Electron Impact Excitation and Ionization

  • Ingolf V. Hertel
  • Claus-Peter Schulz
Part of the Graduate Texts in Physics book series (GTP)


In Sect. 8.1.1 we develop – on a somewhat abstract level – the general formalism of close-coupling theory (CC). We then return in Sect. 8.2 once again to Born approximation as the most simple theoretical approach to electron impact excitation. We present – complementary to optical excitation – the concept of the generalized oscillator strength for e-atom collisions. Section 8.4 treats electron impact ionization, beginning with integral cross sections which are of particular importance for practical applications. Singly and doubly differential cross sections follow, while finally, triply differential cross sections contain the maximum information about any (e,2e) process. This is further elaborated in Sect. 8.4.6 with a brief excursion into (e,2e) spectroscopy, which may be understood as complementary to photoelectron spectroscopy (see Sect.  5.8) in the VUV and XUV spectral region. Finally, in Sect. 8.5 we discuss an example for electron ion recombination – the inverse process to photoionization.


Differential Cross Section Scattered Electron Initial Kinetic Energy Dielectronic Recombination Electron Impact Excitation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Acronyms and Terminology


‘Atomic, molecular and optical’, physics.


‘atomic units’, see Sect.  2.6.2 in Vol. 1.


‘B-spline atomic R-matrix codes’, a general program (beyond CCC) to calculate atomic continuum processes, including electron-atom and electron ion scattering and radiative processes (Zatsarinny 2006).


‘Close-coupling’, calculations, computation of scattering cross sections by solving (part of) the coupled integro-differential equations (see Sect. 8.1.1).


‘Convergent close-coupling’, calculations, special solutions of the coupled integro-differential equations for collisions (see Sect. 8.1).


‘Centre of mass’, coordinate system (or frame) (see Sect.  6.2.2).


‘Cold target recoil ion momentum spectroscopy’, see Appendix B.4.


‘Differential cross section’, see Sect.  6.2.1.


‘Double-differential cross section’, in e,2e ionization processes (see Sect. 8.4).


‘Distorted wave’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.1.2).


‘Distorted wave Born approximation’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.1.2).


‘Exterior complex scaling’, method for solving the close-coupling equations for scattering problems.


‘Electron energy loss spectroscopy’, see Sect. 8.3.2.


‘Electron momentum spectroscopy’, method to determine the momentum distribution of electrons in atoms and molecules exploiting e,2e processes (see e.g. McCarthy and Weigold 1991).


‘First order Born approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects.  6.6 and  5.5.2, Vol. 1, respectively).


‘Full width at half maximum’.


Glauber approximation’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.4.5).

good quantum number

‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the Hamilton operator (see Sect.  2.6.5 in Vol. 1)’.


‘Generalized oscillator strength’, characterizes the strength of electron impact excitation in analogy to the optical oscillator strength see Sect. 8.3.2.


‘Generalized oscillator strength density’, characterizes the strength of electron impact ionization per energy interval in analogy to the optical oscillator strength density (see Sect. 8.4.4).


‘Highly charged ions’, see Sect.  7.5.


‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation.


‘Ordinary differential equation’.


‘Optical oscillator strength density’, characterizes the strength of photoionization per energy interval (see Sect.  5.5.1 in Vol. 1).


‘Photoelectron spectroscopy’, see Sect.  5.8.


‘Plane wave impulse approximation’, basic assumption for EMS (see Sect. 8.4.6).


‘Second order Born approximation’, second order term in the Born series (see Sect.  6.6).


‘Single-differential cross section’, in e,2e ionization processes (see Sect. 8.4).


‘Three body approximate scattering wave function’, accurate method for calculating triple-differential cross section for ionization (Brauner et al. 1989).


‘Time dependent close-coupling calculations’, a method, in principle accurate, for solving the Schrödinger equation for scattering problems (Colgan and Pindzola 2006).


‘Triple-differential cross section’, in e,2e ionization processes (see Sect. 8.4).


‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between \(100\operatorname{nm}\) and \(400\operatorname{nm}\) according to ISO 21348 (2007).


‘Visible’, spectral range of electromagnetic radiation. Wavelengths between \(380\operatorname{nm}\) and \(760\operatorname{nm}\) according to ISO 21348 (2007).


‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B).


‘Vacuum ultraviolet’, spectral range of electromagnetic radiation, part of the UV spectral range. Wavelengths between \(10\operatorname{nm}\) and \(200\operatorname{nm}\) according to ISO 21348 (2007).


‘Soft x-ray (sometimes also extreme UV)’, spectral wavelength range between \(0.1\operatorname{nm}\) and \(10\operatorname{nm}\) according to ISO 21348 (2007), sometimes up to \(40\operatorname{nm}\).


‘Zero kinetic energy’, photoelectron spectroscopy (see Sect.  5.8.3).


  1. Al-Hagan O., A. J. Murray, C. Kaiser, J. Colgan and D. H. Madison: 2010. ‘Electron-impact-ionization cross sections of H2 for low outgoing electron energies from 1 to 10 eV’. Phys. Rev. A 81, 030701. CrossRefADSGoogle Scholar
  2. Amaldi, U., A. Egidi, R. Marconer and G. Pizzella: 1969. ‘Use of a 2 channeltron coincidence in a new line of research in atomic physics’. Rev. Sci. Instrum., 40, 1001–1004. CrossRefADSGoogle Scholar
  3. Andersen, N. and K. Bartschat: 2003. Polarization, Alignment and Orientation in Atomic Collisions. Berlin, Heidelberg: Springer. Google Scholar
  4. Andersen, N., K. Bartschat, J. T. Broad and I. V. Hertel: 1997. ‘Collisional alignment and orientation of atomic outer shells: 3. Spin-resolved excitation’. Phys. Rep., 279, 252–396. CrossRefADSGoogle Scholar
  5. Andersen, N., J. W. Gallagher and I. V. Hertel: 1988. ‘Collisional alignment and orientation of atomic outer shells: 1. Direct excitation by electron and atom impact’. Phys. Rep., 165, 1–188. CrossRefADSGoogle Scholar
  6. Avaldi, L., R. Camilloni, E. Fainelli and G. Stefani: 1987. ‘Absolute double-differential ionization cross-section for electron-impact: He’. Nuovo Cimento D, 9, 97–113. CrossRefADSGoogle Scholar
  7. Baertschy, M., T. N. Rescigno and C. W. McCurdy: 2001. ‘Accurate amplitudes for electron-impact ionization’. Phys. Rev. A, 64, 022709. CrossRefADSGoogle Scholar
  8. Bartlett, P. L. and A. T. Stelbovics: 2004a. ‘Differential ionization cross-section calculations for hydrogenic targets with Z≤4 using a propagating exterior complex scaling method’. Phys. Rev. A, 69, 040701. CrossRefADSGoogle Scholar
  9. Bartlett, P. L. and A. T. Stelbovics: 2004b. ‘Threshold behavior of e-H ionizing collisions’. Phys. Rev. Lett., 93, 233201. CrossRefADSGoogle Scholar
  10. Bartschat, K.: 1998. ‘The R-matrix with pseudo-states method: Theory and applications to electron scattering and photoionization’. Comput. Phys. Commun., 114, 168–182. CrossRefzbMATHADSGoogle Scholar
  11. Bartschat, K. and I. Bray: 1996. ‘Electron-impact ionization of atomic hydrogen from the 1s and 2s states’. J. Phys. B, At. Mol. Phys., 29, L577–L583. CrossRefADSGoogle Scholar
  12. Berger, M. J., J. S. Coursey, M. A. Zucker and J. Chang: 2005. ‘ESTAR, PSTAR, and ASTAR: Computer programs for calculating stopping-power and range tables for electrons, protons, and helium ions (version 1.2.3)’, Physical Measurement Laboratory., accessed: 9 Jan 2014.
  13. Bethe, H.: 1930. ‘Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie’. Anal. Phys., 397, 325–400. CrossRefGoogle Scholar
  14. van Boeyen, R. W. and J. F. Williams: 2005. ‘Multidetection (e, 2e) electron spectrometer’. Rev. Sci. Instrum., 76. Google Scholar
