Calculated Electron Impact Ionization Cross Sections of Excited Ne Atoms Using the DM Formalism

Contrib. Plasma Phys. 45, No. 7, 494 – 499 (2005) / DOI 10.1002/ctpp.200510055

Calculated Electron Impact Ionization Cross Sections of ExcitedNe Atoms Using the DM Formalism

H. Deutsch1, K. Becker2, A. N. Grum-Grzhimailo3, M. Probst4, S. Matt-Leubner4, and T.D.Mark∗4

1 Institut fur Physik, Ernst Moritz Arndt Universitat, Domstr. 10a, D-17487 Greifswald, Germany2 Department of Physics and Center for Environmental Systems, Stevens Institute of Technology, Hoboken, NJ

07030, USA3 Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia4 Institut fur Ionenphysik, Leopold Franzens Universitat, Technikerstr. 25, A-6020 Innsbruck, Austria

Received 14 February 2005, accepted 6 April 2005Published online 26 September 2005

Key words Electron ionization, cross sections, excited states, rare Gases.PACS 34.80 Dp and 52.20 Fs

We used the semi-classical Deutsch-Mark (DM) formalism to calculate absolute electron-impact ionizationcross sections of excited Ne atoms from threshold to 1000 eV. Excited states of Ne where the outermost va-lence electron is excited to states with principal quantum numbers up to n = 7 and orbital angular momentumquantum numbers up to l = 2 have been considered and systematic trends in the calculated cross section dataare discussed.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Electron-induced formation of positive ions is a fundamental collision process. Ionization cross sections of atomshave been measured and calculated since the early days of collision physics [1] because of their basic importanceand because of their relevance in practical applications such as gas discharges, plasmas, radiation chemistry, plan-etary atmospheres, and mass spectrometry. Considerable progress in the quantitative determination of ionizationcross sections for atomic (and simple molecular) targets [2, 3] has been made in the past decade. The ionizationof rare gas atoms has been studied extensively, in particular the ionization of ground-state rare gas atoms whoseionization cross sections are considered benchmark data [2]. Much less effort has been devoted to the ionizationof atoms in excited states. Ionization cross sections have been measured for the metastable rare gas atoms He[4, 5], Ne [6, 7], and Ar [6] and several calculations using different approximations have been carried out for thesetargets [8, 9, 10] including an earlier calculation based on the semi-classical Deutsch-Mark (DM) formalism [11].

Recently, we applied the semi-classical Deutsch-Mark (DM) to the calculation of absolute electron-impactionization cross sections for excited Ar atoms [12]. In this paper, we extend these calculations to Ne using themost advanced variant of the DM approach [13]. Specifically, we consider 30 different excited states where thevalence electron is excited to states characterized by principal quantum numbers up to n = 7 and orbital angularmomentum quantum numbers up to l = 2. Excited rare gas atoms play an important role in the ionization balancein gas discharges and in low-temperature plasmas because of the importance of step-wise ionization processes[14]. For instance, Bogaerts et al. [15] developed extensive collisional radiative (CR) models for rare gas glowdischarges which require - among others - ionization cross sections for many atomic levels. Other environmentswhere the ionization of excited states is of importance include discharges [16], solar and stellar atmospheres[17] and astrophysical media [18]. Most notably, MacAdam and co-workers [19] reported the first experimentaldetermination of ionization cross section of highly excited Na Rydberg atoms (35 < n < 51, where n denotesthe principal quantum number) and found serious disagreement between their measured data and the predictions

∗ Corresponding author: e-mail: [email protected] adjunct professor at: Dept. Plasmaphysics, Univerzita Komenskeho, Mlynska dolina, 842 48 Bratislava 4, Slovak Republic


Contrib. Plasma Phys. 45, No. 7 (2005) / www.cpp-journal.org 495

of simple theoretical models scaled to such high values of the principal quantum number n. Lastly, the ionizationof highly excited states of Ne is of particular relevance in the context of fusion plasmas [20].

