Contrib. Plasma Phys. 35 (1995) 4-5, 421-431

Calculated Cross Sections for Double and Triple Ionization of Atoms by Electron Impact

H. DEUTSCH (a), K. BECKER (b), and T. D. MARK (c)

(a) Fachbereich Physik, Ernst-Moritz-Arndt-Universitidt, D-17487 Greifswald, Germany; (b) Physics Department, City College of C.U.N.Y., New York, NY 10031 USA; (c) Institut f i r Ionenphysik, Leopold Franzens Universitit, Technikcrstrak 25, A-6020 Innsbruck, Austria

Abstract

The semi-empirical Deutsch-Mark (DM) formalism for the calculation of absolute cross sections for the single ionization of atoms has been extended to the calculation of cross sections for the double and triple ionization of ground state atoms. Detailed comparisons are made with available experimental data from threshold to 200 eV. In general, we find good agreement between the present calculations and the available experimental data. On that basis, we feel confident that the present formalism can be applied to predict double and triple ionization cross sections for those atoms, for which no experimental data are currently available, with an accuracy in the 30- 50% range for impact energies where direct ionization processes dominate and where contributions from tightly bound inner shells can be neglected. This level of precision should be sufficient for modelling purposes and many other practical applications.

1 Introduction

The ionization of an atom in its ground state by electron impact is one of the most important fundamental collisional interactions in. atomic physics. Ionization processes play an important role in many applications including discharges and plasmas, gas lasers, planetary, cometary and stellar atmospheres, and radiation chemistry [ 11. Atomic ionization processes have been studied extensively since the early years of this century. The need for measurements of atomic ionization cross sections in a systematic fashion, e.g. along isoelectronic sequences or for groups of atoms of similar structure and with similar physical and chemical properties, was first expressed by FREUND [2]. Two recent reviews by TARNOVKSY and BECKER [3, 41 revealed that single ionization cross sections for ground state atoms have been measured for about half the atoms in the periodic table. The list of target atoms for which cross sections for multiple ionization have ben measured reliably is much smaller. Even though multiple ionization processes typically have cross sections that are smaller than the cross sections for single ionization by at least one order of magnitude (the only exception to this being Mg [5 ] ) , multiple ionization processes are nonetheless of considerable interest from a basic viewpoint as well as in many applications. The simultaneous removal of two or more electrons from an atom caused by the impact of a single electron is conceptually a challenging fundamental collision problem. Double and triple ionization processes, which have comparatively high threshold energies, are important in environments where energetic electrons are abundant e.g. in moderate- and high-temperature plasmas and in some planetary and stellar atmospheres.

Ab initio, fully quantum mechanical calculations of atomic single and multiple ionization cross sections are difficult for all but the simplest targets [l, 61. Semi-empirical and

422 Contrib. Plasma Phys. 35 (1995) 4-5

semi-classical approaches are often preferred by the practitioner in the field, since they can predict ionization cross section functions of sufficient accuracy for many applications such as e.g. modelling purposes with much less computational effort than rigorous theoretical models [7,8]. The present paper extends the semi-empirical Deutsch-Mark (DM) formalism [8], which was originally developed for the single ionization of ground state atoms, to the double and triple ionization of ground state atoms by electron impact. A detailed comparison is made with existing experimental double and triple ionization cross section data. The most extensive series of atomic single and multiple ionization cross sections measurements by a single group using the same apparatus was carried out by Freund and collaborators. These authors measured single ionization cross sections from threshold to 200 eV for 27 atoms, double ionization cross sections for 22 atoms and triple ionization cross sections for 15 atoms using the fast-neutral beam technique [5,9 - 1 I]. In almost all cases, good agreement was found between the present calculation and the available experimental data. In those cases where the same cross section was measured more than once and where the agreement between the different experiments is poor (i.e. discrepancies of 50% and more in the absolute cross section values and differences in the cross section shape from threshold to 200 eV) our calculations generally favor the cross sections measured by the fast-beam technique [5 , 9- 111

2 Theoretical Background

Starting point for the present calculations is the DM formula for the total single ionization cross section 0 of an atom [8, 121

where (r,,,)' is the mean square radius of the nl-shell (to be taken from the tables of Desclaux [13]), tn, is the number of electrons in the nl-shell, g,, is a weighting factor (see refs. [8] and [12] for further details), and the functions f(V) has the form

f ( U ) = (U)- ' . [ ( U - 1)/(U + l)]". { b + c[l - (2U)-'] In [2.7 + (V - l)0.5]}.

