K-, L- and M-shell x-ray production cross sections by 1–1.3 MeV protons

Nuclear Instruments and Methods in Physics Research B 325 (2014) 84–88

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K-, L- and M-shell x-ray production cross sections by 1–1.3 MeV protons E. Batyrbekov a,1, I. Gorlachev b,⇑, I. Ivanov b,2, A. Platov b,3 a b

National Nuclear Centre, Krasnoarmeyskaya 2, Kurchatov 071100, Kazakhstan Institute of Nuclear Physics, Ibragimov 1, Alatau 050032, Almaty, Kazakhstan

a r t i c l e

i n f o

Article history: Received 5 November 2013 Received in revised form 25 December 2013 Available online 13 February 2014 Keywords: X-ray production cross section PIXE Electrostatic accelerator

a b s t r a c t In 2012, our institute had initiated a series of research activities aimed to measure the characteristic x-ray production cross-section, arising from interaction of accelerated particles with target atoms. This paper presents the data of x-ray production cross-sections under excitation of K-, L- and M-shells of target atoms in the range of mass from Mg to Bi by protons of the energies in the range of 1–1.3 MeV. We used the approach based on calculation of x-ray production cross sections through the cross-section of Rutherford backscattering, which can be calculated with high accuracy from the Rutherford formula. Such approach reduces the measurement errors of x-ray cross-sections and thus improves the accuracy of the obtained data. It is further planned to expand the area of research to protons of other energies and heavy charged particles in the energy range of 0.5–1.7 MeV/nucleon. Ó 2014 Published by Elsevier B.V.

1. Introduction The task of improving the accuracy of quantitative determination of elemental composition of the samples by the method of xray fluorescence with ion excitation (PIXE) requires obtaining the most reliable data, used as the fundamentals of the method, such as the cross section production of characteristic radiation produced by interaction of accelerated particles with target atoms [1]. Currently, there are several theoretical models with various reliability, describing the processes occurring in the interaction of accelerated ions with the inner electron shells of the target atoms. Since the generation of the characteristic x-ray radiation is the result of several processes at the atomic level, starting from the primary ionization by charged particles up to the subsequent filling of vacancies by the electrons of outer shells, it is necessary to provide the accurate description in the theoretical calculations. An approximation of the perturbed stationary state (ECPSSR theory) [2–4] allows to obtain the most satisfactory description of the x-ray production cross-sections for the range of proton energies 0.5–1.7 MeV and the ratio of charges of incident particle atoms and target atoms 0.03 < Z1/Z2 < 0.3. ECPSSR model is the result of plane-wave born approximation (PWBA) improvements, taking into account the energy loss of the incident particle, ⇑ Corresponding author. Tel.: +7 (727) 3866855. E-mail addresses: [email protected] (E. Batyrbekov), [email protected] (I. Gorlachev), [email protected] (I. Ivanov), [email protected] (A. Platov). 1 Tel.: +7 (2251) 23333. 2 Tel.: +7 (727) 3866800. 3 Tel.: +7 (727) 3866855. http://dx.doi.org/10.1016/j.nimb.2013.12.025 0168-583X/Ó 2014 Published by Elsevier B.V.

coulomb deflection of ion, polarization and the energy change of electrons bond in atoms (using the method of perturbed stationary states) and relativistic correction of electrons mass. Using ECPSSR theory the satisfactory results were obtained on cross section ionizations for K-shell of target atoms [5]. The less reliable calculated values of cross section ionizations for L-shells due to greater complexity are presented in [2,6]. At the same time, considering the importance of increasing the accuracy of the x-ray cross section production determination, the work on obtaining empirical data using the ion accelerators is continued up to the present time in various analytical laboratories.

