Ionisation energy losses of highly relativistic charged particles in thin silicon layers

356

IONISATION ENERGY LOSSES IN THIN SILICON LAYERS

Nuclear Instruments and Methods in Physics Research 220 (1984) 356-362 North-Holland, Amsterdam

OF HIGHLY RELATIVISTIC

CHARGED

PARTICLES

G. HALL

Blackett Laboratory, Imperial College, London, England Received 25 July 1983

The energy loss distribution of highly relativistic charged particles is calculated for thin silicon detectors using the theory of Landau and incorporating effects due to the binding of atomic electrons. These are calculated using a semiempirical model which relates the absorption of photons in matter to the ionisation cross-sections. The model predicts a broadening of the distribution as well as the rate of increase of the average energy loss with incident momentum and is shown to be in substantial agreement with experiment.

1. Introduction

The energy loss of charged particles in thin absorbers has been of interest for many years and recently, with the more widespread use of silicon active targets and microstrip detectors, has been studied in detail. Several experiments with high energy beams [1,2] have observed that while the general features of the earliest theoretical formulation by Landau [3] of the energy loss process are in agreement with measurements the width of the distribution of energies is often broader than expected. This is not explained by the resolution of the read-out electronics, or other sources of noise, and in some cases the experimenters attributed their result to the escape of energetic ionisation electrons from the detector [1,4]. Since the width of the energy loss distribution is sometimes used to specify the quality of a thin detector to the manufacturer and the ultimate achievable energy resolution is also of interest for very thin devices like C C D s attempts have been made to calculate the effects due to electron binding in the material. Recent developments in research using intense sources of synchrotron radiation have produced detailed data on the absorption of photons in matter and have made possible a semiempirical approach to the energy loss calculation. The method has been used to investigate the resolution of gaseous detectors [5], and in this work the same model has been employed to compute the energy loss distribution in silicon targets.

derived the expected energy loss distribution by solving an integral transport equation df

~ x (X, A) = fo~w( E ) [ f ( x , A -- E ) - f ( x , A)] d E ,

(1) Here f ( x , A) represents the distribution function, the probability that the incident particle will lose an amount A of energy on traversing a layer of thickness x. w ( E ) d E denotes the probability per unit path length of a collision transferring energy E to an electron in the material. Eq. (1) can be solved by the Laplace transform method and the result can be written 1

f(x'a)=2~iJ,

The question of energy loss by fast charged particles in thin layers was first solved by Landau [3] who 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

• i~

1

edp,

(2)

where

I =pA -- X f o ~ w ( E ) ( 1 - e -pL) dE.

(3)

The function w(E) is not generally known but Landau was able to derive an approximate solution by using the free electron (Rutherford) cross section

w ( E ) = 3" ~ E~' 1

(4)

with

2~re4 NoZ mc2fl 2 p" A

x 2. The energy loss distribution

rc+i~

--

1.78

× 10- 2 k e V / ~ m - l in silicon.

(5)

B2 In eq. (5) e is the electronic charge, assumed to be the same as the incident particle, N o is Avogadro's

G. Hall / lonisation energylosses number, Z and A are the atomic number and weight of the material and # is the density, m is the mass of the electron and tic the velocity of the incident particle. Subsequently Vavilov [6] derived an improved solution by taking into account kinematic constraints on the energy transfer E. For a heavy incident particle, e.g. ~r, K, p, this is limited to

2mcZ/3zY2 2vm/M)'

(6)

Emax - (1 +

M being the incident particle mass and y2 = 1/1 - 132. For the collision cross section Vavilov used the form 1

-

'

(7)

which takes into account the spin of the incident particle [7]. A solution of eq. (1) can be obtained analytically by expanding the integrand in eq. (3) as a Taylor series I = p ( a - A) -

XfoEm"w(E)(1 - e -pE - p E ) de,

(8)

and ignoring terms of order p2 and higher. A is the average energy loss in the layer. Vavilov also demonstrated that the solution coincides with the Landau form for K < 0.01, where K = ~/Em~, and the result can then be written

fL(x, A) = ~qJ(X),

(9)

with q~(h) a universal function of the variable X only. , ( X ) =-1

Ce_,,y/2cos(ylny+Xy)dy '

(10)

and X

~

(1 +/32 - C) - In K,

(11)

C = Euler's constant = 0.5772... The Landau distribution, fL, is asymmetric with a tail extending to Em~,; it peaks at ~, = -0.223 and has a full width at half-maximum of 4.02 in units of h [8]. z~can be calculated from the well known Bethe-Bloch formula [9] ,~= 2 , [In(

