Abstract

The energy level structure, relative line strengths, and Landé <i>g</i> factors of two-electron configurations are discussed for four important types of pure coupling: <i>LS</i>, <i>LK</i>(<i>Ls</i>), <i>jK</i> (<i>jl</i>), and <i>jj</i>. Transitions from one type of coupling to another are discussed in detail, the configuration <i>pf</i> being used as an example. The appropriateness of <i>LS</i>- and <i>jj</i>-coupling notation in two-electron spectra is quite limited for atoms of medium atomic weight, where nearly all excited configurations show a strong tendency toward pair (<i>LK</i> to <i>jK</i>) coupling. For other atoms, pair coupling occurs mainly for high values of orbital angular momentum of the excited electron: the coupling may then be near <i>LK</i> for small values of the principal quantum number of this electron, approaching pure <i>jK</i> as this quantum number increases. Either <i>LK</i> or <i>jK</i> notation can serve unambiguously to identify levels throughout the range of intermediate pair couplings, but neither will correctly designate the nature of the quantum states in all cases because of exchanges of the <i>L</i> (and of the <i>j</i>) compositions of certain states which occur as the coupling conditions change from pure <i>LK</i> to pure <i>jK</i>. Examples are discussed from the spectra of N, P, Ge, and the rare gases.

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  1. G. Racah, J. Opt. Soc. Am. 50, 408 (1960).
  2. A. G. Shenstone, Phil. Trans. Roy. Soc. (London) A235, 195 (1936).
  3. G. H. Shortley and B. Fried, Phys. Rev. 54, 749 (1938).
  4. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge, England, 1935), especially Chaps. 6, 7, 10, 11, 13.
  5. J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), especially Chaps. 12, 13, 24. For brief discussions of the theory, see also H. G. Kuhn, Atomic Spectra (Academic Press, Inc., New York, 1962), Chap. V, and Ref. 25, Sec. II.
  6. K. B. S. Eriksson, Phys. Rev. 102, 102 (1956); Arkiv Fysik 13, 303 (1958).
  7. L. J. Radziemski and K. L. Andrew, J. Opt. Soc. Am. 55, 474 (1965).
  8. Y. G. Toresson, Arkiv Fysik 18, 389 (1960).
  9. W. C. Martin, J. Opt. Soc. Am. 49, 1071 (1959).
  10. K. L. Andrew and K. W. Meissner, J. Opt. Soc. Am. 49, 146 (1959); C. J. Humphreys and K. L. Andrew, ibid. 54, 1134 (1964).
  11. A. M. Crooker (private communication).
  12. W. G. Brill, Ph.D. thesis, Purdue University (1964); J. Opt. Soc. Am. 54, 566A (1964).
  13. B. Edlén and J. W. Swensson, reported at Atomic Spectroscopy Symposium, Argonne National Laboratory, June 1961.
  14. J. Sugar, J. Opt. Soc. Am. 55, 33 (1965).
  15. K. B. S. Eriksson and I. Johansson, Arkiv Fysik 19, 235 (1961).
  16. K. B. S. Eriksson, Arkiv Fysik 19, 229 (1961).
  17. L. Minnhagen, Arkiv Fysik 18, 97 (1960).
  18. L. Minnhagen, J. Opt. Soc. Am. 51, 298 (1961).
  19. J. L. Tech, J. Res. Natl. Bur. Std. 67A, 505 (1963).
  20. L. Minnhagen, Arkiv Fysik 21, 415 (1962); L. Minnhagen and C. J. Humphreys, J. Opt. Soc. Am. 54, 1403A (1964).
  21. G. Racah, Phys. Rev. 61, 537 (1942).
  22. See Ref. 4, p. 123.
  23. A. M. Gutman and I. B. Levinson, Astron. Zh. 37, 86 (1960) [English transl.: Soviet Astron.—AJ 4, 83 (1960)].
  24. N. H. Möller, Arkiv Fysik 18, 135 (1960).
  25. B. Edlén, "Atomic Spectra," in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, pp. 93 and 123.
  26. We shall not be much interested in M degeneracy in this paper and will speak only in terms of J degeneracies, considering there to be only one state per value of J.
  27. G. Racah, Phys. Rev. 63, 367 (1943).
  28. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957).
  29. B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill Book Company, Inc., New York, 1963).
  30. M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).
  31. See Ref. 28, p. 41, or Ref. 29, Eq. (1–21).
  32. G. H. Shortley and B. Fried, Phys. Rev. 54, 739 (1938).
  33. K. Smith and J. W. Stevenson, Argonne National Laboratory Reports ANL-5776 and ANL-5860.
  34. Actually, these are already available in the literature for most cases of interest (see Refs. 4, 5).
  35. Small values of ζf would, of course, also remove the degeneracy (see Table II).
  36. G. H. Shortley, Phys. Rev. 44, 670 (1933).
  37. Note that this is different from the definition of purity used in discussing Figs. 4 and 5, and accounts for the increase in apparent LS and LK purity in going from the 4f to the 6f configuration.
  38. F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963).
  39. Ref. 25, Tables 38 and 42 and Fig. 12.
  40. L. Minnhagen, Arkiv Fysik 1, 425 (1949).
  41. F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).
  42. G. Racah, Phys. Rev. 62, 438 (1942).
  43. G. Racah, Phys. Rev. 62, 438, Eqs. (44) (1942); or Ref. 28, Eqs. (7.1.7) and (7.1.8).
  44. See Ref. 29, Eq. (4-4).
  45. The diagonal elements in any representation can be calculated directly by means described by J. C. van den Bosch in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1957), Vol. XXVIII, pp. 305 ff, but the off-diagonal elements are readily found only by transforming the g matrix from the LS representation.
  46. Data from C. E. Moore, Natl. Bur. Std. (U.S.) Circ. No. 467 1 (1949), 2 (1952). Data for Xe I have been omitted because they are badly perturbed: J. B. Green, E. H. Hurlburt, and D. W. Bowman, Phys. Rev. 59, 72 (1941).
  47. J. B. Green, Phys. Rev. 52, 736 (1937).