2. Experiment The tandem type accelerator was used in the experiments for production of accelerated proton beam [7], with the possibility of energy variations in the range from 300 keV to 1.7 MeV. The target plane was oriented perpendicular to the incident beam direction in the experiments. Samples were prepared in the form of metal films deposited by magnetron plasma deposition on an organic substrate. The thickness of the used films ranged from 29 to 200 lg/cm2. X-ray production cross-sections in the experiments were calculated through the Rutherford backscattering cross-section, therefore x-ray photons and backscattered particles were simultaneously detected. X-ray irradiation was detected by Si (Li) detector of 12 mm2 area and 145 eV resolution at 5.9 keV located at 135° with respect to the incoming beam direction. The detector is provided with 8 lm thick protective beryllium window. For detection of K-lines of Zr, Nb,

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Mo, Ag, Cd, In, Sn, Sb and L-lines of Ta, W, Pb, Bi the 25 lm thick Al absorber was located before the detector to reduce the contribution of low-energy x-ray radiation to the measured spectrum. To detect all M-lines, L-lines of Cu, Zn, Zr, Nb, Mo, Ag, Cd, In, Sn, Sb and K-lines of Al, Mg, Ti, Cr, Cu, Zn absorber was not used. To calculate the efficiency of x-ray detection system the set of calibration sources Eu-152, Eu-154, Eu-155, Na-22, Mn-54, Ba-133, Cd-109, Co-57, Zn-65, Am-241 and Fe-55 was used. X-ray fitted detection efficiency without absorber is presented in Fig. 1. For detection of backscattered particles we used the surface-barrier detector of 20 mm2 area, located at 142° angle to the incoming beam direction. The solid angle subtended by the surface-barrier detector was 9.1 msr in the experiments. The calculation of the x-ray production cross section is given by:

rX ¼

Fig. 1. X-ray fitted detection efficiency without absorber.

  NX DX XR rR lt t ; eX NR DR 1  elt t

ð1Þ

Table 1 The measured K-shell x-ray production cross sections (in barns). Element

Ep [MeV]

Ktot

Ratio measured to the reference Ktot cross section, [reference]

Mg

1.0 1.1 1.2 1.3

525 ± 37 575 610 638

0.82 0.79 0.79 0.73

[8] [8] [8] [8]

Al

1.0 1.1 1.2 1.3

463 ± 33 494 528 561

0.85 0.80 0.79 0.78

[8], [8], [8], [8],

Ti

1.0

37.2 ± 3.5

4.8 ± 0.4

42.0 ± 3.9

1.1 1.2 1.3

45.0 55.0 65.5

5.8 6.8 8.5

50.8 61.8 74.0

1.13 [15], 0.66 [16], 0.69 [17], 1.01 [18], 0.77 [19], 0.74 [20], 0.86 [21], 0.80 [22], 0.82 [23], 0.86 [24], 0.86 [25], 0.85 [26], 0.84 [53] 0.64 [16], 0.74 [19], 0.83 [21] 0.64 [16], 0.74 [19], 0.84 [21], 0.72 0.78 [22], 0.76 [23], 0.81 [24], 0.85 [25] 0.66 [16], 0.73 [16], 0.83 [21]

Cr

1.0 1.1 1.2 1.3

21.6 ± 1.8 28.1 33.2 41.5

2.6 ± 0.2 3.4 4.2 5.5

24.2 ± 2.0 31.5 37.4 47.0

0.67 0.67 0.68 0.69

Cu

1.0

5.6 ± 0.7

0.80 ± 0.09

6.4 ± 0.7

1.1 1.2

7.6 9.2

1.0 2.7

8.6 11.9

Ka

Kb

1.04 1.02 1.01 1.02

[16], [16], [16], [16],

[9], 1.04 [10], 0.97 [11], 0.69 [12], 0.97 [13], 0.77 [14], 0.97 [53] [10] [10], 0.86 [11], 0.86 [13] [10]