2mc2/32"t21eft],/- f12 _

8/2],

(12)

taking into account the shielding due to the polarisation of the atoms in the material in response to the electric field of the incident particle (density effect). This effect has been calculated for many media including silicon by Sternheimer [10] and is represented by the correction 8. Other effects, e.g. shell corrections for slowly moving projectiles, are negligible for the relativistic energies considered here [9]. I, ff is the average ionisation potential of atoms in the material estimated to be 172 eV for silicon from a semiempirical formula. It is expected to

357

equal a logarithmic mean of the atomic ionisation potentials but in practice it is necessary to scale the observed ionisation energies by a factor for each element to obtain agreement with experiment. In the case of sificon this factor has a value of 1.774 [10]. Further corrections to the theory taking into account the fact that the electrons in the material are not free have been attempted by Blunck and Leisegang [11], Shulek et al. [12] and Bichsel [13,14]. In the case of solid media these calculations are aimed at the experimental observation that while the most probable energy loss agrees rather well with the prediction of the theory the width of the distribution is broader than expected and cannot be accounted for by electronic noise or imperfect resolution. The effect is particularly noticeable for very thin absorbers [1], of the order of a few hundred # m or less. The method of calculation is to expand the integral, eq. (3), to next order in p and to replace the function I of eq. (2) by I ' = 1 + 81 where

81= --x foEm'"Iw'( Z ) -- w ( e ) ] ( 1

= xp2foE" [ w ' ( E )

- w(e)]E

-

e-PE- pe ) d e

2 de

= ½xeZ82,

(13)

with w(E) the assumed form of the electron cross-section in the material and w'(E) the actual cross-section. Clearly this approach requires either theoretical or experimental knowledge of the behaviour of electrons in the material in response to the passage of a charged particle. It has been pointed out [9,15] that this approach is particularly convenient when the Vavilov and Landau solutions coincide, i.e. K < 0.01. We can easily show, using the convolution properties of Laplace transforms, that

/(x,a)

1

fT/L(a--y)e-Y2/2X*2dy.

(14)

In other words the experimentally observed spectrum of energy loops can be calculated by convoluting the Landau distribution with a normal distribution of variance x82, which can be computed with relative ease. The results of the convolution of a broader distribution with the peak value increased (usually) by a small amount compared to the Landau theory. This supposes that there are no further effects which give rise to additional broadening of the distribution. Bichsel [14] has observed that further calculation using higher orders is not expected to contribute greatly; one test of this is whether the estimates are consistent with experimental measurements. Other experimental effects are discussed later.

G. Hall / lonisation energy losses

358 Shulek et al. propose the form

8 ~

81 = 5 x ~ I ' f '

In

2mc2fl 2

(15)

I,

as an estimate of the effect, where I i is the effective ionisation potential of the i th shell and f, is the fraction of electrons in that shell. In silicon this approaches a value 1.8 × 10 3 keV 2 cm -1 for fl = 1 (using the ionisation potentials of Sternheimer). Fano [9] derives a similar formula differing only in the absence of the factor 8/3. Bichsel and Yu have made estimates using a heavy hydrogen model for heavy atoms and predict t~2 -----104 keY 2 cm -1 [16]. We have taken another approach which makes use of the accurate experimental data on absorption of photons in matter to model the energy loss process. This is discussed in the following section.

82=

3. The photoabsorption ionisation model

rE~ do = -]o NE~-EdE,

(16)

with the single electron scattering cross section ddo E

a { ~ 2~rc f12--~ l n [ (2l - f 1[, 2cl)2

Cl )

+fl4c2]-,/2

( fl2C2 )

2~r( + ~ f i E _ _ _ tan-1 _ _ Icl2 1 --C1• 2 2~rc2 1 { 2mcEf12 ]

J 1 ,'E r ~ + ~ 1 °v(E ) d E ' f tCl-ZL-'0

f0~o~(E) dE- ~'e2hZ mc

(18)

Over most of the energy range the dielectric constant is nearly real and c 1 - 1 so the imaginary part has a simple relation to the photoabsorption cross-section

at

27rZE % Nhc c]/2

2*rZE Nhc c2"

(19)

The detailed behaviour of the real part can be calculated from the Kramers-Kronig relation and takes the form

