NOM B Nuclear Instruments and Methods in Physics Research B 83 (1993) 21-30 North-Holland Beam Interactions

with Materials 8 Atoms

Production of L, subshell and M shell vacancies following inner-shell vacancy production

S. Puri ‘, D. Mehta a, B. Chand a, Nirmal Singh a, J.H. Hubbell b and P.N. Trehan a a Department of Physics, Panjab Uniuersity, Chandigarh-160014, India b National Institute of Standardr and Technology, Ionizing Radiation Diuision, Gaithersburg, MD 20899, USA

Received 2 November 1992 and in revised form 25 March 1993

The probabilities for transfer of vacancies from the K shell to an L, subshell (TJ =,) and to the M shell (T&, and from the L, subshell to the M shell (nL,+,) are evaluated for elements with atomic numbers 18 I Z rz 96 using the theoretical radiative transition rates of Scofield [Phys. Rev. A 9 (1974) 1041; At. Data Nucl. Data Tables 14 (1974) 1211 and radiationless transition rates tabulated by Chen et al. [At Data Nucl. Data Tables 24 (1979) 13; Phys. Rev. A 21 (1980) 4421. The calculated vacancy transfer probabilities are least-squares fitted to polynomials to obtain analytical relations that represent these probabilities as a function of atomic number.

1. Introduction

The ionisation of an inner shell electron leaves the atomic system in a very unstable electronic configuration which may be followed by either a radiative transition or an Auger process. In either of these two alternative decay modes, the initial inner shell vacancy is transferred to a higher shell or subshell and additional higher shell vacancies may be created. This vacancy cascade process continues until all vacancies reach the outermost occupied shell. Theoretical modeling of the complete atomic rea~angement processes is very complex and requires understanding of many effects, which are often obscured because of the many possible pathways.

In this paper, we consider only the L (M) she11 vacancies produced following the decay of a K (K or L) shell vacancy. These data regarding the vacancy transfer probabilities (vii) are very essential in the basic studies of nuclear and atomic processes, e.g. in the study of the L (M) shell X-ray emission following the excitation by incident photons having energy above the K (K or L> edge of the element or following radio-active decays fll. In the text, the symbol 77ij represents the number of jth shell vacancies resulting from the decay of an ith shell vacancy through single step processes. The tota number of jth she11 vacancies resulting from all the transitions initiated by the filhng of an ith shell vacancy, i.e. including multistep and Coster-Kronig (CK) transitions, is denoted by the symbol Fjiij.

The probabilities for transfer of vacancies to L and M shells, following the de-excitation of singly ionised atoms in the K shell or one of the L, subshells were reported for elements in the atomic number region 20 12: I 94 by Rao et al. 121. In these calculations of K to L, subshell vacancy transfer probabilities, nKLt, the contributions due to the Auger and radiative transitions were derived using the best fitted experrmental data on the K shell fluorescence yields and intensity ratios of different components of K-LX (X= L, M, N, . . * > Auger electrons and the K X-rays, available upto that time. The vacancy transfer probabilities -qLiM and nKM were evaluated 121 using the X-ray emission rates tabulated by Scofield [3I and the Auger transition probabilities calculated in j-coupling using nonrela- tivistic hydrogenic wave functions. The other set of vaIues of Li to M shell vacancy transfer probabilities,

Correspondence to: P.N. Trehan, Department of Physics, Panjab University, Chandigarh 160014, India.

0168-583X/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

22 S. Pun’ et al. / Production of outer-shell vacancies

?TjLiM, for eight elements in the atomic number region 50 I Z I 90 were reported by McGuire 141. These probabilities were deduced using radiative and radiationless transition rates calculated in LS coupling using the nonrelativistic approximate reman-S~llman Q&IS) central potential. Recently, we have reported the experimentally deduced vacancy transfer probabilities 77xr_ [5] and qrM [6] for elements in the atomic number regions 37 I Z I 42 and 70 I Z I 92, respectively, at 22.6 keV incident photon energy.

More complex theoretical calculations and Monte Carlo simulations for the determination of probabil- ities of differently ionised products or, equivalently, for the number of Auger electrons emitted following the cascade decay of an inner she11 vacancy, for a few low-Z elements are available in the literature [7-lo]. Measurements on various charge distributions produced as a result of inner-shell photoionisation have been reported, in the case of inert elements [lo].

