Ionization energy loss by charged particles, Part II: Effects of atomic structure

.s____ ll!B ELSEVIER

Nuclear Instruments and Methods in Physics Research A 401 (1997) 263-274

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SectionA

Ionization energy loss by charged particles, Part II: Effects of atomic structure D.H. Wilkinson”-b%* “TRIUMF, 4004 Weshrook Mall. Vancouoer: BC. Canada V6T 2243 bUnirrrsi~. qf Sussex. BI-i&on. BNI 9QH. CJK

Received 21 July 1997

Abstract The Landau distribution for the probability of given energy loss by charged particles on passage through thin absorbers requires modification with regard to the atomic properties of the sample. This modification is evaluated for nine representative elements from hydrogen through krypton and for 12 values of fi from 0.3 to 0.9999. Easily evaluated approximate, but accurate, parameterizations for this Z-dependent energy loss are presented.

1. Introduction The Landau by

distribution

ionization,

through meterized

thin

of

absorbers

in Part

[l]

charged

for the energy

particles

has been

I [2] (referred

on

loss,

passage

accurately to as I). This

paradis-

only a limiting form, appropriate for circumstances under which the most probable energy loss is well below the maximum that can be sustained in a single collision between the incident particle and a (free) electron (which we refer to as the “head-on” energy loss) but under which the overall energy loss is considerable in relation to usual energy losses in such single collisions. In practical circumstances we must have regard for the explicit atomic structure of the abtribution

is,

however,

* Correspondence Eastbourne, BN20 + 44 1323 423329.

address: OBA. UK.

Gayles Orchard, Friston, Tel.: + 44 1323423333: fax:

016%9002/97/517.00 #I_’ 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00970-4

sorber and make appropriate allowance for the binding of the orbital electrons. In this paper, we make a broad exploration of these atomic structure influences on the energy-loss distribution and present the results, of good accuracy, in easily applicable form. We do not concern ourselves with finer details such as the condensation effect, the influence of plasmon excitations and other solid-state effects that may be important for work of the highest accuracy but which are not needed for purposes of general reliable initial orientation such as we aim at here. Nor do we have regard for mechanisms of energy loss, bremsstrahlung and nuclear collisions for example, other than those directly due to the atomic electrons. We treat representative elements from hydrogen through krypton. 1 I Z I 36, for projectiles of representative values of fl in the range 0.3 I fi 5 0.9999. Atomic-structure effects were first treated, semiquantitatively, by Blunck and Leisegang [3], who showed that they might substantially modify the

Landau distribution. These modifications were fully spelt out by Talman [4], who followed Chechin et al. [S] in relating the probability of ionization energy loss in single ion-electron collisions to the photo-absorption cross section. It is necessary, for purposes of practical evaluation in the present context, to replace the continuous photo-absorption distribution by a number of effective discrete lines that one naturally relates to the edge structure and other prominent features of the photo-absorption. Specifically. represent the ionization energy loss by a number of discrete lines at energies E, whose strengthsf; are proportional to the areas under the associated portions of the photo-absorption cross section curves a,(~) [6]; normalize

In Eq. (6) (7) where 11~= 0.577.. . being Euler’s constant. .lU is close to, but is not equal to, the most probable energy loss although it is sometimes confusingly referred to as such. We shall, in this paper, be chiefly concerned with @(.x[S],A), which is evidently normalized to unity - TX I A < E,. and that measures on integration the dispersion in energy loss: in what follows. we suppress the thickness index v[S] and understand Q(R) to refer to whatever conditions of absorber are under consideration.

s= N, pi

(1)

= 1.

2. The distribution of energy loss

where N, is the number of tranches into which a,(e) is cut. The effective ionization potential I may then be defined by s= N, In I = 1 ,fSIn a,. s= 1 Consider

From

Talman

[4] we have

(3

now

(8) f(x, A) = + @,(x,A),

(3)

wheref(?c, d) dd is the probability of energy loss in the range A to A + dd in the absorber of thickness x where, in obvious notation,

where si and Ci are, respectively, sine and cosine exponential integrals in standard notation [7] and where p =

2rcNe4p Z <= 2-x mv A

s

(4)

01

0.1536 Z (5) for an absorber of Smglcmzfor whichf(x, f(S, A) and where 1, is given by A-A0 A=----t’

.

