respectively. If corrections of these systematic errors have to be taken into account, for instance with a digital table on a ROM, it depends on the achievable resolution which in turn depends on the chamber size and on the time resolution required.

AVALANCHE

89

LOCALIZATION

5. Numerical example for a large M3VPC,

50 × 50 cm~ Let us evaluate, for a particular case, the expected resolution for a M W P C cathode readout for the following geometry and operating conditions D = I cm, R = 1 5 # m , s = 5 m m (b=0.5), 0 . 5 m of cathode length, avalanche size 2 × 10 6 ion pairs. From fig. 7b it seems reasonable to choose a / D = 0.6 for the cathode width ( 2 a = 1.2 cm): this choice leads to ~2/V~ - 0.22,

b=(15

(24)

(for the worst ca~c: avalanche position in front of one cathode center 2 = 0).

~o%1.\~-_\I --3

0

,

~

.2

D

.4

.6

.8

1.0

1.2

1,4

Fig. 12. Systematic errors in logarithmic scale of avalanche position estimation through the centroid of the number of cathodes shown as parameter in the curves, as a function of the cathode

A =0.25 - ~ b=0.5

width, for avalanche position ;t = 0.25 (a/D), 0.5 (a/D), 0.75(a/ D). The sign of (;tb-2) is shown by the symbols ® and @. -I

From eq. (9) (25)

q MRE,,,~, v®=2c

and from eq. (6) -2

~2

1 1

2

(26)

Inserting in to eq. (24) V. and ~2 as calculated from eqs. (25) and (26), we get

~o%laf.~l -3

=

0

•2

.4

.6

.8

1.0

1.2

1.4

,.~ A =0.5 ..~.b=0.5

2

<

ex2 = 4.545 2 ~1

4C z

1

½e~ qZ M2(RE,,,~) z"

(27)

Inserting R E , . calculated by eq. (8) and table 1 or read from fig. 3, 3 × 1 0 -~s VVHz for e], 50pF for C (25 pF for the cathode +25 pF of stray capacitances and input capacitance of the FET), To= 2.5 ns and M = 1.59 for 2m----100 as read from fig. 2, we have e] = 4.87 × 10 -4 , ca = 2.2 × 10 -2 ,

(28)

and consequently

-2

ex = 220 #m.

-3 8

•2

.4

.6

.8

1.8

1.2

1.4

Useful criticism in discussions with B. Chase, V. Radeka and A . H . Walenta are gratefully aknowledged.

90

E. GATTI et al.

E. Gatti is grateful for the hospitality at the Instrument Division of B.N.L. Appendix

+ao

The potential V~ at the midplane y = D is da 1 - tgh2 (¼n2) V~ - 2 D 1 + tgh 2 (¼n2)'

2 = x/D.

(30)

Let us now consider at the midplane the array of anode wires parallel to x of radius R spaced s. We can take into account their presence by representing them as line charges q(2) at y = D together with their images - D , - 5 D .... -q(2) at y = 3D, 7D .... and at

5D, 9D, ... y = [ - 3 D , - 7 D ....

We calculate the potential due to all these line charges V. and we find q(2) by imposing

(31)

vs+v~ = o

at the surface of the wires. We get with some approximations R ,~D) +09

1 f _~ +® 4neo

Va(l+p, 2 , 0 ) = - -

q(2')d2'

~' n =

x

where p = R / D , b = s / D , which can be written 200

(

k=--cC

ZL × n=--oo

cos (nk) x ,v/{[1 + p _ ( 2 k + l ) ] 2 + 2z + (nb)2}.

?D

.~

(34)

Eq. (31) gives

da l - t g h 2 (¼n2) 2D 1 +tgh 2 (¼n2) -

4neo

q(2),f(2).

(35)

The weighting field on the wire Ew(2) can be written as a function of q(2)

REw(2) - q(2) 27reo'

(36)

SO that eq. (35) can be written: REw(2) 1 - tgh 2 (¼n2) da/--~D *f(2) - 1 +tgh2(¼n2)"

(37)

AS f0-) is a known function as the one at the second member of eq. (37) REw(2)/(da/D) can be solved by anticonvolution. The analytical expression of Ew(2) is, for b - oo and R--,0

REw (2) _ 1 1 - tgh 2 (¼n2) da/O 4 In (0.88585 x/p) 1 +tgh2(¼n2)"

(38)

Ew(2) is also analytically known for b ~ 0 when the anode wires are so close to form a conducting plane: in fact in this case the geometry becomes again that of fig. 13 and the complex potential is: ~ =

tgh ~--~,

z = x +iy,

(32)

nda [1 - tgh 2 (½n2)],

Ew - 2 D 2

(39)

_I_

Fig. 13. Ideal geometry with a cathode of width 2 da.

(40)

which formally can be written

REw (~.) = ½n p [1 - tgh 2 (½n2)J da/D

J

q

Yl

+oo

2

which gives for y = D an electric field Ew at the now conducting midplane

-oo

cos (rck)

,

f(2) =

+~

Z k=-oo

(R ,~ s,.

x x / { [ l + p _ ( 2 k + 1)]1 + ( 2 - 2 ' ) 2 + (nb)2} '

°

(33)

where

A

Calculation of RE,~(),) By conformal mapping it is easy to find that the complex potential W~ due to the infinite strip of width 2da (at potential 1) in the space between two parallel conducting planes distant 2/9 (see fig. 13) is da z W~ = ~-~ tgh n 4--D' z = x +iy. (29)

q(2)

1

Va = 4ne-----~q ( 2 ) , f ( 2 ) ,

(41)

For every finite b, the behaviour of REw/(da/D) versus b will be within the two limiting cases of eqs. (38) and (41); the following eq. (42), with coefficients depending on b, has proven to fit well REw/ (da/D) obtained, by the computer, by anticonvolution of eq. (37):

91

AVALANCHE LOCALIZATION

R E , (2)

da/D

1 - tanh 2 (K 2 2) = K1 1 + K 3 tanh/(K2).)