In the present work, we have evaluated the probabilities, 7KLj, 7riM and nm, for elements in the atomic number region 18 I Z 5 96, using the radiative transition rates of Scofield fllJ21 and radiation- fess transition rates of Chen et al. [13,14]. Also, the total numbers of M shell vacancies produced from all transitions, initiated by the filling of a K shell vacancy (;i-i& and an Lj subshell vacancy (;rSr.&, including multistep processes have been computed. These probabilities are of direct use in various applications.

2. Basic formaIism

In the present calculations, it is assumed that the energy of the ionising photon or particIe is significantly above the threshold, where the two-step model of excitation-de-excitation is applicable. Following the definition of Rao et al. [2], the probability for the transfer of a vacancy from a K to an L, subshell, qroj, is defined as the number of Lj subshell vacancies produced in the decay of one K shell vacancy through the radiative K-L, transitions or through the Auger K-L,L, and K-LiX (X= M, N, * - * ) transitions. This definition does not include the L, subshe vacancies produced through the CK transitions of the type L,-L,X. The equations defining these probabilities are as follows:

rlKLt = &) [WKL,) + 2UI%W t- UKL,L,) + &(KL,L,) + Tk(KLiX)I T (1)

77KL2 = & [GdFd + v4ww2) + w%Ld + LwbL3) + I’*W2X)l s (2)

%L3 = & [Mu,) + 2Lww4,) + c4ww3) + m%L3) + fx=4)17 (3)

(X=M,N,O;**).

In these equations, r, and I”” are the radiative and the Auger partial widths corresponding to the transitions between the shells written within the brackets, and r is the total level width.

The probability vLzM represents the number of M shell vacancies produced in the decay of a Li subshell vacancy through single step processes, i.e. through the radiative L,-M, transitions and the Auger L,-M,M, L-M,X (X= N, 0, *. * ) and L,-L,M, transitions. The qL,M values have been calculated using the following equations, given by Rao et al. [21,

- 1 C [ r,(LiMj) + 2T,(L,MjMj) + T,(LiMjX) + &(LiL,Mj)] 7 77L,M - f(Li) j

(4)

X=M, (k>j),N, O*** and n>i,

i = 1, 2, 3.

S. Pun’ et al. / Production of outer-shell vacancies 23

The total numbers of M shell vacancies, YjjLiM, that result from all transitions initiated by the filling of an Li subshell vacancy, i.e. also including the contributions due to multistep processes, are given by the following equations:

qL$l= qL,M +.fhL,M + (f-13 +f12f23hL3M~ (5)

?rL,M = qL,M +f23~L3M3 (6)

SjL,M = qL,M 9 (7)

where fij (i = 1, 2, 3; j > i) denotes the Li subshell Coster-Kronig transition probabilities. The probability for the transfer of a vacancy from the K to the M shell, nm, is defined [2] as the

average number of primary M shell vacancies produced per K shell vacancy decay through a radiative K-M transition or an Auger transition of the type K-LM, K-MM or K-MX (X = N, 0, * . .). The M shell vacancies created in multistep processes, in which the decay of a K shell vacancy first leads to production of an L shell vacancy which then decays further to produce one or more M shell vacancies, are excluded from this definition. The expression defining nm is as follows:

1

‘KM= T(K) i - C [ I”,(KM,) + 2r,(KMMj) + T,(KLM,) + r’.(KMiX)I 3

X= M,(j > i), N, 0, * * a.

The total number of M shell vacancies produced by the filling of an initial K shell vacancy including the multistep processes has been calculated using the relation