i”fs ln2mc2~2~2 -j’+l. 8, ES

(

1

Now, the integrand of Eq. (8) oscillates and the evaluation off(S, A) is a matter of able delicacy, particularly for larger values only safe procedure is to effect all the expansions and to use [4]:

violently considerof a,: the relevant

A) becomes

(6)

x [Fzn + G2,(2 In @; - f12)]p’“-‘~‘”

\

D.H. Wilkinsort/Nucl.

xcos

Instr. and .&4&h. iti Phjs. Rex A 301 (1997) 363-274

% 1 (-)n+l n=l

y(A+lny)+

x CF2,+1 + G2,+1(21nB; - ~‘)]~““y’“+

’ dy,

(10)

I where

j-l

and to explore the convergence of thef(S, d) of Eq. (10) as j,, the value ofj through which the summations of Eq. (10) are in practice taken, is increased.

3. Evaluation In the evaluation of Eq. (10) two assurances must be obtained: (i) the adequacy of the representation of a,.(~) by a number of discrete lines; (ii) the convergence of Eq. (10) with increasing j,. In respect of the first required assurance, it will be evident that the least-favourable circumstance is that in which a,(~) is: (i) featureless and when: (ii) it falls most slowly with energy. Our test case, was, therefore, chosen as a smoothed CJ~(E),eliminating strong features and absorption edges, running through the de facto ~JE) for krypton. (Specifically, the de facto lnu;,(F) was fitted to a third-order polynomial in In E taken up to F:= 100 keV.) This smoothed U?(E) was divided into N, tranches in equal divisions of In E. We define the tranche energy E,~,here and in all subsequent analyses, as ln8, =

o;,(C)lncdlnc s

/ aY(Odlncn !I/s

265

Eq. (lo), taken to convergence in j,, gives, via Eq. (3), @(A) from which its maximum value Q0 is obtained and also its FWHM. Fig. 1 shows @,, as a function of N, for 1 I N, I 9 for the smoothed a,.(e) of krypton where the estimated value of @,, as N, -+ ~LCis also shown. The FWHM shows similar convergence with N, (the change of the FWHM between N, = 7 and N, = 9 is only 0.7%). The convergence demonstrated in Fig. 1 is for /I = 0.9; Smglcm~= 1.5 to which the conclusion that N, = 9 is adequate for our present purposes is not sensitive. All results presented in this paper are, unless otherwise stated, obtained using appropriate N, = 9 c,: fs representations of the CJ~(E)data from Ref. [6] with tranche boundaries taken with regard to absorption edges and any other prominent features of the 0.,(E). In respect of the second required assurance we note that convergence with j, becomes rapidly slower as the rate of fall of cry(&)with E decreases viz. as Z increases. Two examples, for fl = 0.9, that makes this point, are shown in Fig. 2 for Q0 = 0.1 which is found for argon at Smgicm~= 4 and for krypton at SmgjcmI= 5. It is seen that the rise, with I,,,, of QO(j,,) towards its saturation value is much slower for krypton than for argon. For practical purposes to secure adequate saturation in the

I

I

2

3

I

I

I

I

I

I

4

5

6

7

8

9

0.145

0.140

0.135 @?I 0.130

0.125

0.120

(13)

where the integrals are taken between the limits of the tranche. The,f, are derived, after normalization, from the integrals of o,(e) with respect to E across the tranches.

1

TRANCHES

Ni

Fig. I. Convergence of Q0 with increasing N,, the number of tranches into which the smoothed a,.(e) for krypton is divided. (The value of @,, at N, = 1 is 0.168.) The dashed line at N, = 9 indicates the estimated value of a0 as N, 4 x.

166

D.H. WilkinsoniNucl.

Instr. and Meth. in Phlbs. Res. A 401 (1997) 263-274

5

0.8

k

0.7

E .Y 20.6 &

0

I 0

0.51 ” 0

p =

0 o

+o(jm--)

0.10

3

0.08

e

0.06

5 CM.ATM.

-DISCRETE

K

0 0 0 CONTINUOUS

0.9 0.04

N 0.10

0.02 0

0 0

20

”

40

:

KRYPTON

”

60

”

80

”

100

”

120

1’

j, Fig. 2. Convergence (Do h 0.1 as j, + x (of s,,,,,zz 5).