C°

wi'th K~, K2, K3 given for 1 / p = 6 6 6 and b --: 0, 0.2, 0.5, 1 by the values of table 1 of the main text. REw/(da/D) is plotted in fig. 4. By integration of eq. (42), i.e. superposing the effects of neighbouring strips of width 2da, we get for a finite width 2a of a cathode strip the expression K1 REw(2) - 2K2 ~/K3 ×

×(arctg{\/K3tanh[K2(2+D)l}-arctg{x/K3tanhIKz(2-D)l} ),

(43)

E

This capacitance is invariant if we open up the rectangle in a t plane, as shown in fig. 15. The planes u and t are related by the functional re.lationship

~dt/ ( l _ k Z t 2 ),

J

(45)

At2 ....

-

]

D° C °

B° A

ec 1 ~Z

C°

B°

Ac A

g/a

C

D

E

H=

t~

_

9 1

.Z 1

.= tl

where F is the elliptic function o f the arguments and ~p, and k = sin ~, k'= cos ~, t = sin ~p. As k = 1/(] +g/a) we get from eq. (44)

C = 4%.

(F/m).

g

Bl~

~

a+g_Fi k2,~z)

~

(46)

Here we have inserted a factor 2 because we want to calculate the capacitance between the strip BB° and the ground plane not only due to the field of the upper semiplane but also due to the mirror image field with respect to the t l axis. Eq. (46) is plotted in fig. 6. Let us now consider the t' plane shown in fig. 16; the capacitance between BB° and ( C H ~

/I

I

1B

Fig. 15. Field of fig. 14 conformally mapped on the upper halfplane of this figure.

L-- K Ik 2)

D°

u,

Fig. 14. Field which will be mapped on the upper half-plane of fig. 15.

Ho H

i.~

..

IA B1 -,,~,K(k)=F(k,1 ) J-

B°

i, u2

DO-- K~Ik2 :

C

k-~ Fig. 16. Electrode arrangement with indicated the points D and E which correspond to bending points in the conformal mapping which leads to the electrode configuration of fig. ]7.

The calculations are done by the method of conformal mappingS). Let us solve a preliminary problem which will be useful for achieving the one of' interest. Let us consider the plane u as shown in fig. 14 and the uniform field inside the rectangle ABCH and its specular one with respect to the u 2 axis. The capacitance between the BB° and CC ° plates is AB C = 2% ~ (F/m) = 2% K(k)/K'(k). (44)

.-0

D

Klk)=F(k:l

c°B °

Calculath)n of the capacitance of the cathode strips.

((l_t/)

H

E°

Jk

B

u = F(k, t) = f o

D°

E°

corresponding to eq. (7) of the main text. Appendix

U2

(42)

D

Fig. 17. Result of a conformal mapping of the electrode configuration shown in fig, 16.

u~

92

E. GATTI et al.

and C ° H ~ ) due to the upper plane field is

-~

,_9_~_

2a ~ 9

,

(47)

C = 2 eo g ( k ) / g ' ( k ) .

+

iOo iOo

Let us now pass to a plane u' folding up the t' plane, as drawn in fig. 17, by the relationship u' = F ( k 2 , t ' )

fr

=

o

dt' ~f(l_t,2) x/(l_k2t,2).

(48)

Fig. 18. Electrode configuration representing the actual cathode geometry in order to evaluate its capacitance.

below the u{ axis of the symmetrical geometry represented in fig. 18, we have K(k)

C = 4% K ' ( k )

We can write: (49)

a _ F(k2,1/z)

Do

K'(k2)

' (50)

a +.___99= F ( k 2 , 1 / k z )

DO

K'(k2)

L _ K(k2) Do

' (51)

K'(k2)

Eq. (51) for L / D o ~ ~ gives k2 = 1. Consequently, eqs. (49) and (50) become respectively a

Do

_

1.5708 '

a +9 _ F(1 , 1/kz) DO 1.5708

(52)

(53)

From eqs. (52) and (53), k can be found and inserted into eq. (47). Inserting a factor 2 in order to take into account the capacitance due to the field

(F/m),

(54)

where k is calculated from eqs. (52) and (53). Do should be put equal to D if the anode wire plane is considered as a continuous conducting surface. For large b, D O should be put equal to 2D in eqs. (52) and (53). The capacitance as given by eq. (54), taking into account eqs. (52) and (53), is plotted in fig. 18. References l) G. Charpak and F. Sauli, Nucl. Instr. and Meth. 113 (1973) 381. 2) F. Sauli, Proc. C.E.A. Journees d'l~tudes, Sac~ay (14-18 March 1977) p. 89. 3) j.M. Durand, Proc. C.E.A. Journees d'F,tudes, Saclay (14-18 March 1977)p. 111. 4) j. Fisher, H. Okuno and A.H. Walenta, to be published. 5) D. H. Wilkinson, Ionization chambers and couaters (Cambridge University Press, 1950) pp. 91-99. 6) j. Fisher, H. Okuno and A. H. Walenta, IEEE Trans. Nucl. Sci. NS-25 (1978) 794. 7) A. H. Walenta, private communication. 8) W.J. Gibbs, Conformal transformation in electrical engineering (Chapmann and Hall, London, 1958).