Calculated Electron Impact Ionization Cross ...

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

Calculated Electron Impact Ionization Cross Sections of Excited Ne Atoms Using the DM Formalism H. Deutsch1 , K. Becker2 , A. N. Grum-Grzhimailo3, M. Probst4 , S. Matt-Leubner4 , and T.D. Märk∗4 1 2

3 4

Institut für Physik, Ernst Moritz Arndt Universität, Domstr. 10a, D-17487 Greifswald, Germany Department of Physics and Center for Environmental Systems, Stevens Institute of Technology, Hoboken, NJ 07030, USA Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia Institut für Ionenphysik, Leopold Franzens Universität, Technikerstr. 25, A-6020 Innsbruck, Austria

Received 14 February 2004, accepted 6 April 2005 Published 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 ionization cross sections of excited Ne atoms from threshold to 1000 eV. Excited states of Ne where the outermost valence electron is excited to states with principal quantum numbers up to n = 7 and orbital angular momentum quantum numbers up to l = 2 have been considered and systematic trends in the calculated cross section data are 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 atoms have been measured and calculated since the early days of collision physics [1] because of their basic importance and because of their relevance in practical applications such as gas discharges, plasmas, radiation chemistry, planetary atmospheres, and mass spectrometry. Considerable progress in the quantitative determination of ionization cross sections for atomic (and simple molecular) targets [2, 3] has been made in the past decade. The ionization of rare gas atoms has been studied extensively, in particular the ionization of ground-state rare gas atoms whose ionization cross sections are considered benchmark data [2]. Much less effort has been devoted to the ionization of 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 these targets [8, 9, 10] including an earlier calculation based on the semi-classical Deutsch-Märk (DM) formalism [11]. Recently, we applied the semi-classical Deutsch-Märk (DM) to the calculation of absolute electron-impact ionization cross sections for excited Ar atoms [12]. In this paper, we extend these calculations to Ne using the most advanced variant of the DM approach [13]. Specifically, we consider 30 different excited states where the valence electron is excited to states characterized by principal quantum numbers up to n = 7 and orbital angular momentum quantum numbers up to l = 2. Excited rare gas atoms play an important role in the ionization balance in 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 glow discharges which require - among others - ionization cross sections for many atomic levels. Other environments where 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 experimental determination of ionization cross section of highly excited Na Rydberg atoms (35 < n < 51, where n denotes the principal quantum number) and found serious disagreement between their measured data and the predictions ∗ Corresponding author: e-mail: [email protected] Also adjunct professor at: Dept. Plasmaphysics, Univerzita Komenskeho, Mlynska dolina, 842 48 Bratislava 4, Slovak Republic c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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of simple theoretical models scaled to such high values of the principal quantum number n. Lastly, the ionization of 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 expresses the total single ionization cross section σ of an atom as X 2 σ= gnl πrnl ξnl f (u) (1) n,l

where rnl is the radius of maximum radial density of the atomic subshell characterized by quantum numbers n and 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 were originally determined from a fitting procedure [23, 24] using reliable experimental cross section data for the rare gases 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) subshell). 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 the modified DM formula which has the explicit form X (q) 2 σ(u) = gnl πrnl ξnl bnl (u)[ln(cnl · u)/u] (2) n,l (q)

where the function bnl (u) has the form (q)

bnl (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 number of electrons in the (nl) subshell. Values of all constants and parameters relevant to the application of the DM formula of equation (2) can be found in the two papers by Deutsch et al. [13, 24] to which we refer the reader for further details. Neon has a (1s)2 (2s)2 (2p)6 electron configuration in its ground state, which is a 1 S0 state. The lower lying excited 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, we need 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 obtained from numerical Hartree-Fock calculations [26]. Table 1 summarizes the values for rnl and Enl used in the present calculation up to values of n = 7 and l = 2. Furthermore, the application of the DM formalism requires knowledge of the weighting factors gnl for the higher-lying (n,l) sub-shells. As these weighting factors were determined from a fitting procedure using experimentally determined reliable cross section data and such data are only available for values of the orbital angular momentum quantum number l up to l = 2 (d-electrons), the lack of reliable weighting factors gnl for higher l-values limits the present calculations to electron configurations where 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 the removal of an electron from a particular (nl) orbital summed over all (nl) configurations, it is illustrative to calculate separately the contribution to the total ionization cross section arising from the removal of an electron from the core and from the (nl) valence electron. The contribution due to the core electrons, which is common to all configurations treated here, peaks at an energy that is slightly above 100 eV with a maximum value of about 0.5 × 10−20 m2 and is in good accordance with our previous results [11] as discussed below in Fig. 1 in more detail. c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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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 the same value for both the primed and unprimed configuration. In the case of the energy values, we used energies averaged over all fine structure components of each (n,l) and (n,l’) configuration, respectively.