  15. Böhm, S. et al.: 2002. ‘Measurement of the field-induced dielectronic-recombination-rate enhancement of O5+ ions differential in the Rydberg quantum number n’. Phys. Rev. A, 65, 052728. CrossRefADSGoogle Scholar
  16. Born, M.: 1926a. ‘Quantenmechanik der Stoßvorgänge’. Z. Phys., 38, 803–840. CrossRefADSGoogle Scholar
  17. Born, M.: 1926b. ‘Zur Quantenmechanik der Stoßvorgänge’. Z. Phys., 37, 863–867. CrossRefzbMATHADSGoogle Scholar
  18. Brauner, M., J. S. Briggs and H. Klar: 1989. ‘Triply-differential cross-sections for ionization of hydrogen-atoms by electrons and positrons’. J. Phys. B, At. Mol. Phys., 22, 2265–2287. CrossRefADSGoogle Scholar
  19. Bray, I.: 2002. ‘Close-coupling approach to Coulomb three-body problems’. Phys. Rev. Lett., 89, 273201. CrossRefADSGoogle Scholar
  20. Bray, I. and D. V. Fursa: 1996. ‘Calculation of ionization within the close-coupling formalism’. Phys. Rev. A, 54, 2991–3004. CrossRefADSGoogle Scholar
  21. Bray, I. and D. V. Fursa: 2011. ‘Benchmark cross sections for electron-impact total single ionization of helium’. J. Phys. B, At. Mol. Phys., 44, 061001. CrossRefADSGoogle Scholar
  22. Bray, I., D. V. Fursa and A. T. Stelbovics: 2003. ‘Electron-impact ionization doubly differential cross sections of helium’. J. Phys. B, At. Mol. Phys., 36, 2211–2227. CrossRefADSGoogle Scholar
  23. Bray, I., D. A. Konovalov and I. E. McCarthy: 1991. ‘Electron-scattering by atomic sodium: 3 2S−3 2S and 3 2S−3 2P cross-sections at 10 to 100 eV’. Phys. Rev. A, 44, 7179–7184. CrossRefADSGoogle Scholar
  24. Buckman, S. J. and J. P. Sullivan: 2006. ‘Benchmark measurements and theory for electron(positron)-molecule(atom) scattering’. Nucl. Instrum. Methods B, 247, 5–12. CrossRefADSGoogle Scholar
  25. Burke, P.: 2006. ‘Electron-atom, electron-ion and electron-molecule collisions’. In: G. W. F. Drake, ed., ‘Handbook of Atomic, Molecular and Optical Physics’, 705–729. Heidelberg, New York: Springer. CrossRefGoogle Scholar
  26. Burke, P. G., C. J. Noble and V. M. Burke: 2007. ‘R-matrix theory of atomic, molecular and optical processes’. In: ‘Advances in Atomic Molecular and Optical Physics’, vol. 54, 237–318. Amsterdam: Elsevier. Google Scholar
  27. Byron, F. W. and C. J. Joachain: 1989. ‘Theory of (e, 2e) reactions’. Phys. Rep., 179, 211–272. CrossRefADSGoogle Scholar
  28. Byron, F. W., C. J. Joachain and B. Piraux: 1986. ‘Theory of coplanar asymmetric (e, 2e) reactions in helium’. J. Phys. B, At. Mol. Phys., 19, 1201–1210. CrossRefADSGoogle Scholar
  29. Colgan, J. and M. S. Pindzola: 2006. ‘Double- and triple-differential cross sections for the low-energy electron-impact ionization of hydrogen’. Phys. Rev. A, 74, 012713. CrossRefADSGoogle Scholar
  30. Coplan, M. A., J. H. Moore and J. P. Doering: 1994. ‘(e, 2e) spectroscopy’. Rev. Mod. Phys., 66, 985–1014. CrossRefADSGoogle Scholar
  31. Cvejanov, S. and F. H. Read: 1974a. ‘New technique for threshold excitation spectroscopy’. J. Phys. B, At. Mol. Phys., 7, 1180–1193. CrossRefADSGoogle Scholar
  32. Cvejanov, S. and F. H. Read: 1974b. ‘Studies of threshold electron-impact ionization of helium’. J. Phys. B, At. Mol. Phys., 7, 1841–1852. CrossRefADSGoogle Scholar
  33. Dal Cappello, C., A. Haddadou, F. Menas and A. C. Roy: 2011. ‘The second Born approximation for the single and double ionization of atoms by electrons and positrons’. J. Phys. B, At. Mol. Phys., 44, 015204. CrossRefADSGoogle Scholar