1 Nhc 1 t E

The physical basis of the model is the idea that an electrically charged particle interacts with atomic electrons as it traverses a material via the emission and absorption of virtual photons. The range of the virtual photon, and thus the cross section, becomes larger as the photon becomes more nearly real and therefore it is plausible to make use of the detailed data on interactions of ultra-violet and X-ray photons with matter to calculate ionisation energy losses. A full description of the basis of the model has been given by Allison and Cobb [5] and they successfully applied it to energy loss calculations for gaseous detectors. Only the main ideas are given here and we refer to Allison and Cobb for full details. The electric field at a moving charged particle in a medium of complex dielectric constant c = ca + icE, is calculated using Maxwell's equations, and the mean rate of energy loss is derived as the effect of the electric field doing work on the particle. The results can be written

dE (~x)

o is the fine structure constant and N is the electron density in the material; E is the kinetic energy of the ionisation electron. At large values of E the last term in eq. (17) is the only significant contribution and then the scattering cross-section converges to the simple form of the Rutherford cross-section, eq. (4), by virtue of the sum rule:

(17)

c, -- 1

rr 2 Z -

E 2 J0 ov(E' ) d E ' .

(20)

These approximations are very good for all energies above the L shell ionisation edge at = 100 eV in silicon but below this is is no longer sufficient to use data from isolated atoms since collective effects within the silicon crystal become important. For the four M shell electrons the dominant energy loss mechanism takes the form of plasma excitations of the valence electron gas. Tlie behaviour of the dielectric constant has been determined by reflectance studies using low energy photons [17] and is parameterised here as a function of frequency co by c(CO) = 1

2 COp

,

(21)

(CO+ i v ) 2 where COpis the characteristic plasma frequency and y is a parameter related to the relaxation time of the medium. Silicon has a single group of valence bands which are energetically well separated from the core states and the plasma frequency observed is in good agreement with naive calculations assuming all four electrons contribute. Except for the very low energy behaviour ( - few eV) the parameterisation provides a good description of the dielectric constant and in particular the behaviour of - I m ( 1 / c ) which is the quantity of interest for energy loss calculations. It has already been observed that the energy loss function is rather insensitive to structure in the dielectric constants at low energy [17] and we expect this to be more valid when considering relativistic particles. Once the behaviour of the dielectric constant has been determined the computation of the average energy loss and the parameter 82 is relatively easily done by numerical integration. The model has the attractive

G. Hall / lonisation energy losses feature of incorporating the density correction naturally and makes the comparison with other methods of calculation and experiment an interesting check. For high incident velocities the kinematics of the energy loss process require that the exchanged photons approach more closely to real particles; the range of allowed energy losses also increases so the finer details of the atomic cross-sections play a less significant role. For these reasons we expect the model to be most appropriate in the region fl = 1. For the photoabsorption data we have relied on twc. sources: Henke et al. [18] who have compiled and fitted experimental data from a large number of primary sources in the range < 100 eV to 10 keV, and Scofield [19] who tabulates the results of theoretical calculations from 1 to 1500 keV in energy. In the overlap region the two sets of cross sections agree typically to within 2% (though at a few points they differ by as much as 4%). The cross section was extrapolated smoothly beyond 1500 keV; this had a negligible effect on the results. Cross section data also exist for 1 to 10 eV [20] and these have also been used. In the region from 10 to 100 eV the values htop/2~r = 16.4 eV and hy/2~r=4.7 eV were used to parameterise the dielectric constant, with ~2 = 0 for E < 5 eV, and the photoabsorption cross section for solid silicon calculated from eq. (19). The cross sections calculated by this method exceed the measured values by about 14% between 5 and 10 eV; therefore the calculated values were reduced to bring them into agreement with experiment. However, the low energy region makes very little contribution to ~ d E / d x ) and 82 so these results are barely affected by this procedure. The sum rule for the integrated cross section ( f o y d E ) is then satisfied to better than 1%, and all values of or were renormalised by this amount. Varying t0p and y by reasonable amounts then had the effect of changing the estimates of ~ d E / d x ) and 82 by a few percent as the integrated cross section changed. The numerical integrations were carried out using a Simpson's rule method with adaptive quadrature to roughly an order of magnitude greater precision than the other uncertainties in the calculation.

