Avalanche fluctuations in a uniform field proportional counter filled with the penning mixture neon-argon

Nuclear Instruments and Methods in Physic> Research 219 119~4i i~34 ",4f "q~,rth-Holland. ,~stcrdaw,

534

AVALANCHE FLUCTUATIONS IN A UNIFORM FIELD PROPORTIONAL WITH THE PENNING MIXTURE NEON-ARGON

J.P. S E P H T O N

*, M . J . L . T U R N E R

COUNTER

FILLED

a n d J.W. L E A K E **

X-ray Astronomy Group, Physics Department, Unmersio' of Leicester, Leicester, LEI 7RH England Received 18 April 1983 and in revised form 1 August 1983

The energy resolution of a proportional counter depends on fluctuations in the primary ionization and subsequent electron avalanche. In general avalanche fluctuations dominate and an improvement in energy resolution can be expected by raising the efficiency of the ionization mechanisms involved. This can be achieved using Penning mixtures. Previous theoretical investigations of avalanche fluctuations predict such an improvement but detailed comparison between theory and experiment requires the use of a uniform field proportional counter. Avalanche fluctuations in the Penning mixture neon-argon have been investigated under uniform field conditions. Comparison between the theory and the experimental results suggests that factors other than ionization efficiency play an important role and limit the achievable energy resolution.

1. Introduction The proportional counter plays an important role in the detection of soft X-rays in applications as diverse as Astronomy, Medicine and Nuclear Physics. Its utility is mainly due to the large proportional signal obtained from the electron avalanche in a counter of relatively simple construction. The energy resolution of the device, typically 16% at about 6 keV is a limitation and for this reason, the more complex gas scintillation proportional counter, with a resolution about twice as good, is preferred in some applications. However, the simplicity and robustness of the proportional counter encourage attempts to improve its energy resolution to comparable values. Seen from a theoretical standpoint the fluctuations in primary ionization created when the X-ray is detected are rather small [1]. It is the fluctuations in the proportional avalanche which lead to large dispersion in the signal. The theory of avalanche fluctuations has been developed, by Alkhazov [2,3] and others. F r o m this it appears that dispersion in avalanche size is related to fluctuating partition of energy between excitation and ionization as the avalanche develops. Thus there is some promise that increased ionization efficiency in the avalanche may lead to improved proportional counter energy resolution. Penning mixtures provide such increased ionization efficiency and Sipila [4] has reported improved energy resolution using such mixtures in a cylindrical counter. However, the rapidly changing field * Atomic Energy Research Establishment, Harwell, Oxon OXll ORA England. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

in a cylindrical counter makes detailed comparison with theory difficult, in fact close to the anode the field varies significantly over a single ionization mean free path. It is thus essential for this purpose that avalanche development in a uniform field is studied; the present work describes an experimental investigation of avalanche fluctuations in Penning mixtures in a uniform field and a comparison of the results with theoretical predictions.

2. Elementary theory The operation of a proportional counter can be explained as follows. The photon when detected ionizes the gas, producing an average number N of ion pairs given by N = E , , / W where E x is the energy of the photon and W is the mean ionization energy of the gas. This initial charge is amplified by electron avalanching. The average multiplication factor K is governed by Townsend's first ionization coefficient a. The energy resolution of the device depends on fluctuations in both processes. The relative variance of N can be expressed ~N2/K 2 = F/U

(1)

where F is the Fano factor (the factor by which the variance is reduced from that of a Poisson distribution [1] and ¢~N is the standard deviation of N. For the gas filling a r g o n - m e t h a n e F is typically about 0.2. If we use f to denote the relative variance of the avalanche size then one can express the theoretical resolution, RTH , of a proportional counter in the follow-

J.P. Sephton et aL / Avalanchefluctuations ing way [4]

RT~ = 236[ ( F ~ ) W ] l / 2 ( % ) .