  34. Deb, N. C. and D. S. F. Crothers: 2002. ‘Electron-impact ionization of atomic hydrogen close to threshold’. Phys. Rev. A, 65, 052721. CrossRefADSGoogle Scholar
  35. Ehrhardt, H., K. Jung, G. Knoth and P. Schlemmer: 1986. ‘Differential cross-sections of direct single electron-impact ionization’. Z. Phys. D, 1, 3–32. CrossRefADSGoogle Scholar
  36. Ehrhardt, H., M. Schulz, T. Tekaat and K. Willmann: 1969. ‘Ionization of helium – angular correlation of scattered and ejected electrons’. Phys. Rev. Lett., 22, 89–92. CrossRefADSGoogle Scholar
  37. Hertel, I. V. and K. J. Ross: 1969. ‘Octupole allowed transitions in the electron energy loss spectra of potassium and rubidium’. J. Chem. Phys., 50, 536–537. CrossRefADSGoogle Scholar
  38. Inokuti, M.: 1971. ‘Inelastic collisions of fast charged particles with atoms and molecules – Bethe theory revisited’. Rev. Mod. Phys., 43, 297–347. CrossRefADSGoogle Scholar
  39. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. Google Scholar
  40. Kessler, J.: 1985. Polarized Elektrons. Berlin, Heidelberg: Springer. CrossRefGoogle Scholar
  41. Kim, Y. K.: 1975a. ‘Energy-distribution of secondary electrons’. Radiat. Res., 64, 96–105. CrossRefGoogle Scholar
  42. Kim, Y. K.: 1975b. ‘Energy-distribution of secondary electrons. 1. Consistency of experimental-data’. Radiat. Res., 61, 21–35. CrossRefGoogle Scholar
  43. Kim, Y. K.: 1975c. ‘Energy-distribution of secondary electrons. 2. Normalization and extrapolation of experimental-data’. Radiat. Res., 64, 205–216. CrossRefGoogle Scholar
  44. Kim, Y. K.: 2007. ‘Scaled Born cross sections for excitations of H2 by electron impact’. J. Chem. Phys., 126. Google Scholar
  45. Krishnakumar, E. and S. K. Srivastava: 1988. ‘Ionization cross-sections of rare-gas atoms by electron-impact’. J. Phys. B, At. Mol. Phys., 21, 1055–1082. CrossRefADSGoogle Scholar
  46. Lassettre, E. N., A. Skerbele, M. A. Dillon and K. J. Ross: 1968. ‘High-resolution study of electron-impact spectra at kinetic energies between 33 and 100 eV and scattering angles to 16’. J. Chem. Phys., 48, 5066–5096. CrossRefADSGoogle Scholar
  47. Lin, C. C. and J. B. Boffard: 2005. ‘Electron-impact excitation cross sections of sodium’. In: ‘Advances in Atomic Molecular, and Optical Physics’, vol. 51, 385–411. Amsterdam: Elsevier, Academic Press. Google Scholar
  48. Lin, C. D.: 1974. ‘Correlations of excited electrons – Study of channels in hyperspherical coordinates’. Phys. Rev. A, 10, 1986–2001. CrossRefADSGoogle Scholar
  49. Lotz, W.: 1967. ‘An empirical formula for electron-impact ionization cross-section’. Z. Phys., 206, 205–211. CrossRefADSGoogle Scholar
  50. Lotz, W.: 1968. ‘Electron-impact ionization cross-sections and ionization rate coefficients for atoms and ions from hydrogen to calcium’. Z. Phys., 216, 241–247. CrossRefADSGoogle Scholar
  51. Lotz, W.: 1970. ‘Electron-impact ionization cross-sections for atoms up to Z=108’. Z. Phys., 232, 101–107. CrossRefADSGoogle Scholar
  52. Macek, J. H.: 1967. ‘Application of Fock expansion to doubly excited states of the helium atom’. Phys. Rev., 160, 170–174. CrossRefADSGoogle Scholar
  53. McCarthy, I. E. and E. Weigold: 1991. ‘Electron momentum spectroscopy of atoms and molecules’. Rep. Prog. Phys., 54, 789–879. CrossRefADSGoogle Scholar
