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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 1, Iss. 2 — Apr. 1, 1984
  • pp: 246–254

Parametric model for f-shell configurations. I. The effective-operator Hamiltonian

H. M. Crosswhite and H. Crosswhite  »View Author Affiliations

JOSA B, Vol. 1, Issue 2, pp. 246-254 (1984)

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A parametric model for lanthanide and actinide atomic and crystal energy levels is presented that correlates trends in Hartree–Fock calculations with empirically determined atomic parameters in such a way that predictions for unclassified complex cases can be made from the analysis of simpler ones. When appropriate effective operators, including electrostatic operators up to the third rank for fN configurations and magnetic operators up to the second rank for fN and fNd shells, are used in the parametric Hamiltonian, statistical errors of 10–20 cm−1 are typical for the simpler examples presented.

© 1984 Optical Society of America

H. M. Grosswhite and H. Crosswhite, "Parametric model for ƒ-shell configurations. I. The effective-operator Hamiltonian," J. Opt. Soc. Am. B 1, 246-254 (1984)

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  1. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, Cambridge, 1935, 1951).
  2. G. Racah, "Theory of complex spectra. II," Phys. Rev. 62, 438–462 (1942); "Theory of complex spectra. III," 63, 367–382 (1943); "Theory of complex spectra. IV," 76, 1352–1365 (1949). [CrossRef]
  3. U. Fano and G. Racah, Irreducible Tensorial Sets (Academic, New York, 1959).
  4. B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill, New York, 1963).
  5. B. G. Wybourne, Spectroscopic Properties of Rare Earths (In-terscience, New York, 1965).
  6. Z. B. Goldschmidt, "Atomic properties (free atom)," in Handbook on the Physics and Chemistry of Rare Earths, K. A. Gschneidner, Jr., and L. Eyring, eds. (North-Holland, Amsterdam, 1978), Vol. I, Chap. I, pp. 1–247. [CrossRef]
  7. R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981).
  8. See, for instance, C. F. Fischer, The Hartree-Fock Method for Atoms (Wiley, New York, 1977).
  9. G. H. Dieke, Spectra and Energy Levels of Rare Earths Ions in Crystals (Interscience, New York, 1966).
  10. H. M. Crosswhite, "Systematic atomic and crystal-field parameters for lanthanides in LaCl3 and LaF3," in Spectroscopic des Éléments de Transition et des Éléments Lourdes dan Les Solides (Editions du CNRS, Paris, 1977), pp. 65–69.
  11. S. Hüfner, Optical Spectra of Transparent Rare Earth Compounds (Academic, New York, 1978).
  12. The program that we are using was adapted for us by M. Wilson from C. Froese-Fischer's general multiconfiguration code [C. Froese-Fischer, "A multi-configuration Hartree-Fock program," Comput. Phys. Commun. 1, 151–166 (1969)]. [CrossRef]
  13. R. D. Cowan and D. C. Griffin, "Approximate relativistic corrections to atomic radial wave functions," J. Opt. Soc. Am. 66, 1010–1014 (1976). [CrossRef]
  14. P. F. A. Klinkenberg and R. Lang, "The spectrum of trebly ionized thorium, Th IV," Physica 15, 774–788 (1949). [CrossRef]
  15. P. F. A. Klinkenberg, "Analysis of double ionized thorium, Th III," Physica 16, 618–650 (1950); G. Racah, "On the spectrum of Th III," Physica 16, 651–666 (1950). [CrossRef]
  16. J. Sugar, "Analysis of the third spectrum of praseodymium," J. Opt. Soc. Am. 53, 831–839 (1963); J. Res. Nat. Bur. Stand. (US) 73A, 333–351 (1969). [CrossRef]
  17. C. E. Moore, Atomic Energy Levels as Derived from the Analysis of Atomic Spectra, U.S. Natl. Bur. Stand. Circ. 467, Vol. III (1958); reissued as NSRDS-NBS 35 (U.S. Government Printing Office, Washington, D.C., 1971).
  18. V. Kaufman and L. J. Radziemski, Jr., "The sixth spectrum of uranium (U VI)," J. Opt. Soc. Am. 66, 599–600 (1976). [CrossRef]