2 Theoretical Background

The original DM formalism [21] has been modified and extended several times. The original DM formula ex-presses the total single ionization cross section σ of an atom as

σ =∑

n,l

gnlπr2nlξnlf(u) (1)

where rnl is the radius of maximum radial density of the atomic subshell characterized by quantum numbers nand l (as listed in column 1 in the tables of Desclaux [22]) and ξnl is the number of electrons in that subshell.The sum extends over all atomic sub-shells labelled by n and l. The factors gnl are weighting factors which wereoriginally determined from a fitting procedure [23, 24] using reliable experimental cross section data for the raregases and uranium. The function f(u) describes the energy dependence of the cross sections, where u = E/Enl

(with E denoting the incident energy of the electron and Enl referring to the ionization energy in the (n,l) sub-shell). Recently, Deutsch et al. [13] replaced the previously used modified Gryzinski energy dependence f(u)by an energy-dependent function that has the ”correct” Born-Bethe high-energy form. This now results in themodified DM formula which has the explicit form

σ(u) =∑

n,l

gnlπr2nlξnlb

(q)nl (u)[ln(cnl · u)/u] (2)

where the function b(q)nl (u) has the form

b(q)nl (u) =

A1 − A2

1 + (u/A3)p+ A2 (3)

The 4 quantities A1, A2, A3, and p are constants that were determined (in conjunction with the constant cnl)from reliable measured cross sections for the various values of n and l. The superscript q refers to the numberof electrons in the (nl) subshell. Values of all constants and parameters relevant to the application of the DMformula of equation (2) can be found in the two papers by Deutsch et al. [13, 24] to which we refer the reader forfurther details.

Neon has a (1s)2(2s)2(2p)6 electron configuration in its ground state, which is a 1S0 state. The lower lyingexcited states of Ne result from the promotion of one of the six (2p)-electrons into higher orbitals such as 3s, 3p,3d, etc. In order to apply the DM formula to the calculation of ionization cross sections of excited Ne atoms, weneed to know the values of the 2 parameters rnl and Enl for the higher-lying (n,l) sub-shells. The values for Enl

were obtained from the well-known energy level diagram of the Ne atom [25]. The radii rnl for Ne were obtainedfrom numerical Hartree-Fock calculations [26]. Table 1 summarizes the values for rnl and Enl used in thepresent calculation up to values of n = 7 and l = 2. Furthermore, the application of the DM formalism requiresknowledge of the weighting factors gnl for the higher-lying (n,l) sub-shells. As these weighting factors weredetermined from a fitting procedure using experimentally determined reliable cross section data and such dataare only available for values of the orbital angular momentum quantum number l up to l = 2 (d-electrons), thelack of reliable weighting factors gnl for higher l-values limits the present calculations to electron configurationswhere the outermost valence electron is in higher-lying s-, p-, and d-orbitals.

All excited Ne electron configurations are of the form ”core + nl”, where the core is of the form (1s)2(2s)2(2p)5.Since the DM formalism calculates the ionization cross section as the sum of the partial cross sections for theremoval of an electron from a particular (nl) orbital summed over all (nl) configurations, it is illustrative tocalculate separately the contribution to the total ionization cross section arising from the removal of an electronfrom the core and from the (nl) valence electron. The contribution due to the core electrons, which is common toall configurations treated here, peaks at an energy that is slightly above 100 eV with a maximum value of about0.5 × 10−20m2 and is in good accordance with our previous results [11] as discussed below in Fig. 1 in moredetail.


496 H. Deutsch, K. Becker, and A.N. Grum-Grzhimailo et al.: Calculated Ionization Cross Sections for Excited Ne Atoms

Table 1 Values of the radii rnl and the energies Enl for outermost Ne valence electron. In the case of the radii, we used thesame value for both the primed and unprimed configuration. In the case of the energy values, we used energies averaged overall fine structure components of each (n,l) and (n,l’) configuration, respectively.