Andrew, K. L.

L. J. Radziemski and K. L. Andrew, J. Opt. Soc. Am. 55, 474 (1965).

K. L. Andrew and K. W. Meissner, J. Opt. Soc. Am. 49, 146 (1959); C. J. Humphreys and K. L. Andrew, ibid. 54, 1134 (1964).

Bivins, R.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

Brill, W. G.

W. G. Brill, Ph.D. thesis, Purdue University (1964); J. Opt. Soc. Am. 54, 566A (1964).

Condon, E. U.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge, England, 1935), especially Chaps. 6, 7, 10, 11, 13.

Crooker, A. M.

A. M. Crooker (private communication).

Edlén, B.

B. Edlén, "Atomic Spectra," in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, pp. 93 and 123.

B. Edlén and J. W. Swensson, reported at Atomic Spectroscopy Symposium, Argonne National Laboratory, June 1961.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957).

Eriksson, K. B. S.

K. B. S. Eriksson and I. Johansson, Arkiv Fysik 19, 235 (1961).

K. B. S. Eriksson, Phys. Rev. 102, 102 (1956); Arkiv Fysik 13, 303 (1958).

K. B. S. Eriksson, Arkiv Fysik 19, 229 (1961).

Fried, B.

G. H. Shortley and B. Fried, Phys. Rev. 54, 749 (1938).

G. H. Shortley and B. Fried, Phys. Rev. 54, 739 (1938).

Green, J. B.

J. B. Green, Phys. Rev. 52, 736 (1937).

Gutman, A. M.

A. M. Gutman and I. B. Levinson, Astron. Zh. 37, 86 (1960) [English transl.: Soviet Astron.—AJ 4, 83 (1960)].

Herman, F.

F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963).

Johansson, I.

K. B. S. Eriksson and I. Johansson, Arkiv Fysik 19, 235 (1961).

Judd, B. R.

B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill Book Company, Inc., New York, 1963).

Levinson, I. B.

A. M. Gutman and I. B. Levinson, Astron. Zh. 37, 86 (1960) [English transl.: Soviet Astron.—AJ 4, 83 (1960)].

Martin, W. C.