ARGON

0.12

1

of B&j,,,) with j, for a converged value for argon (of S,,.,,,: h 4) and for krypton

evaluations to be presented in this paper, it has been necessary to permit j, to range up towards 200 in some instances. Now, as Talman [4] has emphasized, as S mg/cm- 1 + 0 we cannot expect Eq. (10) to reproduce the fictitious discrete energy-loss spectrum upon which it is based. In practical cases this question arises chiefly in respect to the highest &,-values; Talman handled this problem, in the case of argon at /I = 0.913 for an absorber of 5 cm atm. which was the only one that he considered in detail, by treating the K-shell discretely and the remainder continuously using Eq. (10). For sufficiently thin absorbers the problem does not arise because K-shell ionization is then rare and for thick absorbers we also have no worry because K-shell absorption is frequent and then it also can be treated continuously. But if the average number of K-shell absorptions is about unity, more detailed consideration is needed. However, we find that by carrying j,,, to sufficiently-high values, it is not necessary to separate out the K-shell in this way for discrete treatment. This is illustrated in Fig. 3 which compares Talman’s result for discrete treatment of the K-shell with the present continuous treatment resulting from direct application of Eq. (10) using a convergence value for j,. (For this comparison we use, for our own continuous treatment, Talman’s 4-component ~,:f,-spectrum rather than our own

0.0

LL -6

-4

-2

0

2

4

6

8

10

12

h

Fig. 3. G(n) for the stated conditions of argon using the 4component z,:f, of Talman [4] evaluated with discrete treatment of the K-shell and with continuous treatment for a convergence value of j,,,.

9-component spectrum for argon that we use for all other illustrations in this paper. Talman’s continuous treatment used only jm = 10 which was adequate for convergence in view of the fact that the highest E,-value that he used in his a,:f,-spectrum was about 4 keV whereas our own 9-component spectrum, as used in the evaluation of Fig. 2, ranges up to over 10 keV and accordingly requires higher values of j, for convergence as is seen in Fig. 2.)

4. Results After construction of appropriate N, = 9 .ss:fispectra, following Ref. [6], detailed evaluation of Eq. (10) over the range - 6 I i 5 15 has been carried out for hydrogen, helium, carbon, neon, silicon, argon, calcium, copper and krypton each for p-values of 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99, 0.999 and 0.9999. We give in Fig. 4 just two illustrations, of the comparison between experiment and the present theoretical results. Both illustrations, with experimental data for silicon [8] and argon [9], are chosen for their wide departures from the Landau distribution: in both cases Q0 is only about 65% of that of the Landau distribution. (We are, in this present study, concerned chiefly with the spread in energy loss: in Fig. 4 slight adjustments have been

D.H. Wilkinson/Nucl.

2 G&l/c 32 pm

261

Instr. and Meth. in Phys. Res. A 401 (1997) 263-274

PIONS SILICON

0 25 2 g

i

I

I

I

I

I

I

I

4

6

8

10

12

14

16

ENERGY

LOSS

0.20

(keV) n

“3

4

5

6

7

8

9

10

11

12

A (keV) Fig. 5. Distributions of energy loss for several elements each with its own value of S,,+,,,Z such as to give a most probable energy loss of 6 keV.

4 (b)

6

8

10

ENERGY

12

LOSS

14

16

18

(keV)

Fig. 4. (a) Probability ofgiven energy loss in silicon of the stated particles under the stated conditions. Full line: experiment [S]; dots: the present theory; dashed line: Ca(0.9),. (b) Probability of given energy loss in argon of the stated particles under the stated conditions. Full line: experiment [9]; dots: the present theory; dashed line: Ca(0.9),.

made along the energy-loss axes so that the peaks of experimental and theoretical distributions coincide in energy.) There is some interest in noting the Z-dependence of the energy-loss distribution for a given most probable energy loss. Fig. 5 shows such distributions for some elements in the Z-range that we study all evaluated for a most-probable loss of 6 keV. In order to make presentation and application of our results tractable they have all been referred to those for calcium at /? = 0.9 which we name Ca (0.9). Call Smgicm2[Ca(0.9)] the thickness of calcium

