Abstract

Selection rules and electric-dipole transition moments between rotational substates in excited vibrational states of spherical-top molecules have been derived. The explicit dependence of these moments on rotational and vibrational quantum numbers has been calculated. These results, which are generally applicable, have been used to analyze the concept of a multiple-photon pathway in the 3 “ladder” of SF6. It will be shown that some ladder schemes must have “rung gaps” in their sequence of proposed strongly allowed electric-dipole transitions.

© 1977 Optical Society of America

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  1. For a recent review, see C. K. Rhodes, C. D. Cantrell, in The Significance of Nonlinearity in the Natural Sciences, A. Perlmutter, B. Kursunoglu, L. F. Scott, eds. (Plenum, New York, 1977).
  2. C. D. Cantrell, H. W. Galbraith, Opt. Commun. 18, 513 (1976), and references therein.
    [CrossRef]
  3. C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
    [CrossRef]
  4. For a recent review, see E. Yablonovitch (Ref. 1).
  5. See, for example, G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, New York, 1945), p. 336.
  6. K. T. Hecht, J. Mol. Spectrosc. 5, 355 (1960).
    [CrossRef]
  7. See, for example, A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960).
  8. K. Fox, J. Mol. Spectrosc. 9, 381 (1962).
    [CrossRef]
  9. K. Fox, Phys. Rev. Lett. 27, 233 (1971).
    [CrossRef]
  10. See, for example, H. H. Nielsen, in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1959), Vol. 37, Pt. 1.
  11. W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
    [CrossRef]
  12. K. Fox (to be published).
  13. K. Fox, W. B. Person, J. Chem. Phys. 64, 5218 (1976).
    [CrossRef]
  14. G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
    [CrossRef]
  15. K. Fox, in Ref. 1.
  16. H. W. Galbraith and C. D. Cantrell, in Ref. 1.
  17. That this subcase might have simple selection rules had been suggested in a private communication from P. F. Moulton, by analogy with known results6 for case (i). However, the extreme simplicity deduced here from Eqs. (13) and the selection rules had not been anticipated.
  18. K. Fox, Opt. Commun. 19, 397 (1976). In that work, Eq. (9) should have an additional factor of (T0/T) on the right-hand side, and the exponent in Eq. (11) should be 3/2 instead of ½. (I am grateful to C. D. Cantrell and R. S. McDowell, who arrived at the modifications independently, for bringing these points to my attention.) As a consequence, the values in the third column of Table 1 should be multiplied by a factor of (T/T0) ≈ 1.09.
    [CrossRef]
  19. F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.
  20. E. D. Hinkley, Appl. Phys. Lett. 16, 351 (1970).
    [CrossRef]
  21. J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
    [CrossRef]

1977 (2)

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

1976 (3)

K. Fox, Opt. Commun. 19, 397 (1976). In that work, Eq. (9) should have an additional factor of (T0/T) on the right-hand side, and the exponent in Eq. (11) should be 3/2 instead of ½. (I am grateful to C. D. Cantrell and R. S. McDowell, who arrived at the modifications independently, for bringing these points to my attention.) As a consequence, the values in the third column of Table 1 should be multiplied by a factor of (T/T0) ≈ 1.09.
[CrossRef]

C. D. Cantrell, H. W. Galbraith, Opt. Commun. 18, 513 (1976), and references therein.
[CrossRef]

K. Fox, W. B. Person, J. Chem. Phys. 64, 5218 (1976).
[CrossRef]

1975 (1)

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

1971 (1)

K. Fox, Phys. Rev. Lett. 27, 233 (1971).
[CrossRef]

1970 (1)

E. D. Hinkley, Appl. Phys. Lett. 16, 351 (1970).
[CrossRef]

1962 (1)

K. Fox, J. Mol. Spectrosc. 9, 381 (1962).
[CrossRef]

1960 (1)

K. T. Hecht, J. Mol. Spectrosc. 5, 355 (1960).
[CrossRef]

1939 (1)

W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
[CrossRef]

Aldridge, J. P.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Cadot, J.

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

Cantrell, C. D.

C. D. Cantrell, H. W. Galbraith, Opt. Commun. 18, 513 (1976), and references therein.
[CrossRef]

For a recent review, see C. K. Rhodes, C. D. Cantrell, in The Significance of Nonlinearity in the Natural Sciences, A. Perlmutter, B. Kursunoglu, L. F. Scott, eds. (Plenum, New York, 1977).