  54. McCarthy, I. E. and X. Zhang: 1989. ‘Distorted-wave Born approximation for electron-helium double-differential ionization cross-sections’. J. Phys. B, At. Mol. Phys., 22, 2189–2193. CrossRefADSGoogle Scholar
  55. Müller, A.: 2008. ‘Electron-ion collisions: fundamental processes in the focus of applied research’. In: E. Arimondo et al., eds., ‘Advances in Atomic, Molecular, and Optical Physics’, vol. 55, 293–417. Amsterdam: Elsevier, Academic Press. Google Scholar
  56. Müller-Fiedler, R., K. Jung and H. Ehrhardt: 1986. ‘Double-differential cross-sections for electron-impact ionization of helium’. J. Phys. B, At. Mol. Phys., 19, 1211–1229. CrossRefADSGoogle Scholar
  57. NIFS and ORNL: 2007. ‘Atomic & molecular numerical databases’, NIFS, National Institute for Fusion Science, Japan., accessed: 9 Jan 2014.
  58. Ning, C. G. et al.: 2008. ‘High resolution electron momentum spectroscopy of the valence orbitals of water’. Chem. Phys., 343, 19–30. CrossRefADSGoogle Scholar
  59. Oda, N.: 1975. ‘Energy and angular-distributions of electrons from atoms and molecules by electron-impact’. Radiat. Res., 64, 80–95. CrossRefGoogle Scholar
  60. Ovchinnikov, S. Y., G. N. Ogurtsov, J. H. Macek and Y. S. Gordeev: 2004. ‘Dynamics of ionization in atomic collisions’. Phys. Rep., 389, 119–159. CrossRefADSGoogle Scholar
  61. Percival, I. C. and M. J. Seaton: 1957. ‘The partial wave theory of electron-hydrogen atom collisions’. Proc. Camb. Philol. Soc., 53, 654–662. CrossRefzbMATHADSGoogle Scholar
  62. Percival, I. C. and M. J. Seaton: 1958. ‘The polarization of atomic line radiation excited by electron impact’. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci., 251, 113–138. CrossRefADSGoogle Scholar
  63. Pickup, B. T.: 1977. ‘Theory of fast photoionization processes’. Chem. Phys., 19, 193–208. MathSciNetCrossRefADSGoogle Scholar
  64. Rau, A. R. P.: 1971. ‘2 electrons in a Coulomb potential – double-continuum wave functions and threshold law for electron-atom ionization’. Phys. Rev. A, 4, 207–220. CrossRefADSGoogle Scholar
  65. Ray, H., U. Werner and A. C. Roy: 1991. ‘Doubly differential cross-sections for ionization of helium by electron-impact’. Phys. Rev. A, 44, 7834–7837. CrossRefADSGoogle Scholar
  66. Ren, X. et al.: 2011. ‘Electron-impact ionization of helium: A comprehensive experiment benchmarks theory’. Phys. Rev. A, 83, 052711. CrossRefADSGoogle Scholar
  67. Ren, X., T. Pfluger, J. Ullrich, O. Zatsarinny, K. Bartschat, D. H. Madison and A. Dorn: 2012a. ‘Low-energy electron-impact ionization of argon: Three-dimensional cross section’. Phys. Rev. A, 85, 032702. CrossRefADSGoogle Scholar
  68. Ren, X., T. Pfluger, S. Xu, J. Colgan, M. S. Pindzola, A. Senftleben, J. Ullrich and A. Dorn: 2012b. ‘Strong molecular alignment dependence of H2 electron impact ionization dynamics’. Phys. Rev. Lett., 109, 123202. CrossRefADSGoogle Scholar
  69. Röder, J., M. Baertschy and I. Bray: 2003. ‘Measurements of the ionization of atomic hydrogen by 17.6-eV electrons’. Phys. Rev. A, 67, 010702. CrossRefGoogle Scholar
  70. Rudd, M. E.: 1991. ‘Differential and total cross-sections for ionization of helium and hydrogen by electrons’. Phys. Rev. A, 44, 1644–1652. CrossRefADSGoogle Scholar
  71. Saenz, A., W. Weyrich and P. Froelich: 1996. ‘The first born approximation and absolute scattering cross sections’. J. Phys. B, At. Mol. Phys., 29, 97–113. CrossRefADSGoogle Scholar
