Energy resolution of thin silicon semiconductor ΔE detectors for alpha particles and heavy ions

NUCLEAR

INSTRUMENTS

AND METHODS

I3I

(I975) 61-68;

©

NORTH-HOLLAND

PUBLISHING

CO.

ENERGY R E S O L U T I O N OF T H I N SILICON S E M I C O N D U C T O R AE D E T E C T O R S FOR ALPHA PARTICLES A N D H E A V Y IONS V. V. AVDEICHIKOV, E . A . GANZA and O. V. LOZHKIN

V. G. Khlopin Radium Institute, Leningrad, 197022, U.S.S.R. Received 13 August 1975 The energy resolution of high performance silicon A E detectors with thicknesses 3.8, 10.7, 15.2 and 38.0/tin has been measured for 4He, 12C, 14N, 15N, 160, 2°Ne, 22Ne and 40Ar ions in the energy interval 0.5-5 MeV/amu. The results are compared with

calculations according to Tschal~ir with "binding effect" corrections. The discrepancy between the theory and experiment is discussed.

1. Introduction Thin silicon semiconductor AE detectors are widely used for the identification of charged particles as a front element in telescope-detector systems, for example, in the A E - E technique. The feasibility of A E - E identification depends primarily on the characteristics of the AE detectors, such as the thickness uniformity, microscopic homogeneity, the energy-loss distribution function. In order to carry out an experiment in optimal conditions from the point of view of particle mass resolution it is necessary to know the quantitative criteria for the preselection of the AE detectors. The most important characteristics for such detectors are the energy resolution for heavy charged particles and the energy-loss distribution function. In an ideal case the energy resolution of a very thin AE detector is defined by energy-loss fluctuation of charged particles that pass through the detector sensitive region. For the case when the rate of the energy loss is much smaller than the primary particle energy the theory of the mechanism for the energy-loss process has been developed by Landau') and Vavilov2). The basic assumption of the abovementioned theories is that electrons in the stopping material can be considered to be free and that the charge and the velocity of the incident ion are constant throughout the whole stopping process. The corrections due to the "binding effect" of the electrons in the stopping material 3 5) and due to the escape of 6-electrons from the active volume of the "thin absorber"*) have been introduced in Vavilov's theory. Recently Bichsel 7) has put forward the energy-loss straggling problem for the case of "very thin absorbers". Within the limits of the experimental errors, there is rather good agreement between the measured energy-loss distributions and those predicted by theories for solid absorbers. So, the

limiting value of energy resolution of the d E detectors can easily be predicted. Neither of the abovementioned assumptions of Vavilov's theory is valid for heavy ions passing through the "thick absorbers", when the particle energy-loss AE is of the same order of magnitude as the incident energy E. In this case the distribution function of energy losses f(A) is determined not only by statistical fluctuations in the number of large-energy-transfer collisions, but also depends essentially on the change of the average rate of particle energy losses during the slowing-down process. The contribution of the nonstatistical term to the distribution function of energy losses may dominate for "thick absorbers". The study of energy-loss distributions of moderate and low energy charged particles passing through matter has a long history, but the subject has not been exhausted of its possibilities up to now because of various experimental refinements and a new theoretical approach. More complete calculations made recently by Tschalfir s) and Payne9), independently, replace approximate results of early theories of Bohr and others"-'2). The experiments with proton and a-particle beams at moderate energies 13, ~6) have given evidence for the validity of Tschal~ir's calculations. But in the region of low energies of the incident ions and for the case of "thick absorbers", that is of special interest in nuclear physics experiments, the information is quite poor26'27). In our preceding paper 17) the energy-loss fluctuation of ~-particles and some heavy ions in silicon absorbers has been studied. Rather good agreement with the theory of Tschal/ir has been found for a-particles, but at the same time there was a great discrepancy for heavy ions. The present study was intended to establish the 61

62

v.v.

