An extended parametrization of gas amplification in proportional wire chambers

210

Nuclear Instruments and Methods m Physics Research A260 (1987) 210-220 North-Holland, Amsterdam

AN EXTENDED PARAMETRIZATION OF GAS AMPLIFICATION IN PROPORTIONAL WIRE CHAMBERS Sean P. BEINGESSNER * and R.K . CARNEGIE

Carleton University Physics Department, Ottawa, Ontario, Canada KIS 5B6

C.K. HARGROVE

National Research Council, Ottawa, Ontario, Canada KJA OR6 Received 27 March 1987 It is normally assumed that the gas amplification in proportional chambers is a function of Townsend's first ionization coefficient, a, and that a is a function of the anode surface electric field only. Experimental measurements are presented demonstrating the breakdown of the latter assumption for electric fields, X, greater than about 150 V/cm/Torr on the anode wire surface for a gas mixture of 80/20 argon/methane. For larger values of X, the parametrization of the proportional gas gain data requires an additional term related to the gradient of the electric field near the wire. This extended gain parametrization remains valid until the onset of nonproportional contributions such as positive ion space charge saturation effects. Furthermore, deviations of the data from this parametrization are used to measure the onset of these space charge effects. A simple scaling dependence of the gain data on the product of pressure and wire radius over the whole proportional range is also demonstrated . 1. Introduction We have made detailed measurements of the gas amplification in a proportional counter as a function of wire radius, electric field and pressure in an 80/20 argon/methane gas mixture . These measurements have been used to investigate the accuracy of the standard parameterizations of gas gain on these variables. The proportional counter gas gain, G, is given by the integral of Townsend's first ionization coefficient, a ; In G=

J

~a(E) dr,

where ra is the anode wire radius. Experimentally it has been well determined that a/P is a function of the single variable E/P, where E is the electric field and P is the pressure, for a uniform electric field. If it is assumed that a(E) also applies to the case of the varying electric field near the anode wire, one may then parametrize the integral of eq . (1) in a simple way as a function of the variable Ea/P where Ea is the electric field on the anode wire surface. The standard empirical formulae for proportional counter gas amplification have been reviewed by Charles [1]. Charles demonstrates that the difference between the existing empirical formulae Present address: University of Victoria, Victoria, BC, Canada . 0168-9002/87/$03 .50 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)

are only due to different choices in parametrizing this function . We refer to the region where this parametrization is valid as the Townsend region . Our results show that these empirical formulae for the gas gain are only valid for part of the proportional gain region . We have found that for operating conditions involving high electric field gradients near the anode wire, an additional phenomenon occurs which modifies the gain. This region of modified gain defines a second proportional region beyond the Townsend region, yet before the onset of nonproportional effects like space charge saturation . We call this second region the non-Townsend proportional region. Furthermore, we have been able to develop an extended parametrization of a involving an additional term depending on the electric field gradient which fits all of our data in the non-Townsend proportional region. This supports our assumption that the measured effect is due to the fact that the effective electron swarm temperature at a point in the high gradient field region has not been able to reach the equilibrium temperature it would have had in a uniform field of the same value. The data also demonstrate the scaling dependence of the gas gain on the product of pressure and wire radius for the entire Townsend and non-Townsend proportional regions. This scaling follows from more general arguments based on the similarity principle [2]. Finally, this scaling is used to measure the onset of nonproportional effects on the gain, specifically the attenuation of

S. P. Bemgessner et al / Extended parametrization of gas amplification

the gain due to positive ion space charge effects. Sect . 2 reviews the theory of the gas gain for proportional wires . Sect . 3 describes the experimental technique . Sect . 4 gives a description of the data set . Sect . 5 describes the analysis of the data . Sect . 6 discusses effects of possible systematic errors . Sect . 7 explains the measurement of the onset of space charge saturation effects and sect . 8 gives the conclusions .

In the first case where the ionization coefficient z is no longer a function of X alone, for each value of K (counter voltage), z can be a different function of s, i.e . z=z(K . s) . The integral In G=

f,s, z(K, s) ds,

=1n G(K, sa) . 2 . Theory

then X = K/s . For cylindrical geometry, K = V/ln(rc/ra), where r. is the cathode radius . For fixed K one then has

This functional parametrization of the ionization coefficient follows from the similarity principle [2] or scaling of the gas gain with respect to the product of pressure and wire radii. This amounts to changing the geometry of the counter in proportion to the change in the average mean free path between collisions such that the energy gained from the field between collisions is the same . For example, for the same value of K = Eat = Eaz, the gain of an rl = 10 jim anode wire at P l = 2 atm equals the gain of an r2 = 20 f,m wire at P2 = 1 atm . However, the gain cannot be reduced to depend on a single variable . The gain now depends on the two variables K and s, . Some form of pressure scaling should be maintained since z is still determined by the average energy gained from the field and lost in collisions . However, the electron avalanche gain now depends on the gradient of E near the wire as well as on the field at the anode wire surface . As X,=Klsa and dX/ds(X=Xa )=-K/s 2 , any function of K and s may be rewritten as a function of X and dX/ds and so

ds= -(K/X2 ) dX,

z=z(X, dX/ds) .