(2)

This expression does not take account of secondary processes such as electric field distortion, electron attachment to impurities and electronic noise. Using the conventional gas filling of 90% argon and 10% methane, and a very uniform anode wire, Charles and Cooke [5] have obtained an energy resolution of 14.2% at 5.89 keV. This implies a n f v a l u e of about 0.65. ( W has a value of about 26 eV with such a mixture [2]). It is clear from eq. (2) that the resolution at a given X-ray energy can be improved by reducing the quantity ( F + f ) W . Sipila [4,6,7], has shown that the use of Penning mixtures leads to reductions in ionization and avalanche fluctuations and has obtained an energy resolution of about 12% at 5.89 keV with the Penning mixture n e o n - a r g o n in a cylindrical counter. With Penning mixtures the ionization efficiency is increased be-

535

cause energy spent exciting metastable levels in the host gas is used to ionize the admixture gas. The two gases are chosen so that the first metastable level of the host gas is slightly higher than the ionization level of the admixture gas. Cross ionization can therefore occur, diverting energy which would otherwise be lost as excitation, into ionization, and reducing fluctuations.

3. Avalanche models

3.1. Snyder's model Avalanche fluctuations were first considered by Snyder [9] and later by Wijsman [10], Legler [11], Kharitonov [12] and Frisch [13]. They assumed that the instantaneous probability of ionization in the avalanche development is independent of the distance travelled by the electron since its previous ionizing collision. Such an assumption leads to the following probability distribu-

I

I

I

10-6

10-6

P(K)

P(K)

10-7

10-7 I

2 x

106

I

1

I

;~xlO 6

L

2x10 6

K

K I

10-6

I0-6

P(K)

P(K)

10-7

_ff

lO-'t 0 L

il I

"'1 I K I

:}xlO6

0

K i

_

I0-6 1

I

10- 6 ~ ~ M , _ "

P(K)

I

4

(f)

10-7 lO-71-

I K

2x

106

0

,

,

I K

Fig. 1. The distribution of avalanche sizes in methane as a function of ionization efficiency (X) (a) E/p = 51.3; X = 0.034. (c) E/p = 78.9; X = 0.088. (d) E/p =120; X = 0.13. (e) E/p =156; X = 0.19. (f) Cookson and Lewis [14]).

2x106 E/p E/p

= 48.2; X = 0.026. (b) = 218; X = 0.26 (from

5 ~6

J.P. Sephton et aL /' A calanche.fluctuattons

tion for values of K >> 1:

P(K) = (l/F`) exp(-K/F,

),

(3)

where P ( K ) is the probability that the avalanche contains K electrons. For such a distribution f takes the value unity. Avalanche fluctuations have been measured by many experimenters. Some data obtained by Cookson and Lewis [14] for methane is shown in fig. 1. The data was obtained using a uniform electric field (E). For each curve the quantity X = aUi/E is shown where U, is the first ionization potential: X is in effect a measure of the ionization efficiency. It can be seen in fig. 1 that in weak uniform fields, where X is low; the exponential distribution described above is observed. However, with higher fields the distributions depart from this form and exhibit a pronounced maximum and reduced dispersion. Sharper distributions are also observed with cylindrical devices. Curran et al. [15] found that the pulse height spectrum from a cylindrical counter could be described approximately by the empirical relation: P( K)=

(K/F`)'/2exp(-

3K/2F`

).

For this distribution f takes the value 2 / 3 . To explain the discrepancy between the theory of Snyder and others and the experimental results at high values of E/p, (the ratio of electric field to gas pressure, or reduced electric field), one can examine the validity of the main assumption; viz. the probability that an electron will ionize is independent of its distance from the cathode. After each ionizing collision the electron must travel a distance Ui/E before ionization is again possible. If Ui/E is much smaller than the mean distance between ionizing collisions ( I / a ) then the idea of a constant ionization probability seems reasonable - the electron velocities are able to achieve equilibrium in non-ionizing collisions. This condition holds for low values of E / p because the ionization efficiency X is small. Avalanche fluctuations are therefore large as observed. However, at high values of E / p a much greater fraction of the energy gained by electrons is now spent on ionization as opposed to excitation. With high values of ionization efficiency, ie. aUi/E = 1 (which implies Ui/E = l / a ) the assumption is no longer valid. The ionization mean free path is now so reduced that it has become comparable with U~/E. The instantaneous probability of ionization will now depend on the previous history of the electron, in particular on the distance which this electron has covered towards the anode after the previous ionization.

