Calculated Cross Sections for the Multiple Ionization of Krypton and Chromium Atoms by Electron Impact

Contrib. Plasma Phys. 41 (2001) 1, 73−83

Calculated Cross Sections for the Multiple Ionizationof Krypton and Chromium Atoms by Electron Impact

H. Deutscha), K. Becker

b), T.D. Mark

c,d)

a)Institut fur Physik, Ernst-Moritz-Arndt Universitat, D-17487 Greifswald,Germany

b)Department of Physics, Stevens Institute of Technology, Hoboken,NJ 07030, USA

c)Institut fur Ionenphysik, Leopold-Franzens Universitat, A-6020 Innsbruck,Austria

d)Also adjunct professor at Dept. Plasmaphysics, Comenius UniversityBratislava, Slovak Republice-mail: [email protected]

Received 19 May 2000, in final form 23 June 2000

Abstract

The Deutsch-Mark (DM) formalism has been used to calculate cross sections for the single-step formation of Krm+ (m = 2 - 18) and Crm+ (m = 2 - 12) ions by electron impact onthe neutral atoms. Our calculated cross sections for these multiple ionization processes arecompared with available experimental and other theoretical data and systematic trends inthe data are highlighted.

1 Introduction

The formation of a multiply charged atomic ion Am+ by single-step multiple ionizationof a neutral atom A

A+e− → Am++(m+1)e− (1a)

is one of the basic collisional interactions between electrons and atoms. Ionizationprocesses play an important role in many applications such as discharges and plasmas,gas lasers, planetary, cometary, and stellar atmospheres, radiation chemistry, massspectrometry, and chemical analysis [1, 2]. The formation of a particular multiplycharged ion Am+ via process (1) is always in competition with a variety of step-wiseionization processes leading to the same Am+ ion

An+ +e− → Am++([m−n] + 1) e− (1b)

with n < m. Even though the cross sections for the single-step multiple ionizationof an atom are significantly smaller than cross sections for single ionization and theionization of a singly or multiply charged ion [1] and decline rapidly with the stageof ionization [3 - 6], the density of neutral atoms in most environments mentionedabove is typically much larger than the density of multiply charged ions, so that thecontributions of process (1a) and processes (1b) to the formation of a particular ionAm+ may well be comparable.

Calculations of multiple atomic ionization cross sections using rigorous quantummechanical methods are difficult for all but the simplest atoms [1, 7, 8] because of

74 Contrib. Plasma Phys. 41 (2001) 1

s-electrons: a = 1.06 b = 0.23 c = 1 d = 1.1

p-electrons: a = 2 b = 1 c = 1 d = 1

d-Electrons: a = 3/2 b = 3 c = 2/3 d = 1

f-Electrons: a = 3/2 b = 1 c = 2/3 d = 1

Tab. 1: Parameters a, b, c and d for the energy dependent functionin Equ. (3) for s-, p-, d-, and f-electrons

- among other things - the need to consider two or more continuum electrons and theirmutual interaction in the exit channel. Experimental data for the formation of highlycharged ions are scarce for most atoms [3 - 6, 9] because of among other reasons the factthat the cross sections for the single-step formation of highly-charged multiple ions arecomparatively small. Modellers and practitioners rely heavily on semi-empirical andsemi-classical methods to determine multiple ionization cross sections for modellingpurposes and for other applications [1, 2, 7, 10]. Recently, the Deutsch-Mark (DM)formalism [11] has been applied to the calculation of cross sections for the formationof multiply charged ions Am+ for several atoms such as high-Z atoms with a nuclearcharge Z of 10 and above and various stages of ionization [6, 12, 13] as well as severallow-Z targets [3 - 5, 14]. The DM formalism has the following advantages over othersemi-empirical methods that are used for the calculation of cross sections for multipleionization of neutral atoms [15, 16]: (i) its application requires fewer semi-empiricalparameters which, in addition, can easily be related to physical quantities and (ii)the energy-dependent function (derived from classical considerations) is the same forall stages of ionization. It should be noted, however, that all three semi-empiricalmethods cannot distinguish between direct ionization and indirect ionization processesvia intermediate autoionizing states.