The ionization coefficient, a, is a macroscopic parameter that depends on the average effect of many microscopic electron-molecule collisions in the presence of an external electric field . In this way it is similar to the electron drift velocity and diffusion . These latter quantities are known to scale in a simple way with the ratio of electric field to pressure, E/P. The gas gain is complicated only because it depends on the integral of a and not on a directly. It is convenient to introduce new variables ; X=E/P,

z(X)=a/P,

s=Pr .

In a proportional chamber, the electrostatic field falls off as 1/r in the vicinity of the anode wire . Define K by, E=K/r,

If the effects of the dX/ds dependence are not too large, they may be described by adding to z(X) a dependence on powers of X and dX/ds, i .e .

and then In G- f~adr ra _ - Kf

0

Xa (z(X)IX 2 ) d X,

= KIP(X.),

z(X, dX/ds) =z(X) + Y_z, , X'(dX/ds)j,

,1 #z0 .

After integrating one has, (2)

or In G/K= 41 (X,) . Therefore the quantity In G/K is expected to be a function of the single variable Ea /P or Xa . Two processes that disrupt this relationship are considered in this paper. The first is the breakdown in the assumption that the ionization coefficient z is a function of X alone . This is caused by the electric field changing at a rate such that the avalanche electron energy distribution does not attain the equilibrium distribution expected for the given E field, and thus, the ionization coefficient is lower than one would expect. The second process is the familiar positive ion space charge modification of the gain at the high gain end of the proportional region .

In G=K~P(X)+Y_a,,X'+J/sj-1, a,, -_

I)J+iz

(4)

/(i+2 j -1) .

In sect . 5 it will be seen how one or two of these additional terms dramatically improves the representation of the data for large values of X . This more general two variable scaling symmetry is destroyed by the onset of positive ion space charge effects . This is because the above scaling depends on the field falling off as 1/s from the surface of the wire . In the case of the space charge cloud the dependence is more complicated and breaks the similarity principle. To see this on an intuitive level consider Poisson's equation for the potential V, 02V= _P1/(O'

212

S. P. Bemgessner et al. / Extended parametrization of gas amplification

where p' is the charge per unit volume. Rewriting this to terms of the scaled distance s in cm Torr gives 2 2 , 3 lIlp 10 V= _Pp /e o p , 02 V= _Pp'IEO,

CATHODE

where p is the charge per scaled volume . It is immediately apparent that the effect of space charge increases with pressure for fixed K and s. Thus to the presence of space charge the gain becomes a function of three separate variables K, P, ra . Because the effect of space charge increases with pressure for fixed K and s, it will effect the gain of a 10 ftm radius wire at a given pressure P at a lower gain than a 100 pin radius wire at a pressure of P/10 . This effect is demonstrated in sect . 7.

LeCROY qVt MULTICHANNEL ANALYSER

PDP-I I

3. Experimental measurements The gas amplification in an 80/20 argon/methane gas mixture was measured using simple proportional 55 counter tubes and a Fe X-ray source at the midpoint . The proportional tubes were 40 em long of the cathode constructed from a 6.0 cm inner diameter aluminum tube . The measurements were made using 9.4, 100 and 406 pin radius wires. The 9.4 pin wire was nominally 10 pin and so is referred to as such . The proportional counters were mounted in a general purpose pressure vessel . Gas pressures between 0.1 and 2 atm were used . During the data taking process a fourth tube with a 62 .5 pin radius wire was run in the same gas volume to independently monitor the gas properties . Gas pressure and temperature were recorded for each data point. The electronics used to make the measurements is shown in fig. 1. The connections to both the anode and cathode were made through high voltage feedthroughs in the pressure vessel end plate. For all measurements the cathode was at negative high voltage and the anode at ground . The low pass filter was introduced just before the cathode in order to reduce external noise. The current from the high voltage supply was monitored to check for breakdown in the chamber or at the connection . This was done on the power supply side of the filter . The anode signal went directly into an Ortec 120-2B preamplifier mounted at the outside face of the end plate for measurements of low proportional wire gas gain . It was replaced by an Ortec 113 preamplifier for high gas gain measurements. The input of the Ortec 113 preamplifier was paralleled with a 1 MQ resistor to maintain the anode wire at ground. The Berkeley Nucleonics PB-4 pulse generator was connected to the test input of the preamplifier in order to calibrate and monitor the electronics gain of the entire system . The pulser fed a voltage into the preamp

