Abstract

The energy level structure, relative line strengths, and Landé g factors of two-electron configurations are discussed for four important types of pure coupling: LS, LK(Ls), jK(jl), and jj. Transitions from one type of coupling to another are discussed in detail, the configuration pf being used as an example. The appropriateness of LS- and jj-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 (LK to jK) 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 LK for small values of the principal quantum number of this electron, approaching pure jK as this quantum number increases. Either LK or jK 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 L (and of the j) compositions of certain states which occur as the coupling conditions change from pure LK to pure jK. Examples are discussed from the spectra of N, P, Ge, and the rare gases.

© 1965 Optical Society of America

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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).
    [Crossref]
  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).
    [Crossref]
  7. L. J. Radziemski and K. L. Andrew, J. Opt. Soc. Am. 55, 474 (1965).
    [Crossref]
  8. Y. G. Toresson, Arkiv Fysik 18, 389 (1960).
  9. W. C. Martin, J. Opt. Soc. Am. 49, 1071 (1959).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  19. J. L. Tech, J. Res. Natl. Bur. Std. 67A, 505 (1963).
    [Crossref]
  20. L. Minnhagen, Arkiv Fysik 21, 415 (1962); L. Minnhagen and C. J. Humphreys, J. Opt. Soc. Am. 54, 1403A (1964).
    [Crossref]
  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).
    [Crossref]
  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, 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).
    [Crossref]
  33. K. Smith and J. W. Stevenson, .
  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).
    [Crossref]
  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).
    [Crossref]
  42. G. Racah, Phys. Rev. 62, 438 (1942).
    [Crossref]
  43. G. Racah, Phys. Rev. 62, 438, Eqs. (44) (1942); or Ref. 28, Eqs. (7.1.7) and (7.1.8).
    [Crossref]
  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), Natl. Bur. Std. (U.S.) Circ. No. 467 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).
    [Crossref]
  47. J. B. Green, Phys. Rev. 52, 736 (1937).
    [Crossref]

1965 (2)

1963 (1)

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

1962 (1)

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

1961 (3)

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

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

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

1960 (5)

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

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).

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

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

1959 (3)

1956 (1)

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

1949 (2)

Data from C. E. Moore, Natl. Bur. Std. (U.S.) Circ. No. 467 1 (1949), Natl. Bur. Std. (U.S.) Circ. No. 467 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).
[Crossref]

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

1943 (1)

G. Racah, Phys. Rev. 63, 367 (1943).
[Crossref]

1942 (3)

G. Racah, Phys. Rev. 61, 537 (1942).

G. Racah, Phys. Rev. 62, 438 (1942).
[Crossref]

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

1938 (2)

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

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

1937 (1)

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

1936 (1)

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

1933 (1)

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

Andrew, K. L.

Bivins, R.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, 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 and J. W. Swensson, reported at Atomic Spectroscopy Symposium, Argonne National Laboratory, June 1961.

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

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, Arkiv Fysik 19, 229 (1961).

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

Fried, B.

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

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

Green, J. B.

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

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.

Meissner, K. W.

Metropolis, N.

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

Minnhagen, L.

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

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

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

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), Natl. Bur. Std. (U.S.) Circ. No. 467 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).
[Crossref]

Racah, G.

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

G. Racah, Phys. Rev. 63, 367 (1943).
[Crossref]

G. Racah, Phys. Rev. 61, 537 (1942).

G. Racah, Phys. Rev. 62, 438 (1942).
[Crossref]

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

Radziemski, L. J.

Rohrlich, F.

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

Rotenberg, M.

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, 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, 749 (1938).
[Crossref]

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

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

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.

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, .

Stevenson, J. W.

K. Smith and J. W. Stevenson, .

Sugar, J.

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).
[Crossref]

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, J. K.