In this paper, we present the results of the application of the DM formula to thecalculation of cross sections for the formation of multiply charged ions of krypton,Kr (Z = 36; outer electron configuration ...(3d)10(4s)2(4p)6) and chromium, Cr (Z= 24; ...(3d)5(4s)1) up to the formation of the ions Kr18+ and Cr12+. In the case ofKr, we can compare the present results obtained with the DM formula with availableexperimental data. We also compare the present results with cross sections derivedfrom the method of Shevelko and Tawara [16] and discuss systematic trends in the twocalculated cross section data sets; the method of Fisher et al. [15] was found to showvery large deviations from the data for higher charge states (see e.g. Fig. 5 in Ref. [6])and is, therefore, not considered in more detail here.

Targets of importance in Tokamak fusion plasmas have been discussed in severalreviews [2, 17, 18], and according to Post [19] and Janev [20] the elements that requireparticular attention include Be, B, C, N, Al, Si, Mg, Ti, Cr, Fe, Ni, Cu, Mo, Nb,Ta and W as possible impurities in the plasma edge. In addition, Ne, Ar, Kr, Xeand N atoms may be present in cases where these gases are injected into the plasmavolume for cooling the plasma scrape-off layer [14]. Although electron temperaturesin these plasma regions are relatively low, well below 1 keV, the high energy part ofthe energy distributions may lead to the production of more highly charged speciesand thus significantly influence the plasma properties. Corresponding electron impactcollision data for multiple ionization up to the formation of fully stripped ions of Be,

H. Deutsch et al., Calculated Cross Sections for Multiple Ionization 75

Atom, Nuclear Charge Z Weighting Factor g2+ Weighting Factor g3+

Cr, 24 0.1315 8.85 ×10−3

Kr, 36 8.08 ×10−2 1.14 ×10−2

(a) Values of the weighting factors g2+ and g3+ for Kr and Cr

Atom, Nuclear Charge Z Parameter a Parameter b

Cr, 24 0.85 1.65Kr, 36 1.00 1.46

(b) Parameters a and b in equation (4) to calculate theweighting factors gm+ (m > 3) for the atoms Cr and Kr

Tab. 2: Weighting factors for the calculation of the cross sections for Krand Cr according to the DM formalism as expressed in equation (2)

B, C, N, O, Ne, Si, and Ar have been calculated recently [3, 4, 14]. Here we add tothis list the relevant data for Kr and Cr which were selected (i) to extend the previousdata for Ne and Ar to the next heavier rare gas Kr and (ii) to respond to data needsfor Cr as a result of recent advances in the development of new chromium-based alloysfor use as structure breeding blanket material in next generation tokamaks [21]. Theadvantageous material properties of chromium include a low activation probabilityunder neutron irradiation and good mechanical stability at high temperatures [21].

2 Theoretical background

The DM formalism was originally developed for the calculation of cross sections forthe single ionization of atoms [11, 22]. The extension of the DM formalism to thecalculation of cross sections for the formation of multiply charged ions by electronimpact on the neutral atom has been described in detail in previous publications [3 -6, 12 - 14]. Briefly, the cross section σm+ for the formation of an ion Am+, which inprinciple is a product of m independent terms each describing the removal of a singleelectron, can be simplified to an expression of the form

σm+ = gm∑

k

π(rk)2ξkfk(U) (2)

where the summation extends over the various atomic sub-shells with k = 1 referringto the outermost sub-shell, k = 2 to the second outermost sub-shell, etc. In the aboveequation, (rk)

2 is the radius of maximum radial density of the atomic sub-shell labeledby k as listed in column 1 in the tables of Desclaux [23], ξk is the number of electronsin that sub-shell, and gm are weighting factors (see refs. [3, 4] for further details). Thefunctions fk(U) describe the energy dependence of the ionization cross section [11, 22]

fk(U) = d

(1

U

) [U − 1

U + 1

]a [b + c

(1− 1

2U

)]ln

[2.7 + (U − 1)0.5

](3)