  19. W. C. Martin, R. Zalubas, and L. Hagan, Atomic Energy Levels—The Rare-Earth Elements, NSRDS-NBS 60 (U.S. Government Printing Office, Washington, D.C., 1978).
  20. J. F. Wyart, V. Kaufman, and J. Sugar, "Analysis of the spectrum of four times-ionized uranium (U V)," Phys. Scr. 22, 389–396 (1980). [CrossRef]
  21. C. H. H. Van Deurzen, K. Rajnak, and J. G. Conway, "Uranium five (U V), the 1S0 level and a paramagnetic analysis of the 5ƒ2 configuration," J. Opt. Soc. Am. B 1, 45–47 (1984). [CrossRef]
  22. H. H. Marvin, "Mutual magnetic interactions of electrons," Phys. Rev. 71, 102–110 (1947). [CrossRef]
  23. The Fischer program that we are using includes the Blume, Freeman, and Watson corrections to the spin-orbit integrals; see M. Blume and R. E. Watson, "Theory of spin-orbit coupling in atoms I. Derivation of the spin-orbit coupling constant," Proc. R. Soc. Lond. Ser. A 270, 127–143 (1962); "II. Comparison of theory with experiment," Proc. R. Soc. London Ser. 271, 565–578 (1963); M. Blume, A. J. Freeman, and R. E. Watson, "III.," Phys. Rev. 134, A320–A327 (1964). [CrossRef]
  24. These results are similar to those produced by R. D. Cowan's HFR program. See Chap. 8 of his book.7
  25. R. E. Trees, "Configuration interaction in Mn II," Phys. Rev. 83, 756–760 (1951); "Term values in the 3d54s configuration of Fe III," Phys. Rev. 84, 1089–1091 (1951). [CrossRef]
  26. K. Rajnak and B. G. Wybourne, "Configuration interaction effects in lN configurations," Phys. Rev. 132, 280–290 (1963). [CrossRef]
  27. B. R. Judd, J. E. Hansen, and A. J. J. Raassen, "Parametric fits in the atomic d shell," J. Phys. B 15, 1457–1472 (1982). [CrossRef]
  28. B. R. Judd and H. Crosswhite, "Orthogonalized operators for the ƒ shell," J. Opt. Soc. Am. B 1, 255–260 (1984). [CrossRef]
  29. B. R. Judd, H. M. Crosswhite, and H. Crosswhite, "Inter-atomic magnetic interactions for ƒ electrons," Phys. Rev. 169, 130–138 (1968). [CrossRef]
  30. Z. B. Goldschmidt, A. Pasternak, and Z. H. Goldschmidt, "Magnetic interactions in heavy atoms," Phys. Lett. A 28, 265–266. (1968). [CrossRef]
  31. B. R. Judd, "Three-particle operators for equivalent electrons," Phys. Rev. 141, 4–14 (1966). [CrossRef]
  32. J. C. Morrison, "Effect of core polarization upon the ƒ-ƒ interactions of rare-earth and actinide ions," Phys. Rev. A 6, 643–650 (1972). [CrossRef]
  33. D. J. Newman and C. D. Taylor, "Configuration interaction in rare-earth ions. I. Three-particle correlation in Pr2+," J. Phys. B 4, 241–247 (1971). [CrossRef]
  34. G. M. Copland, D. J. Newman, and C. D. Taylor, "Configuration interaction in rare-earth ions. II. Magnetic interactions," J. Phys. B 4, 1388–1392 (1971). [CrossRef]
  35. G. M. Copland, D. J. Newman and C. D. Taylor, "Configuration interaction in rare-earth ions. III. Trees parameters for Pr2+ and Pr3+," J. Phys. B 4, 1605–1610 (1971). [CrossRef]
  36. H. Casimir, Proc. K. Ned. Akad. Wet. 34, 844 (1931).
  37. R. E. Trees, "The 4ƒ3 and 4ƒ25d configurations of doubly ionized praseodymium (Pr in)," J. Opt. Soc. Am. 54, 651–657 (1964). [CrossRef]
  38. Z. B. Goldschmidt, "Spin-dependent interactions in U V 5ƒ2" Phys. Rev. A 27, 740–753 (1983). [CrossRef]
  39. N. Spector, "Configurations 4ƒ36s and 4ƒ36p in doubly ionized Pr," J. Opt. Soc. Am. 54, 1359–1362 (1964). [CrossRef]
  40. K. Rajnak, "Configuration interaction in the 4ƒ3 configuration of Pr III," J. Opt. Soc. Am. 55, 126–132 (1965). [CrossRef]
  41. H. Crosswhite, H. M. Crosswhite, and B. R. Judd, "Magnetic parameters for the configuration ƒ3," Phys. Rev. 174, 89–94 (1968). [CrossRef]
  42. B. R. Judd, "Developments in atomic shell theory," in The Structure of Matter, B. G. Wybourne, ed. (U. Canterbury Press, Christchurch, New Zealand, 1972); B. R. Judd and M. A. Suskin, "Complete set of orthogonal scalar operators for the configuration ƒ3," J. Opt. Soc. Am. B 1, 261–265 (1984). [CrossRef]
