Multiple ionisation of atoms

The multiple ionization of a neutral atom by electron impact (single-step multiple ionization)

A + e ® A²⁺ + (2+1) e

(12a)

A + e ® A³⁺ + (3+1) e

(12b)

A + e ® A^m+ + (m+1) e

(12c)

is a collision process of fundamental interest. The cross sections for the single-step multiple ionization of an atom are significantly smaller than cross sections for single ionization [16]

and decline rapidly with the stage of ionization [108, 109, 128, 129]. Nonetheless, multiple ionization processes are important in then tokamak edge plasmas [1] and in other environments with an abundance of energetic electrons. Although electron temperatures in these plasma regions are relatively low, well below 1 keV, the high energy part of the energy distributions may after all lead to the production of more highly charged species and thus influence the plasma properties. As mentioned above the elements that need special consideration include Li, Be, B, C, N, Al, Si, Mg, Ti, Cr, Fe, Ni, Cu, Mo, Nb, Ta and W as possible impurities in the plasma edge and Ne, Ar, Kr, Xe as gases which are injected from the outside into the plasma volume for cooling the plasma scrape off layer [1, 5].

Calculations of multiple atomic ionization cross sections using rigorous quantum mechanical methods are difficult for all but the simplest atoms [16, 73–75]. This is due to — among other things — the need to consider two or more continuum electrons and their mutual interaction in the exit channel. Experimental data for the formation of highly charged ions are scarce for most atoms [108, 109, 128–130] because of among other reasons the fact that the cross sections for the single-step formation of highly-charged multiple ions are comparatively

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small. Modellers and practitioners rely heavily on semi-empirical and semi-classical methods to determine multiple ionization cross sections for modelling purposes and for other applications [1, 16, 76]. A simple variant of the Deutsch-Märk (DM) formalism described above has been applied successfully to the calculation of cross sections for the formation of multiply charged ions A^m+ for several atoms such as high-Z atoms with a nuclear charge Z of 10 and above and various stages of ionization [106–108] as well as several low-Z targets [109, 128, 129].

This simple DM variant has the following advantages over two other semi-empirical methods which have been recently proposed by Fisher et al. [131] and by Shevelko et al. [132], respectively, to calculate cross sections for multiple ionization of neutral atoms: (i) the application of the DM variant requires fewer semi-empirical parameters which, in addition, can easily be related to physical quantities and (ii) the energy-dependent function of the DM variant (derived from classical considerations, see [95, 111]) is the same for all stages of ionization. So far the following targets and respective ionization stages have been investigated with this DM variant: Be (m up to 4) [128], B (m up to 5) [128], C (m up to 6) [128], N (m up

In the following we will shortly summarize the theoretical background for this DM variant and give some illustrative examples. Where possible, we compare the results obtained with the DM formalism with experimental data and results obtained by using the semi-empirical method (ST) of Shevelko and Tawara [132], the other semi-empirical method [131] was found to show very large deviations from the data for higher charge states, see e.g. Fig. 5 in Ref. [108]) and is, therefore, not considered in more detail here.

Starting from the DM formalism described above, which was originally developed for the calculation of cross sections for the single ionization of an atom [95, 96], the cross section s^m+ for the formation of an ion A^m+, which in principle is a product of m independent terms each describing the removal of a single electron, can be simplified to an expression of the form

where the summation extends over the various atomic sub-shells with k = 1 referring to the outermost sub-shell, k = 2 to the second outermost sub-shell, etc. In the above equation, (r_k)² is the square of the maximum charge density of the atomic sub-shell labelled by k, x_k is the number of electrons in that sub-shell, and g^m are weighting factors (see Refs. [109, 128] for further details). The functions f_k(U) describe the energy dependence of the ionization cross section and are identical to the function given in equ. (8), however U refers to the reduced impact energy, U = E/Em, where E is the energy of the incident electron and Em is the ionization energy required for the simultaneous removal of m electrons from atom A, which is larger than the binding energy Ek of electrons in the sub-shell labelled by k. We further note that for impact energies above about 10⁵ eV the energy dependence has to be modified to include relativistic effects. This has been discussed in a previous paper [98]. The weighting factors g^m for equ. (13) have been determined from a fitting procedure (for details, see the

previous papers by Deutsch et al. [109, 128]) and can be approximated by an exponential function of the form:

m Z a Z e

g = ^-

(14)

where Z is the nuclear charge and a(Z) and b(Z) are two empirically determined functions (see Fig. 20).

Figure 20. Parameter a from equation (14) as a function of the nuclear charge Z (left hand side) and parameter b from equation (14) as a function of nuclear charge Z (right hand side) after Ref. [109].