1.12 1.15 1.04 1.13

[18], [27], [27], [27],

0.97 0.88 0.85 0.89

[27], [21], [21], [21],

0.86 0.83 0.84 0.82

[21], 0.85 [24], 0.91 [25], 0.86 [28], 0.83 [53] [28] [24], 0.87 [25], 0.76 [28] [28]

1.3

12.1

1.6

13.7

0.74 [29], 1.47 [30], 1.02 [31], 0.73 [32], 1.19 [15], 1.07 [33], 0.88 [34], 0.74 [16], 0.89 [19], 0.81 [27], 0.88 [22], 0.90 [23], 0.86 [35], 1.06 [13], 0.89 [36], 0.88 [24], 1.12 [37], 0.93 [25], 0.93 [14], 0.91 [53], 0.74 [29] 1.21 [54], 1.08 [33], 0.74 [16], 0.68 [27], 1.05 [55], 0.88 [35], 1.09 [37] 0.74 [29], 1.27 [31], 1.11 [33], 0.83 [16], 0.98 [19], 0.68 [27], 0.95 [22], 1.03 [23], 0.98 [35], 1.11 [13], 0.98 [24], 1.23 [37], 1.03 [25] 1.26 [54], 1.02 [33], 0.75 [16] 0.74 [27], 0.93 [35], 1.05 [37]

Zn

1.0 1.1 1.2 1.3

4.2 ± 0.5 5.5 7.3 8.9

0.52 ± 0.05 0.7 1.0 1.2

4.7 ± 0.5 6.3 8.3 10.1

0.73 1.18 0.91 1.21

[38], [54], [33], [54],

Zr

1.0 1.1 1.2 1.3

0.41 ± 0.05 0.56 0.76 0.97

0.080 ± 0.014 0.10 0.15 0.20

0.49 ± 0.06 0.66 0.91 1.17

1.58 1.45 1.55 1.51

[15], 1.51 [40], 1.30 [23], 1.36 [53] [40] [40] [40]

Nb

1.0 1.1 1.2 1.3

0.30 ± 0.03 0.43 0.58 0.73

0.062 ± 0.009 0.09 0.11 0.14

0.36 ± 0.03 0.51 0.69 0.88

1.24 [53] 1.42 [45]

Mo

1.0 1.1 1.2 1.3

0.27 ± 0.02 0.39 0.52 0.68

0.049 ± 0.010 0.08 0.11 0.15

0.32 ± 0.02 0.47 0.63 0.83

1.39 [29], 1.10 [31], 1.47 [41], 1.39 [42], 1.45 [37], 1.26 [43], 1.45 [53], 1.39 [31] 1.52 [37] 1.5 [29], 1.19[31], 1.57 [41], 1.5 [19], 1.47 [37], 1.39 [43] 1.54 [37]

Ag

1.0

0.079 ± 0.007

0.014 ± 0.003

0.094 ± 0.009

1.1 1.2 1.3

0.117 0.157 0.208

0.023 0.034 0.044

0.141 0.191 0.252

2.63 [30], 1.18 [32], 1.45 [15], 1.47 [44], 1.33 [19], 1.19 [45], 1.49 [42], 1.20 [23], 1.52 [46], 1.29 [36], 1.24 [37], 1.40 [53], 1.08 [48] 1.43 [19], 1.64 [55], 1.18 [37] 1.12 [31], 1.47 [44], 1.43 [19], 1.36 [45], 1.44 [42], 1.38 [23], 1.27 [37] 1.42 [19], 1.33 [37]

0.84 0.90 0.90 0.89

[39], [33], [35], [33],

0.92 [35], 0.86 [24], 0.89 [25], 0.89 [28], 0.87 [53] 0.85 [35], 0.86 [28] 0.90 [24], 0.95 [25], 1.0 [28] 0.84 [35], 0.99 [28]

1.35 [45]

(continued on next page)

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Table 1 (continued) Element

Ep [MeV]

Ka

Kb

Ktot

Ratio measured to the reference Ktot cross section, [reference]