AVDEICHIKOV

reasons of these discrepancies and to obtain more complete experimental data concerning the energy resolution of silicon A E detectors and the distribution function of energy losses of heavy ions in the energy region ~ 0.5-5 MeV/amu using the new experimental method. 2. Principles of the method 2.1.

EXPERIMENTAL ARRANGEMENT

The schematic diagram of the experimental arrangement and associated electronics is shown in fig. 1. in our method the variable quantity is the incident particle energy, not the absorber thickness, as in traditional experiments on energy straggling of charged COLLIMATOR DETECTORS ~--o.8mm ~E E

FOILS

\

7

et al.

particles. The heavy ion beam was slowed down to the necessary energy by AI-foils and was registered by the A E - E telescope system. The collimator with diameter 0.8 mm and 0.8 mm thick was placed in front of the telescope in order to minimize the effects of possible macroscopic non-uniformity of the AE-detector. In such a method the incident ions energy dispersion is very large due to the Al-foils, but for our purpose a beam with a minimum energy dispersion was needed. The problem of forming the effective incident beam with low energy dispersion has been solved in the following way. The signals from the thin front A E detector and the thick back E detector (thick enough to stop the particles with highest energy after penetrating the front detecto0 were sent to preamplifiers, amplifiers and analog-todigital converters. Then the digitized signals were sent to the summator to get the value of Etota I = Eres. q-zJE. The digital window was set on E .... + A E signals. Its position and the width, usually (0.2-0.4)% Etot,~, was varied by the operator and defined the effective incident energy, and only such events of A E that satisfied the window limits were sent to the P H A analyser. In such a way the energy-loss distribution function of charged particles in the A E detector for a large interval of the incident energies from Eto,~l A E to Etot~~= E~.......... was obtained. An energy calibration for each detector was obtained using a 226Ra a-source in conjunction with a precision pulse generator. The gain of the A E and E systems were matched via pulser and the ~-source. By the same method the linearity of the system has been checked. 2.2. DETECTORS

pH

A - 4096

]

Fig. 1. Schematic diagram of the experiment and the associated electronics. P is preamplifier, ADC is amplitude-to-digital converter, PHA-4096 is pulse-height analyser.

High quality semiconductor surface-barrier Sidetectors with thicknesses 3.8/~m, 10.7pro, 15.2~tm and 38.0/~m were used. A low noise semiconductor Si-detector served as back E detector*). The nominal silicon resistivity was 0.5 0 . 6 k Q . c m . The silicon wafers from which the detectors were made were cut at 5 ~' in relation to the (111) axis in order to reduce the channeling effects when the detectors were oriented perpendicularly to the particle beam. In these measurements we applied a bias voltage ~ 1V//~m, considerably higher than necessary to deplete the detectors completely, so the thickness of the silicon dead layer should be the same for all detectors. The gold layer in all cases was about 30/~g/cm 2. The energy resolution * All detectors were manufactured by the authors in the V. G. Khlopin Radium Institute.