that at /J = 0.9 gives the same value for Q0 as does j [Z(j)] for the element in question at the va ue in question. Fig. 6 displays Smg,c.m~ [Ca(O.9)]/ P;S mg,cml [Z(B)]; they also display the amount, A).,,. by which Q(L) evaluated for Ca (0.9) at Smg,‘cm~ [Ca(0.9)] must be shifted to the right along the i-axis so that its maximum matches that of the element in question in position along that axis as well as in its @,-value. Call this final so-shifted Q(L), derived from calcium at {j = 0.9, Ca(0.9),: the Ca(0.9) surrogate. The question naturally arises as to how well the Q(i) of Ca(0.9), matches the @(jL)of Z(p) of which it is the surrogate. An example is given in Fig. 7; the quality of the match does not depend much on Z, /L Smgicmz.Other illustrations of the match of the precisely computed a(n) with Ca(0.9), are given in Fig. 8. Fig. S(a) shows the dependence upon b of the precisely computed @(i’) for argon for 0.3 I fi _< 0.9999 (@(i’) differing from Q(n) only by slight shifts along the i-axis so that, for ease of inter-comparison, all the @(A’)peak at n’ = 0); all evaluations are for a0 = 0.12. Fig. 8(a) also shows the associated Ca(0.9), (which is found for Smgjcm: = 9.62). Fig. S(b) shows the similar dependence upon Z, for p = 0.99, for all elements studied, helium through krypton (hydrogen lies outside our range for pbO= 0.12 for which Fig. 8(b) is also constructed) again with the associated Ca(0.9),.

5. Parameterization

which we are here concerned. However. the computation of Ca(0.9),, for whatever Smg.‘cmZ[Ca(0.9)]value is required for it by Figs. 6, remains a heavy task: a more easily applicable method is needed. This is provided by a parameterization of Ca(0.9),.

All investigations, of which we have presented only illustrations, confirm that Ca(0.9), is an adequate surrogate for the precisely computed Q(L), certainly for the purposes of orientation with

ti

KY

t

11 ’

\

\ -03

““‘I

I

I

1

/

‘\

i

1/1111

I

I

1

10 S mg/cm’

(c)

10

S mg/cm

*

CARBON

t

1

I

IllIll

I

I

10

S mg/cm

100

’

Fig. 6. S,,/,,. [Ca(0.9)]/S,,!,,: [Z(/I)] for the elements studied in detail labelled by the B-values (unringed): (a) hydrogen (presented in terms of the Smpicm2for deuterium for approximate uniformity of Z/A with the other elements);(b) helium; (c) carbon; (d) neon; (e) silicon: (f) argon; (g) calcium; (h) copper; (i) krypton. The other full lines give the amount A& (ringed) by which the @(A) for Ca(0.9). thus determined. should be shifted to the right along the i-axis so that its peak matches in &value as well as in magnitude that of the element in question. The dash-dotted lines indicate equality of do and the “head-on” energy loss.

D.H. Wilkinsm/Ntrcl.

(e)

269

Imtr. arzd Meth. in Phys. Res. A 401 (1997) 263-274

SILICON

(f)

ARGON

i

1

s mg/cm' 100

’ ’ ’ “‘I’

’

’ ’ I I””

(4)

10

100

Smg/cm’ ’ ’ ’ ’ 11”

CALCIUM

(h)

COPPER

S mg/cm

2

Fig. 6. (Continued).

In Fig. 9 we present, as a function of Smg,cm~rQ0 for Ca(0.9) and also A0 which is the value of 1, for which @(A) finds its maximum: call this latter &[Ca(0.9)]. We now find that an adequate parameterization of @(l.) for Ca(0.9),, which we call @(A),, is given by @(A), = GOexp

- d2 n + b6 + cc?’ + dh3 1 ’

(14)

where 6 = i, - Aos and E.,, = i. [Ca(0.9)] + A&. The coefficients a, b. c, d, are determined from the values of 6, namely ii-,,, and L,.,, at which Q(i), falls, respectively. to one-half and one-tenth of its peak value to the left of the peak and, similarly, a,,, and 6,., to the right of the peak. L,.,, 6-0.r, & and dO,r are presented in Fig. 10. For convenience we note, writing 6 _ 6 6_ 0.5 = 62, 6_ 0.1- -6 a,., = 63, 1, 0.5 4

270

(i)

KRYPTON

S mg/cm

2

+

0.06

( -

s:s;-

s1s2s; - S,d,S,S, + 6,&dl

(15c) 11= (St&S, + S1S,S4 - SIS,S, - S2S,S,)(y He

0

Ne

0

Ar

0

x)/D. (15d)

where D = (6, - cS3)(b3- cS2)(d4- d,)(ci, - d2).~y.