Corice, R. J.

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

Cupp, J. D.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.

Danielson, B. L.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.

Edmonds, A. R.

See, for example, A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960).

Filip, H.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Flicker, H.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Fox, K.

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

K. Fox, W. B. Person, J. Chem. Phys. 64, 5218 (1976).
[CrossRef]

K. Fox, Opt. Commun. 19, 397 (1976). In that work, Eq. (9) should have an additional factor of (T0/T) on the right-hand side, and the exponent in Eq. (11) should be 3/2 instead of ½. (I am grateful to C. D. Cantrell and R. S. McDowell, who arrived at the modifications independently, for bringing these points to my attention.) As a consequence, the values in the third column of Table 1 should be multiplied by a factor of (T/T0) ≈ 1.09.
[CrossRef]

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

K. Fox, Phys. Rev. Lett. 27, 233 (1971).
[CrossRef]

K. Fox, J. Mol. Spectrosc. 9, 381 (1962).
[CrossRef]

K. Fox (to be published).

Galbraith, H. W.

C. D. Cantrell, H. W. Galbraith, Opt. Commun. 18, 513 (1976), and references therein.
[CrossRef]

Hecht, K. T.

K. T. Hecht, J. Mol. Spectrosc. 5, 355 (1960).
[CrossRef]

Herzberg, G.

See, for example, G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, New York, 1945), p. 336.

Hinkley, E. D.

E. D. Hinkley, Appl. Phys. Lett. 16, 351 (1970).
[CrossRef]

Holland, R. F.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Jensen, C. C.

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

Krohn, B. J.

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

McDonald, D. G.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.

McDowell, R. S.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Nereson, N. G.

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Nielsen, H. H.

W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
[CrossRef]

See, for example, H. H. Nielsen, in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1959), Vol. 37, Pt. 1.

Overend, J.

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

Person, W. B.

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

K. Fox, W. B. Person, J. Chem. Phys. 64, 5218 (1976).
[CrossRef]

Petersen, F. R.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.

Pierre, G.

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

Rhodes, C. K.

For a recent review, see C. K. Rhodes, C. D. Cantrell, in The Significance of Nonlinearity in the Natural Sciences, A. Perlmutter, B. Kursunoglu, L. F. Scott, eds. (Plenum, New York, 1977).

Shaffer, W. H.

W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
[CrossRef]

Thomas, L. H.

W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
[CrossRef]

Appl. Phys. Lett. (1)

E. D. Hinkley, Appl. Phys. Lett. 16, 351 (1970).
[CrossRef]

Can. J. Phys. (1)

G. Pierre, J. Cadot, R. J. Corice, K. Fox, Can. J. Phys. 55, 473 (1977).
[CrossRef]

J. Chem. Phys. (1)

K. Fox, W. B. Person, J. Chem. Phys. 64, 5218 (1976).
[CrossRef]

J. Mol. Spectrosc. (3)

K. T. Hecht, J. Mol. Spectrosc. 5, 355 (1960).
[CrossRef]

K. Fox, J. Mol. Spectrosc. 9, 381 (1962).
[CrossRef]

J. P. Aldridge, H. Filip, H. Flicker, R. F. Holland, R. S. McDowell, N. G. Nereson, K. Fox, J. Mol. Spectrosc. 58, 165 (1975).
[CrossRef]

Opt. Commun. (3)

K. Fox, Opt. Commun. 19, 397 (1976). In that work, Eq. (9) should have an additional factor of (T0/T) on the right-hand side, and the exponent in Eq. (11) should be 3/2 instead of ½. (I am grateful to C. D. Cantrell and R. S. McDowell, who arrived at the modifications independently, for bringing these points to my attention.) As a consequence, the values in the third column of Table 1 should be multiplied by a factor of (T/T0) ≈ 1.09.
[CrossRef]

C. D. Cantrell, H. W. Galbraith, Opt. Commun. 18, 513 (1976), and references therein.
[CrossRef]

C. C. Jensen, W. B. Person, B. J. Krohn, J. Overend, Opt. Commun. 20, 275 (1977), and references therein.
[CrossRef]

Phys. Rev. (1)

W. H. Shaffer, H. H. Nielsen, L. H. Thomas, Phys. Rev. 56, 895 (1939).
[CrossRef]

Phys. Rev. Lett. (1)

K. Fox, Phys. Rev. Lett. 27, 233 (1971).
[CrossRef]

Other (10)

See, for example, H. H. Nielsen, in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1959), Vol. 37, Pt. 1.