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

Arkiv Fysik (7)

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

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

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

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

Astron. Zh. (1)

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

Astrophys. J. (1)

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

J. Opt. Soc. Am. (6)

J. Res. Natl. Bur. Std. (1)

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

Natl. Bur. Std. (U.S.) Circ. No. 467 (1)

Data from C. E. Moore, Natl. Bur. Std. (U.S.) Circ. No. 467 1 (1949), Natl. Bur. Std. (U.S.) Circ. No. 467 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).
[Crossref]

Phil. Trans. Roy. Soc. (London) (1)

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

Phys. Rev. (9)

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

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

G. Racah, Phys. Rev. 61, 537 (1942).

G. Racah, Phys. Rev. 63, 367 (1943).
[Crossref]

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

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

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

G. Racah, Phys. Rev. 62, 438 (1942).
[Crossref]

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

Other (20)

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.

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.

K. Smith and J. W. Stevenson, .

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).

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, The 3-j and 6-j Symbols (Technology Press, Cambridge, Massachusetts, 1959).

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

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.

See Ref. 4, p. 123.

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.

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.

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Figures (5)

Fig. 1
Fig. 1

Development of the level structure of the configuration pf for both LS-coupling (ξ≅0) and LK-coupling (ξ≅1) conditions, and the transition from one extreme to the other for the special case F2 = 1, 50G2+ζp =3, G4 =ζf=0. Note the inverted 3D, and that as ξ increases the triplet level which splits off to pair with the singlet is the level J = L+1 for 3G and the level J = L−1 for 3D and 3F.

Fig. 2
Fig. 2

The development of the level structure of the configuration pf for jK-coupling (ξ≅0) and jj-coupling (ξ≅1) conditions, and the transition between the two extremes for the special case ζp= 1, 5F2+ζf=0.15, G2 = G4=0. Note that the two highest J=4 levels, and the two lowest J=3 levels, would cross if it were not for the interaction between them.

Fig. 3
Fig. 3

The energy-level structure of the configuration pf in pair coupling (Gk = ζf = 0). The 12 J states of this configuration correspond to six doubly degenerate levels, each characterized by a value of [K] and with J = K ± 1 2. In the limit ξ → 1, pure LK coupling exists and the three quadruply degenerate levels correspond to L = F, G, D; in the limit ξ → 1, the coupling is pure j1K, with one quadruply degenerate j 1 = 1 2 level and one eightfold degenerate j 1 = 3 2 level. As examples, the positions of the levels of P ii 3p4f and of N ii 2p4f-6f are plotted at the appropriate ξ values.

Fig. 4
Fig. 4

The composition of the lower-energy K = 7 2 states (either J = 3 or 4, and any M value) of the configuration pf. The solid curves give the composition in the LK representation (e.g., at ξ = 0.2, these states are each 96% F [ 7 2 ] and 4% G [ 7 2 ]). Similarly, the dashed curves give the composition for jK basis functions (e.g., at ξ = 0.2 the states are 40% 1 2 [ 7 2 ] and 60% 3 2 [ 7 2 ]). (The compositions of the higher-energy K = 7 2 states are the complements of the low-energy compositions; i.e., they are given by interchanging the F and G labels, and interchanging the j1 values, in the figure.)

Fig. 5
Fig. 5

The average purity of the eigenvectors of the pf configuration (averaged over the ten levels J ≠ 1, 5), for each of the four representations LS, LK, jK, jj.

Tables (9)

Tables Icon

Table I Transformation matrices for pf, p5f, or f13p. (Matrices for the LSjK, LKjj, and LSjj transformations are readily obtainable by multiplication of those given here.)

Tables Icon

Table II Coefficient matrices for pf. (Coefficients are for F2, G2, G4, ζp, ζf, in that order.)

Tables Icon

Table III Coefficient matrices for p5f. Coefficients for F2, ζp, and ζf are, respectively, −1, −1, and +1 times those for pf. The only nonzero coefficients for G2 and G4 are as shown below.

Tables Icon

Table IV Calculated compositions of the 4p4f states of Ge i.