The parameters a, b, c are given by [8]: a = 7/4, b = 1, and c = 1 for s-selections, a = 2, b = 1, and c = 1 for p-electrons, a = 312, b = 3, and c = 213 for d-, and f-electrons, and U = E/E,, denotes the reduced impact energy, where E is the energy of the incident electrons and EnI refers to the binding energy of electrons in the nl-shell.

For the calculation of cross sections for multiple ionization, om+ (m = 2, 3, .. .), eq. (1) has to be expressed as the product of two or more independent sums, each describing the removal of one electron. In a first attempt to simplify the calculational procedure as much as possible we modified equation (1) as follows

where the weighting factors g"+ are different from the g,, in eq. (1) and, of course, different for double, triple, etc. ionization. However, we use the same weighting factor gm+ for the different nl-shells that can contribute to a particular ionization cross section. Also, no additional weighting factors for the differnt contributing subshells are used. The energy- dependent function f( U ) is different for the different contributing subshells and is labelled according to the orbital angular momentum quantum number 1 (see eq. (3)). Since we usually limit our calculations to the energy regime below 200 eV where direct ionization

H. DEUTSCH, K. BECKER, T. D. MARK, Electron Impact Ionization 423 2rl I I

,€ 0

' 0 .\ . -.- OO 20 40 60 80

NUCLEAR CHARGE 2

Fig. 1. Product of weighting factor g"' and ionization energy Em+ as a function of the nuclear charge Z for double ionization (top curve (rn = 2), 0 ) and triple ionization (bottom curve (m = 3), M). See text for further details. Two typical error bars are indicated which reflect the range of cross section values that were used in the determination of the repective two data points.

processes dominate and where contributions from more tightly bound inner shells can be neglected, it is sufficient to extend the sum in equation (3) over only those sub-shells nl with a binding energy En, which is smaller than or comparable to the corresponding ionization energy for m-fold ionization, EY+. For example, if we consider the double ionization of Ne with the ground state electron configuration ( 1 ~ ) ~ (2s)' ( 2 ~ ) ~ and a threshold for double ionization of E:+ = 61.25 eV, we find that we have to consider contributions to the double ionization cross section from the 2p-shell (binding energy E Z p = 21.65 eV) and from the 2s-shell (binding energy E,, = 48 eV). Contribution from the 1s-shell can be neglected, since the binding energy E = 480 eV is much larger than E:+ and larger than the highest energy for which we carry out our calculations (typically 200 eV). In this case, eq. (3) has the explicit form

(4) In some cases, particularly when &electrons are involved, sub-shells with a binding energy slightly above the corresponding ionization energy can contribute significantly. As an example, we mention the double ionization of Pb where the ionization E 2 + has a value of E 2 + = 22.45eV. In this case, one has to take into account the 6p, 6s, and 5d sub-shells with binding energies of 7.42 eV, 10.0 eV, and 26.5 eV, respectively.

The factors g"' in eq. (3) were determined from a fitting procedure similar to what was done in the case of the single ionizatioo of atoms [S] using the well-known and reliable cross sections for the double and triple ionization of the rare gases Ne, Ar, Kr, and Xe [5 , 14- 161 and the double and triple ionization cross sections for one atoms with higher nuclear charge, in this case Bi with 2 = 83 [5 ] . The rare gas measurements carried out by the four groups [9, 14-16] show agreement in the absolute value of the double ionization cross sections for the various rare gases at the 7% level and at the 12% level for the triple ionization cross section (for impact energies away from the threshold). The rare gas cross section data of KRISHNAKUMAR and SRIVASTAVA[~~] where not used in our fitting procedure, since some of their cross section values for multiple ionization of the rare gases appear to be systematically higher than the values reported by the other four groups [5, 14- 161, even though their single ionization cross sections are in very good agreement with the other measurements. In the particular case of Ar", it has been argued [16, 181 that the cross section value reported KRISHNAKUMAR and SRIVASTAVA [ 171 is too high by about 25%. Fig. 1 shows the factors g"' in the reduced form as the product g"' . Emf (as was done

.It,' = (gz,. 2p)2 + . [n . (r2,)' . 5 2 p . fP(W + 71 . (rzs)' . (2s . L(Wl .