3.2. Legler's model Legler [16] was the first to develop an avalanche model which did not assume a constant ionization prob-

ability. Legler suggested that for high values of X. a local ionization coefficient a~ can be defined which depends on the distance ~ an electron has travelled since the previous ionization in the following way:

a,(~)=

0

for ~ < Uo/E.

ao

for~>~ Uo/E.

where a 0 = constant = a / ( 2 e x p ( - a U o / E ) - 1). U0 is the so called model parameter. For the model to fit as Legler supposed, b~ should take the value Ui. Fig. 2 shows the step function variation of a i with ~ (model B). The probability distribution of the avalanche size was calculated for the model gas with the following result: P(K)=

),

(1/K)gJ(K/F`

(5)

where q, is a function which depends on aUo/E but which can only be evaluated numerically. At high values of E / p Legler's model predicts a reduced dispersion of avalanche size - as measured experimentally. Legler compared calculated and measured avalanche distributions and found that U0 agreed with the expected value U~ to within 20%,

3.3. Byrne, and Lansiart and Morucci's model To explain the results obtained at high E/p by Curran and others, an alternative model was proposed by Byrne [17] and Lansiart and Morucci [18]. In this model the local ionization coefficient is assumed to decrease as the number of electrons in the avalanche

O'i T

MODEL A

/ 1

tait

LJ

, 6

~

~ E/Uo

MODEL B ,

2

~

,

3

t,

t t I I

1

ai ' ~

, 3

I 1

I

= 2

2

3

~

~ E/Uo

~ ~ E/Uo

4

MODEL E

Fig. 2. The ionization coefficient for an electron as a function of the distance travelled since the last ionizing collision in units of Ui/E where E is the field and Ui the ionization potential. Model A is Snyder's model, model B is Leglers model, models D and E are from Alkhazov.

J.P. Sephton et al. /Aualanchefluctuations increases in the following way: a i = a(1 + I ~ / K ) , where /~ is an empirically defined parameter which depends on conditions in the gas. This model gives rise to a Polya distribution of the form: b

537

expression for the moments of the avalanche size. The variation of f0 with X for four avalanche models is shown in fig. 3. (f0 is the value to which f tends as/~ tends to infinity). According to Alkhazov fo is related to in the following way: f ( K " ) = f 0 ( 1 - 1/~" ).

b-1

(7)

where b = 1 + ~. This model is unsatisfactory for a number of reasons although it fits observed avalanche fluctuations well [141. In particular the proposed decrease in ionization efficiency with avalanche size is not easy to reconcile with measured values of this parameter [14].

3.4. Alkhazov' s analysis With the exception of Snyder's model (for which f = 1) the models just described do not give explicit expressions for f as a function of X. Non-empirical calculations of proportional counter resolution are not therefore possible. Alkhazov [3] has developed a technique whereby explicit expressions for f can be obtained for a given model relationship between a i and distance. Alkhazov's method involves the derivation of a general

f0

For large values of ~', f = f0- Note that this expression implies that improved energy resolution can, in principle, be obtained by operating a proportional counter at low values of K'. In practice, electronic noise becomes significant and degrades the observed resolution. By using fig. 3 with published Townsend coefficient data one can obtain theoretical values off0 for a chosen gas and E / p value. The theoretical resolution may be evaluated from eq. (2) using published values of F and W. With cylindrical geometry account has to be taken of the varying electric field strength (see Alkhazov [3]). High values of X can be obtained using the Penning mixture neon-argon. With 99.9% Ne + 0.1% Ar at an E / p of 3 V / c m . Torr a value of X of 0.61 has been measured by Kruithof and Penning [19]. Table 1 shows theoretical values of f, and resolution at 5.89 keV, predicted by various avalanche models, for n e o n - a r g o n under the above conditions. The electric field is uniform and h ' = 100. Very low signal dispersions are predicted by models B and D. Kruithof and Druyvesteyn [20] have determined ex-