Here U refers to the reduced impact energy, U = E/Em, where E is the energyof 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 ofelectrons in the sub-shell labeled by k. A detailed discussion of the functions fk(U),whose exact form differs for s-, p-, d-, and f-electrons, has been given in a recent topicalreview article [24] and the parameters are summarized in Tab. 1 for the convenienceof the reader. We further note that for impact energies above about 105 eV theenergy dependence has to be modified to include relativistic kinetic energy effects asdiscussed in a previous paper [25]. The weighting factors gm for m = 2 and 3 weretaken from Deutsch et al. [12] (see Table 2a) and the weighting factors for m > 3 weredetermined from a fitting procedure (for details, see Deutsch et al. [3, 4]) accordingto an exponential function of the form

gm(Z) = a(Z) exp[−b(Z)m] (4)

where Z is the nuclear charge and a(Z) and b(Z) are two empirically determinedfunctions (see Refs. [3, 4] for details). The values of the parameters a and b for thetwo atoms studied in this paper have been summarized in Tab. 2b.

3 Results and discussion

As was done in our previous papers [3 - 5, 14] we omit the cross sections for the singleionization of the respective target atoms from the discussion, since these cross sec-

tions have been presented

Fig. 1: Calculated cross sections for the formation of Kr2+

ions by electron impact on Kr as a function of electron en-ergy. The solid line represents the present calculation (DMcross section) and the dashed line represents the ST crosssection. The experimental data are from Wetzel et al. [26](×), Stephan et al. [27] (+), Lebius et al. [28] (open invertedtriangles), and Syage [29] (open triangles).

and discussed in detail ear-lier [22]. Experimental datafor the formation of highlycharged ions Am+ (for m >3) produced by electron im-pact on the neutral atom Aare available for only a fewatoms, primarily for the no-ble gases Ne, Ar, Kr, andXe. In particular, multi-ple ionization cross sectionsfor Kr have been reportedby 8 groups [26 - 33]. Ex-perimental data for the for-mation of multiply chargedions of Cr are not availablein the literature to the bestof our knowledge.

Fig. 1 shows the calcu-lated cross sections for theformation of Kr2+ ions us-ing the present method andthe method of Shevelko andTawara [17] (which will bereferred to as DM cross sec-

tions and ST cross sections, respectively, throughout the remainder of this article) incomparison with the experimental data of Lebius et al. [28], Syage [29], Wetzel et al.


Fig. 2: Calculated cross sections for the for-mation of Kr3+ ions by electron impact onKr as a function of electron energy. Thesolid line represents the present calculation(DM cross section) and the dashed line rep-resents the ST cross section. The exper-imental data are from Lebius et al. [28](open inverted triangles), Syage [29] (opentriangles), Krishnakumar and Srivastava [30](stars), Schram [31] (filled squares), Almeida[32] (open squares), and Nagy et al. [33](open circles).

Fig. 3: Calculated cross sections for the for-mation of Kr4+ ions by electron impact onKr as a function of electron energy. Thesolid line represents the present calculation(DM cross section) and the dashed line rep-resents the ST cross section. The exper-imental data are from Lebius et al. [28](open inverted triangles), Syage [29] (opentriangles), Krishnakumar and Srivastava [30](stars), Schram [31] (filled squares), Almeida[32] (open squares), and Nagy et al. [33](open circles).

[26], and Stephan et al. [27]. There is good agreement between the various experimen-tal data sets and both calculations reproduce the maximum cross section value ratherwell, but differ from the experimental data in terms of the energy shape.Both calculated cross section curves show a similar energy dependence, but the DMcross section lies above the ST cross section for all impact energies. Fig. 2 shows theresults for the formation of Kr3+ ions. As before, the two calculated cross sectioncurves are similar in shape, but the DM cross section lies above the ST cross sectionby about 20% for all impact energies. Experimental data from 6 groups [28 - 33] havealso been included in Fig. 2 (for clarity of presentation, the data of Wetzel et al. [26]and Stephan et al. [27], which were limited to impact energies below 200 eV and whichare essentially identical in that energy regime with the data of Syage [29], have beenomitted from the diagram). There is a noticeably larger variation in the experimentaldata for Kr3+ as compared to Kr2+. Our DM calculation predicts a cross section curvein reasonably good agreement with the average of the various experimental data in theenergy from threshold to about 1000 eV. However, at higher impact energies the DMcross section lies below all experimental data. In Fig. 3, we summarize the results for