Cd

1.0 1.1 1.2 1.3

0.059 ± 0.005 0.084 0.105 0.168

0.012 ± 0.002 0.017 0.021 0.031

0.071 ± 0.006 0.101 0.126 0.198

1.45 1.85 1.26 1.83

[44], 1.54 [39], 2.05 [40] [40] [44], 1.4 [39], 1.6 [40] [40]

In

1.0 1.1 1.2 1.3

0.044 ± 0.003 0.065 0.091 0.116

0.0082 ± 0.0013 0.013 0.019 0.025

0.053 ± 0.003 0.078 0.110 0.141

0.87 0.83 0.86 0.83

[27], [27], [27], [27],

Sn

1.0 1.1 1.2 1.3

0.037 ± 0.002 0.055 0.080 0.100

0.0068 ± 0.0014 0.012 0.018 0.017

0.043 ± 0.004 0.067 0.098 0.117

0.73 [31], 1.34 [15], 1.34 [44], 1.65 [39], 1.48 [42], 2.57 [40], 1.26 [37], 1.16 [43], 0.84 [31] 1.4 [37], 1.23 [43] 1.69 [44], 1.75 [39], 1.53 [42], 1.44 [37], 1.31 [43] 1.23 [37], 1.17 [43]

Sb

1.0 1.1 1.2 1.3

0.030 ± 0.003 0.044 0.061 0.073

0.0064 ± 0.0005 0.011 0.010 0.017

0.036 ± 0.004 0.055 0.071 0.090

1.38 [15], 1.38 [44], 1.40 [47]

1.47 [39], 1.05 [45], 2.01 [40], 1.41 [47] 2.0 [40] 1.51 [39], 1.29 [45], 1.93 [40] 1.95 [40]

1.29 [44]

Table 2 The measured L-shell x-ray production cross sections (in barns). Element

Ep [MeV]

Ltot

Ratio measured to the reference Ltot cross section, [reference]

Cu

1.0 1.1 1.2 1.3

La + Ll

1258 ± 58 1343 1388 1495

1.39 [49]

Zn

1.0 1.1 1.2 1.3

967 ± 40 1044 1124 1174

Zr

1.0 1.1 1.2 1.3

350 ± 25 392 427 462

Nb

1.0 1.1 1.2 1.3

288 ± 17 329 348 421

Mo

1.0 1.1 1.2 1.3

266 ± 17 314 363 396

Ag

1.0 1.1 1.2 1.3

164 ± 11 197 226 247

0.53 [53]

Cd

1.0 1.1 1.2 1.3

140 ± 11 169 200 225

0.54 [53]

In

1.0 1.1 1.2 1.3

123 ± 11 147 180 207

0.51 [53]

Sn

1.0 1.1 1.2 1.3

117 ± 10 147 177 195

0.81 [34], 0.56 [53]

Sb

1.0 1.1 1.2 1.3

113 ± 9 134 152 172

0.63 [53]

Ta

1.0 1.1 1.2 1.3

9.7 ± 1.1 12.5 15.7 18.1

1.04 0.80 0.79 0.78

6.1 ± 0.6 7.9 9.8 11.2

Lb

3.2 ± 0.4 4.1 5.2 6.1

Lc

0.40 ± 0.07 0.5 0.7 0.9

[50], 0.78 [52], 0.97 [53] [52] [52] [52]

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E. Batyrbekov et al. / Nuclear Instruments and Methods in Physics Research B 325 (2014) 84–88 Table 2 (continued) Element

Ep [MeV]

La + Ll

Lb

Lc

Ltot

Ratio measured to the reference Ltot cross section, [reference]

W

1.0 1.1 1.2 1.3

6.0 ± 0.2 7.9 9.7 11.9

3.1 ± 0.3 4.0 5.0 6.1

0.40 ± 0.07 0.5 0.7 0.8

9.5 ± 0.5 12.5 15.4 18.8

0.86 0.80 0.91 0.87

[52], 1.00 [53] [52] [52] [52]

