Cathode charge distributions in multiwire chambers: 4. Empirical formula for small anode-cathode separation

602

Nuclear Instruments and Methods in Physics Research A270 (1988) 602-603 North-Holland, Amsterdam

Letter to the Editor CATHODE

CHARGE

DISTRIBUTIONS

IN MULTIWIRE

CHAMBERS:

4. Empirical formula for small anode-cathode separation E. M A T H I E S O N

Physics Department Leicester University, Leicester, LE1 7RH, England Received 2 March 1988

Single parameter values for the Gatti empirical formula describing cathode charge distribution in a symmetrical MWPC are presented, graphically, for small values of anode-cathode separation.

In applications of multiwire chambers where knowledge of the distribution of induced charge is required, the empirical formula proposed by Gatti et al. [1] is generally of quite sufficient accuracy. It has further been shown that this distribution may be conveniently described in terms of a single parameter only [2]. Let p (2t) represent the cathode induced charge distribution in a symmetrical chamber, where X = x / h , the x-axis being either parallel to or normal to the anode wire direction. The a n o d e - c a t h o d e separation is h. Then the single-parameter formula may be written

p(~k) --K1 1 - tanh2(K2X) qa 1 + K 3 tanh2(K2h) '

(la)

where K1=

K2~/t~ 4 tan- ] ~/~

what emphasised at very low values of h/s so that K 3 values for both distributions are shown separately, in figs. 1 and 2. Both distributions tend, as h/s and r J s tend to zero, to the Endo distribution [4,2], P/qa = 0.25 sech(~rX/2), that is K 3 = 1.0. However the rate of convergence is quite different for the two cases, as illustrated by figs. 1 and 2. At normally accessible values of r J s the two distributions are not greatly different, as seen in fig. 3. It must be stressed of course that the simple empirical formula above can only represent an average behaviour since it does not recognise any angular localisation of the avalanche. These aspects have already been discussed in refs. [2,5,6]. For illustration, at a very small value of h/s, fig. 3 compares, for a particular geometry, the empirical distribution, eq. (1),

1.00

I

I

I

T

I

I

I

I

I

I

I

I

I

I

I

and

4)

lb,

0.90

0

080

Here qa is the net anode charge. Values of the single parameter K 3 have been presented (fig. 2, ref. [2]) as a function of chamber parameters, ra/s and h/s, where r a is the anode wire radius and s is the anode wire pitch. The values of h/s covered the range 1.4 to 6.0. However, in certain applications, in order to increase the relative cathode charge at a particular time, considerably smaller values of h/s are advantageous [3]. The purpose of this present note therefore, is to extend the information on K 3 down to these smaller values of h/s. It was pointed out in ref. [2] that the calculated (approximate) empirical distributions 01 (parallel to the anode wire direction) and P2 ( n o r m a l t o the anode wire direction were slightly different. This difference is some0168-9002/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

0.70 0.60

0-50

0

0.40

2 250 x 10-3 3 375 x 10-'

0.30

.

4 5

0.0

5.25 x 10 -3 ~ 7.50 x 10-3

. . 02

. . 0.4

~

~ ~ ' - ~

2 3

5

.

. . . . . . 0.6 0.8 1.0

. . 1.2

. 1.4

1.6

his

Fig. l. Values of K 3 as a function of chamber geometrical parameters h/s and ra/s, for empirical distribution Pl (parallel to anode wire direction).

E. Mathieson / Cathode charge distributions in multiwire chambers 1, O0

I

I

•

I

I

I

I

I

I

I

I

I

0.90

I

"

"

I

I

"

o

w i t h t h e m o r e e x a c t d i s t r i b u t i o n s as c a l c u l a t e d b y t h e m e t h o d s o f refs. [2,6]. T h e c o n d i t i o n s a n d g e o m e t r y for t h e s e c a l c u l a t i o n s a r e g i v e n in t h e f i g u r e a n d c a p t i o n . It is s e e n f r o m eq. (1) t h a t t h e F W H M , relative to t h e a n o d e - c a t h o d e s e p a r a t i o n h , is g i v e n b y

0.80

K3

603

0.70

4 t a n h - 1 ( 2 + K 3) 0.60

FWHM

r

0-50 t

0

o.z,oqt

2 2 5 o , ,o'

\ ' ~ -

a a75~,o'

~

=

rr(1 - 0.5V/~-3 )

1/2 (2)

1-0 x 10-m

4

5

5

7.50 x 10-3

- 10-3 2

5

~

5

2

x

0.30 ! . . . . . . . . . . . . . . . O. 0 0.4 0.8 1.2

1- 6

h/s

Fig. 2. Values of K 3 as a function of chamber geometrical parameters h / s and ra/S, for empirical distribution P2 (normal to anode wire direction).

i

i

I

i

i

I

i

i

F o r t h e t w o a p p r o x i m a t e d i s t r i b u t i o n s s h o w n in fig. 3 K 3 = 0.70, F W H M = 1.55 f o r 01, a n d K 3 = 1.0, F W H M = 1.68 for 02- F o r t h e m o r e e x a c t d i s t r i b u t i o n s t h e v a l u e s o f F W H M are, for a = + ¢ r / 2 , 1.13 f o r Pl a n d 1.29 for P2 a n d for a = - ~ r / 2 1.88 f o r Pl a n d 2.38 for P2.

i

ra Is = 1.50 x 10-3

O. 50 -

References

h/s : 0.20

[1] E. Gatti, A. Longoni, H. Okuno and P. Semenza, Nucl. Instr. and Meth. 163 (1979) 83. [2] E. Mathieson and J.S. Gordon, Nucl. Instr. and Meth. 227 (1984) 277. [3] G.C. Smith, J. Fischer and V. Radeka, IEEE Trans. Nucl. Sci. NS-32 (1985) 521. [4] I. Endo, T. Kawamoto, T. Mizuno, T. Ohsugi, T. Taniquchi and T. Takashita, Nucl. Instr. and Meth. 188 (1981) 51. [5] J.S. Gordon and E. Mathieson, Nucl. Instr. and Meth. 227 (1984) 267. [6] J.R. Thompson, J.S. Gordon and E. Mathieson, Nucl. Instr. and Meth. A234 (1985) 505.

0.40

p(X)

0-30

0.20

O.lO

0.00

,

-2.0

-I.0

GO

1.0

2.0

?,. = x / h

Fig. 3. Comparison of empirical distributions, G, with exact distributions; Pl (full curves) and P2 (broken curves). The exact distributions were calculated at a time 1.0 /zs, for avalanche angular positions a = + ~r/2 and - ~r/2, with rms angular spread o = 40 o, and for s = 3.0 mm, Va = 0.8 kV and /z = 1.9 cm2/Vs. The total cathode charges, relative to the net anode charge, are - 0 . 7 1 (Pl and P2) for + rr/2, 0.29 for - ~ r / 2 and necessarily 0.5 for G. The empirical distributions represent K 3 = 0.70 for Pl and 1.0 for P2-