Fig. 4: Calculated cross sections for the for-mation of Kr5+ ions by electron impact onKr as a function of electron energy. Thesolid line represents the present calculation(DM cross section) and the dashed line rep-resents the ST cross section. The experimen-tal data are from Lebius et al. [28] (open in-verted triangles), Syage [29] (open triangles),Schram et al. [31] (filled squares), Almeida[32] (open squares), and Nagy et al. [33](open circles).

Fig. 5: Calculated cross sections for the for-mation of Kr6+ ions by electron impact onKr as a function of electron energy. Thesolid line represents the present calculation(DM cross section) and the dashed line rep-resents the ST cross section. The experimen-tal data are from Lebius et al. [28] (open in-verted triangles), Syage [29] (open triangles),Schram et al. [31] (filled squares), Almeida[32] (open squares), and Nagy et al. [33](open circles).

Kr4+. The various experimental data [28 - 33] diverge even more, by almost a factorof 2 in terms of the cross section maximum. The difference between the two calculatedcross section curves has also increased to about a factor of 2. All experimental dataexcept for the measurement of Krishnakumar and Srivastava [30] (which are believedto be too high [4, 5]) are bounded by the two calculated cross section curves with theDM cross section serving as the upper bound and the ST cross section as the lowerbound.

Fig. 4 shows the comparison between the calculated and measured Kr5+ cross sec-tions. The experimental data of Almeida [32], Schram [31], and Lebius et al. [28] arein reasonably good agreement with the DM cross section up to an energy of 3000 eV.The experimental data of Syage [29] are smaller than all other measured data and thecalculated ST cross section is considerably smaller (by a factor of 2.5) than the DMcross section and all measured data except for the data of Syage [29]. In the case ofKr6+ (Fig. 5), the DM cross section is in reasonably good agreement with the measureddata of Almeida [32] for impact energies above about 800 eV, whereas the DM crosssection at lower energies lies above the measured data. The data of Schram [31] are


below the DM cross section for all impact energies. The data of Syage [29] and Lebiuset al. [28], which are limited to energies below 750 eV lie below the DM cross sectionin that energy range. The ST cross section is smaller than the DM cross section byabout a factor of 2.5.

Fig. 6 shows the exper-

Fig. 6: Calculated cross sections for the formation of Kr7+

ions by electron impact on Kr as a function of electronenergy. The solid line represents the present calculation(DM cross section) and the dashed line represents the STcross section. The experimental data are from Lebius et al.[28] (open inverted triangles), Schram [31] (filled squares),Almeida [32] (open squares), and Nagy et al. [33] (opencircles).

imental data for Kr7+ ofAlmeida [32], Schram [31]and Lebius et al. [28] incomparison with the calcu-lated DM and ST cross sec-tions. The ST cross sec-tion, which is lower than theDM cross section by a factorof 2, is in good agreementwith the data of Schram[31] and the few data pointsreported by Lebius et al.[28], whereas the data ofAlmeida [32] lie above theDM cross section for im-pact energies above 1000eV. The DM and ST crosssections for Kr8+ (Fig. 7)which differ by a factor of 2in the maximum cross sec-tion value (the DM calcula-tion yields the higher max-imum cross section value)agree to within a factor of2 or better with the mea-sured data of Schram [31].The data of Lebius et al.[28], which cover only theenergy range up to 1000 eVare consistently below bothcalculated cross sections inthis energy regime. Kr9+ isthe most highly charged Krion for which cross sections

have been measured [31]. These data lie below both calculated cross sections curves.As with all other Krm+ (m = 2 - 9) cross sections, the DM curve lies above the STcurve, though by less than a factor of 2 in this particular case. In general, we foundthat the difference in the calculated Krm+ (m = 2 - 9) cross sections between the DMformalism and the ST formalism to be much less than what was observed for the low-Z atoms [3 - 5, 14], where the discrepancies in the predictions from the two methodsreached several orders of magnitude in some cases.