Fig 1. Block diagram of the electronics. with a 50 ns rise time and a 200 ps fall time through a precision 1.0 pF capacitor. The calibration procedure is treated in more detail in sect . 6. The output of the preamp went to the normal input of an Ortec 450 research amplifier. The research amplifier had a discretely variable course gain between 5 and 2000 and a continuously variable fine gain between 0.5 and 1 .5 . The integration and differentiation times were fixed at 1 .5 ps for these measurements . It was necessary to correct the measurements to take into account these fixed time constants. This is described to sect . 6. There were three outputs from the amplifier; the fast bipolar output went to the discriminator, the delayed unipolar output to the LeCroy qVt multichannel analyzer and the bipolar output to a Tektronix oscilloscope for signal observation . 4. The data The gain measurements were made using the 5.9 keV photopeak events produced in the counter gas by the 55 Fe radioactive source . These events produce an average of 226 ionization electrons in the counter gas. For each apparatus configuration defined by wire radius, gas pressure, and counter voltage, the counter pulse height distribution produced by the 55 Fe source was recorded together with pulser calibration data . Each measured photopeak pulse distribution was then fitted with a Gaussian function to determine the centroid . A typical spectrum is shown in fig. 2. The centroid value was then converted to an equivalent charge based on the pulser calibration and corrected for the effects of

S. P . Beingessner et al. / Extended parametrization of gas amplification

C O

213

104

O U

a E 0 0

C7

10 3

10 2

100 125 150 175 200 225 250 275 300 K Volts

Fig. 2. Typical 55 Fe spectrum taken using the 100 pin radius wire . The pressure was 61 Torr and the cathode voltage was -1 .1 kV . The fit to the photopeak is also shown. C O

the amplifier time constant attenuation of the signal . The error on individual gain measurements was 4%, with the dominant source of error being due to the calibration . The data sample consists of 325 separate gain measurements. For each of the three anode wire radii, several different gas pressure values were selected, and then data recorded as a function of counter voltage. The range of pressures used and the number of data points obtained for each anode wire are listed in table 1 . In this paper all pressures (and hence the variable X) are normalized to a temperature of 0 ° C. The common way of displaying gain measurements is to plot the gain versus counter voltage for data taken at constant pressure. In terms of the present parameters the equivalent approach is to plot gain versus K (units of volts) for fixed s (in cm Torr units) . Figs . 3a and 3b show the gain measurements versus K made with the 10 and 100 pin anode wires respectively for several different gas pressures (s = Pr) . The curves are polynomial fits to the data that then permit interpolation between Table 1 Data points Anode wire radius [pin] 10 100 406

Pressures [m Torr at 0°C] 141-1398 56-1390 142- 562

X range [V/cm Torr]

No . of data points

131 < X < 1075 50 < X < 416 44 < X < 117

78 200 47

10 4

O U

° 103 ó

10 2

10

100 200 300 400 500 600 700 800 K Volts

Fig. 3 . (a) Data for the 10 p m wire . The lines are lines of equal pressure or s for s = 0.13, 0.26, 0.39, 0.53, 0.66, 0.79, 0.92, 1 1, 1 .2, 1.3 . The symbols are for the data points . The lines are the result of polynomial interpolation between the data points . (b) The same graph for the data taken with the 100 tLm radius wire for s values of 0.56, 0.61, 1 .4, 1 .45, 2.79, 2 .8, 4 .2, 4 .9, 5.1, 5.5, 6.7, 6.9, 7 .0, 8 .3, 9.7, 11.1, 12 .4 and 13 .9 . data points . Only those data points with a gas amplification G < 2 x 10 4 have been included in this figure . The data, at fixed s, show the usual approximately exponential increase of gain with counter voltage. The full set of curves in figs. 3a and 3b is one method of clearly showing that the gas gain depends smoothly on the two variables K and s. An alternative, and a particularly useful method, is

214

S. P. Bemgessner et al. / Extended parametrization

to plot the contours for fixed gas amplification in the two-dimensional s vs K plane. In figs . 4a and 4b, the contours of fixed gas gain are shown in decade steps from 10 2 to 10 6 for the 10 and 100 pin anode wire data respectively . The dotted lines included in fig. 4 correspond to lines of constant X(= E/P = K/s) . Naively one might expect that as one increased the pressure for a fixed wire radius the gas amplification might scale with E/P such that for the same gain one would 14 12 10

of gas

amplification

require the same value of E/P at the surface of the wire. This would mean that the lines of constant gain would also be lines of constant E/P. Clearly the data show that this is not the case . However, using fig. 4 it is easy to extract the dependence of the gain on pressure and voltage. As an example of this one may consider the constant gain = 10 ° line of fig. 4a . For s = 0.2 cm Torr, P = 200 Torr and K is about 150 V; for s = 1 .2 cm Torr, P = 1200 Torr and K is only 275 V, thus while the pressure has increased by a factor of 6 the voltage has less than doubled. After it has been shown in sect . 5 that the curves of one wire predict the gains of other radius wires, fig. 4 becomes even more useful. This is because, ignoring space charge, the measurements at s = 1 cm Torr or P = 100 Torr on the 100 ft m wire give the expected gain at P = 1000 Torr for the 10 pin wire .