Pb

1.0 1.1 1.2 1.3

2.4 ± 0.2 3.1 3.9 4.5

1.3 ± 0.1 1.8 2.3 2.7

0.20 ± 0.04 0.3 0.4 0.4

3.9 ± 0.2 5.3 6.6 7.6

0.99 1.18 1.22 1.07

[51], 1.13 [52], 0.98 [53] [52] [52] [52]

Bi

1.0 1.1 1.2 1.3

2.1 ± 0.2 2.6 3.4 4.2

1.1 ± 0.2 1.6 2.1 2.5

0.20 ± 0.04 0.3 0.3 0.4

3.4 ± 0.4 4.5 5.8 7.2

1.43 1.12 1.07 0.95

[50], 1.01 [52], 0.94 [53] [52] [52] [52]

Table 3 The measured M-shell x-ray production cross sections (in barns). Element

Ep [MeV]

Mtot

Mtot, [reference]

Ta

1.0 1.1 1.2 1.3

705 ± 58 800 900 957

0.69 0.70 0.70 0.68

[52] [52] [52] [52]

W

1.0 1.1 1.2 1.3

647 ± 53 745 815 914

0.75 0.71 0.86 0.77

[52] [52] [52] [52]

Pb

1.0 1.1 1.2 1.3

438 ± 50 470 513 605

0.87 0.92 1.02 0.87

[52] [52] [52] [52]

1.0 1.1 1.2 1.3

384 ± 30 435 494 554

0.84 [52] 0.94 [52] 0.93 0.82 [52]

Bi

where NX and NR are the numbers of detected x-ray photons and the backscattered particles, respectively; DX and DR – dead time corrected coefficients; XR (sr) is particle detector solid angle; eX (EX) is the Si (Li) detector efficiency for x-ray energy EX; lt (cm2/g) (EX) is the mass coefficient of x-ray attenuation in the irradiated sample, which depends on the energy of x-ray photon EX; t(g/ cm2) is the thickness of the deposited film and rR – backscattering cross section for the given angle and the energy of incoming beam.

The term in the square brackets in the expression (1) takes into account the self-absorption of x-ray radiation in the target. In case of using the filter the expression (1) takes the following form:

rX ¼

  N X DX XR rR lt t ;  l q x  l t eX NR DR e f f 1  e t

ð2Þ

where lf(cm2/g) (EX) is the mass coefficient of x-ray radiation attenuation in the filter which depends on the energy of x-ray photon, qf(g/cm3) is the density of the filter material and x(cm) is the filter thickness. 3. Results and discussion To determine the ratio uncertainty of the detected x-ray photons number to the number of backscattered particles used in the expression (1), a series of five measurements was carried out at different integrals of the beam current at the target and different conditions of x-ray spectra and backscattered particles processing. The results of the performed studies were the average values of fluorescence production cross sections for 14 series of K-lines, 14 series of L-lines and 4 M-lines under excitement of target atoms by the proton beam with the energies of 1, 1.1, 1.2 and 1.3 MeV. Tables 1 (K-line), 2 (L-line) and 3 (M-line) present either individual or total x-ray productions of the series (Ktot, Ltot and Mtot respectively), being the sum of the individual lines. For protons with the energy of 1 MeV the tables show the measurement uncertainties and data on x-ray production cross-sections obtained

Fig. 2. X-ray production cross sections for Ktot, Ltot and Mtot as a function of atomic number of the target nucleus. (1) Ep = 1 MeV, (2) Ep = 1.1 MeV, (3) Ep = 1.2 MeV, (4) Ep = 1.3 MeV.