0.9

030

0B 0.7

005

0.6

020

05 0t~

005

03

010 02 0.1 0

0.05 I

i

i

t

i

01

0"2

03

04

05

• MODEL B 0 MODEL C • MODEL D

"r ~-....._1~

0"6

Or7

08

~:ocUi

T

• MODEL E

Fig. 3. The theoretical relative variance of the avalanche size (fo) as a function of ionization efficiency (X) for four models. The variation of ionization probability with distance from the last ionization is given in fig. 2 for models B D and E. Snyder's model gives f = 1 and model C is Byrnes model.

0

15

2"-0 25 30v ~V Fig. 4. The probability that an electron will excite (a) or ionize (b) a neon atom as a function of the potential drop through which it has fallen. Note that the excitation probability is very high and in the Penning mixture neon-argon excited neon atoms ionize argon atoms with almost 100% efficiency. From Kruithof and Druyvesteyn [20].

53g

J.P. Sephton et al. /' A t,alanche .fluctuations

Table 1 Values of fo and resolution predicted by various models and the experimental probability data obtained by Kruithof and Druyvesteyn 20]; for 99.9% Ne+0.1% Ar at E / P = 3 V/cm-Torr. The best values of measured intrinsic (noise subtracted) resolution and f~, 3btained in the present work are also shown. Theoretical models A fo fwhm resolution (%)

B

l

0.01

15.9

3.8

C

D

0.83

E

0.08

14.4

0.54

5.6

In order to make a valid comparison between experiment and the various theoretical models the electric field in the avalanche region needs to be uniform and

Tensioning Ceramic Bolt Brock Beryllium / /

--I wi°d°w / /

Ceramic

8.7 + 1.5

/

Experimental results 0.75 13

maintained at a value where the ionization efficiency X is a maximum. Under uniform field conditions Townsend's formula for gain applies so K = e x p ( a x ) where x is the depth the the avalanche region. For a fixed field, the gain will increase with x, and the practice it is necessary to limit x to a rather small value ( - 1 mm) to avoid breakdown; this in turn imposes strict tolerance requirements on the construction of the detector. This was achieved by grinding the U-shaped avalanche cell into a block of ceramic material, the anode being evaporated onto the base and the cathode grid being supported on the walls which were lapped accurately parallel to the anode. The use of an unquenched noble gas mixture requires extreme levels of gas purity to avoid charge attachment which would otherwise affect the results. A schematic diagram of the chamber used to undertake the measurements is shown in fig. 5. Separate absorption and avalanche regions are employed. A low drift field exists in the absorption region to transport the electrons to the high field avalanche region, which is separated from the absorption region by a fine wire grid. The grid was formed from 8 / z m diameter wires at

4. Experimental approach

Indium Wire

0.27 ± 0.11

11.8

perimentally the probability that an electron will excite or ionize a neon atom - see fig. 4. The probability of excitation is much higher. The areas under the excitation (a) and ionization (b) curves are 0.98 and 0.02, respectively. For this mixture the probability that an excited neon atom ionizes an argon atom can be taken as unity [3]. We can therefore use curve (a) in fig. 4 as the ionization probability function for n e o n - a r g o n under the conditions outlined above. The work of Alkhazov [3] allows theoretical values of f and R to be calculated from such distributions. Applying this technique gives the values shown in the sixth column of table 1. The error limits arise from the numerical calculation employed. The predicted values of resolution in table 1 are compared with our experimental results in section 5.

Locating ~otf

Prediction based on Kruithof and Druyvesteyn's data

I

/

/

I

Grid Frame (stainless steel) Stainless sleet body

Base PLate

/

/

Support Fig. 5. Schematic diagram of the experimental apparatus showing the ceramic channel supporting the wire grid, and the stainless steel chamber with a brazed in beryllium window, sealed with indium wire.

