Abstract

Quantum-mechanical expressions for nonresonance-dipole light-scattering cross sections are related to corresponding expressions for oscillator strengths and refractive indices. In particular, Rayleigh and Raman cross sections for atoms are expressed in terms of oscillator strengths and vector-coupling coefficients. Several of our results, including the relationship between Rayleigh scattering and refractive index, differ in general from the corresponding results of classical dispersion theory. We show that these differences arise from antisymmetric components Ckja = (CkjCjk)/2 of the polarizability tensor, which have been neglected in the classical analyses. Calculated Rayleigh cross sections for cesium and aluminum atoms, and Raman cross sections for aluminum atoms are presented to illustrate our results. The antisymmetric contribution is found to be substantial in all of these cross sections; its most obvious effect is to cause the depolarization (for linearly polarized incident light) to exceed 34 over extended wavelength ranges away from resonance. On the other hand, for atoms initially in states of zero angular momentum, and molecules under conditions which allow the use of Placzek’s polarizability approximation, our results agree with those of classical dispersion theory.

© 1969 Optical Society of America

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References

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  1. Rayleigh, Phil. Mag. 47, 375 (1899).
  2. M. Born, Optik (Julius Springer-Verlag, Berlin, 1933).
    [Crossref]
  3. A. Dalgarno, J. Opt. Soc. Am. 53, 1223 (1963); Douglas W. O. Heddle,  54, 264 (1964). Both Dalgarno and Heddle based their calculations on Rayleigh’s result for isotropic particles.
    [Crossref]
  4. D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963); P. Gill and D. W. O. Heddle,  53, 847 (1963).
    [Crossref]
  5. See for example S. Bhagavantam, Scattering of Light and the Raman Effect (Chemical Publishing Co., Inc., Brooklyn, N. Y., 1942); H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  6. G. Placzek, Handbuch der Radiologie, VI (Akademische Verlagsgesellschaft, Leipzig, 1934), Vol. 2, p. 209.
  7. P. A. M. Dirac, Quantum Mechanics (Oxford University Press, London, 1958).
  8. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, London, 1957).
  9. Note that ρ differs from the depolarization ρ′ of light scattered through 90° from an unpolarized incident beam. The latter quantity, which is employed frequently in the literature but is perhaps less convenient in an age of laser sources, is related to ρ by ρ′ = 2ρ/(1 + ρ).
  10. S. P. S. Porto, J. Opt. Soc. Am. 56, 1585 (1966).
    [Crossref]
  11. Here we propose to ignore any nuclear angular momentum. A treatment including the nuclear angular momentum reveals that the scattering integrated over any fine details that arise from hyperfine splitting of initial and final states does not depend significantly on nuclear spin at separations from resonance that are large compared to the hyperfine splitting of the (initial, intermediate, and final) states involved. The coupling of nuclear angular momentum to the scattering nearer resonance is discussed in Ref. 6, Ch. 12. See also A. Ellett, Phys. Rev. 35, 588 (1930).
    [Crossref]
  12. See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).
  13. Albert Messiah, Quantum Mechanics (North-Holland Publishing Co., Amsterdam, 1962) Vol. 2, Appendix C.
  14. P. M. Stone, Phys. Rev. 127, 1151 (1962).
    [Crossref]
  15. N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR) 14, 5 (1964).
  16. The symmetric scattering vanishes for transitions J=12→J′=12. This is one of the selection rules derived by Placzek in Ref. 6.
  17. Born presents a discussion which relates to this point in Ref. 2, Art. 73.

1966 (1)

1964 (1)

N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR) 14, 5 (1964).

1963 (2)

1962 (1)

P. M. Stone, Phys. Rev. 127, 1151 (1962).
[Crossref]

1930 (1)

Here we propose to ignore any nuclear angular momentum. A treatment including the nuclear angular momentum reveals that the scattering integrated over any fine details that arise from hyperfine splitting of initial and final states does not depend significantly on nuclear spin at separations from resonance that are large compared to the hyperfine splitting of the (initial, intermediate, and final) states involved. The coupling of nuclear angular momentum to the scattering nearer resonance is discussed in Ref. 6, Ch. 12. See also A. Ellett, Phys. Rev. 35, 588 (1930).
[Crossref]

1899 (1)

Rayleigh, Phil. Mag. 47, 375 (1899).

Bhagavantam, S.

See for example S. Bhagavantam, Scattering of Light and the Raman Effect (Chemical Publishing Co., Inc., Brooklyn, N. Y., 1942); H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Bivins, R.

See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).

Born, M.

M. Born, Optik (Julius Springer-Verlag, Berlin, 1933).
[Crossref]

Condon, E. U.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, London, 1957).

Dalgarno, A.

Dirac, P. A. M.

P. A. M. Dirac, Quantum Mechanics (Oxford University Press, London, 1958).

Ellett, A.