See, for example, A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960).

For a recent review, see E. Yablonovitch (Ref. 1).

See, for example, G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, New York, 1945), p. 336.

K. Fox (to be published).

For a recent review, see C. K. Rhodes, C. D. Cantrell, in The Significance of Nonlinearity in the Natural Sciences, A. Perlmutter, B. Kursunoglu, L. F. Scott, eds. (Plenum, New York, 1977).

K. Fox, in Ref. 1.

H. W. Galbraith and C. D. Cantrell, in Ref. 1.

That this subcase might have simple selection rules had been suggested in a private communication from P. F. Moulton, by analogy with known results6 for case (i). However, the extreme simplicity deduced here from Eqs. (13) and the selection rules had not been anticipated.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Proceedings of the Vail (Colorado) International Conference on Laser Spectroscopy, R. G. Brewer, A. Mooradian, eds. (Plenum, New York, 1974), p. 555.

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Figures (1)

Fig. 1
Fig. 1

A possible nν3 multiple-photon ladder (schematic, not to scale) for spherical-top molecules. Only sublevels with l = 0 or 1 are included here. The Coriolis splittings shown for n = 1 and 3 involve the implicit assumption that ζ3 > 0, as in SF6, for example.21 The transition dipole moments shown for Q-branch transitions come from Eqs. (7), (16), and (18) in the text.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

J = R + l
R = J l .
| nlm , R K R , J K > = | nlm > * | J K > ( lJmK | lJR K R ) ,
μ Z = λ Z x μ x + λ Z y μ y + λ Z z μ z ,
μ Z = A 3 z 3 + A 4 z 4 + A 3 x 3 y 3 + A 4 x 4 y 4 + ,
μ z A 3 z 3 , etc .
n l m , R K R , J K | μ Z | nlm , R K R , J K ,
μ l + 1 n + 1 2 = A 3 2 ( R + J + l + 2 ) ( R + J + l + 1 ) ( R J + l + 1 ) ( R + J l ) ( n + l + 3 ) ( 2 J + 1 ) 8 J ( J + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) ,
μ l 1 n + 1 2 = A 3 2 ( R + J + l + 1 ) ( R + J + l ) ( R J + l ) ( R + J l + 1 ) ( n l + 2 ) ( 2 J + 1 ) 2 J ( J + 1 ) ( 2 l 1 ) ( 2 l + 1 ) .
μ 1 1 2 = ½ A 3 2 ( 2 J + 1 ) .
μ 01 2 = ½ A 3 2 ,
μ 1 1 2 = μ 01 2 ( 2 J + 1 ) ,
μ 2 2 2 = A 3 2 ( R + J + 3 ) ( R + J + 2 ) ( R J + 2 ) ( R + J 1 ) ( 2 J + 1 ) 24 J ( J + 1 ) ,
μ 0 2 2 = A 3 2 ( R + J + 2 ) ( R + J + 1 ) ( R J + 1 ) ( R + J ) ( 2 J + 1 ) 3 J ( J + 1 ) ,
μ 0 2 2 = 4 / 3 A 3 2 ( 2 J + 1 ) ,
μ 0 2 2 = 0 .
μ 1 3 2 = A 3 2 ( 2 J + 1 ) .
μ 1 n + 1 2 = A 3 2 ( R + J + 2 ) ( R + J + 1 ) ( R J + 1 ) ( R + J ) ( n + 3 ) ( 2 J + 1 ) 24 J ( J + 1 ) .
μ 1 n + 1 2 = [ ( n + 3 ) / 3 ] μ 01 2 ( 2 J + 1 ) ; n = 0 , 2 , 4 , .
μ 1 n + 1 2 = A 3 2 ( R + J + 2 ) ( R + J + 1 ) ( R J + 1 ) ( R + J ) ( n + 1 ) ( 2 J + 1 ) 6 J ( J + 1 ) .
μ 0 n + 1 2 = 4 / 3 ( n + 1 ) μ 01 2 ( 2 J + 1 ) ; n = 1 , 3 , 5 , .

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