0 8

E

5. Experimental analysis

0 6

u

0 4 02

30

120

160 200 240 280

320 360 400 K Volts

1

14

-

102

r

1

10 3

12 10

_ó E u

8 6 4 2 0

1 1 1 I --1 1 1 1 1 100 200 300 400 500 600 700 800 900 K Volts

Fig. 4. (a) This is a graph of the gain data taken with the 10 pin radius wire plotted in the K s plane. The solid lines represent contours of equal gain m logarithmic steps from 10 2 to 10 6. The dotted lines are lines of equal X or E/P for X = 200, 400, 500, and 600 V/cm Torr . (b) Data from the 100 pin radius wire as in fig. 4a . However, the dotted lines of equal X are now for X= 50, 100, 150, 200, and 250 V/cm Torr .

The formalism presented in sect . 2 indicates that in order to most concisely parameterize the gas amplification data, one should examine the quantity In G/K. The full set of measurements of In G/K versus E/P for the 80/20 Ar/CHQ gas mixture is shown in fig. 5a . The pressure P has been normalized to a temperature of 0 ° C. Data from all three wire radii are included . It is immediately apparent that for low values of E/P the data appear to be a simple function of the single variable E/P. However, the simple functional relationship is lost at larger values of the variable X. This is a sign that processes other than those described strictly by the Townsend ionization coefficient are present. In order to remove the effect of space charge for the initial analysis, the gas amplification is constrained to moderate values . Fig. 5b shows the experimental values of In G/K versus X subject to a gain cut of less than 2 x 10 °. This value is justified in sect . 7. It is important to make the empirical observation that In G/K is an approximate function of X for the data included in fig. 5b . In this sense, a totally empirical parametrization of the gain is possible . This allows, for example, a simple estimation of approximate operating voltage for MWPCs at the design stage. This, in itself, is valuable m evaluating the interplay of the electric field in drift regions with that of the amplification region, quantities often determined by the same externally applied voltages . However, it is also apparent that for large X the dispersion of the data points increases. In order to avoid theoretical bias or the discussion of which empirical formula is best it was decided to fit the data to a fifth order polynomial in X: 5

In G = K Y_ a,X`, 1=0

(5)

S. P. Berngessner et al / Extended parametrization of gas amplification #10 -3 80

a

70 60 _ U~

ó 50 v

v

40

N

c

Y 30 ç 20 10 0 0

200

400

200

400

600

800

1000

E/P Volts/cm/torr

-3

ff10 70

b

60

ó

50 40

m

c

Y w

30

ç 20 10 0 0

600

800

1000

E/P Volts/cm/torr

Fig. 5. (a) Graph of In G/K as a function of X or K/s for the full experimental data sample . This plot contains the experimental data without any cut on the gas amplification, thus it includes those data points modified by space charges as well as those which deviate from the pure Townsend region . (b) Graph of In G/K vs X as in fig. 5a, except with a cut of 2 X 10 4. This cut removes those points that have space charge saturation . The remaining deviation at high X is due to the dependence of the ionization coefficient on dX/d s. Initially only values of X < 400 V/cm Torr were considered in order to avoid the influence of the very large X values where the data are sparse . The resultant fit is given in column A of table 4 with a X z = 342 for 205