J.P. Sephton et aL /Avalanche fluctuations a pitch of 25 ~m. The grid was designed using the criteria of Bunemann et al. [21] for high electron transmission, good electrical shielding and a uniform electric field in the avalanche region. The tolerance on the grid anode distance was chosen to keep gain variation to within _+1/2%. The chamber was designed for very high gas purity operation. Low outgassing materials (stainless steel and ceramic) were used throughout and the chamber was leak tight to 10-s T o r r - 1 / s and could be ion pumped to a pressure of 10 -9 Torr following a bake out. B.O.C. Research grade gas was used for the measurements. The mixture was neon 99.9% argon 0.1%. The gas was purified further using a Vacuum Generators titanium sublimation pump operating as a getter. The gas was stored in the pump chamber following titanium sublimation for 72 h prior to being admitted to the test chamber. The final pressure was one atmosphere. The anode pialses due to the avalanches were detected using a pre-amplifier of the type described by Smith and Cline [22]. This has a very low noise and enables measurements to be obtained at small gas gains. An Ortec 450 main amplifier and a Harwell ® 6000 pulse height analyzer were also employed. Because of the rather slow anode pulse rise time in the uniform field the detector was sensitive to acoustic interference and was operated in a sound insulated chamber. X-rays from an 55Fe radioactive source (5.9 keV) were collimated into a beam - 2 mm wide and used to test the chamber.

539

GAS GAIN 10s EXPERIMENTAL GAS GAIN

.

°• °•"

•

10'

1;o

I

1,o

.." / • /

I

PRIMARY AVALANCHE

GAIN

I

1;o

1;o

GRID -ANODE VOLTAGE

Fig. 6. The proportional gas gain of the counter as a function of anode potential. The full line is the theoretical gain based on the Townsend avalanche formula. Note the step like increases in gain at intervals of approximately 16 V.

5. Experimental results 5.1. Gas gain The variation of gas gain with grid-anode voltage (Vg,) is shown in fig. 6. The theoretical curve is based on the Townsend equation and the value of X derived from the work of Penning and Kruithof. The anode voltage was varied in 1 V steps to give a detailed impression of small scale ariations in the multiplication process. The absolute value of gain is in very good agreement but the detailed variation with grid-anode voltage is clearly not predicted by the Townsend formula. The stepwise increase of gain is a result of high ionization efficiency in this gas mixture and was first observed by Kruithof and Penning [19] in a Townsend discharge experiment. The measured ionization efficiency X = aUi/E can be derived from fig. 6 using the relationship between gain and a / E given by Kruithof and Penning [19],

K.=

e x p [ ( e ~ / E ) ( V - Vs) ] 1 + V ( e x p [ ( a / E ) ( V - Vs) ] - 1} '

where V~ and y are determined empirically and relate to the acceleration potential of the electrons ( - 20 V) and to secondary electrons released from the cathode. X = aUi/E is plotted in fig. 7 as a function of Vg~; Ui is taken as the first metastable level of neon (16.53 eV). The maxima of the curve, reflecting the discontinuities in fig. 6 occur at integral values of Ui. This is what one would expect from the ionization probability function obtained by Kruithof and Druyvesteyn (fig. 4). The probability of an electron exciting a neon atom reaches a sharp maximum when it has fallen through a potential of 16.54 V. In the Penning mixture this excitation is converted to ionization with almost 100% efficiency. Following excitation collisions the electrons are all at very low energy and have to accelerate through a further 16 V before excitation can again occur. So the ionization shows peaks at 16 V intervals in the grid anode voltage.