Experimental data for the formation of multiply charged ions A^m+ (for m > 3) produced by electron impact ionisation of a neutral atom A are available for only a few atoms, primarily for the noble gases Ne, Ar, Kr, and Xe (e.g., [50, 51, 129, 134–141]]. In contrast experimental data for the formation of multiply charged ions from other neutral atomic targets mentioned above (e.g., N etc.) are not available in the literature to the best of our knowledge.

The cross section for the double ionization of Ar has been measured by more groups than any other double ionization cross section. This is due to a controversy regarding the cross section ratio Ar⁺/Ar²⁺ which appeared in the literature in the late 1980s [50]. As a consequence, many groups using different experimental techniques revisited the measurement of this cross section ratio, e.g., [51, 137, 142]. This resulted in a level of confidence of the experimentally determined Ar²⁺ ionization cross section by now of better than 10% which renders it perhaps the best known atomic double ionization cross section. Fig. 21 shows as an example the DM calculation for the formation of Ar²⁺ in comparison with some of the experimental data that have been reported for this cross section [134, 143, 144] and in comparison with the ST cross section [145]. It is obvious that the experimental data sets are in excellent agreement with each other (in particular those of Ref. [134] and [139]) and that the DM cross section is in good agreement with the measured data. The ST cross section lies significantly below the measured data for impact energies up to 500 eV (up to a factor of 4 near the maximum of the cross section) and approaches the measured data only in the high energy region. Fig. 22 shows as another example the calculated DM cross sections for Ar^m+ (m=3–5) in comparison with selected experimental data [138, 141, 144] and with the ST cross sections. For more

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calculated DM cross sections and the calculated ST cross sections as m increases) where experimental data are available up to m= 8 see [133].

Figure 21. Cross sections for the formation of Ar2+ ions by electron impact ionisation of Ar after [133]. The experimental data are from Märk [142] (open triangles), Wetzel et al. [134] (filled dots), and McCallion et al. [144] (crosses). The solid line represents the DM cross section and the dashed line represents the ST cross section.

Figure 22. Cross sections for the formation of Arm+ (m=3–5) ions by electron impact ionisation of Ar after [133]. The experimental data are from Almeida et al. [138] (open dots), Koslowski et al. [141]

(stars), and McCallion et al. [144] (filled dots). The solid line represents the DM cross section and the dashed line represents the ST cross section.

In Ne, experimental data are available up to m = 5. Ne cross sections based on the DM formalism with m up to three have been presented and discussed in [106] and excellent agreement was observed between our calculations and experimental data of Lebius et al.

[136]. Figure 23 shows as an example a comparison between the available experimental data for the formation of Ne⁴⁺ and Ne⁵⁺ and the results of calculations based on the DM-formalism (dashed line) and the ST method (solid line). Two observations are apparent, (i) there is satisfactory agreement between the experimental data and the present predictions of the DM-formalism for Ne⁴⁺ over the entire range of impact energies and for Ne⁵⁺ in the regime of higher impact energies (above about 2 keV) and (ii) the predictions of Shevelko and Tawara [132] overestimate the experimental data in both cases and the overestimation increases with increasing stage of ionization.

Figure 23. Upper part: Calculated cross sections after [109] for the formation of Ne⁴⁺ by electron impact ionisation of Ne using the DM formalism (dashed line) and the ST method (solid line) in comparison with the experimental data points designated in [109]. Lower part: Same as upper part for Ne⁵⁺.

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Figure 24 summarizes the results given in [109] of the calculations for the formation of Ne^m+

ions for m = 1–10. It is apparent from this figure 4 that the cross sections for the formation of multiply charged ions decreases rapidly with the degree of ionization m. This is to be expected, since the probability that a collision of an atom with a single electron results in the simultaneous, single-step removal of m electrons decreases rapidly with increasing m.

Moreover, the position of the maximum of the cross section curve for the formation of the multiply-charged ions shifts strongly with the degree of ionization. This again is to be expected as the threshold for ionization increases strongly with the degree of ionization and thus the cross section is shifted to higher energies. In order to understand this in a more quantitative manner it is interesting to rewrite equ. (13) in the reduced form, i.e.,

U

Figure 24. Calculated cross sections using the DM formalism for the formation of Ne^m+ (m = 1–10) ions by electron impact ionisation of Ne as a function of electron energy after [109].

In this case the dependence of the cross section on the reduced energy is determined only by the function fk(U). For Ne ions up to m = 8, the cross section dependence on U is independent on m, i.e. given by

It follows, that the position of the maximum of the cross section scales with the ionization energy Em because Umax = Emax/Em is independent of m. This scaling law is also valid (to within about 5%) for the production of ninefold and tenfold charged Ne ions as the additional