215

data points . The X z and the standard deviation of the deviations of the data points from this fit for two ranges of X values are shown in table 2, where N is the number of data points per X range. The quality of the fit of In G/K to the polynomial function ~P(X) is good at low X, but is unsatisfactory for the higher X data. In order to examine this effect at the higher X values further, the data sample was split approximately in two by cutting at X = 100 V/cm Torr . There are 110 points with G < 2.0 X 10 4 and X < 100 and 106 points with the same gain cut and X > 100. The result of a polynomial fit, eq . (5), dust to the X > 100 data including the data above X= 400 is given in column B of table 4, with XZ = 346 for the 106 points. The deviations between the measured gains and the gam predicted from the fit expressed as a percent deviation were calculated and are shown in fig. 6 . The above results demonstrate that the present measurements cannot be fully described by the usual empirical proportional chamber gain formulae for X > 200 V/cm Torn It will now be shown that the discrepancies may be accounted for by assuming that the ionization coefficient begins to depend on the derivative of the electric field and so is less than the Townsend coefficient measured in a uniform field. There is therefore a nonTownsend proportional region of wire counter operation. To demonstrate this, the XS term in the above fit was removed and a number of non-Townsend alternative terms having the form proposed in sect. 2 were tried. The same 106 data points, for which 100 < X < 1025 and G < 2 .0 X 10 4 , were again chosen and many different fits were done to the data in each case using a fourth order polynomial in X plus one or two additional non-Townsend terms. The non-Townsend terms investigated were restricted to the six containing a power of X less than or equal to 3 . The agreement of the individual data points with the best fit were then compared for each case investigated. The X Z of the fits are given in table 3 for each non-Townsend parametrization studied. The diagonal entries correspond to just adding the single non-Townsend term listed in the row or column heading. The nondiagonal entries give the results when the two non-Townsend terms referred to are included . For comparison, a fit to a fourth order polynomial in X excluding any non-Townsend terms has a X2 = 378. The table shows that the addition of one or more Table 2 Results of Townsend fit X

40 < X < 200 200 < X < 400

N

168 37

XZ

XZ/N

(a)

197 170

4.9

8.5%

1.2

4.4%

S.P Bemgessner et al. / Extended parametrization of gas amplification

216 40

O IONM " IOONM

ó Z 20

0

Q

(a) _

A0 0 2A,00 o

0

w

° -20

0

0

0 0

0 0 0

0

0

0

0 0

-40

100

300

200

14

(b)

12

IOPM

w

8

0 w

6

(C)

12

w á w

i

1000 VOLTS/CM/TORR

L IOO um

10 r Z ó

> 10

V

X

500

6

w 4

w

-30-20 -10 0 10 20 30 DEVIATION %

-30-20 -10 0 10 20 30 DEVIATION

Fig. 6. The top diagram shows the deviation of the experimental points from their fit to eq . (5), fit B of table 4. The dark points are 100 pin radius wire data, the open circles 10 pin wire data. The frequency spectrum of the deviations of the 10 and 100 pin wire data are shown m figs . 6b and 6c respectively . non-Townsend terms is sufficient to completely describe the expenmental data over all X values when the gains are limited to the proportional region . There is no clearly preferred choice for a particular extra term or terms . The most economical fit, within the precision of the experiment, is the one that adds only the single X2 non-Townsend term, 4

In G=KY_ a,X ' +bX2 . r=0

X 2 X

X 2 /s X3

X 3/s X 3 /s 2

179

98 102

97 95 124

ao X10 3 a, X 10 4 a 2 x10' a 3 X10 9 a 4 X1012 a 5 X10' S b X10 6 X range N X

2

A

-6 .2 ±0.2 2.44±0.07 0.0 ±7 .0 -3 .3 ±0 .6 11 .0 ±2 .0 -11 .0 +2 .0 -

X < 400 205 342

B

-7 .0 ±0.4 2.90±0.06 -7 .0 ±0 .4 1 .0 ±0 .1 0.8 ±0 .1 0.28±0.05 -

X < 100 106 346

C

-4.9 ±0.02 2.51+0.03 -4.5 ±0.1 0.46±0.02 -0 .15±0.01 -3 .7 ±0 .2 X < 100 106 102

-e and is negative as expected for a (-3.7+0.2) X 10 process which decreases the gas avalanche gain . The deviations of the data from this fit are shown in fig. 7. It should be noticed that eq . (6) includes the same number of parameters as eq . (5). The non-Townsend term has the effect of accounting for the observed spread in the gain values at fixed X. This spread to gain is about an 8% effect at X= 150 and a 100% effect at X= 400. A 10 pin radius wire would normally operate between X = 350 and 400 at atmospheric pressure . The X2 term corresponds to the z, I coefficient in eq . (4), sect . 2 ; i.e., the coefficient that multiplies the term X(dX/ds) .

40 0 20 00 [-

0

-40

(6)

For X > 100 V/cm Torr, the Xz of this fit is 102 for the 6 parameters and 106 points . The parameters are listed in column C of table 4. The final non-Townsend term is Table 3 X2 using non-Townsend terms 2 X 2 /s X X

Table 4 Resultant parameters from the three gam fits

175

12

r 2l25 w n 10

U z 8 w

15

0

X3

93 93 102 112

X3/s 103 93 112 100 187

X3/ s 2

115 93 113 102 121 241

14

20

w

0 6 LLJ

75

w 4

5

2

25 0

-30 -20 -10 0 10 20 30 DEVIATION %

0 -30 -20 -10 0 10 20 30 DEVIATION %

Fig. 7. The deviation of the data fr m eq . (6), fit C of table 4 This shows the improvement due to the single non-Townsend term

