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Group Theory for Atomic Shells

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Springer Handbook of Atomic, Molecular, and Optical Physics

Part of the book series: Springer Handbooks ((SHB))

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Abstract

The basic elements of the theory of Lie groups and their irreducible representations (IRs) are described. The IRs are used to label the states of an atomic shell and also the components of operators of physical interest. Applications of the generalized Wigner-Eckart theorem lead to relations between matrix elements appearing in different electronic configurations. This is particularly useful in the f shell, where transformations among the seven orbital states of an f electron can be described by the unitary group U(7) and its sequential subgroups SO(7), G2, and SO(3) with respective IRs [λ], W, U, and L. Extensions to groups that involve electron spin S (like Sp(14)) are described, as are groups that do not conserve electron number. The most useful of the latter is the quasispin group whose generators Q connect states of identical W, U, L and seniority v in the f shell. The symmetries of products of objects (states or operators) that themselves possess symmetries are described by the technique of plethysms.

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Abbreviations

CG:

Clebsch-Gordan

IR:

irreducible representations

References

  1. G. Racah: Group Theory and Spectroscopy, Springer Tracts in Modern Physics, Vol. 37 (Springer, New York 1965)

    Google Scholar 

  2. L. P. Eisenhart: Continuous Groups of Transformations (Dover, New York 1961)

    MATH  Google Scholar 

  3. E. Cartan: Sur la Structure des Groupes de Transformations Finis et Continus, Thesis (Nony, Paris 1894)

    Google Scholar 

  4. B. R. Judd: Group Theory and Its Applications, ed. by E. M. Loebl (Academic, New York 1968)

    Google Scholar 

  5. E. U. Condon, H. Odabasi: Atomic Structure (Cambridge Univ. Press, Cambridge 1980)

    Google Scholar 

  6. G. Racah: Phys. Rev. 76, 1352 (1949)

    Article  MATH  ADS  Google Scholar 

  7. B. R. Judd: Operator Techniques in Atomic Spectroscopy (Princeton Univ. Press, Princeton 1963)

    Google Scholar 

  8. E. P. Wigner: Group Theory (Academic, New York 1959)

    MATH  Google Scholar 

  9. B. R. Judd: Phys. Rev. 162, 28 (1967)

    Article  ADS  Google Scholar 

  10. L. Armstrong, B. R. Judd: Proc. R. Soc. London Ser. A 315, 27 and 39 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  11. B. R. Judd, G. M. S. Lister: J. Phys. A 25, 2615 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. B. G. Wybourne: Classical Groups for Physicists (Wiley, New York 1974)

    MATH  Google Scholar 

  13. B. G. Wybourne: Symmetry Principles and Atomic Spectroscopy (Wiley, New York 1970)

    Google Scholar 

  14. W. G. McKay, J. Patera: Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras (Dekker, New York 1981)

    MATH  Google Scholar 

  15. E. B. Dynkin: Am. Math. Soc. Transl. Ser. 2 6, 245 (1965)

    Google Scholar 

  16. D. E. Rutherford: Substitutional Analysis (Edinburgh Univ. Press, Edinburgh 1948)

    MATH  Google Scholar 

  17. M. Moshinsky: Group Theory and the Many-Body Problem (Gordon Breach, New York 1968)

    Google Scholar 

  18. H. Weyl: The Theory of Groups and Quantum Mechanics (Dover, New York undated)

    Google Scholar 

  19. W. G. Harter: Principles of Symmetry, Dynamics and Spectroscopy 8, 2819 (1973)

    Article  ADS  Google Scholar 

  20. W. G. Harter: Principles of Symmetry, Dynamics and Spectroscopy (Wiley, New York 1993)

    Google Scholar 

  21. W. G. Harter, C. W. Patterson: A Unitary Calculus for Electronic Orbitals, Lect. Notes Phys., Vol. 49 (Springer, Berlin, Heidelberg 1976)

    Google Scholar 

  22. G. W. F. Drake, M. Schlesinger: Phys. Rev. A 15, 1990 (1977)

    Article  ADS  Google Scholar 

  23. R. D. Kent, M. Schlesinger: Phys. Rev. A 50, 186 (1994)

    Article  ADS  Google Scholar 

  24. D. E. Littlewood: The Theory of Group Characters (Clarendon, Oxford 1950)

    MATH  Google Scholar 

  25. H. A. Jahn: Proc. R. Soc. London Ser. A 201, 516 (1950)

    Article  MATH  ADS  Google Scholar 

  26. B. H. Flowers: Proc. R. Soc. London Ser. A 212, 248 (1952)

    Article  MATH  ADS  Google Scholar 

  27. B. R. Judd, H. T. Wadzinski: J. Math. Phys. 8, 2125 (1967)

    Article  ADS  Google Scholar 

  28. C. L. B. Shudeman: J. Franklin Inst. 224, 501 (1937)

    Article  Google Scholar 

  29. B. R. Judd, G. M. S. Lister: Phys. Rev. Lett. 67, 1720 (1991)

    Article  ADS  Google Scholar 

  30. B. G. Wybourne: Spectroscopic Properties of Rare Earths (Wiley, New York 1965) p. 15

    Google Scholar 

  31. C. W. Nielson, G. F. Koster: Spectroscopic Coefficients for the p n , d n , and f n Configurations (MIT Press, Cambridge 1963)