Tables Icon

Table V Least-squares parameter values and coupling type in Ge i (s and parameter values in cm−1).

Tables Icon

Table VI Relative line strengths for pfpg (or p5fp5g) transitions in LK coupling.

Tables Icon

Table VII Relative line strengths for pfpg (or p5fp5g) transitions in jK coupling.

Tables Icon

Table VIII J = 1 g matrices for pp′ or p5p′ in the representations LS, LK, jK, jj.

Tables Icon

Table IX Landé g factors, least-square parameters, and coupling type in p5p′.

Equations (42)

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[ ( l 1 l 2 ) L , ( s 1 s 2 ) S ] J ,
[ ( l 1 s 1 ) j 1 , ( l 2 s 2 ) j 2 ] J ;
[ ( ( l 1 s 1 ) j 1 , l 2 ) K , s 2 ] J .
[ ( ( l 1 l 2 ) L , s 1 ) K , s 2 ] J .
[ ( l 1 l 2 s 1 ) K , s 2 ] J ,
L S J M ) ,             L K J M ) ,             j 1 K J M ) ,             and             j 1 j 2 J M ) ,
β J M ) = L S L S J M ) ( L S J β J ) L S L S J M ) B L S J β ,
U j 1 K , j 1 j 2 ( ( j 1 l 2 ) K , s 2 J j 1 , ( l 2 s 2 ) j 2 , J ) = ( - 1 ) j 1 + l 2 + s 2 + J [ K ] 1 2 [ j 2 ] 1 2 { K s 2 J j 2 j 1 l 2 } ,
U L K , L S ( ( L s 1 ) K , s 2 J L , ( s 1 s 2 ) S , J ) = ( - 1 ) L + s 1 + s 2 + J [ K ] 1 2 [ S ] 1 2 { K s 2 K S L s 1 } ,
[ K ] 2 K + 1 , etc .
( j 1 j 2 ) j 3 ) = ( - 1 ) j 1 + j 2 - j 3 ( j 2 j 1 ) j 3 ) ,
U L K , j 1 K = ( - 1 ) - l 1 - l 2 + L + j 1 + l 2 - K ( ( l 2 l 1 ) L , s 1 K l 2 , ( l 1 s 1 ) j 1 , K ) = ( - 1 ) L + j 1 + l 2 + s 1 [ L ] 1 2 [ j 1 ] 1 2 { L s 1 K j 1 l 2 l 1 } .
U L S , j 1 j 2 = K U L S , j 1 K U j 1 K , j 1 j 2 = ( [ L ] [ S ] [ j 1 ] [ j 2 ] ) 1 2 K ( - 1 ) 2 K [ K ] { K s 2 J j 2 j 1 l 2 } { K s 2 J S L s 1 } { L s 1 K j 1 l 2 l 1 } = ( [ L ] [ S ] [ j 1 ] [ j 2 ] ) 1 2 { l 1 l 2 L s 1 s 2 S j 1 j 2 J }
H = H el + H mag = j > i e 2 r i j + i ξ i ( r i ) × ( l i · s i ) ,
( L S H el L S ) = k ( f k F k + g k G k ) .
f k = ( - 1 ) L ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) × ( l 1 l 1 k 0 0 0 ) ( l 2 l 2 k 0 0 0 ) ( l 1 l 2 L l 2 l 1 k ) ,
g k = ( - 1 ) S ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) × ( l 2 l 2 k 0 0 0 ) 2 { l 1 l 2 L l 1 l 2 k } ,
( j 1 j 2 H mag j 1 j 2 ) = i d i ζ i .
