Contrib. Plasma Phys. 34 (1994) 1, 19-24

Calculation of Absolute Outer-Shell Electron Impact Ionization Cross Sections

H. DEUTSCH '), T. D. MARK

lnstitut fur lonenphysik, Leopold Franzens Universitbt, A 6020 Innsbruck. Austria

Abstract

The recently developed semiempirical DM-approach for the calculation of electron impact single ionization cross sections is applied here to the ionization of the rare gases (Ne, Ar, K r and Xe) via ejection of an outer s-shell electron. The results obtained are compared with previous experimental and theoretical s-shell ionization cross section functions.

1. Introduction

Although most of the experimental and theoretical studies concerning electron impact partial ionization cross sections of atoms have been devoted to the determination of integrated partial ionization cross sections (that is taking into account contributions from all electron shells of the target atom under consideration) or to inner-shell ionization cross sections, there exist also a few recent reports on outer-shell ionization. Unfortunately, large discrepancies exist among these outershell ionization cross sections reported so far. As single ionization of rare gas atoms proceeds mainly by the removal of an outer-shell electron, the determination of outer-shell electron cross sections is, however, of basic importance in elucidating electron impact ionization.

Based on the Born Bethe approximation and the classical binary encounter approximation DEUTSCH and MARK [ I ] proposed recently an ionization cross section formula, where contributions from the various shells are explicitely taken into account. In a series of recent papers it could be demonstrated that this formulation (which consists of a simple analytic expression depending only on basic (known) atomic properties) may be successfully used to predict not only integrated electron impact ionization cross section function for single ionization of ground state atoms [ l , 21, excited state atoms [3], radicals [4], molecules [5] and clusters [6], but also for specific inner-shell ionization processes [7]. Based on this success we have extended here the use of this semi-empirical DM formulation to ionization of the rare gases (Ne, At-, Kr and Xe) via ejection of an outer s-shell electron, e.g. in case of argon via

Ar(3s23p6) + e -+ Arf'(3s3p6) + 2e.

The present results are compared with previous experimental and theoretical s-shell ionization cross section functions.

') Guest professor at the Institut for Ionenphysik, Universitat Innsbruck. Permanent address: Fachbereich Physik, Emst-Moritz-Arndt-Universitat, D- 17 489 Greifswald, Germany

2'

20

2. Theoretical Consideration

Contrib. Plasma Phys. 34 (1994) 1

Using classical mechanics, THOMSON [S] was the first to derive a formula for the electron impact single ionization cross section. This classical treatment has been modified by several authors (e.g. see refs. [9, lo]) using different initial conditions, e.g. GRYZINSKI [l l] introduced the assumption of a continuous velocity distribution for atomic electrons, leading to an expression for the integrated cross section

with

1 u - 1 f ( u ) = - u (-y u + l jb + c (1 - i) In (2.7 + (u - l)’’z) (3)

where a = 3/2, b = 1, and c = 213, a, the Bohr radius, Cn, the number of electrons in the nl subshell, EnI the ionization energy in the corresponding nl-subshell, E y the ionization energy of hydrogen, u = E/Enl, and E the energy of the incident electron. Despite a significant improvement over previous formulae, the Gryzinski formula fails in case of rather simple atoms such as N, Ne and F [l].

Based on a comparison between this classical formula and the quantum mechanical equation derived by BETHE [12]

(with M,, dipole matrix element and cnI collisional parameter), DEUTSCH and MARK [ I ] suggested recently to replace the Bohr radius a, in the classical formula (2) by the radius of the corresponding subshell rnk This step is in line, (i) with the results of BETHE’S calculation [12] that the ionization cross section of an atomic electron with quantum numbers (n, r ) is approximately proportional to the mean square radius ( r i , ) In of the electron shell (n, 1) [13, 141, and (ii) with the experimental observation of a correlation between the maximum of the atomic cross section and the sum of the mean square radii of all outer electrons [14- 161. Following this suggestion, MARGREITER et al. [2, 31 successfully applied the following formula (DM-formulation) to a large number of ground state and excited state atoms

where r i l is the mean square radius of the n, 1 subshell, and g,, are weighting factors. MARGREITER et al. determined these generalized weighting factors g,, via a fitting procedure using reliable experimental data for the rare gases and uranium as test cases. In a first approximation these weighting factors were taken to be 3 for s electrons and 0.5 for electrons other than s electrons [3]. In a more sophisticated analysis it turned out that g,, = gn,(n, I , En,) in such a way that the product of g,, times En, is independent of Z [2] for complete subshells. Moreover, the parameters a, b, c in f ( u ) had to be slightly revised [2] in order to improve the agreement in the energy dependence, i.e. for s-electrons a = 7/4, b = 1, c = 1, for p-electrons a = 2, b = 1, c = 1, for d electrons a = 312, b = 3, c = 2/3 and forfelectrons a = 3/2, b = 1, c = 2/3.

In the present study we have extended this DM approach outlined above to the calculation of cross sections, c,,, for removal of an electron from a specified outer subshell nl using

H. DEUTSCH, T. D. MARK, Ionization Cross Sections 21

for rf, values reported by DESCLAUX [ 171 (obtained by quantum mechanical calculations taking into account relativistic effects) and for En, values summarized in the literature [18]. The corresponding value for the product g,, En, is 20, 14, 10 and 7.5 eV for 23, 3s, 4s and 5s electrons, [2 ] , respectively.