424 Contrib. Plasma Phys. 35 (1995) 4-5

I- I

q 70'20

J

200 400 600 ELEC TRON ENERGY Ce V7-

162'

Fig. 2. Comparison of our calculated Ne2+ and Ne3+ cross section functions (solid lines) with the experimental data of Lebius et al. [15] ( x ) from threshold to 700eV.

before in the. case of the single ionization [8]) for the double and triple ionization as a function of Z, where Em+ denotes the respective ionization energy. It is apparent from Fig. 1 that the product g2+ . E 2 + is essentially constant above Z = 10, which is similar to what was found in the case of the single ionization [8]. For triple ionization processes, on the other hand, the product g 3 + . E 3 + r eaches a constant value at a significantly higher value of Z and, surprisingly, shows a weak minimum at low values of around Z = 18. The values of g"+ (m = 2,3) for any atom in the periodic table where determined from the curves indicated by the dashed lines in Fig. 1, which were obtained by extrapolation using the fixed g"+ values for the rare gases and Bi. The rare gas data points in Fig. 1 are based on four independent measurements which agree at the 10% level, whereas the Bi data points are only based on a single experiment. While it could be argued, that, as a consequence, the extrapolated curves in Fig. 1 are less reliable in the high-Z region, one should keep in mind, however, that all g'"' . Em+ curves tend to be rather flat in that region. Moreover, the only element between Xe (Z = 54) and Bi (Z = 83) for which experimental data are available is Pb (Z = 82). The excellent agreement between calculated and measured PbZ+ and Pb3+ cross sections (see below) adds further confidence to the reliability of the shapes of the two curves in Fig. 1 in the high-Z region. Since the fitting procedure to determine the weighting factors g"+ was done using only the maximum values of the experimentally determined rare gas cross sections, it is important to demonstrate how the present calculation can reproduce the entire ionization cross section funcitons. This is shown in Fig. 2 where we compare our calculated cross section functions for Ne2+ and Ne3+ from threshold to 700 eV with the experimental data of Lebius et al. [15]. The agreement is excellent over the entire energy range for both cross sections with a maximum deviation of less than 15% near the maximum of the Ne2+ cross section.

3 Results and Discussion

3.1 Double Ioniza t ion We calculated absolute double ionization cross sections from threshold to 200 eV according to eq. (3) with the empirically determined weighting factor g 2 + as discussed in the previous section for the following atoms: Si, P, S, C1, Fe, Cu, Ga, Ge, As, Se, Ag, In, Sn, Sb, Te, Pb,

H. DELJTSCH, K. BECKER, T. D. M ~ R K , Electron Impact Ionization 425

0.5

__._._. --.-- _I**'-' ,$ *""'....'.. ....-.-.

<i.

0 5 0 100 150 200

Electron energy(eV1 - Fig. 3. Comparison of experimentally determined and calculated cross sections for S2 +, Cu2+, and Ga2+. The dots are the measurements of Freund and collaborators [9, 101 and the dash-dot line represents our calculations. The solid lines refer to experimental results obtained by other groups (see text for details).