,~. ”_

Kr

-5

0

5

10

15

x Fig. 7. Comparison betw-een the precisely-computed @(A)for the elements in question at /I = 0.9999 (the circles). all for such S.,,I,,2-values as yield Do 2 0.1, and the @(A)for Ca(0.9). for the same respective @,-values, shifted along the A-axis to match at the peaks viz. the Ca(0.9), (the full lines).

(15a)

(1%)

(15e)

The adequacy of @(i,), as a representation of the Q(i) of the surrogate Ca(0.9),, as also as of the precisely computed Q(R), is seen in Figs. 4 and 8. The values of @,, and & are correct by construction so the question of adequacy refers chiefly to the form of @(;.),. One measure of form is the FWHM. As may be expected, the fit between @(i.), and the @(,I) precisely computed for 2, /I, Smg;cm: is poorest for elements remote from calcium. Fig. 11 shows the error in the FWHM resulting from the use of @(j*), to represent helium and krypton over the range of /I and Smg,cmzcovered: it is seen that the maximum error is about 6% and that it is usually considerably less; errors become the smaller the closer to Z = 20 that one goes.

D.H. WilkinsonjNucl.

and Meth.

Phys. Res.

401 (1997)

171

0.12 0.10

_

03ip10.9999

0.10

0.08

_

'x z,

0.08

'4 0.06

z

0.06

0.04

0.04

0.02

0.02

0

-6

-4

-2

0

2

(a)

4

6

8

10

12

14

0

16

A1

-6

-4

-2

0

2

(b)

4

6

8

10

12

14

16

A’

Fig. 8. (a) The @(i’) for argon for Q0 = 0.12 and for all /I: 0.3 4 p 2 0.9999 all lie between the lines. The @(L’) are the G(L) shifted along the i,-axis so as to peak at i’ = 0. The circles show @(I) for Ca(0.9), for Q0 = 0.12, also shifted to i’ = 0 and the triangles similarly the corresponding parameterized @(I), from Eq. (14). (b) As in (a) but with the lines enclosing the @(i.‘) for all elements studied from helium through krypton for j = 0.99

008

3

10

100

S

mg/cm’

Fig. 9. a,, and & for Ca (0.9) as a function

s mg/cm

of S,,+,,,:.

Fig. 6 has presented results for discrete B-values and for a limited number of Z-values: how do we deal with intermediate values of fi and of Z? Interpolation on fi for a given Z is not a problem: it is smooth and is easily effected from Fig. 6. However, the purpose of our present exercise has been to make allowance for atomic effects which are by no means always smooth with Z owing to the atomic shell structure so that we should not expect interpolation in Z itself to be smooth. Fig. 12 taken from Fig 6 present the Z-dependence of Smg:cm~[Ca(0.9)]/

Fig. 10. The d_,,,, s mg,‘Cll+

K,,,,

6,, 5, 6,.

’

, for Ca(0.9) as a function of

S mg,cmz[Z(/$] for the usual range of p-values for some Smg,cmAvalues. It is seen that the overall trends of Smgicm’ [Ca(0.9)]/SmgjcmL [Z(p)] with Z, which are themselves easily understood, are indeed modulated, in a systematic although not in a simple way, by shell effects. Interpolation on the eye-guide lines should give a reasonable estimate but its quantitative reliability is unknown. To summarize the recipe: (i) for a given S mg,cmz[Z(/$] obtain from Fig. 6, directly or by appropriate interpolation (the latter being reliable

D.H. Wilkinson ~Nucl. Instr. and Meth. in Pfys. Res. A 401 (1997) 263-274

212

in /J but more unsure in Z), Smgjcm~ [Ca(0.9)]/ s ,,_,_,~ [Z(j)] and hence the SmElcm~[Ca(0.9)] for Ca(0.9) that gives the same Q0 as does Smgicmz [Z(/?)] for the Z(p) in question; (ii) from

8 !S

6

I

[a,_]

Fig. 9 obtain @,, and the & [Ca(0.9)] of this s mg/cm:[Ca(0.9)1; (“‘) 111a g ain directly or by interpolation from Fig. 6 obtain Ai0 hence & = i.,, CCa(0.91 + AA,,; this JoS of the surrogate Ca(0.91, is the same as the directly computed i, of Smg_,z [Z(p)]; (iv) from Fig. 10 using Smg,‘_,z [Ca(0.9)] obtain &0.5, (5_,.,, ii,,, and bO,r and hence. via Eqs. (15). the a, b, c and (1 to use in Eq. ( 14). where 6 = 2 - &, for the final Q(i),,.