W. C. Martin, J. Opt. Soc. Am. 49, 1071 (1959).

Meissner, K. W.

K. L. Andrew and K. W. Meissner, J. Opt. Soc. Am. 49, 146 (1959); C. J. Humphreys and K. L. Andrew, ibid. 54, 1134 (1964).

Metropolis, N.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

Minnhagen, L.

L. Minnhagen, Arkiv Fysik 18, 97 (1960).

L. Minnhagen, Arkiv Fysik 21, 415 (1962); L. Minnhagen and C. J. Humphreys, J. Opt. Soc. Am. 54, 1403A (1964).

L. Minnhagen, J. Opt. Soc. Am. 51, 298 (1961).

L. Minnhagen, Arkiv Fysik 1, 425 (1949).

Möller, N. H.

N. H. Möller, Arkiv Fysik 18, 135 (1960).

Moore, C. E.

Data from C. E. Moore, Natl. Bur. Std. (U.S.) Circ. No. 467 1 (1949), 2 (1952). Data for Xe I have been omitted because they are badly perturbed: J. B. Green, E. H. Hurlburt, and D. W. Bowman, Phys. Rev. 59, 72 (1941).

Racah, G.

G. Racah, J. Opt. Soc. Am. 50, 408 (1960).

G. Racah, Phys. Rev. 62, 438, Eqs. (44) (1942); or Ref. 28, Eqs. (7.1.7) and (7.1.8).

G. Racah, Phys. Rev. 62, 438 (1942).

G. Racah, Phys. Rev. 63, 367 (1943).

G. Racah, Phys. Rev. 61, 537 (1942).

Radziemski, L. J.

L. J. Radziemski and K. L. Andrew, J. Opt. Soc. Am. 55, 474 (1965).

Rohrlich, F.

F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).

Rotenberg, M.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

Shenstone, A. G.

A. G. Shenstone, Phil. Trans. Roy. Soc. (London) A235, 195 (1936).

Shortley, G. H.

G. H. Shortley and B. Fried, Phys. Rev. 54, 739 (1938).

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge, England, 1935), especially Chaps. 6, 7, 10, 11, 13.

G. H. Shortley, Phys. Rev. 44, 670 (1933).

G. H. Shortley and B. Fried, Phys. Rev. 54, 749 (1938).

Skillman, S.

F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963).

Slater, J. C.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), especially Chaps. 12, 13, 24. For brief discussions of the theory, see also H. G. Kuhn, Atomic Spectra (Academic Press, Inc., New York, 1962), Chap. V, and Ref. 25, Sec. II.

Smith, K.

K. Smith and J. W. Stevenson, Argonne National Laboratory Reports ANL-5776 and ANL-5860.

Stevenson, J. W.

K. Smith and J. W. Stevenson, Argonne National Laboratory Reports ANL-5776 and ANL-5860.

Sugar, J.

J. Sugar, J. Opt. Soc. Am. 55, 33 (1965).

Swensson, J. W.

B. Edlén and J. W. Swensson, reported at Atomic Spectroscopy Symposium, Argonne National Laboratory, June 1961.

Tech, J. L.

J. L. Tech, J. Res. Natl. Bur. Std. 67A, 505 (1963).

Toresson, Y. G.

Y. G. Toresson, Arkiv Fysik 18, 389 (1960).

van den Bosch, J. C.

The diagonal elements in any representation can be calculated directly by means described by J. C. van den Bosch in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1957), Vol. XXVIII, pp. 305 ff, but the off-diagonal elements are readily found only by transforming the g matrix from the LS representation.

Wooten, Jr., J. K.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

Other (47)

G. Racah, J. Opt. Soc. Am. 50, 408 (1960).

A. G. Shenstone, Phil. Trans. Roy. Soc. (London) A235, 195 (1936).

G. H. Shortley and B. Fried, Phys. Rev. 54, 749 (1938).

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge, England, 1935), especially Chaps. 6, 7, 10, 11, 13.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), especially Chaps. 12, 13, 24. For brief discussions of the theory, see also H. G. Kuhn, Atomic Spectra (Academic Press, Inc., New York, 1962), Chap. V, and Ref. 25, Sec. II.