‘is, = 7)kM + C9KL,GLiM. (9) i

3. Evaluation procedure and discussions

The probabilities for transfer of a vacancy from the K shell to an L, subshell vKL, (i = 1,2,3) and qKL (= C qioi) for elements in the atomic region 18 I Z -< 96 have been evaluated using eqs. (1-3). In these evaluations, the K shell radiative rates based on the relativistic Hartree-Slater theory are taken from the tabulations of Scofield [11,121. Two sets of radiationless transition rates based on the RDHS model are chosen. The first is calculated using the jj-coupling scheme [131 and the second one is calculated in the intermediate coupling with configuration interaction (ICCI) 1141. The corresponding probabilities are denoted by vKLi ( jj) and qIKL_ (ICCI), respectively. The radiationless transition rates in the ICC1 scheme [14] are available only for the K-L,L, transitions. These transitions contribute 70-G% to the total radiationless transition rates for all the elements. In the present evaluation of rjkr, (ICCI), the transition rates for the dominating Auger K-LiLj transitions based on the ICC1 scheme [14] and for the K-L,X (X= M, N, * * . ) transitions based on the jj-coupling scheme 1131 have been used. It may be noted that the sum of intensities of groups of radiationless transitions that include all the possible jj-configurations which correspond to a given LS-configuration is independent of the coupling scheme 1131. Therefore, the sum of the major contributing K-LiLI transition rates and the sum of K-L,X (X = M, N, . . . > transition rates are independent of the couplmg scheme. This validates the use of K-L,X (X = M, N, * . . > transition rates calculated in the jj-coupling scheme 1131 in the evaluation of Q,+ (ICC11 to a reasonable extent.

The value of vacancy transfer probabilities, qKLi (ICCI) and vu, (jj>, for the elements in the region 18 5 Z f 96 are plotted in fig. 1 along with the values for these parameters tabulated by Rao et al. [2]. It is clear from fig. 1 that the vro, (ICCI) and nKL, (jj) values are in general agreement with each other, while the values of qm, g iven by Rao et al [2] are lower by upto 30% for elements below Z = 35 and above Z = 80. In the case of the qkr, and vKL3 parameters (fig. 0, the three sets of values are in

24 S. Pun’ et al. / Production of outer-shell vacancies

l TKLi (ICC1 1

’ 7~ (Rae I

A TKLi fii )

- Fitted Curve

.2tl -

-16 -

*oa -

.OL -

0 I I I 1 I I t , I

18 26 XL L2 50 58 66 ?L 82 $0 98

ATOMIC NUMBER. -

Fig. 1. K shell to L, subshell vacancy transfer probabilities as a function of atomic number.

agreement with each other, except for the elements below 2 = 40, where the qxr, (jj) and qro, (jj) values deviate significantly from the other two sets. This is because of the fact that the electrostatic interaction starts competing with the spin-orbit interaction and the simple jj-coupling scheme alone is inadequate to explain the Auger transition rates in this atomic region 2141. The present evaluated total

S. Puri et al. / Production of outer-shell vacancies 25

l Ttj fleer 1

0 qij IRoo)

- FItled Curve

ATOMIC ~UM6ER -

Fig. 2. K to L shell vacancy transfer (q& and to M shell vacancy transfer Gj KM1 probabilities as a fmxtion of atomic number.

vKL values, which are independent of the coupling scheme used, agree remarkably with the values of Rao et al. [2] in the atomic region 18 I 2 I 96 (see fig. 2).

it has been observed that the major contributing K-LL Auger transition probabilities calculated using ICC1 [14] and the radiative transition rates based on relativistic Hartree-SIater theory (123 lead to good agreement with the experimental data in the entire atomic range 18 I 2 I; 96 [11,14]. Therefore, the set of qKLi (XXI) values given in table 1 can be used in various applications.

In the present calculations of Li subshell to the M shell vacancy transfer probability, rfLiM li = 1, 2, 3) (eq. (411, the RDHS model based L shell radiative emission rates tabulated by Scofield El21 and the Auger transition rates calculated in the $-coupling scheme assuming frozen orbitals by Chen et al. [13] have been used. The probabilities, FjtiM, have been further deduced using eqs. (S-7) and the CK transition probabilities based on the RDHS model [15,16]. It is worthy to mention that the nonradiative parts of the qlL.M and the CK transition probabilities are also independent of the coupling scheme as mentioned earher in this paper 1131. Therefore, the values of Auger rates based on the RDHS mode1 in the jj coupling scheme [13’j are applicable for the evaluation of Q+ and -ijLtM, for all the elements over the atomic number region 25 I 2 I 96. The presently calculated probabilities ?J,+~ and ?jLiM are listed in table 2.