Quantum numbers (n,l)

Value of rnl

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

−9

Value of Enl

7s, 7s’

2.74 × 10

3p, 3p’

2.71 × 10−10 m

m

−10

0.42 eV, 0.32 eV 2.81 eV, 2.73 eV

4p, 4p’

7.38 × 10

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

−9

m

1.36 eV, 1.23 eV

4d, 4d’

1.14 × 10

m

0.85 eV, 0.76 eV

5d, 5d’

1.88 × 10−9 m

0.54 eV, 0.45 eV

−9

6d, 6d’

2.92 × 10

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 attributable to direct ionization processes only. Multiple ionization processes and indirect processes such as autoionization 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 3 P0 state in Ne (see Fig. 1) with the experimental results of Johnston et al. [27] and the Born calculation of Ton-That and Flannery [28]. The Ne 3 P0 state is the higher-lying of the two metastable states corresponding to the first excitedstate configuration of Ne, (1s)2 (2s)2 (2p)5 [3 P1/2 ]3s. Our DM cross section is in good agreement with the Born calculation except for the low energy region up to about 20 eV, where the Born cross section lies above our data and the measured cross section data, which is not uncommon for Born calculations. The DM cross section lies consistently slightly above the experimental data, but the discrepancy is within the quoted roughly 40% error margin of the measured data. Hyman [29] using a symmetric binary encounter model in conjunction with a semi-empirically determined momentum distribution function for the bound excited electron calculated electron impact ionization cross sections for the lowest excited states of the rare gases, cadmium, and mercury. His results in Ne are in good agreement with our data for the 3s level, but are smaller than our data for the 3p level (shown below 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 cross sections for excited Ne atoms with the electron configurations . . . (2p)5 ns and . . . (2p)5 ns′ (n = 3 − 7). Here, the unprimed configurations correspond to those states whose core has a total angular momentum of 3/2, whereas the primed configurations denote the states whose core has a total angular momentum of 1/2. We note that the c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Ionization Cross Section (10 -20 m2 )


497

Σ

1

1

(3s) 5

(2p)

0.1 2

(2s)

0.01

10

100

Electron Energy (eV)

Fig. 1 Calculated electron impact ionization cross section of the 3 P0 state in Ne using the DM formalism as a function of electron energy using the DM formalism. The solid curves refer to the contributions from the various sub-shells and have been labelled accordingly. The DM cross section is given as the sum of the various subshell contributions and has been labelled by the symbol ”Σ”. Also shown as the dashed line is the Born calculation of Ref. [28] and the filled squares represent the experimental data of Ref. [27]. The combined statistical and systematic error in the experimental data is about 40% .


curves in Figs. 2-4 represented by the connected symbols denote the contribution to the ionization cross section attributable to the removal of the outermost valence electron (”valence contribution”). Also shown as the solid line is 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 ”core contribution” and the respective ”valence contribution”. It is apparent that the ”core contribution” is appreciable only for the 3s and 3s’ levels. It is somewhat surprising that the ”core” contribution is comparatively small in all cases, 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 in n 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 all cases, the cross section curve corresponding to the primed state lies slightly above the cross section curve for the unprimed state and has a slightly lower ionization energy. This is to be expected as the unprimed configurations lie energetically slightly below the primed configuration and have a slightly higher ionization energy. The difference in the absolute ionization cross section values between any two curves corresponding to the primed and unprimed configuration, respectively, is generally very small and becomes noticeable only for the larger values of n and there only in the low-energy region. 10000 1000

core 3s 3s' 4s 4s' 5s 5s' 6s 6s' 7s 7s'