K. B. S. Eriksson, Phys. Rev. 102, 102 (1956); Arkiv Fysik 13, 303 (1958).

L. J. Radziemski and K. L. Andrew, J. Opt. Soc. Am. 55, 474 (1965).

Y. G. Toresson, Arkiv Fysik 18, 389 (1960).

W. C. Martin, J. Opt. Soc. Am. 49, 1071 (1959).

K. L. Andrew and K. W. Meissner, J. Opt. Soc. Am. 49, 146 (1959); C. J. Humphreys and K. L. Andrew, ibid. 54, 1134 (1964).

A. M. Crooker (private communication).

W. G. Brill, Ph.D. thesis, Purdue University (1964); J. Opt. Soc. Am. 54, 566A (1964).

B. Edlén and J. W. Swensson, reported at Atomic Spectroscopy Symposium, Argonne National Laboratory, June 1961.

J. Sugar, J. Opt. Soc. Am. 55, 33 (1965).

K. B. S. Eriksson and I. Johansson, Arkiv Fysik 19, 235 (1961).

K. B. S. Eriksson, Arkiv Fysik 19, 229 (1961).

L. Minnhagen, Arkiv Fysik 18, 97 (1960).

L. Minnhagen, J. Opt. Soc. Am. 51, 298 (1961).

J. L. Tech, J. Res. Natl. Bur. Std. 67A, 505 (1963).

L. Minnhagen, Arkiv Fysik 21, 415 (1962); L. Minnhagen and C. J. Humphreys, J. Opt. Soc. Am. 54, 1403A (1964).

G. Racah, Phys. Rev. 61, 537 (1942).

See Ref. 4, p. 123.

A. M. Gutman and I. B. Levinson, Astron. Zh. 37, 86 (1960) [English transl.: Soviet Astron.—AJ 4, 83 (1960)].

N. H. Möller, Arkiv Fysik 18, 135 (1960).

B. Edlén, "Atomic Spectra," in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. XXVII, pp. 93 and 123.

We shall not be much interested in M degeneracy in this paper and will speak only in terms of J degeneracies, considering there to be only one state per value of J.

G. Racah, Phys. Rev. 63, 367 (1943).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957).

B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill Book Company, Inc., New York, 1963).

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

See Ref. 28, p. 41, or Ref. 29, Eq. (1–21).

G. H. Shortley and B. Fried, Phys. Rev. 54, 739 (1938).

K. Smith and J. W. Stevenson, Argonne National Laboratory Reports ANL-5776 and ANL-5860.

Actually, these are already available in the literature for most cases of interest (see Refs. 4, 5).

Small values of ζf would, of course, also remove the degeneracy (see Table II).

G. H. Shortley, Phys. Rev. 44, 670 (1933).

Note that this is different from the definition of purity used in discussing Figs. 4 and 5, and accounts for the increase in apparent LS and LK purity in going from the 4f to the 6f configuration.

F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963).

Ref. 25, Tables 38 and 42 and Fig. 12.

L. Minnhagen, Arkiv Fysik 1, 425 (1949).

F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).

G. Racah, Phys. Rev. 62, 438 (1942).

G. Racah, Phys. Rev. 62, 438, Eqs. (44) (1942); or Ref. 28, Eqs. (7.1.7) and (7.1.8).

See Ref. 29, Eq. (4-4).

The diagonal elements in any representation can be calculated directly by means described by J. C. van den Bosch in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1957), Vol. XXVIII, pp. 305 ff, but the off-diagonal elements are readily found only by transforming the g matrix from the LS representation.

Data from C. E. Moore, Natl. Bur. Std. (U.S.) Circ. No. 467 1 (1949), 2 (1952). Data for Xe I have been omitted because they are badly perturbed: J. B. Green, E. H. Hurlburt, and D. W. Bowman, Phys. Rev. 59, 72 (1941).

J. B. Green, Phys. Rev. 52, 736 (1937).

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