Fig. 9 summarizes the DM calculation for the formation of Krm+ (m = 10 - 18) ionsfor which no experimental results exist. For clarity of presentation, we omitted thecalculated ST cross sections. As one would expect, the absolute cross section declines


Fig. 7: Calculated cross sections for the for-mation of Kr8+ ions by electron impact onKr as a function of electron energy. The solidline represents the present calculation (DMcross section) and the dashed line representsthe ST cross section. The experimental dataare from Lebius et al. [28] (open invertedtriangles), Schram [31] (filled squares).

Fig. 8: Calculated cross sections for the for-mation of Kr9+ ions by electron impact onKr as a function of electron energy. The solidline represents the present calculation (DMcross section) and the dashed line representsthe ST cross section. The experimental dataare from Schram [31] (filled squares).

rapidly with increasing stage of ionization m and the thresholds in the cross sectioncurves shift to larger impact energies. We note that the change in the slope of the crosssection curves at higher impact energies (above 105 eV) is caused by relativistic kineticenergy effects as discussed before. Cross sections for higher stages of ionization werenot calculated as the cross section values are becoming exceedingly small for m > 18which correspond to the removal of electrons from the tightly bound (3p) sub-shelland the even more tightly bound lower-lying sub-shells.

Figs. 10 - 12 summarize the calculated DM cross sections for the formation of Crm+

(m = 2 - 12) ions. No experimental data are available for comparison for these ions.Also shown in figs. 10 and 11 are the ST cross sections for m = 2 - 9. For clarity ofpresentation, the curves representing odd and even values of m are shown in separatedfigures up to m = 9. As was the case for Krm+ (m = 2 - 9), the DM and the ST crosssections are generally in fair agreement (to within a factor of 2 or better), the absolutecross section values decline with m, and the thresholds shift to higher energies. Lastly,figure 12 summarizes the calculated DM cross sections for the formation of Crm+ (m= 10 - 12) ions. Ionization thresholds are above 1000 eV, maximum cross sectionsdecline from 2 × 10−23 cm2 (m = 10) to 1 × 10 −24 cm2 (m = 12), and relativistickinetic energy effects are beginning to make their presence felt in the cross sectionshapes.


Fig. 9: Calculated cross sections for the for-mation of Krm+ (m = 10 - 18) ions by elec-tron impact on Kr as a function of electronenergy. The solid lines represent the presentcalculation (DM cross section).

Fig. 10: Calculated cross sections for the for-mation of Crm+ (m = 2, 4, 6, 8) ions byelectron impact on Cr as a function of elec-tron energy. The solid lines represent thepresent calculation (DM cross section) andthe dashed lines represent the ST cross sec-tion.

Acknowledgements

This work has been carried out within the Association EURATOM-OAW and waspartially supported by the FWF Wien, Austria. One of us (KB) acknowledges partialsupport by the Division of Chemical Sciences, Office of Basic Energy Sciences, Officeof Science, U.S. Department of Energy.

References

[1] Mark, T.D., Dunn, G.H. (editors), Electron Impact Ionization, Springer Verlag,Wien (1985)

[2] Janev, R.K. (editor), Atomic and Molecular Processes in Fusion Edge Plasmas,Plenum Press, New York (1995)

[3] Deutsch, H., Becker, K., Matt, S., Mark, T.D., Plasma Phys. Control. Fusion40 (1998) 1721


Fig. 11: Calculated cross sections for the for-mation of Crm+ (m = 3, 5, 7, 9) ions byelectron impact on Cr as a function of elec-tron energy. The solid lines represent thepresent calculation (DM cross section) andthe dashed lines represent the ST cross sec-tion.