PHOTOABSORPTION CROSS SECTION E" 10 7~_

100[

° 105[ 10

/

/

\a

(

\

10 3[[. /

\

X

~, 1°-1g 1:

10 - 3

,,coo 10 - 2

10 -1

\ 1

10

10 2

10 3

E, (keV) Fig l The photoproduction cross section for silicon used in the calculation. Between 100 eV and 10 keV the data points are taken from experiment, while above 10 keV they are the results of theoretical calculations, both sets for isolated atoms. From 10 to 100 eV the plot shows the cross section derived from a parameterisation of the dielectric constant described in the text. Below 10 eV data from measurements on solid silicon have been used.

the L-shell region. The sensitivity to the range of energy losses considered was studied by varying the lower limit of integration [eqs. (13) and (16)] and normalising the values of ~ d E / d x ) and 82 to those obtained with a lower limit of zero. This is shown in fig. 3 and it can be seen that neither quantity is very sensitive to the details of the low energy photoabsorption cross section. In the case of 82 deviations from the Rutherford form at high energies could change the result by a few percent (note that 82 results from the difference of two cross sections weighted by the energy squared) although the major

ELECTRON CROSS

SECTION

10

"~ 10 5 10 10

4. Results and comparison with experiment

359

10

4 3 2

10

The results of the calculations are summarised in figs. 1-5. Fig. 1 shows the photoabsorption cross section used as input to the calculation with the values below 100 eV being the results of the parameterisation of the dielectric constant. The resulting electron cross section ( N d o / d E ) for a fl = 1 incident particle is shown in fig. 2. The calculated cross section is compared in the figure with the Rutherford formula and the large deviations from the 1 / E 2 form are evident, particularly for

1

10-1 10 - 3

10 - 2

10 -1

1

10

10 2

(EaV)

Fig. 2. The energy loss cross section, Ndo/dE, resulting from the calculation as a function of the ionisation electron energy. N is the electron density in silicon. The dashed curve shows the Rutherford cross section for free electrons.

360

o

o~

G. Hall / lonisation energy losses

~.0

1.

3

LL

3.5

0.8

I"",

3. 0.6

'~

* Hancocket al. [21] ',

___. Shulek etal. [12,22]

2.5

__

This calculation

0.4 1.5

0.2

1. 0.5

O.

10

102

103 104 105 Lower Integration Limit (eV)

Fig. 3. The fractional values of the computed quantities shown as a function of the lower limit of integration. Curve (a) shows the total cross section, (b) the mean rate of energy loss ( d E / d x ) and (c) the width parameter 32.

contribution comes from the region where the photoabsorption cross section is best known; between - 100 eV and 10 keV. Fig. 3 also shows the behaviour of the total electron cross section; as expected the very large number of low energy transfer collisions leads to a rapid fall as the minimum energy is increased. A comparison of the average energy loss from the calculation and that computed from the Bethe-Bloch formula is shown in fig. 4. For the solid curve the values suggested by Stemheimer [10] for silicon for the effective ionisation potential (172 eV) and the parameterisation of the density correction were used. The excellent agreement is evident. This quantity is not easily mea-

ENERGY LOSS

70 60 50 40 30 20

. . . . Landau most probable

10 0

Average

_ _

J"J 10 2

i

I

I

,

,i,ii

i

10 3

i

i

i

iiiii

i

i

10 4 p

(MeV/c)

Fig. 4. A comparison of the average energy loss in a 100 ~tm layer of silicon calculated using the Bethe-Bloch formula with Stemheimer's density correction (solid line) and the results of this calculation (open squares). The dashed curve shows the most probable energy loss. Note that while the mean energy loss scales linearly with absorber thickness the most probable value does not.

O.

, ,=JJl

,

1

,

~

, ~,,,I

. . . . . . . .

10

I

102

,

,

By

Fig. 5. The width parameter 82 computed as a function of fly = p / m . The dashed curve shows the result of the calcu-

lation of Shulek et al. The data points are from ref. 21.