Quantum numbers (n,l) Value of rnl Value of Enl

of valence electron (same for both configurations)

3s, 3s’ 1.87 × 10−10 m 4.92 eV, 4.78 eV

4s, 4s’ 5.40 × 10−10 m 1.88 eV, 1.79 eV

5s, 5s’ 1.14 × 10−9 m 1.05 eV, 0.90 eV

6s, 6s’ 1.88 × 10−9 m 0.62 eV, 0.52 eV

7s, 7s’ 2.74 × 10−9 m 0.42 eV, 0.32 eV

3p, 3p’ 2.71 × 10−10 m 2.81 eV, 2.73 eV

4p, 4p’ 7.38 × 10−10 m 1.36 eV, 1.23 eV

5p, 5p’ 1.38 × 10−9 m 0.77 eV, 0.67 eV

6p, 6p’ 2.27 × 10−9 m 0.50 eV, 0.40 eV

7p, 7p’ 3.31 × 10−9 m 0.35 eV, 0.30 eV

3d, 3d’ 4.76 × 10−10 m 1.52 eV, 1.42 eV

4d, 4d’ 1.14 × 10−9 m 0.85 eV, 0.76 eV

5d, 5d’ 1.88 × 10−9 m 0.54 eV, 0.45 eV

6d, 6d’ 2.92 × 10−9 m 0.38 eV, 0.28 eV

7d, 7d’ 4.24 × 10−9 m 0.28 eV, 0.18 eV

Lastly, we note that the calculated DM cross sections represent total single ionization cross sections at-tributable to direct ionization processes only. Multiple ionization processes and indirect processes such as au-toionization are not described by the DM formalism.

3 Results and Discussion

We begin our discussion with a comparison of the calculated DM ionization cross section of the 3P0 state inNe (see Fig. 1) with the experimental results of Johnston et al. [27] and the Born calculation of Ton-That andFlannery [28]. The Ne 3P0 state is the higher-lying of the two metastable states corresponding to the first excited-state configuration of Ne, (1s)2(2s)2(2p)5[3P1/2]3s. Our DM cross section is in good agreement with the Borncalculation except for the low energy region up to about 20 eV, where the Born cross section lies above our dataand the measured cross section data, which is not uncommon for Born calculations. The DM cross section liesconsistently slightly above the experimental data, but the discrepancy is within the quoted roughly 40% errormargin of the measured data. Hyman [29] using a symmetric binary encounter model in conjunction with asemi-empirically determined momentum distribution function for the bound excited electron calculated electronimpact ionization cross sections for the lowest excited states of the rare gases, cadmium, and mercury. His resultsin Ne are in good agreement with our data for the 3s level, but are smaller than our data for the 3p level (shownbelow in fig. 3) by a factor of 3.

The results of the present calculations are shown in 3 figures. Fig. 2 summarizes the calculated ionization crosssections for excited Ne atoms with the electron configurations . . . (2p)5ns and . . . (2p)5ns′ (n = 3 − 7). Here,the unprimed configurations correspond to those states whose core has a total angular momentum of 3/2, whereasthe primed configurations denote the states whose core has a total angular momentum of 1/2. We note that the



10 100

0.01

0.1

1 (3s)1

(2p)5

(2s)2

Σ

Ioniz

atio

n C

ross

Sect

ion (10

-20 m

2)

Electron Energy (eV)

Fig. 1 Calculated electron impact ionization crosssection of the 3P0 state in Ne using the DM formal-ism as a function of electron energy. The solid curvesrefer to the contributions from the various sub-shellsand have been labelled accordingly. The DM crosssection is given as the sum of the various subshellcontributions and has been labelled by the symbol”Σ”. Also shown as the dashed line is the Born calcu-lation of Ref. [28] and the filled squares represent theexperimental data of Ref. [27]. The combined statis-tical and systematic error in the experimental data isabout 40% .

curves in Figs. 2-4 represented by the connected symbols denote the contribution to the ionization cross sectionattributable to the removal of the outermost valence electron (”valence contribution”). Also shown as the solid lineis the contribution to the ionization cross section arising from the core electrons (”core contribution”, see above).For each electron configuration, the total single ionization cross section is thus given as the sum of the ”corecontribution” and the respective ”valence contribution”. It is apparent that the ”core contribution” is appreciableonly for the 3s and 3s’ levels. It is somewhat surprising that the ”core” contribution is comparatively small in allcases, since the 2s core ionization contributes significantly to the simultaneous ionization-excitation of Ne [30].The cross section maxima increase with increasing principal quantum number n. The increase per unit change inn is roughly an order of magnitude from 6×10−20m2 for n = 3 to 1×10−16m2 for n = 7. As one would expect,the cross section maxima and the thresholds of the cross section curves shift to lower energy as n increases. In allcases, the cross section curve corresponding to the primed state lies slightly above the cross section curve for theunprimed state and has a slightly lower ionization energy. This is to be expected as the unprimed configurations lieenergetically slightly below the primed configuration and have a slightly higher ionization energy. The differencein the absolute ionization cross section values between any two curves corresponding to the primed and unprimedconfiguration, respectively, is generally very small and becomes noticeable only for the larger values of n in thelow-energy region.