3. Results and Discussion

Using formula (5) given above it is possible to calculate outer-she11 ionization cross sections for any given nl-subshell. We have applied here this approach to the single ionization of rare gases via the ejection of outer-shell s electrons. Fig. 1 to 4 show the calculated cross section functions for Ne, Ar, Kr and Xe, respectively. Also shown for comparison are previous theoretical and experimental determinations.

Electron energy Lev1

Fig. 1

Fig. 1. Absolute partial outer-shell ionization cross section in Ne for the reaction Ne(Zs'2p6) + e -+ Ne+(2s2p6) + 2e. Experimental results: x LUYKEN et al. [I9], 0 ZAPESOCHNYI et ak. [20], DIJKAMP and DE HEER [21] (see also similar results by ECKHARDT et al. reported and discussed in ref. [21]). Theoretical results: Born approximations by PEACH [22] (designated - - -), OMIDVAR and KYLE [23] (designated 0 ) and M c GUIRE [24] (designated --.--); semiclassical BEA by VRIENS [25] (designated . - . - . ); classical BEA [ l l ] (designated +); empirical LOTZ formula [26] (designated - . -)

Fig. 2. Absolute partial outer-shell ionization cross section Ar for the reaction Ar(3s23p6) + e -f Arf(3s3p6) + 2e. Experimental results: x LUYKEN et al. [19], o ZAPESOCHNYI et al. [20], LI et al. [27] (see also similar results by RAN et al. [28] and MENTALL and MORGAN [29]) Theoretical results: Hartree Fock approximation by AMUSIA and SHEINERMAN [30] (designated .---.--); Born approximations by PEACH [22] (designated - - --) and by MC GUIRE [24] (designated __.-- ); semiclassical BEA by VRIENS [25] (designated . - . - ' ); empirical LOTZ formula [26]

and present results (designated - ).

(designated - . . -) and present results (designated - ).

22 Contrib. Plasma Phys. 34 (1994) 1

1.0 1

/.--. '& o s t - O H --. I

20 30 50 too 150 300 600 Electron energy Lev1

Fig. 3. Absolute partial outer-shell ionization cross section in Kr for the reaction Kr(4s24p6) + e + Kr+(4s4p6) + 2e. Experimental results: x LUYKEN et al. [I91 and o ZAPESOCHNYI et al. [20]. Theoretical results: Born approximation by OMIDVAR et al. [3 I] (designated -. -. - ); OCHKUR approximation [32] (designated - - - -) and present results (desingated - 1.

Unfortunately, up to now only two groups i.e. that of DE HEER and coworkers [19] and ZAPESOCHNYI et al. [20], reported complete sets of outer s-shell ionization cross section data for Ne, Ar, Kr and Xe. In addition, TAN et al. [28], MENTALL et al. [29] and LI et al. [27] published results on outer s-shell ionization of Ar. Moreover, the group of DE HEER [21] recently revised their own earlier results on Ne downwards by a factor of 3.7. The results of ZAPESOCHNYI et al. are in general one order of magnitude larger than those of the group of DE HEER and coworkers. The data of TAN et al. [28], MENTALL et al. [29] and LI et al.

1 0 - I '.

- .

20 30 50 100 150 300 600 Electron energy [ d l

Fig. 4. Absolute partial outer-shell ionization cross section in Xe for the reaction Xe(5s25p6) + e -+ Xe+(SsSp6) + 2e. Experimental results: x LUYKEN et al. [I91 and 0 ZYPESOCHNYI et al. [20] Theoretical results: Born approximation by OMIDVAR et al. [3 I] (designated -. - . - .); OCHKUR approximation [32] (designated - - - -) and present results (designated - 1.

H. DEUTSCH, T. D. MARK, Ionization Cross Sections 23

[27] agree with each other, but lie even a factor of appr. 2 below the argon data of DE HEER and coworkers [19]. The reason for these large discrepancies are due to difficulties in the absolute normalization of the experimental results. As has beeq pointed out and discussed in detail by LI et al. [27], different methods have been used to calibrate experimental data, all of which appear to yield data not reliable enough to serve as standards for a meaningful comparison with the present results (which lie somewere between the extremes of the experimental data sets).

Nevertheless, it is interesting to compare the present results with earlier theoretical calculations. Not surprisingly, calculations performed with the Born approximation (which are only accurate at higher electron energies) yield results at the maximum which are much larger than the present data (the only exception being the calculation by MCGUIRE [24] using different final-state wave functions). Moreover, as already discussed in the introduc- tion semiclassical and classical concepts (i.e. VRIENS [25] and GRYZINSKI [ l 11) cannot be used for systems like Ne (see ref. [l]) and thus it is not surprising that in Ne these methods yield results which are much larger than the present ones. Conversely, there appears to be a rather good agreement between the present results and those obtained with the empirical Lotz formula (as shown for Ne and Ar in Fig. 1 and 2, respectively), with those of MCGUIRE [24] (Born approximation with improved wave functions) in Ne and Ar, with those of VRIENS [25] in Ar and with those of OCHKUR [32] at higher energies in Kr and Xe.

Acknowledgements

This work was partially supported by the Osterreichischer Fonds zur Forderung der Wissenschaftlichen Forschung and the Bundesministerium fur Wissenschaft und Forschung, Wien.

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Received May 10. 1993