100 10 1 0.1 0.01

1

10

100


1000

Fig. 2 Calculated electron impact ionization cross section of excited Ne atoms with ....(3p)5 ns and ...(3p)5 ns’ (n=3-7) electron configurations as a function of electron impact energy up to 1000 eV using the DM formalism. The various curves are labelled in the figure and represent the contribution to the ionization cross section arising from the removal of the electron from the designated (nl) sub-shell. Also shown as full line is the contribution to the ionization cross section arising from the core electrons (see text for details).

Fig. 3 shows the calculated cross sections for the . . . (2p)5 np and . . . (2p)5 np′ (n = 3 − 7) configurations. The trends that were found before in the case of the ns and ns’ configurations are also observed here. The core contribution is negligible for all values of n. The maximum values of the cross sections which range from about 5 × 10−19 m2 (n = 3) to 7 × 10−16 m2 (n = 7) are roughly one order of magnitude larger than those for the ns and ns’ configurations. As one goes to the . . . (2p)5 nd and . . . (2p)5 nd′ (n = 3 − 7) configurations (fig. 4), the previously observed trends are also found, although the increase in the absolute cross section per unit change of n is less than an order of magnitude for n = 5 and above. The maximum cross section values range from c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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


3 × 10−18 m2 (n = 3) to 6 × 10−16 m2 for the (7d’) level, which exceeds the maximum cross section of the (7d) level by about 50%. 1 00000 1 0000 core 3p 3p' 4p 4p' 5p 5p' 6p 6p' 7p 7p'

1 000 1 00 10 1 01 001

0.1

1

10

100

1000

Ionization Cross Section (10 -20 m 2 )


Fig. 3 Calculated electron impact ionization cross section of excited Ne atoms with ....(3p)5 np and ...(3p)5 np’ (n=3-7) electron configurations as a function of electron impact energy up to 1000 eV using the DM formalism. The various curves are labelled in the figure and represent the contribution to the ionization cross section arising from the removal of the electron from the designated (nl) sub-shell. Also shown as full line is the contribution to the ionization cross section arising from the core electrons (see text for details).

1 0000 core 3d 3d' 4d 4d' 5d 5d' 6d 6d' 7d 7d'

1 000 1 00 10 1 01 001

0.1

1

10

100


1000

Fig. 4 Calculated electron impact ionization cross section of excited Ne atoms with ....(3p)5 nd and ...(3p)5 nd’ (n=3-7) electron configurations as a function of electron impact energy up to 1000 eV using the DM formalism. The various curves are labelled in the figure and represent the contribution to the ionization cross section arising from the removal of the electron from the designated (nl) sub-shell. Also shown as full line is the contribution to the ionization cross section arising from the core electrons (see text for details).

If one looks at the l-dependence of the calculated cross sections for a given value of the principle quantum number n, the following trends are apparent: (1) the position of the cross section maximum shifts towards lower impact energies as l increases and (2) the maximum cross section increases roughly by a factor of 8 as one goes from 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 our calculations for Ar [12]. Acknowledgements This work has been carried out within the Association EURATOM-?AW. The content of the publication is the sole responsibility of its publishers and it does not necessarily represent the views of the EU Commission or its services. It was partially supported by the FWF, Wien, Austria. K. Becker would like to acknowledge partial financial support of 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. Märk, 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). c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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[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 Electronic 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. Märk, J. Phys. B 32, 4249 (1999). [12] Deutsch, K. Becker, A.N. Grum-Grzhimailo, M. Probst, S. Matt-Leubner, T.D. Märk, Int. J. Mass Spectrom. 233 39 (2004). [13] H. Deutsch, P. Scheier, K. Becker, T.D. Märk, 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. Märk, 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. Märk, Int. J. Mass Spectrom. Ion Proc. 139, 127 (1994). [24] H. Deutsch, K. Becker, S. Matt, T.D. Märk, 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, Bristol (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).

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