S. P. Beingessner et al. / Extended parametrization of gas amplification

This analysis shows clear evidence for the breakdown of the standard empirical representation of the gas amplification in a proportional counter when (Ea/P) > 150 V/em Torn This discrepancy is easily accounted for by the inclusion of a term representing a dependence of the ionization coefficient on the derivative of the electric field, a non-Townsend term . With the inclusion of this term, the full set of gain data obtained with different anode wire radii, different gas pressure and counter voltages can be accounted for with a single 5-parameter function . This parametrization is based on the fact that the gas gain scales with respect to the product of pressure and wire radius and thus the result also confirms that the scaling principle is valid . 6. Discussion of possible systematic effects In this section, three possible effects in the measurement technique that could simulate a non-Townsend effect are considered . (1) Errors m the measurement of pressure, temperature and voltage would tend to destroy the observed agreement in the gas gain between the different wire radii for fixed K and s. This has not been observed, implying satisfactory measurement and control of the variables, P, T and V. (2) Consider next the possibility that either the initial number of electrons or absolute amplifier calibration is not accurately known. Each of these would contribute a possible multiplicative error e into the individual gain measurements. Thus the measured gain, G', would equal e times the true gain, G, and so, In G'/K = In G/K + In e/K,

or,

In G' = K~P(X)+Ine . This effective extra term In e would have a spreading effect on the gain, but only a constant effect that does not vary with X. Treating In e as an extra parameter together with the fourth order Townsend polynomial function used above and then refitting the same 106 data points satisfying X > 100 and G < 2.0 X 10 ° as above gives a value Ine=-0 .143+_0.014 . However, the XZ of this fit is 282 compared to 102 using the non-Townsend XZ extra term . The hypothetical In e correction apparently is partially compensating for the non-Townsend effect . If the fourth order polynomial plus In e term is used to fit the 110 points with X < 100 and G < 2.0 X 10 ° , where the contribution of the non-Townsend term is negligible, one obtains the result In e= +0 .022±0 .015 . This is consistent with e = 1 or no multiplicative error

217

being present in the experimental gain measurements . The X 2 is 96 for 6 parameters and 110 points . If the factor e was a real effect it should be equal in both regions which is incompatible with the results . (3) A third possible source of error that might simulate the non-Townsend effect comes from the calibration of the charge measurement. We have normalized our measured pulse heights to a known charge fed into the preamplifier from a precision pulse generator . The response of the electronics changes with pulse shape and since the pulse generator produced a square wave to a good approximation while the wire response had a logarithmic rise one had to apply a correction to take this into account. The rise time of the pulse from the proportional counter depends on the motion of the positive ions and hence on their mobility, w . Their drift velocity may be expressed as d r/d t = N E/P = p K/YP . This equation may be used to calculate for our geometry the charge as a function of time, Q(t), released from the anode for Qo total charge as Q (t) = Q,C ln(1 + tlto), where C = 0.5/ln( r~/r, ), and IIt,, = (2[tlra)(Xa) .

C is a normalizing factor containing the effect of geometry on the amplitude of the charge signal and it is therefore convenient to factor it out. Following Breyer and Cimermann [3], the proportional wire pulse shape, ln(1 + t/to), was then convoluted with the response of the active filter in the Ortec 450 amplifier for 1500 ns time constants, T, to give the pulse height of the proportional pulse. This is then compared to that expected from the pulser step function to obtain the ratio of the two pulse heights for the same charge, called R . Qn, is the charge calculated from the pulser voltage to charge calibration of the pulse height analyser system . The total charge in the avalanche Qo is then Qo = Qm/R . For the present work the following parametrization accurate to better than 0.1% was used to analyse the present data . R = C[0.3998 + 0.9208 ln (1 + ,r/i(,)],

4 .8 < T/t o < 16,

R = C[0.3280 + 0.94241n(1 + T/to) ],

20 < rlto < 200,

R = C[0 .1396 + 0 .9763 In(1 + ,r/to ) ], 500 < T/to < 5000 .