and Bi corresponding to values of the nuclear charge 2 of 2 = 14- 17,26,29, 31 - 34,47, 49-52, 82, and 83, respectively. The most extensive comparison is made between our calculated cross sections and the experimental cross sections of Freund and collaborators [5, 9-11] for Si2+. P2+, Sz+, C12+, FefC, Cu2+, Ga2+, Ge2+, AsZC, Se2+, Ag2+, Inz+, Sn2+, Sb2+, Te2+, Pb2+, and Bi2+. Other experimental cross sections used for comparison are the S2+ cross section of ZIEGLER et al. [19], the Cu2+ cross sections of CRAWFORD [20] and of SCHROEER et all. [21], the Ag2+ cross section of CRAWFORD and WANG [22], the Pb2+ cross section of PAVLOV and STOTSKII 1231, the Ga2+ and In2+ cross sections of VAINSTHEIN et al. [24], and the Oz+ cross section of ZIEGLER et al. [25]. The recently published Cuz+ and Fez+ cross sections of GILBODY and collaborators [26, 271 do not represent independent measurements, since these authors used the Cu2 + and Fe2 + cross sections of FREUND et al. [5 , 111 to normalize their measured relative cross sections.

Fig. 3 shows the double ionization cross sections for S, Cu, and Ga. Each diagram shows two experimental cross section Functions which are compared with the result of our calculation shown as the dash-dot line. The full circles in each diagram represent the fast-beam measurements of FREUND et al. [5,11], while the solid liner in each figure denotes the experimental results obtained by other groups (see list above). In the case of S2+, the two experimental data sets show a similar shape, but differ in the absolute value by about 35%. Our calculations yield an absolute value which is close to the fast-beam data of Freund, but the calculated shape differs from both measured cross section functions. The calculated shape shows a much broader maximum at a higher impact energy compared to both measured shapes. The comparatively sharp rise of the cross section at low impact energies observed in both experimental data sets could indicate the presence of indirect

29 Contrib. Plasma Phys. 35 (19994-5

426 Contrib. Plasma Phys. 35 ( I 995) 4 - 5

Elec t ron energy(eV1-

Fig. 4. Sameas Fig. 2forAgZ+, Inz f , and Pb2+. Alsoshown isthecornparison for Pb3'

ionization processes in the S2+ cross section. The two sets of measured Cu2+ cross sections differ drastically, both in shape and in magnitude. The fast-beam data are lower by almost a factor of 5. Our calculations is somewhat larger than the measured fast-beam cross section, but only by about 25-30% and it reproduces the cross section shape quite well. Our calculations are still a factor of 3.5 lower than the experimental data of CRAWFORD [20] and SCHROEER et al. [21]. The third diagram in Fig. 3 displays a situation where the two sets of experimental data agree very well with each other (deviations of 15% or less) as well as with the result of our calculation, both in terms of absolute value and cross section shape.

Fig. 4 summarizes the results for Ag2+, In2+, Pb2+, and Pb3+ (which will be discussed later). The situation for Ag2+ is somewhat similar to what was observed for Cu2+. The two available experimental data sets differ significantly in magnitude, this time by a factor of 2 with the fast-beam experiment again yielding the lower cross section values. In contrast to Cu2+, our calculated Ag2+ cross section are now even lower than the measurement of FREUND et al. [5, 1 I]. However, the difference of about 15% lies within the quoted margin of uncertainty of the measurement. The second diagram in Fig. 4, the results for In2+, depicts a situation where the two experimental data sets agree in terms of the absolute cross section value near the maximum, but show a different energy dependence. Our calculated cross section yields a maximum which is roughly 20% lower than the two measured maximum values. The calculated cross section shape agrees very well with the fast-beam data of FREUND et al. [5, 111. The double ionization of Pb represents a situation similar to Ga, where both experimental data sets and the calculation all agree very well with one another, both in the cross section shape as well as in the absolute value. Fig. 5 shows the experimentally determined O2 + cross section [25] in comparison with our calculations from threshold to 400 eV. We included the errors bars (systematic + statistical uncertainties) of the measurements as quoted by the authors. There is very good agreement between experiment and theory over the entire range of impact energies. As a general trend, the

H. DEUTSCH, K. BECKER, T. D. MARK, Electron Impact Ionization 427

6 -

5 -

- 1 -

E -4

< 3 - 0 -

2 - G

1 -

0 100 200 300 Elec t ron e n e r g y ( e v )

Fig. 5. Comparison of the experimentally determined cross section for O2 + [24] (0) with the present calculations (solid line).

calculation lies slightly above the measured data at low and high impact energies and reaches a maximum which is somewhat smaller than the measured maximum.