26 S. Pun’ et al. / Production of outer-shell vacancies

Table 1 K Shell to Li subshell and K shell to M shell vacancy transfer parameters

Z Q=, -=z q=, %I_ TKM GM 18 0.318 0.77 0.64 1.72 20 0.298 0.72 0.63 1.65 25 0.246 0.59 0.66 1.50 30 0.190 0.50 0.65 1.34 35 0.141 0.41 0.63 1.19 36 0.132 0.40 0.63 1.16 40 0.104 0.36 0.61 1.07 42 0.093 0.35 0.60 1.04 45 0.078 0.33 0.57 0.99 47 0.070 0.32 0.58 0.97 49 0.063 0.32 0.57 0.95 50 0.060 0.31 0.56 0.94 52 0.054 0.31 0.55 0.92 54 0.049 0.31 0.55 0.90 56 0.045 0.30 0.54 0.89 60 0.039 0.30 0.53 0.87 63 0.035 0.30 0.53 0.86 67 0.031 0.30 0.52 0.84 70 0.029 0.29 0.51 0.84 74 0.026 0.29 0.50 0.83 75 0.025 0.30 0.50 0.82 77 0.024 0.30 0.50 0.82 78 0.024 0.30 0.50 0.82 80 0.023 0.30 0.49 0.81 83 0.022 0.30 0.49 0.81 88 0.021 0.30 0.48 0.80 90 0.021 0.30 0.48 0.80 92 0.021 0.30 0.47 0.79 96 0.021 0.30 0.47 0.79

0.18 3.38 0.17 3.01 0.17 2.64 0.18 2.50 0.17 2.25 0.17 2.16 0.17 2.00 0.17 1.92 0.17 1.81 0.17 1.78 0.17 1.72 0.17 1.67 0.17 1.62 0.17 1.56 0.17 1.51 0.17 1.46 0.17 1.42 0.17 1.37 0.17 1.35 0.17 1.34 0.17 1.33 0.17 1.30 0.17 1.25 0.17 1.19 0.17 1.17 0.17 1.16 0.18 1.15

The present results for TI~.~ are compared with the available nonrelativistic calculations of McGuire [4] and those of Rao et al. [2]‘in fig. 3. The comparison shows that the present values of n,+, and nr+, agree with both tabulations, and are smooth functions of Z in the atomic ranges 36 s Z I 96 and 25 s Z 5 96, respectively. For the q?L,M p arameter, we find good agreement among the present values and that of McGuire [4] and Rao et al. [2] except for the pronounced discrepancies around Z = 74 and Z = 50, where the cutoff and onset of L,-L,Me, Coster-Kronig transitions are expected [17].

Recently, a detailed analysis of L X-ray spectra, induced by 2.5 MeV protons for the elements with Z = 47-53 and by 50 keV electrons for the elements with Z = 73-83, has been performed by Xu et al. [18,19]. This analysis was based on a model that takes into account all the physical events taking place during the initial ionisation and inner-vacancy processes. The results of this analysis demonstrate that the L,-L,M,,, Coster-Kronig transitions (i) are fully operative up to Z = 49 and their intensity decreases abruptly at Z = 50, and (ii) are forbidden in Ta and W (Z = 73, 74) and are allowed in Ir, Pt (Z = 77, 78. The L,-L,M, and L,-L,M, CK transitions are probably located at Z = 75 and 78, respectively. Our experimental measurements of sjLM for the elements with atomic numbers 71 I Z I 92 [6], also support these results. These observations are also in accord with the predictions of L shell CK energy calculations made by Chen et al [17]. The present values of the probabilities nr_+,, and SjLIM have been evaluated by considering the onset and cutoff of various Auger transitions in accordance wrth the predictions of Chen et al. [17].

It may be noted that the values of YjL,M (i = 1, 2, 3) for an element differ significantly from each other, i.e. the average sjLM (= Cn, X ;iiL,M, Cn, = 1) parameter depends upon the number of initial Li

S. Pun‘ et al. / Production of outer-shell vacancies 27

Table 2 Li subshell to M shell vacancy transfer parameters

2 qL,M 1)L,M 1)L,M a SjLIM +iL2M

25 1.02 1.98 1.98 2.92 1.98

30 1.03 1.93 1.98 2.92 1.98

35 1.03 1.93 1.97 2.83 1.97

36 1.02 1.73 1.91 2.75 1.91

40 1.00 1.62 1.87 2.56 1.86

45 0.97 1.51 1.79 2.44 1.78

47 0.95 1.47 1.76 2.38 1.75

49 0.94 1.44 1.73 2.51 1.72

50 0.83 1.42 1.72 1.70 1.70

52 0.80 1.38 1.69 1.68 1.67

54 0.78 1.35 1.66 1.65 1.65

56 0.75 1.32 1.64 1.62 1.62

60 0.72 1.30 1.60 1.58 1.58

63 0.70 1.29 1.58 1.56 1.54

67 0.66 1.26 1.54 1.51 1.50

70 0.68 1.24 1.52 1.48 1.46

74 0.65 1.20 1.47 1.43 1.40

75 0.81 1.19 1.43 1.85 1.39

77 0.81 1.17 1.43 1.83 1.36

78 0.85 1.16 1.42 1.96 1.34

80 0.84 1.13 1.39 1.91 1.31

83 0.84 1.10 1.35 1.87 1.27 90 0.82 1.03 1.27 1.72 1.16 92 0.82 1.01 1.25 1.71 1.19 96 0.81 0.98 1.21 1.64 1.24

a TLxM = SjL3M.