63

THIN S I L I C O N S E M I C O N D U C T O R DETECTORS

(fwhm) for Etota I = E .... + A E was measured using a 216Ra a-source and was found to be 30-35 keV for all detectors with the exception of the 3.8/~m detector. In the latter case the energy reso!ution was about 45 keV due to detector capacity noise. The energy resolution of the AE detector depends primarily on the thickness uniformity of the sensitive volume. One should distinguish two kinds of inhomogeneity: 1) Macroscopic inhomogeneity - this is the gradual variation of the detector thickness that may be determined, for example, by scanning its surface with a beam of collimated a-particles. In our experiment this sort of inhomogeneity of the AE detectors was minimized by the special technology of their production and with the help of a 0.8 m m copper collimator, placed in front of the AE-E-telescope. 2) Microscopic inhomogeneity is a sort of small distortion of the ideal planar surface that may severely damage the energy resolution of the AE detectors. Such inhomogeneities, "pinholes" for example, are inherent to Al-foils, and this was the reason for the " a n o m a l i t y " in the magnitude of the energy-loss straggling of a-particles for A1 absorbers 18). In case of silicon absorbers (wafers of the AE detectors) the second kind of inhomogeneity arises from the small surface destructions that were not totally diminished by polishing and etching processes. The microscope scanning of the transparent silicon wafers that have been used for our previous experiment 17) has revealed a tooth-like structure on the back surface. This kind of inhomogeneity can not be excluded by collimating of the incident beam. Therefore special care was taken in fabrication of the silicon wafers that were to be made into AE detectors for the present experiment. The microscopic analysis which was used for the control of the wafers production technology did not reveal a surface destructive structure for the fabricated detectors. Furthermore, the effects of microscopic inhomogeneities should influence the resolution less if AE detector output signals are used for analysis, contrary to the remaining energy, as is usually done in energy-loss straggling measurements. The microscopic inhomogeneity can be included in this case in the AE detector dead layers. Indeed, it cannot be conceived that the active surfaces of the detector sensitive volume will represent directly the inhomogeneity profile of the Si wafer in all details. Accurate determination of the detector thicknesses was obtained by exposing them to 226Ra a-particles; the range-energy relation from the tables published by Northcliffe and Schilling 19) has been used.

TABLE 1 Characteristics of heavy-ion beams. Ion

4He

Incident energy max. (MeV)

8.78

/)

/;

zeZ h-1

ZeZh-1

4.5

0.65

12C

42.8

1.92

0.84

14N 160

50.0 26.6

1.65 0.97

0.83 0.57

~°Ne 4°Ar 15N

30.6 33.4 107.0

0.82 0.31 2.35

0.62 0.41 1.2

22Ne

142.5

1.56

1.14

Source

Cyclotron FTI Academy of Science, USSR

Cyclotron JINR, Dubna

2.3. SOURCES The experiments were performed using a-particles of natural a - s o u r c e s (228Th, 226Ra, 239pu) and external ion beams of the heavy-ion cyclotrons of J I N R , Dubna, and A.F. loffe FTI, Leningrad. The characteristics of the beams are listed in table 1. In column 2 the maximum incident energy is presented. The minimum incident energy was restricted by the thickness of the AE detectors. The values in columns 3 and 4 are calculated for the case of maximum incident energy. In the measurements with 15N and 22Ne ions the AE detectors were placed in the output focus of the magnetic analyser, set of lab. angle 40 ° to the incident beam scattered on a Th target2°). The absorbers consisting of Al-foils were used to slow down the incident ions, and were set in front of the Th target. The energy spread of the ion beam (0.16%) that passed through the AE detector was defined by the collimator of 1 mm placed in front of the AE detector. The ionbeam divergence in the analyser output focus was about 6 °, so the small correction due to this effect has been introduced in the experimental data. The ion energy was determined by measuring the magnetic field value. During the experiment continuous control of incident beam energy and its energy dispersion was made by an additional E detector which was manufactured in the same manner as the AE one. There were two kinds of corrections that were introduced in the experimental results. One is for the noise of the detectors and electronic circuits, and the second is the correction due to the incident beam

64

v.v.

AVDEICHIKOV et al.

energy dispersion. The latter has a form

6 (AE)/AE = c 6Einc/Einc ()'~;), where 0 < c < 1 has been defined from our experimental AE=f(E~,~) relation. Both corrections were within the limits of the statistical errors for all detectors with the exception of the AE detector of 3.8/ira thickness, where the correction was somewhat larger because of detector high capacity noise. 3. Results and discussion

3.1. A L P H A

PARTICLES

Fig. 2 shows the alpha particle energy resolution (fwhm) plotted against the part of energy lost by a-particles within the A E detector of 15.2/~m thickness. The theoretical curves are calculated following Bohr~°), Williams 2~) and Tschal/irS). In Tschal~ir's theory the decrease of particle velocity within a path length Ax is taken into account. However, it is suggested, that the particle velocity should be higher than that of all atomic orbital electrons of the incident particle z and of the absorber material Z, i.e.