d i = { 1 2 l i , j i = l i + 1 2 , - 1 2 ( l i + 1 ) , j i = l i - 1 2 .
g k = 0 , S = 1 or k L ; g L = 2 ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) 2 L + 1 ( l 1 l 2 L 0 0 0 ) 2 , S = 0. }
ξ = ζ p / ( 18 F 2 + ζ p ) ,
s = { ( error ) 2 No . levels - No . parameters } 1 2 ,
S M M = x y z ( β J M M x β J M ) 2 = q ( β J M P q ( 1 ) β J M ) 2 , = ( β J P ( 1 ) β J ) 2 q | ( - 1 ) J - M ( J 1 J - M q M ) | 2 .
S = M M S M M = ( β J P ( 1 ) β J ) 2 ,
D β ( β J P ( 1 ) β J ) = a b a b ( B a b J β ) * ( a b J P ( 1 ) a b J ) B a b J β ,
D L S ( α 1 L 1 S 1 , l 2 L S , J P ( 1 ) α 1 L 1 S 1 , l 2 L S , J ) = δ S S ( - 1 ) S + J - L 1 - l 2 ( [ J ] [ J ] [ L ] [ L ] ) 1 2 { L J S J L 1 } { l 2 L L 1 L l 2 1 } P ,
D L K ( ( α 1 L 1 l 2 ) L , S 1 K , s 2 J P ( 1 ) ( α 1 L 1 l 2 ) L , S 1 K , s 2 J ) = ( - 1 ) K + s 2 + J + S 1 + K - L 1 - l 2 - 1 ( [ J ] [ J ] [ K ] [ K ] [ L ] [ L ] ) 1 2 { K J s 2 J K 1 } { L K S 1 K L 1 } { l 2 L L 1 L l 2 1 } P ,
D J 1 K ( α 1 L 1 S 1 J 1 , l 2 K , s 2 J P ( 1 ) α 1 L 1 S 1 J 1 , l 2 K , s 2 J ) = δ J 1 J 1 ( - 1 ) s 2 + J - J 1 - l 2 ( [ J ] [ J ] [ K ] [ K ] ) 1 2 { K J s 2 J K 1 } { l 2 K J 1 K l 2 1 } P ,
D J 1 j 2 ( α 1 L 1 S 1 J 1 , l 2 s 2 j 2 , J P ( 1 ) α 1 L 1 S 1 J 1 , l 2 s 2 j 2 , J ) = δ J 1 J 1 ( - 1 ) J 1 + J - l 2 - s 2 ( [ J ] [ J ] [ j 2 ] [ j 2 ] ) 1 2 { j 2 J J 1 J j 2 1 } { l 2 j 2 s 2 j 2 l 2 1 } P .
P ( n 2 l 2 - e r 2 n 2 l 2 ) = ( - 1 ) l 2 ( [ l 2 ] [ l 2 ] ) 1 2 ( l 2 1 l 2 0 0 0 ) 0 ( - e r ) R l 2 ( r ) R l 2 ( r ) r 2 d r ,
= δ l 2 , l 2 ± 1 ( - 1 ) l 2 + l > l > 1 2 0 ( - e r ) R l 2 ( r ) R l 2 ( r ) r 2 d r ,
l 2 = l 2 ± 1 ,
{ a b c d e f } = 0
Δ J = 0 , ± 1             ( J = J = 0 not allowed )
Δ L = 0 , ± 1 ( L = L = 0 not allowed ) Δ K = 0 , ± 1 ( K = K = 0 not allowed ) Δ j 2 = 0 , ± 1 ( j 2 = j 2 = 0 not allowed ) } .
J [ J ] [ J ] { X J Y J X 1 } 2 = [ J ] [ X ] ,
J J [ J ] [ J ] { X J Y J X 1 } 2 = [ X ] - 1 J [ J ] = [ Y ] .
S = [ S 1 ] [ L 1 ] [ s 2 ] ( l 2 P ( 1 ) l 2 ) 2 ,
H mag = e H 2 m c i ( l z i + 2 s z i ) = e H 2 m c ( J z + S z ) .
g k = i j ( B i k ) * ( g i j ) ( B j k ) ,
g i j ( i J z + S z j ) / M
g i i g L S J = 1 + J ( J + 1 ) - L ( L + 1 ) + S ( S + 1 ) 2 J ( J + 1 ) .