subshell vacancies (n,>, as a result of which the ;ii rM parameter analogous to the average L or M shell fluorescence yields (Wt, and GM) [20,211 cannot be universally defined.

The present calculated values of the probabilities for transfer of a vacancy from the K shell to the M shell, v~, for the elements in the atomic region 25 _< 2 I 96 are given in table 1. These values have been evaluated using the radiative rates of Scofield [11,12] and radiationless transition rates of Chen et al. [13] in eq. (8). As the nonradiative part of the probability qm is also independent of the coupling scheme considered, so the set of the Auger transition rates calculated in the j-coupling scheme [13] is applicable in the evaluation of the nonradiative part of the vW for the elements in the atomic number region 25 I 2 I 96. The qm values are almost constant over this atomic number range and agree well with the available values for the elements in the atomic region 47 I 2 I 70 [2]. The probabilities F,, which are of direct practical use, have also been deduced using presently calculated vacancy transfer probabilities in eq. (9). The results are presented in table 1 and fig. 2.

The present calculated values of the different vacancy transfer probabilities 7ij have been fitted to polynomials in Z as

qij = CanZn. (10) n

The resulting coefficients a,, and the atomic number range of validity for each of the polynomials are listed in table 3. Precaution was taken to avoid any undesirable oscillations in the regions between the calculated points. The fitting regions for the probabilities 7rIM and SjL,M were subdivided because of sudden changes in their values at Z = 49, 75 and 78. The accuracy of the fit was checked for each element and was in most cases in agreement with fitting errors of less than 2%.

The fact that the relativistic K and L shell yields (fluorescence and CK) and the level widths calculated using the relativistic radiative [11,12] and radiationless [13,14] transition rates explain the available

28 S. Pun’ et al. / Production of outer-shell vacancies

l TLiM (Chen)

o TLiM (McGuire)

A ‘r\LiM ( ROO)

- Fitted Curve

(cl

(b)

0.601 I

2L 32 LO LB 56 6L 72 80 08 96

ATOMIC NUMBER -

Fig. 3. Li subshell to M shell vacancy transfer probabilities, qLiM, as a function of atomic number.

S. Pun’ et al. / Production of outer-shell vacancies 29

Table 3 Coefficients of the fitted polynomials (C,e,Zn) for the vacancy transfer parameters

Para- meter

Atomic range

aa

lfm, 18-35 0.504 - 1.036 30-96 0.9021 -4.1162 7.459 -6.1582 1.9367 -

7fm* 18-96 1.748 - 7.720 15.358 - 13.555 4.482 -

?J=, 18-96 0.117 5.747 - 21.675 35.661 - 27.515 8.127

7fKL 18-96 2.293 - 1.649 - 14.623 42.701 - 42.930 15.017 77KM 25-96 0.214 - 0.206 0.301 -0.134

?jKh4 25-96 3.302 19.135 - 128.750 217.377 - 246.221 82.380 qL,M 25-49 0.771 1.694 - 2.767

50-74 2.156 -3.967 2.617 75-77 1.076 - 0.350 - 78-96 1.007 - 0.203 -

77L*M 25-35 2.126 - 0.587 -

36-96 6.079 - 24.830 49.882 -45.331 15.154 -

7)L,M = 25-96 - 0.833 28.894 - 107.597 182.316 - 147.57s 46.009 SjL,M 25-49 - 6.223 81.664 - 232.954 209.292

so-74 2.298 - 1.268 0.138 75-77 2.714 - 1.150 -

78-96 4.271 - 3.947 1.255

+L2M 25-96 - 2.485 46.833 - 181.322 326.340 - 282.210 94.236

a TL,M = ?L3M*

experimental data in a better way [11,14,15,22,231 than the nonrelativistic calculations, indicates the reliability of the present calculated vacancy transfer probabilities. It may be mentioned that the radiationless transition rates used in the present work are based on single confi~ration calculations performed within the independent particle model. The configuration interaction and other many-body effects can lead to significant deviations for certain radiationless transitions [24,25]. For example, the RDHS theory overestimates L,-L,,,M,,, rates by about 10% for heavy elements, while for elements with 27 < 2 < 50 this discrepancy can be as large as a factor of 3 1151. The solid state effects and exchange splitting can also contribute to differences between calculated free atom vacancy transfer rates and the me~urements done using solid targets, for low-2 elements [17,26,27].