surable because of difficulties sampling the high energy tail of the Landau distribution but the most probable energy loss, which is directly related, is and has been shown to agree well with the Landau predictions in silicon [1,21]. Finally fig. 5 shows the quantity 82 and its variation with momentum. Given the various uncertainties in the calculation this is unlikely to be estimated to better than about 10% but this is adequate to make comparisons with experiment. Hancock et al. [21] have studied the energy loss distribution in detail using silicon detectors of 300 # m thickness and low noise electronics and have isolated the separate contributions to the shape of the distribution. Data were taken with ~r and proton beams at several momenta from 0.7 G e V / c to 115 G e V / c . Table 1 compares the values from this calculation with the fitted results of ref. 21, and also with the Shulek values [22]. The Bichsel results appear to be inconsistent with the data. The results of Hancock et al. are also displayed in fig. 5 and compared with the calculation where the agreement is seen to be quite good. Bailey et al. [23] have measured the energy deposition by 5 G e V / c ~r- in silicon C C D detectors and because of the small thickness of the active region this makes an interesting comparison. The Landau width is proportional to thickness but the width of the convoluting Gaussian is proportional to (thickness) 1/2 so the additional broadening should be more noticeable for thin layers. The active C C D region was estimated to be 16 # m deep but with an uncertainty of - + 8 # m [24] so we have estimated the peak energy loss and width for three thicknesses of material (table 2). The calculated widths and peak energy losses have been calculated by numerically integrating equation (14) assuming a Gaussian resolution function. Without the atomic binding contribution the distribution is expected to be considerably narrower than observed but when included tea-

G. Hall / lonisation energy losses

361

Table 1 Comparison of experimental observations of broadening and this calculation. Data from ref. 21. Silicon thickness x = 300±5 #m Particle

fl 2

p p p

0.38 0.53 0.81 1.00 1.00 1.00

p

p

O"= X ~ 2

OShulek

Oobserv~l

[GeV/c]

[keY]

[keY]

[keV]

0.736 1.000 1.916 30.0 45.0 115.0

6.2 5.5 4.9 6.5 6.6 6.4

11.0 9.6 8.0 7.3 7.3 7.3

10.2±1.3 7.4±0.6 5.9±1.0 5.7±0.5 5.0±0.9 5.7±1.0

Table 2 Predicted values of the most probable energy loss (Amp) and distribution width ( F = full width at half-maximum) for very thin silicon layers. Incident particle: 5 G e V / c ~-. All values are in keV.

16

22

0.1

0.075

A nap F

2.2 0.7

3.6 1.1

5.1 1.6

B)

A nap

F

2.4 2.0

3.9 2.3

5.4 2.5

A nap F

2,5 3.4

4.1 4.2

5.7 4.9

Amp F

[ k (a)

22 /amSi

0.15

10 A) a)

D)

/N

. 0.175

/,,.._.~.

5 GeV/c lr

0.125

x(~m)

C)

"E

5.3 "t-0.5 5.3+0.5

a) A) Landau theory. B) Landau theory and Gaussian resolution (o = 0.7 keV). C) B + atomic binding from model calculation. D) Experiment. Values estimated from fig. 14 [23]. sonable agreement between the estimates a n d the observations is o b t a i n e d provided the active layer is 1 6 - 2 4 # m deep. Typical distributions expected are s h o w n in fig. 6.

5. Broadening by 8-ray escape One proposed explanation for the b r o a d e n i n g of the energy loss distribution has been the escape of energetic electrons from the detector [1] in which case only a fraction of their energy would b e deposited as ionisation. A detailed calculation has been done for low incident energies b y Laulainen a n d Bichsel [27]; to m a k e a simple estimate of the m a g n i t u d e of this effect we ask the question: what is the practical range of an electron with kinetic energy E a n d what is the probability of p r o d u c i n g such a n electron so close to the edge of the device that escape is possible? By practical range we m e a n the straight line p e n e t r a t i o n distance [25] which is shorter t h a n the total p a t h length travelled because of multiple scattering. This has been para-

0.05 0.025

O.

0

2

4

6

8

10 E

(keV)

Fig. 6. The energy loss distribution predicted for 5 G e V / c ~r's incident on a 22 #m silicon layer. Curve (a) is the simple Landau theory prediction, (b) includes Gaussian resolution (o = 0.7 keV) and (c) incorporates the effects of the atomic binding.

meterised for a l u m i n i u m b y the following empirical form [26]

R=AE(1

B 1 -~CE)'

(22)

with A = 0.55 m g . c m - 3. k e V - t, B = 0.9841 a n d C = 3 × 10 -3 keV -1. If we assume that this a p p r o x i m a t i o n is also valid for silicon we can estimate the probability of p r o d u c i n g an electron with energy > E R in a thickness R. F r o m the properties of the L a n d a u distribution [3] we know t h a t

P ( E > ER) = P()~ > XR) = 1/)H~, where h is defined in eq. (11). F o r the specific case of a 20 G e V / c ~r we can write )~R = 5 6 . 1 - ~ - In R - 10.9, with E R in keV a n d R in # m . Some values are tabulated in table 3 a n d it can be seen that the effect is at the few percent level. Since few 8-rays are emitted along the direction of the incident particle the practical range R actually underestimates the energy required to escape

;

362

G. Hall / lonisation energy losses

Table 3 The probability of a 20 GeV/c ~" producing an ionisation electron in a thickness R of silicon with sufficient energy to escape. Amp = most probable energy loss. R [#m]

ER [keV]

Amp [keV]

hR

5 20 50 100 200 300 400

25 55 92 139 213 276 334

1.0 4.6 12.4 26.0 54.5 84.0 114.0

268 140 88 62 44 35 30

the detector. We conclude that only the tail of the distribution will be substantially affected.