6

Fig. 12: Calculated cross sections for the for-mation of Crm+ (m = 10 - 12) ions by elec-tron impact on Cr as a function of electronenergy. The solid lines represent the presentcalculation (DM cross section).

[4] Deutsch, H., Becker, K., Senn, G., Matt, S., Mark, T.D., Int. J. Mass Spec-trom., 192 (1999) 1

[5] Almeida, D.P., Becker, K., Deutsch, H., Int. J. Mass Spectrom. Ion Proc. 163(1997) 39

[6] Deutsch, H., Becker, K., Almeida, D.P., Mark, T.D., Int. J. Mass Spectrom.Ion Proc. 171 (1997) 119

[7] Younger, S.M., Quantum theoretical methods for calculating ionization cross sections,in: Electron Impact Ionization (Mark, T.D., Dunn, G.H., editors), Springer Verlag,Vienna (1985)

[8] Rudge, M.R.H., Revs. Mod. Phys., 40 (1968) 564[9] Gerdom, K., Puerta, J., Wiesemann, K., J. Phys. B 27 (1994) 747

[10] Younger, S.M., Mark, T.D., Semi-empirical and semi-classical approximations forelectron ionization, in: Electron Impact Ionization (Mark, T.D., Dunn, G.H., edi-tors), Springer Verlag, Vienna (1985)

[11] Deutsch, H., Mark, T.D., Int. J. Mass Spectrom. Ion Proc. 79 (1987) R1


[12] Deutsch, H., Becker, K., Mark, T.D., Contr. Plasma Phys. 35 (1995) 421[13] Deutsch, H., Becker, K., Mark, T.D., J. Phys. B 29 (1996) L497[14] Deutsch, H., Becker, K., Mark, T.D., Plasma Phys. Control. Fusion 42 (2000),

489[15] Fisher, V., Ralchenko, Y., Goldgirsh, A., Fisher, D., Maron, Y., J. Phys. B

28 (1995) 3027[16] Shevelko, V.P., Tawara, H., J. Phys. B 28 (1995) L589[17] Hofer, W.O., Roth, P. (editors), Physical Processes of the Interaction of Fusion

Plasmas with Solids, Academic Press, San Diego (1996)[18] Mohr, P.J., Wiese, W.L. (editors), Atomic and Molecular Data and their Applica-

tions, AIP Conference Proceedings (1998) 434, Woodbury, New York[19] Post, D.E., in Ref. [18], p. 233 - 258[20] Janev, R.K., Chapter 1 in Ref. [2], p. 1 - 13[21] Schedler, B., Lecture at 6th Association Day of Association EURATOM-OAW,

Plansee (1999)[22] Margreiter, D., Deutsch, H., Mark, T.D., Int. J. Mass Spectrom. Ion Proc. 139

(1994) 127[23] Desclaux, J.P., At. Data Nucl. Data Tables 12 (1973) 325[24] Deutsch, H., Becker, K., Matt, S., Mark, T.D., Int. J. Mass Spectrom. 197

(2000) 37[25] Deutsch, H., Margreiter, D., Mark, T.D., Z. Phys. D 29 (1994) 31[26] Wetzel, R.C., Biaocchi, F.A., Hayes, T.R., Freund, R.S., Phys. Rev. A 39

(1987) 559[27] Stephan, K., Helm, H., Mark, T.D., J. Chem. Phys. 73 (1980) 3763[28] Lebius, H., Binder, J., Koslowski, H.R., Wiesemann, K., Huber, B.A., J. Phys.

B 22 (1992) 83[29] Syage, J.A., Phys. Rev. A 46 (1992) 5666[30] Krishnakumar, E., Srivastava, S.K., J. Phys. B 21 (1988) 1055[31] Schram, B.L., Physica 32 (1966) 197[32] Almeida, D.P., Int. J. Mass Spectrom. 184 (1999) 49[33] Nagy, P., Skutlartz, A., Schmidt, V., J. Phys. B 13 (1981) 1249

Documents

Calculated Cross Sections for the Multiple Ionization of Krypton and Chromium Atoms by Electron Impact