HELIUM

6. Higher values of energy loss

S mg/cm’ Fig. 1I. The percentage errors in the FWHM-values, for helium and for krypton with the p-values stated on the curves. on using the parameterized @(j.), for Ca(0.9) rather that the precisely computed G(i) for the Z(B) in question. A positive error means that the FWHM for the precise Q(2) is larger than that for a(?,),.

We have been chiefly concerned with presenting an easily evaluated way of estimating the width of the atomically modified Landau distribution in the region of its peak. What happens for higher values of i? What about the normalization of Q(i) across - x I i. I XI ? The question of practical interest relates to @(I.),. We note that we should not strive for exact normalization, nor for an exact prescription for the extension of Q(2), above the range on the basis of which it has been derived, namely up to @(A), = @,/lo above the peak (name the value

[Z(p)]) as a function of Z for the stated p-values for: (a) S,,,+,,~ = 1: (b) S,,.,,: Fig. 12. Values of Z(S,,:,,> [Ca(O,9)]/S,,+,: = 10; (d) S,,,,,: = 30. The circles show values taken from Fig. 6; the lines joining the circles are eye-guides only. S RIF,‘Crn’

= 3; (c)

Fig. 12. (Cor~tind)

I

’ “““’

I

I

10

IllIll

I

I

100 S

mCl/cm

2

Fig. 13. The sohd line shows the amount by which, as a percentage of oo. 0(i), at & , exceeds the @(i.) of the pure Landau distribution at the point. The dotted line shows the percentage amount by which the integral of Q(i),. over i. up to iL01 plus the integral over the pure Landau distribution from i,, to i = x (the Q(&) of I with its & = i, , ) exceeds unity. (For this figure &, = 0.)

of i at which this is found 3,0,1), since the basis of the Landau distribution itself then becomes unsure on account of the condition related to the “head-on” energy loss. For practical purposes it is sufficient simply, bearing in mind the condi-

tion to which reference has just been made. to use a composite Q(2) consisting of Q(2), up to i,, 1 and the pure Landau distribution above iO,r. It is now of interest to ask: (i) how severe is the mismatch between @(;_), and the pure Landau distribution at AO.r? (ii) what do we find by integrating across Q(i), up to jbO,r and adding to this quantity the integral of the pure Landau distribution above i+r viz. how nearly is our composite @(j.) normalized to unity? For the integral above /,O.l we use Eq. (31) of I and display the result in Fig. 13 (illustratively for i, = 0). We see that the mismatch at i,,, is never more than a few percent of CD,,,that the integral is similarly never more than a few percent away from unity and that the differences reported tend, as expected, towards zero as Smg,‘um2tends to infinity; these results are satisfactory.

References [l] [I?] [3] [4]

L.D. Laudau. J. Exp. Phys. (USSR) 8 (1944) 201. D.H. Wilkinson. Nucl. Instr. and Meth. A 383 (1996) 513. 0. Blunck. S. Leisegang, Z. Phys. 12s (1950) 500. R. Talman, Nucl. Instr. and Meth. 159 (1979) 189.

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D.H. Wilkinsor~iNd.

Instr. and M&h. in Ph?s. Rex A 401 (IYY7) 263-274

51 V.A. Chechin, L.P. Kotenko, G.I. Merson. V.C. Yermilova. Nucl. Instr. and Meth. 98 (1972) 577. j] B.L. Henke. E.M. Gullikson, J.C. Davis Atomic Data and Nuclear Data Tables 54 (1993) 181; E.B. Saloman. J.H. Hubbell. J.H. Scofield. Atomic Data and Nuclear Data Tables 38 (1988) I.

[7] M. Abramowitz, LA. Stegun, Handbook of Mathem Functions, Dover Publications, New York. 1965. [8] J.F. Bak et al., Nucl. Phys. B 288 (1987) 681. [9] E.A. Kopot et al., Sov. Phys. JETP 70 (1976) 397.