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experimentally in other laboratories. In addition to the uncertainty of NX to NR ratio the Tables 1–3 include the errors of determining the solid angle of the charged particle detector, the efficiency of x-ray detection and the cross sections of backscattering. Fig. 2 shows graphically the x-ray production cross sections for Ktot, Ltot and Mtot as a function of atomic number of the target nucleus. From Tables 1–3, it follows that there is a fairly large amount of data for the K-shell ionization. At the same time, the great scattering of the obtained values of the x-ray production cross sections is observed, which confirms the importance of obtaining new data. In the case of L-shell and M-shell the available experimental data are very limited. Acknowledgments The works, the results of which are presented in this article, are made with the financial support of the Ministry of Education and Science of the Republic of Kazakhstan in the framework of the Agreement 0608/GF under the Program of Grant funding of research G.2012. We also express our gratitude to F. Penkov for productive and active participation in the discussions. References [1] S.A.E. Johansson, J.L. Campbell, PIXE: A Novel Technique for Element Analysis, John Wiley, Chichester, 1988. [2] W. Brandt, G. Lapicki, Phys. Rev. A 23 (1981) 1717. [3] G. Lapicki, M. Goldstein, W. Brandt, Phys. Rev. A 23 (1981) 2727. [4] G. Lapicki, X-ray Spectrom. 34 (2005) 269. [5] H. Paul, J. Sacher, Atom. Data Nucl. Data Tables 42 (1989) 105. [6] W. Brandt, G. Lapicki, Phys. Rev. A 20 (1979) 465. [7] Proceedings of XII All-Soviet Union Conference on Charge Particle Accelerators, Joint Institute for Nuclear Research, Dubna, 1990 (p. 139). [8] J.M. Khan, D.L. Potter, R.D. Worley, Phys. Rev. A 139 (1965) 1735. [9] Proceedings of the Third International Conference on Atomic Physics, in: S.J. Smith, G.K. Walters (Eds.), Plenum Press, New York, 1973, p. 155. [10] G. Basbas, W. Brandt, R. Laubert, Phys. Rev. A 7 (1973) 983. [11] H. Tawara, Y. Haciya, K. Ishii, S. Morita, Phys. Rev. A 13 (1976) 572. [12] W.N. Lennard, D. Phillips, Nucl. Instr. Meth. 166 (1979) 521. [13] K. Sera, K. Ishii, M. Kamiya, A. Kuwako, S. Morita, Phys. Rev. A 21 (1980) 1412. [14] M. Geretschlaeger, O. Benka, Phys. Rev. A 34 (1986) 866. [15] R.C. Bearse, D.A. Close, J.J. Malanify, C.J. Umbarger, Phys. Rev. A 7 (1973) 1269. [16] F. Bodart, S. Wilk, G. Deconnik, X-ray Spectrom. 4 (1975) 161. [17] Beam-foil spectroscopy, in: I.A. Sellin, D.J. Pegg (Eds.), Collisional and Radiative Processes, vol. 2, Plenum Press, New York, 1976, p. 519.