Acknowledgements

Two of us (S.P and D.M) wish to ac~owledge the support of the University Grants elision and Council of Scientific and Industrial Research, New Delhi, India, for this work. This work was supported in part by the US-India Foundation.

References

[l] W. Bambynek, B. Craseman, R.W. Fink, H.U. Freund, H. Mark, C.D. Smith, R.E. Price and P.V. Rao, Rev. Mod. Phys. 44 (1972) 716.

[2] P.V. Rao, M.H. Chen and B. Crasemann, Phys. Rev. A 5 (1972) 997. 131 J.H. Scofield, Phys. Rev. A 179 (1969) 9. [4] E.J. MC&ire, Phys. Rev. A 3 (1971) 587. [51 S. Puri, D. Mehta, B. Chand, N. Singh and P.N. Trehau, Nucl. Instr. and Meth. B 73 (1993) 443. 161 S. Puri, D. Mehta, B. Chand, N. Singh and P.N. Trehan, Nucl. Instr. and Meth. B 74 (1993) 347. [7] B.F. Rozsnyai, V.L. Jacobs and J. Davis, Phys. Rev. A 21 (1980) 1798. 181 V.L. Jacobs, J. Davis and B.F. Ronnyai, Phys. Rev. A 21 (1980) 1917.

30 S. Pun’ et al. / Production of outer-shell vacancies

[9] V.L. Jacobs and B.F. Rozsnyai, Phys. Rev. A 34 (1986) 216. [lo] J.C. Levin, C. Biedermann, N. Keller, L. Lilijeby, R.T. Short, LA. Sallin and D.W. Lindle; Phys. Rev. Lett. 65 (1990) 988, and

references therein. [ll] J.H. Scofield, Phys. Rev. A 9 (1974) 1041. [12] J.H. Scofield, At. Data Nucl. Data Tables 14 (1974) 121. [13] M.H. Chen, B. Crasemann and H. Mark, At. Data Nucl. Data Tables 24 (1979) 13. [14] M.H. Chen, B. Craseman and H. Mark, Phys. Rev. A 21 (1980) 442. [15] M.H. Chen, B. Craseman and H. Mark, Phys. Rev. A 24 (1981) 177. [16] S. Puri, D. Mehta, B. Chand, N. Singh and P.N. Trehan, X-Ray Spectrom. 22 (1993) in press. [17] M.H. Chen, B. Crasemann, K. Huang, M. Aoyagia and H. Mark, At. Data Nucl. Data Tables 19 (1977) 97. [18] J.Q. Xu, Phys. Rev. A 43 (1991) 4771, and references therein. [19] J.Q. Xu and E. Rosato; Phys. Rev. A 37 (1988) 1946. [20] M.O. Krause, Phys. Rev. A 22 (1980) 1958. [21] M. Pajek, A.P. Kobzev, R. Sandrik, A.V. Skrypnik and G. Lapicki, Phys. Rev. A. 42 (1990) 261. [22] M.H. Chen, B. Crasemann and H. Mark, Phys. Rev. A 21 (1980) 436. [23] S. Singh, D. Mehta, R.R. Garg, S. Kumar, M.L. Garg, N. Singh, P.C. Mangal, J.H. Hubbell and P.N. Trehan, Nucl. Instr. and

Meth. B 51 (1990) 5. [24] J. Tulkki and T. Aberg, J. Phys. B 18 (1985) L489. [25] K.R. Karim, M.H. Chen and B. Crasemann, Phys. Rev. A 29 (1984) 2605. [26] L.I. Yin, I. Adler, M.H. Chen and B. Crasemann, Phys. Rev. A 7 (1973) 897. [27] J.C. Fuggle and S.F. Alvarado, Phys. Rev. A 22 (1980) 1615.