v > zeZ/h,

(I)

v > ZeZ/h,

(2)

In general both these assumptions are not true both for ~-particles and for heavy ions in the energy region I

I

I

I

I

\ ~

--6

I-

=, o

I

I

15.2pm { {

Z 0

I

{

{{

{ { Tschalar(k~L I{

--t. monr

I,LI n-

-

investigated in this work (see table 1). The correction due to atomic structure was taken into account by Bethe and Livingston l~). They derived the expression for the energy-loss standard deviation under the condition v = const (Ax) as follows:

2

2,/£2

% - L = ~, k2 =

+ -

(3)

In 2

3 i Z2mv 2

(4)

I i /"

In addition to the definition already given, crR is the standard deviation in straggling given by Bohrl°), Z i is the number of effective electrons in the ith shell for which I i < 2 m v 2, Z ' = ~ Z i , li is the average excitation energy of the Z i electrons in the ith shell, m is the mass of the electron and K; is the average kinetic energy of the bound electrons in the ith shell, the sum extending over all shells for which l i < 2 m v 2. When the energy-loss probability distribution is Gaussian, the energy resolution of the A E detector is given by ~2(fwhm) = 2.355 a,

(5)

and in the energy region where the Bohr formula is valid, it will be: ~2 = 10.1 z(Ax,um)~keV.

(6)

The correction term k 2 may be calculated directly from eq. (4). Nevertheless, there are two approximations in the calculation of k z based on the use o f the average ionisation potential of the atom of the stopping material2~), or on the use of the average kinetic energy of atomic electronsZ2). Both these approximations give an increased value of k z, as compared to the one which follows from eq. (4). In fig. 3 we have drawn the binding

~ 4 ~ . .

I

I

I

I

I

I

20

k2

U.I Z 111

d.-PARTICLES

-2

HEAVY I O N S

1.8

1.4 ~ 0.2

I

I

O.l,

I

I

0.6

I

I ENERGY.

0.8

I

I

,tE/E

Fig. 2. Theoretical predictions and experimental data for the :c-particle energy resolution (fwhm) v s / J E / E . The value of AE/E is the part of energy lost by incident z-particles in the AE detector.

1.2

2

I

ois

11o

,15

2io

-

2~

31o

Ion energy. MeV/amu Fig. 3. Corrections due to atomic structure for heavy ions in silicon according to Bethe Livingston 11) and Williams21).

THIN SILICON SEMICONDUCTOR

effect corrections k 2 for the silicon absorber following Williams z~) and Bethe-Livingston [eq. (4)]. In the latter case we have taken K~= 2I~ and I~=2.21J~ (where J, is the ionization potential of the ith shell) following Sternheimer's 23) method of choosing 1~. We have introduced the corrections due to atomic structure in Tschal~ir's calculation in the manner =

"QVsch

(7)

kB-L"

In fig. 2 we presented the results of these calculations. It is seen, that a modification of Tschal~ir's calculation results in an improved description of the energy resolution in the whole interval of the c~-particle incident energy, lit should be noticed that the value of

DETECTORS

the factor k 2 introduced by the abovementioned method is lower than it should have been. To simplify the calculation, the value of 21 was taken in eq. (4) instead of the more correct value of K~. Such an approximation may be considered reasonable for absorbers with Z ~ 7, but for absorbers with higher atomic numbers K~>2I~. Calculation of k 2 with assumptions similar to those accepted by Williams 2t) and Titeica 22) would give a factor k 2 greater than is necessary to describe the experimental data in fig. 2. So, we have come to the conclusion that a satisfactory description of the experimental data on energy resolution for a-particles in A E detectors may be derived from the existing theories. The same extent of agreement I