21 8

S P. Beingessner et al. / Extended parametrization of gas amplification

The three different ranges correspond to the correction necessary for the three different wires, 406, 100 and 10 pin respectively . Thus the values of R do not overlap from wire to wire . Notice that as T/to increases, R approaches the limiting case of C ln(1 + 7/to). Thus the correction is less complex as Xa gets large. In the large X region, the error in the correction is dominated by the error in knowing p, the positive ion mobility. However, any error in [ only effects the gain in a logarithmic way. Thus the correction is not very sensitive to errors in ft . However, it was checked as follows. From the literature, measurements [4] of the positive ion mobility [, for low values of electric field lead to an expected value of ti, of 1 .36 X 10 -6 CM Z Torr/V ns . A study in which ti, was treated as a parameter was carried out with the present data and a value of [ = (1 .45 ± 0.05) X 10 -6 was extracted. Thus mobility result is associated with the motion of the positive ions from a region of much higher field than the previous measurements . This then indicates that the mobility ti, can be treated as a constant m this analysis . As indicated above any error in the calculation of the time constant correction is expected to be most pronounced for low values of Xa . Such an error would lead to a departure of the gain measurements from the single variable X dependence in the low X region that would be wire and X dependent. This is not observed as seen in sect . 5 and fig . 5 . 7. Space charge In this section it will be shown how the present techniques can be used to provide a novel and sensitive method for detecting the onset of space charge processes. As discussed in sect . 2, for large values of the gain and initial ionization, the space charge cloud of the positive ions modifies the value of the electric field enough to change the gain . This causes the gain to depend on pressure P and wire radius ra separately . There are other second order processes such as photon interactions that can also affect the gain. The detection of second order processes in the Townsend region is straightforward. One has merely to identify the divergence of the measured gain from that expected from the parametrization of In G/K as a function of X. In the present experiment the low X Townsend region data are dominated by the large radii wire data . In this case, it was observed that the end of the proportional region is subject to discontinuous behavior similar to the "limited streamer" [5] or "pseudo-geiger" [6] operation observed by others . The case of 80/20 argon/methane gas did not have the stability of gas mixtures containing larger hydrocarbon content. For example, for the case of the 100 p,m radius

wire at P = 702 Torr (1 atm since all pressures are normahzed to 0 ° C), the end of the proportional region occurs for X values near 110 V/cm Torr and avalanche sizes of 1 .5 X 10 8 electrons. This represents a large space charge free gain but it must be remembered that it is accompanied by large instabilities due to fluctuations beyond the proportional region. In the non-Townsend region, secondary processes are slightly more complex to identify. Recall m the non-Townsend region In G/K becomes a function of two variables. In the operating region where the Townsend coefficient is valid, the end of the proportional region could be identified using the data of only one wire, because by changing the pressure it is possible to measure the gain for fixed X on the wire surface for different values of the final total charge . However, in the non-Townsend region, the gain depends on both K and s. In order to identify space charge saturation, it is necessary to measure the gain in the same K and s region with two different wires, because, as shown in sect . 2, the space charge effect destroys the proportional region K, s symmetry in a manner dependent on the pressure . For fixed K and s, linearly decreasing the pressure is effectively the same as linearly decreasing the space charge due to the ions . If space charge attenuation effects are responsible for the end of the proportional region for the 10 ,um radius wire data sample, this fact may be detected and confirmed by comparing the 10 l m wire data with 100 l m wire data taken at 1/10 of the pressure . To illustrate this procedure, the 100 pm wire data which overlapped in the K, s plane with the 10 pm wire data were selected and fit to the functional form of eq . (6) which includes the non-Townsend term . The X region of the overlapping data samples is X > 130. A gain cut of G < 10 5 is used for this 100 ft m wire data sample . The fit had a XZ of 52 using 53 data points . The value of the individual fit parameters differ somewhat from the results given in fit C because the X range and distribution of the input data sample differs. This fit to lust the 100 tim wire data is then compared separately in figs. 8a and 8b to the 100 and 10 Am wire radius data respectively which overlap in terms of s values. Fig. 8 gives the ratio of the measured gam to the gain expected from the fit plotted versus wire gain . Fig. 8a shows that the 100 pm wire data are well described by the fit to these data, whereas in fig. 8b the 10 pin radius wire gains are successfully predicted at low gains, but exhibit large attenuations at high gams . This is consistent with the onset of space charge lowering the measured gain for the thin wire . It should be noted that for these data taken with an "Fe source with its fixed initial charge (226 electrons), a particular value of gain is equivalent to a definite value of total charge in the avalanche. Space charge effects depend on the total size and distribution of the

S.P. Bemgessner et al. / Extended parametrization of gas amplification 12

219

measuring the onset of space charge for a given initial charge situation . 0

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from the

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Fig . 8 . (a) The measured gain/expected gain for the 100 Am wire data from

the fit to the

100 Am wire data . (b) The

measured gain divided by the expected gain for the 10 Am wire data from the fit to the 100 Am wire data .

positive ions cloud . Hence these numerical results cannot be automatically applied to other ionization sources with different magnitudes or distributions of the initial charge. Rather, these results demonstrate a method of

space charge free operation as s or pressure (ra is fixed) initially increases and then it seems to plateau . Presumably this is the result of the two competing processes . First, increasing s a or pressure must cause the avalanche to spread out in space and hence the positive ions are spread out in space causing space charge to be reduced . At the same time as the pressure increases it tends to increase the density of the positive ions canceling the first effect . This leads to the possibility that for different gases there may be an optimal point, but this has not been investigated . Finally it should be mentioned that this approach is only useful for detecting the threshold of space charge effects if it in fact is the dominant process causing the onset of nonproportional gain . This method does not directly indicate whether other second order processes may be affecting the thick wire data at gains higher than the space charge affected points of the 10 Am wire, and which could modify the fit to the 100 A m wire data. Essentially the gain cut of 10 5 on the 100 Am wire data was arbitrary, chosen only to be larger than the point where significant space charge effects were seen on the thin wire .