The data sets displayed Figs. 3 - 5 represent only a small selection of all atoms for which we compared our calculated double ionization cross sections with experimental results. The selection shown in Fig. 3-5 is representative in the sense, that the results for all other atoms listed at the beginning of this section show essentially a very similar behavior. We find very good agreement between our calculations and the available experimental double ionization cross section data. The agreement is particularly good with the fast-beam data of FREUND and collaborators [5, 1 I] with deviations of typically less than 20%. The overall very good agreement between our calculations and the experimental data provides justification for the use of the simplified approach expressed in equation (3) for the calculation of double ionization cross sections of atoms, at least in the energy regime below 200 eV.

3.2 Triple Ioniza t ion

We calculated absolute triple ionization cross sections from threshold to 200 eV according to eq. (3) with the empirically determined weighting factor g 3 + as discussed in the previous section for the following atoms: Ga, Ge, As, Se, Ag, In, Sn, Sb, Te, Pb, and Bi corresponding to values of the nuclear charge Z,of 2 = 31-34, 47, 49-52, 82, and 83 respectively. The most extensive comparison is made between our calculated cross sections and the experimental cross sections of FREUND and collaborators [5 , 111 for Ga3+, Ge3+, As", Se3+, Ag3+, In3+, Sn3+, Sb3+, Te3+, Pb3+, and Bi3+. Other cross sections used for comparison are the Pb3+ cross section of PAVLOV and STOTSKII [23].

The fourth diagram in the previous Fig. 3 shows the comparison between experiment and calculation for the triple ionization of Pb. The two measured data sets differ by about 30-40% in the absolute cross section value, the fast-beam measurement of FREUND et al. [5, 111 again yielding the lower cross section which is in excellent agreement with our calculation from threshold to about 200 eV. There is a slight indication based on the data

428 Contrib. Plasma Phys. 35 (1995) 4-5

I I I I 0 50 100 150 200

E l e c t r o n e n e r g y ( e v )

Fig. 6. Comparison of the experimentally determined cross section Ga3' 19, 10) ( X ) with the present calculation (solid line). Note that the step-like shape of the experimental data is caused by the fact that the results were reported in units of cm2 and only two digits after the decimal point were given.

in the energy regime from 170 - 200 eV that the experimental cross section declines slightly more rapidly towards higher impact energies than our calculation. In all other cases of triple ionization, the only experimentally determined cross section data available for a comparison with our calculations are the fast-beam results of FREUND et al. [ 5 , 111. In general, we find very good (in some cases exceptionally good) agreement between the calculated and the measured triple ionization cross sections. A few examples are shown in

0

;o I

0

I ' I I 50 100 150

E l e c t r o n e n e r g y ( e v ) 0

Fig. 7. Same as Fig. 5 for Ge3+. The open circles (0) refer to the experimental results and the dash-dot line represents the present calculation.

H. DEUTSCH, K. BECKER, T. D. MARK, Electron Impact Ionization 429

0 " 0' 0

?./' O

A s 04'

I

i'. I

0 0 O! I I

I

0 1

03

I I

00 /

1 0

I I

0 ot I

I 9' 1 I 50 100 I50

E l e c t r o p e n e r g y ( e V )

Fig. 8. Same as Fig. 6 for As3+.

Fig. 6-9. As before in the case of the double ionization cross sections, we view the overall very good agreement between our calculations and the experimental data as justification for the use of the simplified approach expressed in eq. (3) for the calculation of atomic triple ionization cross sections, at least in the energy regime below 200 eV.

r x x

E l e c t r o n e n e r g y ( e V )

Fig. 9. Same as Fig. 5 for In3 '.