  72. Schlemmer, P., M. K. Srivastava, T. Rosel and H. Ehrhardt: 1991. ‘Electron-impact ionization of helium at intermediate collision energies’. J. Phys. B, At. Mol. Phys., 24, 2719–2736. CrossRefADSGoogle Scholar
  73. Schow, E., K. Hazlett, J. G. Childers, C. Medina, G. Vitug, I. Bray, D. V. Fursa and M. A. Khakoo: 2005. ‘Low-energy electron-impact ionization of helium’. Phys. Rev. A, 72, 062717. CrossRefADSGoogle Scholar
  74. Senftleben, A., T. Pfluger, X. Ren, B. Najjari, A. Dorn and J. Ullrich: 2012. ‘Tuning the internuclear distance in ionization of H2’. J. Phys. B, At. Mol. Phys., 45, 021001. CrossRefADSGoogle Scholar
  75. Shah, M. B., D. S. Elliott and H. B. Gilbody: 1987. ‘Pulsed crossed-beam study of the ionization of atomic-hydrogen by electron-impact’. J. Phys. B, At. Mol. Phys., 20, 3501–3514. CrossRefADSGoogle Scholar
  76. Sorokin, A. A., L. A. Shmaenok, S. V. Bobashev, B. Mobus, H. Richter and G. Ulm: 2000. ‘Measurements of electron-impact ionization cross sections of argon, krypton, and xenon by comparison with photoionization’. Phys. Rev. A, 61, 022723. CrossRefADSGoogle Scholar
  77. Sun, W. G., M. A. Morrison, W. A. Isaacs, W. K. Trail, D. T. Alle, R. J. Gulley, M. J. Brennan and S. J. Buckman: 1995. ‘Detailed theoretical and experimental-analysis of low-energy electron-N2 scattering’. Phys. Rev. A, 52, 1229–1256. CrossRefADSGoogle Scholar
  78. Taioli, S., S. Simonucci, L. Calliari and M. Dapor: 2010. ‘Electron spectroscopies and inelastic processes in nanoclusters and solids: Theory and experiment’. Phys. Rep., 493, 237–319. CrossRefADSGoogle Scholar
  79. Telega, S. and F. A. Gianturco: 2006. ‘Modelling electron-N2 scattering in the resonant region – Integral cross-sections from space-fixed coupled channel calculations’. Eur. Phys. J. D, 38, 495–500. CrossRefADSGoogle Scholar
  80. Vinodkumar, M., C. Limbachiya, B. Antony and K. N. Joshipura: 2007. ‘Calculations of elastic, ionization and total cross sections for inert gases upon electron impact: threshold to 2 keV’. J. Phys. B, At. Mol. Phys., 40, 3259–3271. CrossRefADSGoogle Scholar
  81. Wannier, G. H.: 1953. ‘The threshold law for single ionization of atoms or ions by electrons’. Phys. Rev., 90, 817–825. CrossRefzbMATHADSGoogle Scholar
  82. Weigold, E. and I. E. McCarthy: 1999. Electron Momentum Spectroscopy. New York: Kluwer/Plenum. CrossRefGoogle Scholar
  83. Williams, J. F., P. L. Bartlett and A. T. Stelbovics: 2006. ‘Threshold electron-impact ionization mechanism for hydrogen atoms’. Phys. Rev. Lett., 96, 123201. CrossRefADSGoogle Scholar
  84. Yates, B. R. and M. A. Khakoo: 2011. ‘Near-threshold electron-impact doubly differential cross sections for the ionization of argon and krypton’. Phys. Rev. A, 83, 042712. CrossRefADSGoogle Scholar
  85. Zatsarinny, O.: 2006. ‘BSR: B-spline atomic R-matrix codes’. Comput. Phys. Commun., 174, 273–356. CrossRefzbMATHADSGoogle Scholar
  86. Zatsarinny, O. and K. Bartschat: 2012. ‘Nonperturbative B-spline R-matrix-with-pseudostates calculations for electron-impact ionization of helium’. Phys. Rev. A, 85, 062709. CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ingolf V. Hertel
    • 1
  • Claus-Peter Schulz
    • 1
  1. 1.Max-Born-Institut für Nichtlineare Optikund KurzzeitspektroskopieBerlinGermany

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