I

10

I

I

l

~N{{~.~

f

I

I

65

1

I

I

I

I

o 4Hev ~50 14N a2ONe

3.8jJm

I

I

15.2um J

I

I

* ~He m~60 ,, 12C =~ONe

A15N ~40A

g6

i

D

"

4He

-

""--L.L

Z

- - - 2 2 2 2 , 2 0

(11

~4N 2ONe

~

I

-

-

02 1

_

tONe

40 A

I

0.4 I

I

06

0.8

I

0i2

t

014

energy, aE/E Fig. 4.

lol

I

I

I

I

I

0i6

I

0i8

J

I

t

energy, aE/E Fig. 5.

I

1

I

I

I

o 4He ,t75N

10.7JJm

• 120

8

=22N

e

[

I

I

I

I

38.0urn ,y

10

• ;2C • ~SN ,~ 14N

8

& I/*N

6

E"

.9 =4

ZHe

~2

14N

~4N

2O~e I

0.2 I

I

0)4

J

0] 6

energy, ,aE/E Fig. 6.

I

0.8 1

I

q

0i2

[

0i4

l

0i6

I

0.18

I

energy, ,~E/E Fig. 7.

Figs. 4-7. Theoretical and experimental energy resolution values for heavy ions in 3.8 Hm, 10.7 Hm, 15.2 Hm and 38.0 Hrn detectors vs AE/E. The value of`dE/E is the part of energy lost by incident heavy ions in the ,dE detector. Solid lines represent the calculations according to Tschal~r with k 2 correction included.

66

v . v . AVDEICHIKOV et al.

between the theory and experiment was observed for other detector thicknesses (see figs. 4-7). 3.2. HEAVY IONS The experimental energy-resolution values (fwhm) for heavy ions in d E detectors are given in figs. 4-7. Solid lines represent the results of theoretical calculations following Tscbal/ir; k2_L corrections are included. It must be noted, that such a direct comparison of the experimental data and theoretical calculations is not correct in the case of heavy ions. Indeed, the theory describes energy-loss straggling distribution f(A), but the experimental energy loss distribution function is determined by the simultaneous action of the straggling process in the AE silicon absorber and the energy resolution inherent to the silicon E detector for heavy ions with energies equal to AE MeV. The latter value has been measured for ~SN and 22Ne ions by the E detector placed in the output focus of the magnetic analyzer and was found to be 0.5-0.8% in the investigated heavy ion energy interval. No attempt was made to correct the observed ionization distribution f ( A E ) for adequate comparison with the theoreticalf(A) (or their fwhm). The following conclusions can be drawn from the experimental data shown in figs. 4-7: l) The experimental energy resolution of the AE detectors for heavy ions does not practically change in the interval of AE detector thicknesses 10-40 pm. For a broad region A E / E = 0 . 2 - 0 . 8 the mean values of experimental resolutions are equal to 3.1% for ~2C; 2.9% for 14'LSN and 2.4% for 2°'22Ne ions. 2) There are systematic discrepancies within the factor 1.1-1.4 in absolute values of experimental and calculated energy resolutions. 3) The energy resolution of the AE detectors improves when the mass of registered ions is increasing, as follows from theoretical predictions. 4) There is a non-significant difference in general behaviour of the experimental dependences and calculated curves for the region AE/E ~ 0.2-0.8. The possible explanation for discrepancies that exist between the experimental data and theoretical calculations in the AE detectors energy resolution can be explained by the lack of detector uniformities (that were mentioned in section 2.2) or by the effect of charge exchange on the energy-loss fluctuation of heavy ions with Z > 2 . The theoretical consideration of the latter effect has been presented recently by Vollmer24). It was found that the contribution of the charge exchange to energy-loss fluctuation may give a

factor of 1.3-1.5 for the investigated heavy ion energy interval and this explains completely the observed discrepancies. However, in order to be able to prove that the existing discrepancy is caused by charge exchange processes it is necessary to be fully confident that the effects of thickness inhomogeneities in the detector sensitive volume are excluded completely. But one may suppose that all observed differences between the experimental and calculated data are caused by the thickness inhomogeneities alone. In this case the fwhm variation in detector sensitive volume thickness Ax can be found from the expression: ~(/[X)

