Abstract: Cross sections for ionization of inner-shell electrons by electron impact
Authors: Powell C. J. 
Published: July 2002

Auger electron spectroscopy (AES) has been used extensively for qualitative surface analyses in recent years. There have, however, been relatively few attempts at quantitative surface analyses on account of lack of knowledge of key parameters (such as electron attenuation lengths, inner-shell ionization cross sections, and backscattering factors) as a function, where appropriate, of material, incident electron energy, Auger electron energy, and instrumental characteristics. A review of experimental electron attenuation length data in the energy range relevant to AES and to x-ray photoelectron spectroscopy (XPS) was given in a previous paper1 together with a method to predict attenuation lengths in other materials. We give a summary here of an analysis2 of available data concerning cross sections for ionization of inner-shell electrons by electron impact. Cross-section data of this type is also required in correction procedures for quantitative analysis by electron probe microanalysis.3 necessity the analysis has been limited to K- and L-shell ionization of light atoms.
Calculated, semiempirical, and experimental cross-section data have been intercompared graphically and through fits to the linearized Bethe equation4,5 for inner-shell ionization. The Bethe equation is (typeset math) where E0=mv2/2 is the energy of the incident electron beam, Enl is the binding energy of electrons in the nl-shell, and Znl is the number of electrons in that shell. The parameter bnl was estimated by Bethe to be between 0.2 and 0.6 and the parameter Bnl to be of the order of Enl. In the derivation of Eq. (1) it has been assumed that E0≫Enl and it then becomes important to ask, as a practical matter, what is the lowest value of E0 for which Eq. (1) will be valid?
It is convenient to write Eq. (1) in the form (typeset math) where cnl=4 Enl/Bnl and where the energy Enl has been expressed in eV. Calculated and measured cross-section data can be used to find ’’effective’’ values of the Bethe parameters bnl and cnl to test, first, the range of validity of Eq. (1) for a particular element and a particular inner shell and, second, the possible variation of the parameters bnl and cnl with Z (f Bethe parameters can be obtained conveniently from a linear least-squares fit to the following equation (a Fano plot5) (typeset math) Almost all of the available data can be satisfactorily fitted over the range 4≲Unl≲30 and values obtained for the effective parameters bnl and cnl.2 Values of the parameter bnl have also been derived from photoabsorption data and were found to be consistent with the ionization data if account was taken of the distribution of differential oscillator strength with respect to excitation energy. Specifically, bnl should properly be regarded as a function of Unl for Unl≲20 although useful empirical fits to the cross-section data can be obtained over a wide range of Unl, as mentioned above, with a different (higher) effective value of bnl and a lower than expected value of cnl. The derived empirical Bethe parameters should therefore not be used outside the range of each fit.
Experimental values of σKEK2 appeared to lie on a common curve when plotted as function of UK. These values agreed quite well with the theoretical results of Rudge and Schwartz6 for 5&lt;UK&lt;26 and the results of McGuire.7 It therefore appears that values of σK for light atoms can be obtained by appropriate scaling.
A greater spread existed in the plot of calculated and measured values of σL23 EL232 as a function of UL23 than for the case of K-shell ionization. A significant variation in the magnitude of σL23 EL232 as a function of Z (for a given value of UL23) is apparent in the experimental data of Vrakking and Meyer8 and in the calculations of McGuire.7 This variation leads to a large variation in the magnitude of the effective bL23 as a function of Z that is not substantiated by the results of many photoabsorption experiments or by the limited number of L-shell ionization cross sections derived from characterization x-ray and Auger electron yield experiments. Further experiments are required to resolve this discrepancy. Although the data base in the range of interest here is limited, it is possible to provide ’’preferred values’’ for the empirical Bethe parameters. For the case of K-shell ionization, the values bK∠0.9 and cK∠0.60–0.75 appear to be useful for light atoms in the range 4&lt;UK&lt;25. The situation for L23-shell ionization is less certain, for the reasons just discussed, but the values bL23 ∠0.5–0.6 23≳20, while the values bL23∠0.6–0.9 seem appropriate for 4≲UL23≲20.
The relatively simple and widely used expressions of Gryzinski,9 Drawin,10 and Lotz11 agree closely with each other (within 10%) in the case of K-shell ionization for UK&lt;30. These results are systematically lower than the experimental measurements by roughly 10% to 15%. For the case of L23-shell ionization, the Lotz expression is substantially different from the Gryzinski and Drawin results (which agree well with each other). The latter results agree closely with the cross-section measurements of Christofzik12 for argon.
The principal conclusions of the analysis are as follows:
(a) Useful empirical fits of inner-shell ionization cross-section data can be made with Eqs. (1) to (3) over a wide range of Unl (typically 4≲Unl≲30) with effective values of the Bethe parameters bnl and cnl. The effective value of bnl is consistent with photoabsorption data taking into account the anticipated functional dependence of bnl on E0.
(b) The present data base does not indicate a significant variation of the Bethe parameters with Z.
(c) A better choice of the Bethe parameters can be made now than those recommended many years ago by Mott and Massey.13 The recent calculations and measurements show that the inner-shell ionization cross sections are appreciably (at least 40% to 50%) higher than those calculated with the Mott and Massey parameters.