1 10 100 10000.01

0.1

1

10

100

1000

10000

Ioniz

atio

n C

ross

Sect

ion (10

-20 m

2)

core

3s

3s'

4s

4s'

5s

5s'

6s

6s'

7s

7s'


Fig. 2 Calculated electron impact ioniza-tion cross section of excited Ne atoms with....(2p)5ns and ...(2p)5ns’ (n=3-7) electronconfigurations as a function of electron im-pact energy up to 1000 eV using the DMformalism. The various curves are labelledin the figure and represent the contributionto the ionization cross section arising fromthe removal of the electron from the des-ignated (nl) sub-shell. Also shown as fullline is the contribution to the ionizationcross section arising from the core elec-trons (see text for details).

Fig. 3 shows the calculated cross sections for the . . . (2p)5np and . . . (2p)5np′ (n = 3 − 7) configurations.The trends that were found before in the case of the ns and ns’ configurations are also observed here. The corecontribution is negligible for all values of n. The maximum values of the cross sections which range from about5 × 10−19m2 (n = 3) to 7 × 10−16m2 (n = 7) are roughly one order of magnitude larger than those for thens and ns’ configurations. As one goes to the . . . (2p)5nd and . . . (2p)5nd′ (n = 3 − 7) configurations (fig. 4),the previously observed trends are also found, although the increase in the absolute cross section per unit changeof n is less than an order of magnitude for n = 5 and above. The maximum cross section values range from


498 H. Deutsch, K. Becker, and A.N. Grum-Grzhimailo et al.: Calculated Ionization Cross Sections for Excited Ne Atoms

3 × 10−18m2 (n = 3) to 6 × 10−16m2 for the (7d’) level, which exceeds the maximum cross section of the (7d)level by about 50%.

0.1 1 10 100 10000 01

0 1

1

1 0

1 00

1 000

1 0000

1 00000

core

3p

3p'

4p

4p'

5p

5p'

6p

6p'

7p

7p'

Ioniz

atio

n C

ross

Sect

ion (10

-20 m

2)


Fig. 3 Calculated electron impact ioniza-tion cross section of excited Ne atoms with....(2p)5np and ...(2p)5np’ (n=3-7) electronconfigurations as a function of electron im-pact energy up to 1000 eV using the DMformalism. The various curves are labelledin the figure and represent the contributionto the ionization cross section arising fromthe removal of the electron from the des-ignated (nl) sub-shell. Also shown as fullline is the contribution to the ionizationcross section arising from the core elec-trons (see text for details).

0.1 1 10 100 10000 01

0 1

1

1 0

1 00

1 000

1 0000

core

3d

3d'

4d

4d'

5d

5d'

6d

6d'

7d

7d'

Ioniz

atio

n C

ross

Sect

ion (10

-20 m

2)


Fig. 4 Calculated electron impact ioniza-tion cross section of excited Ne atoms with....(2p)5nd and ...(2p)5nd’ (n=3-7) electronconfigurations as a function of electron im-pact energy up to 1000 eV using the DMformalism. The various curves are labelledin the figure and represent the contributionto the ionization cross section arising fromthe removal of the electron from the des-ignated (nl) sub-shell. Also shown as fullline is the contribution to the ionizationcross section arising from the core elec-trons (see text for details).

If one looks at the l-dependence of the calculated cross sections for a given value of the principle quantumnumber n, the following trends are apparent: (1) the position of the cross section maximum shifts towards lowerimpact energies as l increases and (2) the maximum cross section increases roughly by a factor of 8 as one goesfrom l = 0 to l = 1 to l = 2.

Lastly we note, that trends in the calculated cross sections for Ne are similar to those found earlier in ourcalculations for Ar [12].