=

2 2 2 ½ (~Qex0--QHl--~theor)

/(dE/dx),

(8)

where dE/dx is the specific energy loss of heavy ions at the degraded energy E - A E . ~?exp and Q,hoor are the experimental and theoretical values of energy dispersion for heavy ions, Qu~ - the energy resolution of the E detector for heavy ions at energy AE, that includes the statistics of the ionization process, the lack of complete collection of electron-hole pairs, etc. The values of f2(pm) determined by this formula are equal to (0.10+0.01)am, (0.15+0.01)pm and (0.18 _+0.01)pm for AE detectors 3.8/tin, 10.7pm and 15.2 pm thick. The value of inhomogeneity was found to be approximately the same for every AE detector in the investigated A E / E = 0 . 2 - 0 . 8 interval. Such small magnitudes of estimated values of Q can be easily ascribed to inhomogeneity due to detector fabrication procedure. Therefore the value of the charge exchange effect on the energy-loss fluctuation in the silicon absorber is, from our point of view, an open question. However, it is necessary to note that combined action of two reasons inhomogeneity and charge exchange seems more probable. 3.3. DISTRIBUTIONFUNCTION OF ENERGY LOSSES According to the straggling theories, the distribution function of energy losses in the investigated energy region should be close to Gaussian. Tschal/ir's theory predicts a small deviation from Gaussian for the region of AE/E ~ 1, but the experimental observation of this effect escapes our experimental possibilities. However, the measurements show that the distribution function is not really a Gaussian, but has a long low-energy tail. Fig. 8 exhibits a typical shape of the experimental distribution function for the case A E / E ~ 0 . 5 . The charge coordinate has been normalized to Z,rr by scaling according to the fwhm of the peaks. In addition, all mass peaks have been normalized in intensity to

THIN

I

SILICON

I

SEMICONDUCTOR

i

I

I

--

67

DETECTORS

I

I

~

I

I

o /.He

v160 O20Ne

.~0A

103

102

Z O

°10

./I

,,"

I

\

'

Op"/f 0ok

102

I

- 0.20Z

I /(/i

-0.15Z

-0.10Z

I

-0.0 5Z

I

Z

0,,05Z

0.10Z

i

(~ISZ

I

02 OZ

NORMALIZED NUCLEAR CHARGE Fig. 8. G r a p h s h o w i n g the extent o f tailing to Z - a Z in the nuclear charge s p e c t r u m constructed from the energy-loss distribution fimctions. T h e yield o f low-Z tailing at true Z value m i n u s one is calculated as N ( a Z = - 1 ) / N ( Z ) . The scale for abscissa is in normalized nuclear charge units for incident heavy ions.

compare directly the shapes of distribution functions for different heavy ions. It can be seen that there is similarity in shapes for all investigated ions (from 2"oft ~ 5 for 12C to Zefr ~ 11 for 4°A). The plots give directly the shapes of the low-Z tailing, the information useful in the AE-E identification method. The yield of low-Z tailing at true Z value minus one is calculated as

N (aZ = - 1)/N (Z).