  43. C. E. Schaffer and C. K. Jørgensen, "The nephelauxetic series of ligands corresponding to increasing tendency of partly covalent bonding," J. Inorg. Nucl. Chem. 8, 143–148 (1958). [CrossRef]
  44. H. Bethe, "Termaufspaltung in Kristallen Electronenterme und Strahlung von Atomen in Kristallen," Ann. Phys. 3, 133–208 (1929); Z. Phys. 60, 218 (1930) [English translation, Splitting of Terms of Crystals (Consultants Bureau, New York, undated). [CrossRef]
  45. K. H. Hellwege, "Electronenterme und Strahlung von Atomen in Kristallen," Ann. Phys. 4, 95–160 (1949).
  46. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford, 1970).
  47. H. M. Crosswhite and H. Crosswhite, "The spectrum of Nd3+: LaCl3," J. Chem. Phys. 64, 1981–1985 (1976). [CrossRef]
  48. W. T. Carnall, H. Crosswhite, H. M. Crosswhite, and J. G. Conway, "Energy level analysis of Pm3+:LaCl3," J. Chem. Phys. 64, 3582–3591 (1976). [CrossRef]
  49. M. S. Magno and G. H. Dieke, "Absorption and fluorescence spectra of hexagonal SmCl3 and their Zeeman effects," J. Chem. Phys. 37, 2354–2363 (1962); K. Rajnak, R. Mehlhorn, and N. Edelstein, "Calculation of the crystal field splitting of Sm3+ levels in LaCl3 with inclusion of J-mixing," J. Chem. Phys. 58, 609–615 (1973); see also Ref. 9. [CrossRef]
  50. H. M. Crosswhite and G. H. Dieke, "Spectrum and magnetic properties of hexagonal DyCl3," J. Chem. Phys. 35, 1535–1548 (1961); see also Ref. 9. [CrossRef]
  51. H. M. Crosswhite, H. Crosswhite, W. T. Carnall, and K. Rajnak, "Parametric energy level analysis of Ho3+:LaCl3," J. Chem. Phys. 67, 3002–3010 (1977). [CrossRef]
  52. G. H. Dieke and S. Singh, "Absorption and fluorescence spectra with magnetic properties of ErCl3," J. Chem. Phys. 35, 555–563 (1961); F. Varsanyi and G. H. Dieke, "Energy levels of hexagonal ErCl3," J. Chem. Phys. 36, 2951–2961 (1962); see also Ref. 9. [CrossRef]
  53. H. M. Crosswhite, H. Crosswhite, W. T. Carnall, and A. P. Paszek, "Spectrum analysis of U3+:LaCl3," J. Chem. Phys. 72, 5103–5117 (1980). [CrossRef]
  54. W. T. Carnall, H. Crosswhite, H. M. Crosswhite, J. P. Hessler, N. Edelstein, J. G. Conway, G. V. Shalimoff, and R. Sarup, "Energy level analysis of Np3+:LaCl3 and Np3+:LaBr3," J. Chem. Phys. 72, 5089–5102 (1980). [CrossRef]
  55. H. Lammerman and J. G. Conway, "Absorption spectrum of Pu3+ in lanthanum trichloride and lanthanum ethylsulfate," J. Chem. Phys. 38, 259–269 (1963); J. G. Conway and K. Rajnak, J. Chem. Phys. 44, 348–354 (1966). [CrossRef]
  56. J. Blaise, Laboratoire Aimé Cotton, Orsay, France; J. G. Conway and E. Worden, Lawrence Livermore National Laboratory, Berkeley, California 94720 (personal communication).
  57. S. Feneuille and N. Pelletier-Allard, "Contribution a l'interpretation des configurations 4ƒ26s, 4ƒ25d et 4ƒ26p de Pr III," Phys. (Utrecht) 40, 347–356 (1968). [CrossRef]
  58. J. F. Wyart, J. Blaise, and P. Camus, "Progres récent dans I'interprétation des configurations 4ƒN(5d + 6s) des lanthanides," Phys. Scr. 9, 325–330 (1974). [CrossRef]
  59. H. M. Crosswhite, "Effective electrostatic operators for two in-equivalent electrons," Phys. Rev. A 4, 485–489 (1971). [CrossRef]
  60. J. Sugar and V. Kaufman, "Spectra and energy levels of three-and four-times ionized hafnium (Hf IV and HF v)," J. Opt. Soc. Am. 64, 1656–1664 (1974). [CrossRef]
  61. V. Kaufman and J. Sugar, "Spectra and energy levels of five-times ionized tantalum (Ta VI)," J. Opt. Soc. Am. 65, 302–309 (1975). [CrossRef]
  62. J. Sugar and V. Kaufman, "Seventh spectrum of tungsten (W vII); resonance lines of Hf v," Phys. Rev. A 12, 994–1012 (1975). [CrossRef]
  63. J. G. Conway, J. Blaise and Jean Verges, "The I.R. spectrum of curium-244," Spectrochim. Acta Part B, 31, 31–47 (1976). [CrossRef]

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