[18] 9-th International Conference on the Physics of Electronic and Atomic Collisions, in: J.S. Risley, R. Geballe (Eds.), University of Washington, 1976, p. 419. [19] M.R. Khan, D. Crumpton, P.E. Francols, J. Phys. B9 (1976) 455. [20] B. Knaf, G. Presaer, J. Stahler, Z. Phys. A282 (1977) 25. [21] J.S. Lopes, A.P. Jevus, G.P. Ferrelra, F.B. Gil, J. Phys. B11 (1978) 2181. [22] K.M. Barfoot, I.V. Mitchell, W.B. Gilboy, H.L. Eschbach, Nucl. Instr. Meth. 168 (1980) 131. [23] E. Laegsgaard, J.U. Andersen, F. Hogedal, Nucl. Instr. Meth. 169 (1980) 293. [24] M.D. Brown, D.G. Simons, D.J. Land, J.G. Brennan, Phys. Rev. A 25 (1982) 2935. [25] M.D. Brown, D.G. Simons, D.J. Land, J.G. Brennan, IEEE Trans. Nucl. Sci. 30 (1983) 957. [26] M. Geretshlager, O. Benks, Phys. Rev. A 34 (1986) 866. [27] E. Kolatay, D. Berenyl, I. Kiss, S. Riez, G. Hock, J. Basco, Z. Phys. A278 (1976) 299. [28] S.O. Olabanji, J.M. Calvert, Nucl. Instr. Meth. 251 (1986) 354. [29] J.M. Hansteen, S. Messelt, Nucl. Phys. 2 (1957) 526. [30] D. Singh, Phys. Rev. 107 (1957) 711. [31] S. Messelt, Nucl. Phys. 5 (1958) 435. [32] Proceedings of International Conference on Inner Shell Ionization Phenomena, U.S. AES Technical Information Center, Oak Ridge, Tennessee, 1973 (p. 1019). [33] R.D. Lear, T.J. Gray, Phys. Rev. A 8 (1973) 2409. [34] F. Hopkins, R. Brenn, A.R. Whittemore, J. Karp, S.K. Blattacherjce, Phys. Rev. A 11 (1975) 916. [35] J.S. Lopes, A.P. Jesus, S.C. Ramos, Nucl. Instr. Meth. 169 (1980) 311. [36] J.U. Andersen, E. Laegsgaard, M. Lund, Nucl. Instr. Meth. 192 (1982) 79. [37] D. Bhattacharya, S.K. Mitra, Pramana 19 (1982) 399. [38] H. Tawara, K. Ishii, S. Morita, H. Kaji, C.N. Hsu, T. Shiokawa, Phys. Rev. A 9 (1974) 1617. [39] M.R. Khan, A.G. Hopkins, D. Crumpton, P.E. Francois, X-ray Spectrom. 6 (1977) 140. [40] C. Magno, M. Milazzo, C. Pizzi, F. Porno, A. Rota, G. Riecobono, Nuovo Cimento A54 (1979) 277. [41] T.L. Criswell, T.J. Gray, Phys. Rev. A 10 (1974) 1145. [42] R. Anholt, Phys. Rev. A 17 (1978) 983. [43] S. Divous, B. Raith, Gonslor, Nucl. Instr. Meth. B3 (1984) 27. [44] N.A. Kheld, T.J. Gray, Phys. Rev. A 11 (1975) 893. [45] S.R. Wilson, F.D. McDanel, J.R. Rowe, J.I. Duggan, Phys. Rev. A 16 (1977) 903. [46] Inner-shell and X-ray Physics of Atoms and Solids/Plenum, in: D.J. Fabian, H. Kleinpoppen, L.M. Watson, New-York, 1981. p. 21. [47] A.P. Jesus, J.S. Lopes, Nucl. Instr. Meth. 192 (1982) 25. [48] G.A. Bissinger, S.M. Shafroth, A.W. Waltner, Phys. Rev. A 5 (1972) 2046. [49] J.M. Khan, D.L. Potter, R.D. Worley, Phys. Rev. 145 (1966) 23. [50] C.N. Chang, J.F. Morgan, S.L. Blatt, Phys. Rev. A 11 (1975) 607. [51] C.E. Busch, A.B. Baskin, P.H. Nettles, S.M. Shafroth, Phys. Rev. A 7 (1973) 1601. [52] M. Goudarzi, F. Shokouhi, M. Lamehi-Rachti, P. Olialy, Nucl. Instr. Meth. B247 (2006) 217. [53] A.A. Kljuchnikov, N.N. Pucherov, T.D. Chesnokova, V.N. Scherbin, Methods for analysis with charge particle beams, Kiev. Naukova Dumka (1987) 132. [54] A. Fahlenius, P. Jauho, Ann. Acad. Sci. Fenn. Ser. A6367 (1971) 3. [55] C. Bauer, R. Mann, W. Rudolph, Z. Phys. A287 (1978) 27.