5.2. Energy resolution The absolute values of X shown in fig. 7 are indeed as high in the uniform field proportional counter

./.P. Sephton et al. /At~alanchefluctua/ions

540

3%=oCLI1 0.61[ E I e•

0601

•

o¢•

•

• •

•

oo

Q•

• •

I•

m

,oB

i

059 o • D

0.58

o g

OQQ

057

056

I

100

110

120

130

140

I 0

10 6RIO ANODE VOLTAGE

Fig. 7. The variation of the ionization efficiency (X) with anode potential derived from the data shown in fig. 6. The peaks are at intervals approximately equal to the first excitation potential of neon (16.53 V). The peaks become wider and shallower as the number of ionization lengths increases. The whole curve is displaced by about 20 V from the origin at zero volts. This is the Cathode 'Retarding Potential' noted by Penning and Kruithof.

avalanche as predicted and one might therefore expect that the energy resolution would be as good as the predicted values from the theoretical models. The resolution for 55Fe X-rays is given in fig. 8 as a function of gas gain or avalanche size. At high gains the resolution deteriorates rapidly, this is mainly due to secondary avalanches in the unquenched mixture, but some part is also played by the decreasing value of X at high fields. As the gain is reduced the dispersion of the avalanches decreases but this is soon dominated by the increasing contribution of preamplifier noise. (The noise can be determined independently by injecting a calibrated charge pulse into the pre-amplifier and observing the

width of the pulse height distribution. This can then be subtracted in quadrature to yield the intrinsic avalanche dispersion.) The best measured intrinsic value of dispersion is - 1 3 % fwhm. This can be compared with the values predicted by the various theoretical models. Table 1 shows the predictions of the models dealt with in section 2. There is a wide variation in predicted resolution, however the models which suggest resolution similar to that measured (models C and E) are mutually inconsistent in their theoretical conception and as noted in section 2 and fig. 2, the chosen variation of ionization probability with distance from the last collision is physically unsatisfactory. In the case of model C it is related

RESOLUTION (%) ~,0

36 32

R• 28 2& 20 16 12 i

I0'

J

i

f

i

,

t

i~

1

10a

*

i

c

*

i

t

l

103 GAS GAIN

Fig. 8. The energy resolution of the uniform field proportional counter (fwhm %) at 5.9 keY, as a function of gas gain. R,, is the measured width of the X-ray line, R I is derived by subtracting the separately measured noise from the measured width.

J.P. Sephton et al. /Aoalanche fluctuations

inversely to the avalanche size, and in E increases smoothly with distance i.e. the converse of model C. It is clear from fig. 6 that some kind of variation of ionization probability similar to that shown under model D (fig. 2) is more appropriate to the present measurements. The agreement in resolution with model C is therefore likely to be spurious. Turning to model D itself there is a large discrepancy with the results which demands explanation. The influence of disturbing effects in the experimental measurements should be considered since these will give an increase in dispersion. Possible contributors would be, electronegative impurities in the gas, and secondary avalanches inthe unquenched noble gas mixture. Particular care was taken to exclude impurities, and a series of tests were conducted showing that as the period of residence of the gas in the gettered intermediate storage vessel increased, so the resolution, and s y m m e t ~ of the peak improved. This is clearly asymptotic (table 2) so that no further improvement from additional purification was observed. The agreement between measured and theoretical gas gain also suggests little effect from charge loss. Secondary effects are evident at the high gain end of fig. 6, but at the low gain end where the resolution is measured these should be greatly reduced. thus it would appear that the discrepancy with model D remains. From the ionization efficiency in this gas mixture derived by Kruithof and Druyvesteyn (fig. 3) one can calculate the expected resolution as described in section 2. Table 1 shows that although this calculated value (8.7%) is closer to our measured resolution the dis-

Table 2 The effect of purification on energy resolution and peak symmetry. The difference between columns 2 and 3 is a measure of peak asymmetry due to charge loss. There is very little improvement between 4 and 5, and the peak is almost symmetrical indicating efficient charge collection. Filling and purificiation

1) Research grade gas; chamber vacuum baked for 3 days 2) Research grade gas; chamber vacuum baked for 10 days 3) As 2, with gas purified by one hour's gettering at half current 4) As 3 but 24 hours at full current 5) As 4 but for 72 hours

Resolution fwhm (%)