8 . Conclusion

The gas amplification of a proportional counter filled with an 80/20 argon/methane gas mixture has been measured as a function of wire radius, gas pressure and counter voltage with a precision of better than 4% . The subsequent analysis of these data shows that the standard functional parametrization of the gas gain based on the Townsend ionization coefficient, a(E), will only describe the gas amplification successfully for a part of the proportional operating region . Specifically, all of the proportional gam data at different pressures, voltage and wire radii can be described by a polynomial function of the single variable Xa = Ea/P as long as Xa is restricted to be below about 150 V/cm Torr . For Xa > 150 and moderate gas gains, G < 2 X 10 4 , the individual measurements of the gain at fixed Xa begin to exhibit a spread in values, with the magnitude of the observed spread increasing as X increases, reaching a factor of 2 at X= 400 . This spread is the gain measurements at fixed X signals the onset of a new type of proportional amplification region which we have called the nonTownsend region . We have assumed that the observed

22 0

S.P . Beingessner et al / Extended parametrization of gas amplification

reduction to gain is due to the fact that the electron swarm temperature at a point in the high gradient field region has not been able to reach the equilibrium temperature it would have had in a uniform field of the same value. The traditional gain parametrization assume such an equilibrium is attained at every point. This new feature of the data can be completely accounted for by assuming that the effective ionization coefficient for the cylindrical geometry of a proportional counter anode wire not only depends on the electric field near the wire, but also depends on the gradient of the field there. With the inclusion of a field gradient term in the parametrization of the ionization coefficient, a, the full set of gain data m the nonTownsend proportional region below the onset of space charge effects can be described with a simple 5-parameter function . It should be noted that these results are not intended to justify using polynomials in X to parametrize the function of X rather than the alternative empirical formulae . The polynomial function is a convenient and successful way to parametrize the data. However, because this function is not unique, it is difficult to completely separate the quantitative effects of the gradient and nongradient contributions from one another m fitting this data sample . One should also note that many proportional wire counter systems used m particle physics experiments operate in this non-Townsend region where the field gradient effect can reduce the gain by as much as a factor of 2. It also has been shown that within the full proportional region, that is for both the Townsend and nonTownsend regions, the gas gain scales with respect to the product of pressure and wire radius at constant K = E,, r, This allows one to use measured or known gains to accurately predict the operating conditions for other wire radii and gas pressures over this full proportional region . Finally, we have used the new parametrization described above to identify the onset of space charge effects on the gain of proportional counter wires . Since

space charge effects set in at lower gas gain on thin wires than thick wires, if one notes the breakdown of the scaling from the thick to thin wire one can ascribe this breakdown as due to space charge effects. We have observed that these space charge effects set in for the 10 pin wire at gains above 10 4 for the 5"Fe source . The work presented here has been carried out for an 80/20 Ar/CH 4 gas mixture. The method should, however, be applicable to any other gas mixture. The measurements should be extended to investigate the full extent of the reduction of gain due to the gradient effect by including measurements at high X and low gradient (i .e . large diameter wires) where the gradient effect is small. This would allow one to make a direct measurement of the magnitude of the gain reduction due to this field gradient effect .

Acknowledgements We would like to thank J. Martin for his helpful comments, G. Findlay and L.R . Sainsbury for their assistance in the design and setup stages of this project and R. Bruce for help in the data taking phase of the project.

References [l] M.W Charles, J. Phys . E5 (1972) 95 [21 G Francis, Handbuch der Physik, vol 22 (1956) 70 ; G Francis, Ionization Phenomena m Gases (Butterworths, London, 1960) p. 59 [3] B Breyer and M. Cimerman, Nucl Instr. and Meth . 92 [4]

(1971) 19 .

G. Charpak and F Sauh, (1977) 67 The data are reproduced 77-09 (1977) p. 20 .

G Schultz,

Rev Phys Appl 12 m F Sauh, CERN

M . Atac et al . Nucl . Instr. and Meth . 200 (1982) 345 . [61 S Brehm et al ., Nucl . Instr. and Meth 123 (1975) 225 [5]