430 Contrib. Plasma Phys. 35 (1995) 4-5

4 Conclusions

We have extended the DM formalism to the calculation of absolute double and triple ionization cross sections of ground state atoms. Detailed comparisons were made with available experimental cross sections for atoms covering a wide range in the periodic table, from oxygen ( Z = 8) to bismuth (2 = 83). In many cases good agreement was found between experimental data and calculations, both in the absolute cross section value and in the cross section shape. On that basis, we feel confident that the present calculations can be applied to the calculation of double and triple ionization cross sections of atoms, for which no experimental data available at this time, with an accuracy in the 30-50% range for impact energies where direct ionization processes dominate and where contribu- tions from more tightly bound inner shells can be neglected. This level of precision should be sufficient for modelling purposes and many other practical applications.

Acknowledgments

This work was partially supported by the Osterreichischer Fonds zur Forderung der wissenschaftlichen Forschung und Bundesministerium fur Wissenschaft und Forschung and by the U.S. Department of Energy under grant DE-FGOZER 14476.

References

[ I ] MARK, T. D., and DUNN, G. H. (editors), Electron Impact Ionization, Springer Verlag, Wien (1 985).

[2] FREUND, R. S., in: PITCHFORD, L. C., et al. (editors), Swarm Studies and Inelastic Electron- Molecules Collision, Springer Verlag, New York (1987).

[3] TARNOVSKY, V., and BECKER, K., Invited Papers of the XVIII International Conference on the Physics of Electronic and Atomic Collision (ICPEAC), Aarhus, Denmark, (l993), ANDERSEN, T. et al.(editors), America1 Institute of Physics Press, New York (1994), p. 234-249.

[4] BECKER, K., and TARNOVSKY, V., Plasma Sources Sci. Technol. 4 (1995) 1. [5] FREUND, R. S., WETZEL, R. C., and SHUL, R. J., Phys. Rev. A 4 1 (1990) 5861. [6] YOUNGER, S. M., Quantum Theoretical Methods for Calculating Ionzization Cross Sections, in:

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[7] YOUNGER, S. M., and MARK, T. D., Semi-Empirical and Semi-Classical Approximation for Electron Ionization, in: MARK, T. D., and DUNN, G. H. (editors), Electron Impact Ionization. Springer Verlag, Wien (1985).

[8] DEUTSCH, H., and MARK, T. D., Int. J. Mass. Spectrom. Ion Proc. 79 (1987) RI. [9] WETZEL, R. C.; BAIOCCHI, F. A., HAYES, T. R., and FREUND, R. S., Phys. Rev. A 35 (1987) 559.

[lo] HAYES, T. R., WETZEL, R. C., and FREUND, R. S., Phys. Rev. A 3 5 (1987) 587. [ 1 I] SHUL, R. J., WETZEL, R. C., and FREUND, R. S., Phys. Rev. A 39 (1989) 5588; A 41 (1990) 5856. [I21 MARGREITER, D., DEUTSCH, H., and MARK, T. D., Int. J. Mass. Spectrom. Ion Proc. 139

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H. DEUTSCH, K. BECKER, T. D. MARK, Electron Impact Ionization 43 1

[20] CRAWFORD, C. K., as reported by KIEFFER, L. I., Atomic Data 1 (1969) 19. [21] SCHROEER, J. M., GUNDUZ, D. H., and LIVINGSTONE, S . , J. Chem Phys. 58 (1973) 5135. [22] CRAWFORD, K., and WANG, K. I., J. Chem Phys. 47 (1967) 4667. [23] PAVLOV, I., and STOTSKII, G. I., Sov. Phys. JEPT 31 (1970) 61. [24] VAINSHTEIN, A., GOLOVACH, D. G., OCHKUR, V. I., RAKHOVSKII, V. I., RUMYANTSEV, N. M., and

[25] ZIEGLER, L., NEWMAN, J. H., SMITH, K. A., and STEBBINGS, R. F., Planet. Space Sci. 30

[26] SHAH, B., MCCALLION, P., OKUNO, K., and GILBODY, H. B., J. Phys. B 26 (1993) 2393. [27] BOLORIZADEH, A,, PATTON, C. J., SHAH, M. B., and GILBODY, H. B., J. Phys. B 27 (1994) 175.

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Received March 3, 1995