Resolution 2 × hwhm high energy half of the peak (%)

541

crepancy is still large; however, this gives a pointer to a possible explanation. Model D assumes that the electrons remain tightly bunched about the mean energy as the avalanche develops, approximating the series of 8 functions shown in fig. 2. In fact the energy distribution gradually broadens with each successive 16 V potential drop as implied by fig. 7 where the peaks in ionization efficiency become successively shallower. Kruithof and Druyvesteyn studied the current between UV illuminated parallel plates as a function of potential difference and based their derived ionization probability distribution on the first 16 V step only, a sufficiently strong current being derived from many simultaneous electrons released from the plates. In our case the signal is derived from an avalanche grown through at least six ionization peaks. The spreading effect noted above would thus cause a degradation of our resolution relative to theirs as observed. One notes that a completely uniform ionization probability as in Snyder's model, would give a resolution of about 16%.

6. Conclusions

The avalanche models described in section 2 give a variety of predictions as to the energy resolution achievable in a gas with very high ionization efficiency such as the Penning mixture neon-argon. Tests of these models in the ideal conditions of a uniform field proportional counter show that, while the general behaviour is indicative of high ionization efficiency, gradual spreading of the electron energy distribution prevents the promised high energy resolution from being realised at practical gas gains. The authors wish to thank Dr E. Mathieson and Dr S. G u r m a n (at Leicester University) and Dr R. Stewart (at Strathclyde University) for helpful discussions. The avalanche chamber and the electrodes were designed by Mr J. Spragg. Financial support was provided by the Science and Engineering Research Council and A E R E (Harwell) under a CASE award.

References

48

28

40

26

24.4

23.6

22.4 22.4

19.1 20.4

[1] U. Fano, Phys. Rev. 72 (1947) 26. [2] G.D. Alkhazov, A.P. Komar and A.A. Vorob'ev, Nucl. Instr. and Meth. 48 (1967) 1. [3] G.D. Alkhazov, Nucl. Instr. and Meth. 89 (1970) 155. [4] H. Sipila, Nucl. Instr. and Meth. 133 (1976) 251. [5] M.W. Charles and B.A. Cooke, Nucl. Instr. and Meth. 61 (1968) 31. [6] H. Sipila, Nucl. Instr. and Meth. 140 (1977) 389. [7] H. Sipila and E. Kiuru, Adv. in X-ray Analysis 21 (1978) 187. [8] H. Sipila, IEEE Transl Nucl. Sci. NS-26 (1979) 181.

542 [9] [10] [11] [12] [13] [14] [15] [16]

J.P. Sephton et al. / Aualanche fluctuations

H.S. Snyder, Phys. Rev. 72 (1947) 181. R.A. Wijsman, Phys. Rev. 75 (1949) 833. W. Legler, Z. Physik 140 (1955) 221. V.M. Kharitonov, V.M., Pribory i Tekh. Eksperim 3 (1956) 45. O.R. Frisch, AECL Report No. 748, Chalk River, Ontario (1959). A.H. Cookson and T.J. Lewis, Brit. J. Appl. Phys. 17 (1966) 1473. S.C. Curran, A.L. Cockroft and J. Angus, Phil. Mag. 40 (1949) 929. W. Legler, Z. Naturforsch. 16a (1961) 253.

[17] J. Byrne, Proc. Roy. Soc. Edinburgh A66 (1962) 33. 118] A. Lansiart and J.P. Morucci, J. Phys. Radium 23 (suppl. no. 6) (1962) 102A. [19] A.A. Kruithof and F.M. Penning, Physica 4 (1937) 430. [20] A.A. Kruithof and M.J. Druyvesteyn, Physica 4 (1~.~37) 450. [21] O. Bunemann, T.E. Cranshaw and J.A. Harvey. ('an. J. Res. A27 (1949) 191. [22] K.F. Smith and J.E. Cline, IEEE Trans. Nucl. Sci. NS-13 (3) (1966) 468. [23] W. Legler, Brit. J. Appl. Phys. 18 (1967) 1275