(9)

Tailing amounted to the value of 0.37% and 0.06% for 4°Ar (Z = 18) and Z°Ne (Z = 10) for the AE detector 15.2/~m thick. The following conclusions may be drawn about the tailing: 1) The relative yield of low-Z tailing is approximate131constant in all investigated AE/E energy intervals for a given AE detector. 2) The relative yield of low-Z tailing decreases when the detector thickness is increased. The origin of this tailing was discussed by Bowman

and others2S). Although the tailing effect in our detectors is much smaller than the one observed in ref. 25 the explanation for this effect based on channeling seems true. We wish to express our gratitude to the Chief of the Laboratory, Prof. N.A. Perfilov for his support and encouragement. We are very much indebted to Prof. I. Kh. Lemberg for his interest in this work. We wish particularly to thank A.G. Artukh, I.N. Chugunov and M.P. Kudojarov for assistance in performing this experiment. We would also like to express our thanks to the operating staffs of the U-300 J I N R cyclotron and A.F. Ioffe FTI cyclotron for their cooperation. References 1) L. L a n d a u , J. Phys. ~) P. V. Vavilov, Zh. z) O. Blunk a n d S. O. Blunk a n d K.

U.S.S.R. 8 Eksperim. Leisegang, Westphal,

(1944) 201. i Teor. Fiz. 32 (1957) 920. Z. Physik 128 (1950) 500; Z. Physik 130 (1951) 642.

68

v.v.

AVDEICHIKOV

4) W. Rosenzweig, Phys. Rev. 115 (1959) 1683. 5) H. Bichsel, Phys. Rev. B1 (1970) 2854. 6) M. Loulainen and H. Bichsel, Nucl. Instr. and Meth. 104 (1972) 531. 7) H. Bichsel, Phys. Rev. A9 (1974) 571. s) C. Tschal~ir, Nucl. Intsr. and Meth. 61 (1968) 141; C. Tschal~r, Nucl. Instr. and Meth. 64 (1968) 237. 9) M. G. Payne, Phys. Rev. 185 (1969) 611. 10) N. Bohr, Phil. Mag. 30 (1915) 581; Kgl Dansk. Vidensk. Selsk. Mat.-Fys. Medd. 18 no. 8 (1948). 11) M. S. Livingston and H. A. Bethe, Rev. Mod. Phys. 9 (1937, 245. 12) K. R. Symon, P h . D . Thesis (Harvard University, 1948) published by B. Rossi, High energy particles (Prentice-Hall, Englewood Cliffs, New York, 1952). 13) j. j. Kolata, T. M. Amoc and H. Bichsel, Phys. Rev. 176 (1968) 484. 14) j. A. Penkrot, B. L. Cohen, G. R. Rao and R. H. Fulmer, Nucl. Instr. and Meth. 96 (1971) 505. 15) H. Nann and W. Sch~ifer, Nucl. Instr. and Meth. 100 (1972) 217.

et al.

16) C. Tschal/ir and H. D. Maccabee, Phys. Rev. BI (1970) 2863. t7) V. V. Avdeichikov, E. A. Ganza and O. V. Lozhkin, Nuc[. Instr. and Meth. 118 (1974) 247. is) G. Weber, L. Quaglia, Nucl. Instr. and Meth. 118 (1974) 573. 19) L. C. Nortbcliffe and R. F. Schilling, Nuclear Data Tables 7 (1970). 2o) A. G. Artukh, V. V. Avdeichikov, J. Er6, G. F. Gridnev, V. L. Mikheev and V. V. Volkov, Nucl. Instr. and Meth. 83 (1970) 72. 21) E. J. Williams, Proc. Roy. Soc. (London) A 135 (1932) 108. 22) S. Titeica, Bull. Soc. Roum. Phys. 38 (1939) 81. 23) R. M. Sternheimer, Phys. Rev. 103 (1956) 511; 117 (1960) 485. o4) O. Vollmer, Nucl. Instr. and Meth. 121 (1974) 373. 25) j. D. Bowman, A. M. Poskanzer, R. G. Korteling and G. W. Butler, Phys. Rev. C9 (1974) 836. 26) D. L. Mason, R. M. Prior and A. R. Quinton, Nucl. Instr. and Meth. 45 (1956) 41. .)7) j. j. Ramirez, R. M. Prior, J. B. Swint, A. R. Quinton and R. A. Blue, Phys. Rev. 179 ([969) 310.