Acknowledgements This work has been carried out within the Association EURATOM-OAW. The content of the publica-tion is the sole responsibility of its publishers and it does not necessarily represent the views of the EU Commission or itsservices. It was partially supported by the FWF, Wien, Austria. K. Becker would like to acknowledge partial financial supportof this work by the US Department of Energy, Office of Science (Basic Energy Sciences).

References

[1] L.J. Kieffer, G.H. Dunn, Rev. Mod. Phys. 38, 1 (1966) and references therein.[2] T.D. Mark, G.H. Dunn, Electron Impact Ionization, Springer, Wien (1985).[3] R.S. Freund, In: Swarm Studies and Inelastic Electron Molecule Collisions (L.C. Pitchford, B.V. McKoy, A. Chutjian,

S. Trajmar, Eds.), Springer, New York pp.329 (1987).[4] A.J. Dixon, A. von Engel, M.F.A. Harrison, Proc. Roy. Soc. London A 34 3, 333 (1975).



[5] A.J. Dixon, M.F.A. Harrison, A.C.H. Smith, J. Phys. B 9, 2617 (1976).[6] A.J. Dixon, M.F.A. Harrison, A.C.H. Smith, Contributed Papers, 8th International Conference on the Physics of Elec-

tronic and Atomic Collisions (ICPEAC), Belgrade p. 405 (1973).[7] M. Johnston, K. Fufii, J. Nickel, S. Trajmar, J. Phys. B 29, 531 (1996).[8] E.J. McGuire, Phys. Rev. A 20, 445 (1979).[9] D. Ton-That, M.R. Flannery, Phys. Rev. A 15, 517 (1977).

[10] H.A. Hyman, Phys. Rev. A 20, 855 (1979).[11] H. Deutsch, K. Becker, S. Matt, T.D. Mark, J. Phys. B 32, 4249 (1999).[12] H. Deutsch, K. Becker, A.N. Grum-Grzhimailo, M. Probst, S. Matt-Leubner, T.D. Mark, Int. J. Mass Spectrom. 233, 39

(2004).[13] H. Deutsch, P. Scheier, K. Becker, T.D. Mark, Int. J. Mass Spectrom. 233, 13 (2004).[14] see e.g. K. Becker, H. Deutsch, M. Inokuti, Adv. At. Mol. Opt Phys. 43, 399 (2000).[15] A.Bogaerts, R.Gijbels, J.Vlcek, J. Appl. Phys. 84, 121 (1998).[16] G.G. Lister, J.J. Curry, J.E. Lawler, Phys. Rev. E 62, 5576 (2000).[17] A.A. Mihalov et al., Astron. Astophys. 324, 1206 (1997).[18] J.W. Ferguson, G.I. Ferland, Astrophys. J. 479, 363 (1997).[19] K. Nagesha, K.B. MacAdam, Phys. Rev. Lett. 91, 113202 (2003).[20] R.K. Janev, ”Atomic and Molecular Processes in Fusion Edge Plasmas”, Plenum Press, New York, 1995.[21] H. Deutsch, T.D. Mark, Int. J. Mass Spectrom. Ion Proc., 79, R1 (1987).[22] J.P. Desclaux, Atom. Data Nucl. Data Tables 12, 325 (1973).[23] D. Margreiter, H. Deutsch, T.D. Mark, Int. J. Mass Spectrom. Ion Proc. 139, 127 (1994).[24] H. Deutsch, K. Becker, S. Matt, T.D. Mark, Int.J.Mass Spectrom. Ion Proc. 197, 37 (2000).[25] A.A. Radzig and B.M. Smirnov, “Reference Data on Atoms, Molecules, and Ions”, Springer Verlag, Berlin 1980.[26] C. Froese-Fischer, T.Brage, P. Jonsson, ”Computational Atomic Structure: an MCHF Approach”, IOP Publishing, Bris-

tol 1997.[27] M. Johnston, K. Fufii, J. Nickel, and S. Trajmar, J. Phys. B 29, 531 (1996).[28] D. Ton-That and M.R. Flannery, Phys. Rev. A 15, 517 (1977).[29] H.A. Hyman, Phys. Rev. A 20, 855 (1979).[30] K. Tachibana, A. Phelps, Phys. Rev. A 36, 999 (1987).


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Calculated Electron Impact Ionization Cross Sections of Excited Ne Atoms Using the DM Formalism