    Google Scholar 

  32. S. Feneuille: J. Phys. (Paris) 28, 61, 315, 701, and 497 (1967)

    Google Scholar 

  33. S. Feneuille: J. Phys. (Paris) 30, 923 (1969)

    Google Scholar 

  34. S. Feneuille, A. Crubellier, T. Haskell: J. Phys. (Paris) 31, 25 (1970)

    Article  Google Scholar 

  35. U. Fano, A. R. P. Rao: Symmetry Principles in Quantum Physics (Academic, New York 1996) Sect. 8.3.3

    Google Scholar 

  36. H. Georgi: Lie Algebras in Particle Physics (Benjamin/Cummings, Reading 1982) Chap. XXV

    MATH  Google Scholar 

  37. B. R. Judd: Phys. Rep. 285, 1 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  38. E. Lo, J. E. Hansen, B. R. Judd: J. Phys. B 33, 819 (2000)

    Article  ADS  Google Scholar 

  39. B. R. Judd, E. Lo: Phys. Rev. Lett. 85, 948 (2000)

    Article  ADS  Google Scholar 

  40. J. E. Hansen, E. G. Ven: Mol. Phys. 101, 997 (2003)

    Article  ADS  Google Scholar 

  41. M. J. Englefield: Group Theory and the Coulomb Problem (Wiley, New York 1972)

    MATH  Google Scholar 

  42. L. Armstrong: J Phys. (Paris) 31, 17 (1970)

    Article  Google Scholar 

  43. C. E. Wulfman: Group Theory and Its Applications, Vol. 2, ed. by E.M. Loebl (Academic, New York 1971)

    Google Scholar 

  44. D. R. Herrick: Adv. Chem. Phys. 52, 1 (1982)

    Article  Google Scholar 

  45. A. G. McLellan: Proc. Phys. Soc. London 76, 419 (1960)

    Article  ADS  Google Scholar 

  46. B. R. Judd: Physica 33, 174 (1967)

    Article  ADS  Google Scholar 

  47. B. R. Judd, H. M. Crosswhite, H. Crosswhite: Phys. Rev. 169, 130 (1968)

    Article  ADS  Google Scholar 

  48. B. R. Judd: Phys. Rev. 141, 4 (1966)

    Article  ADS  Google Scholar 

  49. B. R. Judd, M. A. Suskin: J. Opt. Soc. Am. B 1, 261 (1984)

    Article  ADS  Google Scholar 

  50. B. R. Judd, R. C. Leavitt: J. Phys. B 19, 485 (1986)

    Article  ADS  Google Scholar 

  51. R. C. Leavitt: J. Phys. A 20, 3171 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  52. R. C. Leavitt: J. Phys. B 21, 2363 (1988)

    Article  ADS  Google Scholar 

  53. A. R. Edmonds: Proc. R. Soc. London Ser. A 268, 567 (1962)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. V. L. Donlan: Air Force Material Laboratory Report No. AFML-TR-70-249 (Wright-Patterson Air Force Base, Ohio 1970)

    Google Scholar 

  55. D. D. Velkov: Multi-Electron Coefficients of Fractional Parentage for the p, d, and f Shells. Ph.D. Thesis (The Johns Hopkins University, Baltimore 2000) http://www.pha.jhu.edu/groups/cfp/

  56. P. Nutter, C. Nielsen: Fractional parentage coefficients of terms of f n, II. Direct Evaluation of Racah's Factored Forms by a Group Theoretical Approach, Technical Memorandum T-133 (Raytheon, Waltham 1963) p. 133

    Google Scholar 

  57. M. Rotenberg, R. Bivins, N. Metropolis, J. K. Wooten: The 3- j and 6- j Symbols (MIT Press, Cambridge 1959)

    Google Scholar 

  58. P. H. Butler: Phil. Trans. R. Soc. London Ser. A 277, 545 (1975)

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Astronomy, The Johns Hopkins University, 3400 North Charles Street, 21218, Baltimore, MD, USA

    Brian Judd

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  1. Brian Judd
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Correspondence to Brian Judd .

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Editors and Affiliations

  1. Department of Physics, University of Windsor, 401 Sunset St., N9B 3P4, Windsor, ON, Canada

    Gordon Drake Prof.

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© 2006 Springer-Verlag

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Judd, B. (2006). Group Theory for Atomic Shells. In: Drake, G. (eds) Springer Handbook of Atomic, Molecular, and Optical Physics. Springer Handbooks. Springer, New York, NY. https://doi.org/10.1007/978-0-387-26308-3_3

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