6. Conclusions The broadening of the observed energy loss distribution in thin silicon layers compared to the simple Landau theory can be explained by the effects of binding of atomic electrons. The p h o t o a b s o r p t i o n ionisation model has been applied for the first time to a solid medium and at high incident velocities is in substantial agreement with experimental measurements. The model provides a quantitative understanding of why the effective ionisation potentials of Sternheimer are not the same as the actual ionisation potentials. Similar calculations are possible for germanium and other solid media.

References [1] H. Esbensen et al., Phys. Rev. B18 (1978) 1039. [2] P. Bonamy, P. Borgeaud, J. Movchet and P. Rancoita, Proc. Fermilab Workshop (1981) and CERN Report EP 81--06. [3] L. Landau, J. Phys. (USSR) 8 (1944) 201. [4] E. Heijne et al., Nucl. Instr. and Meth. 205 (1983) 437. [5] W. Allison and J. Cobb, Ann. Rev. Nucl. Sci. 30 (1980) 253. [6] P. Vavilov, Soy. Phys. JETP 5 (1957) 749. [7] B. Rossi, High energy particles (Prentice-Hall, New Jersey, 1952). [8] These values have been calculated by accurate numerical evaluation of eq. (10) and differ shghtly from some earlier estimates. [9] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1.

P ( E > ER)

[%] 0.4 0.7 1.1 1.6 2.3 2.9 3.3

[10] R. Sternheimer, Phys. Rev. 145 (1966) 247. [11] C. Bhinck and S. Leisegang, Z. Physik 128 (1950) 500. [12] P. Shulek, B. Golovin, L. Kulyukina, S. Medved and P. Pavlovich, Soy. J. Nucl. Phys. 4 (1967) 400. [13] H. Bichsel and S. Yu, IEEE Trans. Nucl. Sci. NS-19 (1972) 172. [14] H. Bichsel, Phys. Rev. B1 (1970) 2854. [15] H. Bichsel and R. Saxon, Phys. Rev. All (1975) 1286. [16] This value has been computed using fig. 2 of ref. 13 where the quantity D / E m a x is plotted (D = x82/2~ ) for protons with kinetic energy from 0.1 to 10 GeV. [17] H.R. Philipp and H. Ehrenreich, Phys. Rev. 129 (1963) 1550. [18] B. Henke, P. Lee, T. Tanaka, R. Shimabukuro and B. Fujiwara, At. Data Nucl. Data Tables 27 (1982) 1. [19] J.H. Scofield, UCRL-51326 (1974). The values given have been reduced by about 2% using factors given by the author to allow for the fact that a Hartree-Fock potential should give more accurate results than the Hartree-Slater potential used. [20] W. Dash and R. Newman, Phys. Rev. 99 (1955) 1151; H.R. Philip and E. Taft, Phys. Rev. 120 (1960) 37. [21] S. Hancock, F. James, J. Movchet, P. Rancoita and L. Van Rossum, Phys. Rev. A28 (1983) 615. [221 The values of I, and fi used in eq. (15) were fl = 2/14, f2 = 8/14, f3 = 2/14, f4 = 2/14 and I~ = 3.268 keV, 12 = 0.190 keV, 13 = 0.073 keV and 14 = 0.012 keV. These I, values correspond to the actual ionisation potentials multiplied by a factor of 1.774 to bring the value of Ieff into agreement with experimental observations, as in ref. 10. [23] R. Bailey et al., Nucl. Instr. and Meth. 213 (1983) 201. [24] C. Damerell, private communication. [25] Electron ranges are given in, e.g., Review of particle properties, Phys. Lett. 111B (1982). [26] K.H. Weber, Nucl. Instr. and Meth. 25 (1964) 261. [27] N. Laulainen and H. Bichsel, Nucl. Instr. and Meth. 104 (1974) 531.