Here we propose to ignore any nuclear angular momentum. A treatment including the nuclear angular momentum reveals that the scattering integrated over any fine details that arise from hyperfine splitting of initial and final states does not depend significantly on nuclear spin at separations from resonance that are large compared to the hyperfine splitting of the (initial, intermediate, and final) states involved. The coupling of nuclear angular momentum to the scattering nearer resonance is discussed in Ref. 6, Ch. 12. See also A. Ellett, Phys. Rev. 35, 588 (1930).
[Crossref]

Heddle, D. W. O.

Jennings, R. E.

Messiah, Albert

Albert Messiah, Quantum Mechanics (North-Holland Publishing Co., Amsterdam, 1962) Vol. 2, Appendix C.

Metropolis, N.

See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).

Parsons, A. S. L.

Penkin, N. P.

N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR) 14, 5 (1964).

Placzek, G.

G. Placzek, Handbuch der Radiologie, VI (Akademische Verlagsgesellschaft, Leipzig, 1934), Vol. 2, p. 209.

Porto, S. P. S.

Rayleigh,

Rayleigh, Phil. Mag. 47, 375 (1899).

Rotenberg, M.

See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).

Shabanova, L. N.

N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR) 14, 5 (1964).

Shortley, G. H.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, London, 1957).

Stone, P. M.

P. M. Stone, Phys. Rev. 127, 1151 (1962).
[Crossref]

Woolen, J. K.

See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).

J. Opt. Soc. Am. (3)

Opt. Spectry. (USSR) (1)

N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR) 14, 5 (1964).

Phil. Mag. (1)

Rayleigh, Phil. Mag. 47, 375 (1899).

Phys. Rev. (2)

P. M. Stone, Phys. Rev. 127, 1151 (1962).
[Crossref]

Here we propose to ignore any nuclear angular momentum. A treatment including the nuclear angular momentum reveals that the scattering integrated over any fine details that arise from hyperfine splitting of initial and final states does not depend significantly on nuclear spin at separations from resonance that are large compared to the hyperfine splitting of the (initial, intermediate, and final) states involved. The coupling of nuclear angular momentum to the scattering nearer resonance is discussed in Ref. 6, Ch. 12. See also A. Ellett, Phys. Rev. 35, 588 (1930).
[Crossref]

Other (10)

See, for example, M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Woolen, The 3-J and 6-J Symbols (MIT Press, Cambridge, Mass., 1959).

Albert Messiah, Quantum Mechanics (North-Holland Publishing Co., Amsterdam, 1962) Vol. 2, Appendix C.

See for example S. Bhagavantam, Scattering of Light and the Raman Effect (Chemical Publishing Co., Inc., Brooklyn, N. Y., 1942); H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

G. Placzek, Handbuch der Radiologie, VI (Akademische Verlagsgesellschaft, Leipzig, 1934), Vol. 2, p. 209.

P. A. M. Dirac, Quantum Mechanics (Oxford University Press, London, 1958).

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, London, 1957).

Note that ρ differs from the depolarization ρ′ of light scattered through 90° from an unpolarized incident beam. The latter quantity, which is employed frequently in the literature but is perhaps less convenient in an age of laser sources, is related to ρ by ρ′ = 2ρ/(1 + ρ).

M. Born, Optik (Julius Springer-Verlag, Berlin, 1933).
[Crossref]

The symmetric scattering vanishes for transitions J=12→J′=12. This is one of the selection rules derived by Placzek in Ref. 6.

Born presents a discussion which relates to this point in Ref. 2, Art. 73.

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Figures (4)

Fig. 1
Fig. 1

Rayleigh-scattering cross sections for cesium atoms in ground states calculated from Eqs. (23) and (24) using semi-empirical oscillator strengths given by Stone,14 viz: f8944 = 0.394, f8521 = 0.814.

Fig. 2
Fig. 2

Depolarization of the Rayleigh scattering from cesium atoms in ground states, calculated from the cross sections in Fig. 1.

Fig. 3
Fig. 3

Rayleigh-scattering cross sections for aluminum atoms in 3p2P1/2 and 3p2P3/2 states, calculated from Eq. (23) and (24) using experimental oscillator strengths given by Penkin and Shabanova,15 viz: f3944 = 0.15, f3952 = 0.15.

Fig. 4
Fig. 4

Raman-light-scattering cross sections for aluminum atoms for the transitions 3p2P1/2 → 3p2P3/2 and 3p2P3/2 → 3p2P1/2, calculated from Eqs. (25) and (26) using the oscillator strengths mentioned in the caption to Fig. 3.

Equations (64)

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( σ 12 ) i f = ( ω - ω f i ) 4 c 4 ( C k j ) i f 1 j 2 k * 2 ,
( C k j ) i f = 1 g { f D k g g D j i ω g i - ω + f D j g g D k i ω g f + ω } .
ω f i = ( E f - E i ) / ,
( σ 12 ) T J T J = 1 2 J + 1 M M ( σ 12 ) T J M T J M .
1 j = δ j z
2 k = δ k z cos ψ + δ k x sin ψ ,
δ k l = 1 , k = l , = 0 , otherwise ,
T J M D z T J M = 0 , M M T J M D x T J M = 0 , M M ± 1 ,
( σ 12 ) T J T J = ( σ z z ) T J T J cos 2 ψ + ( σ z x ) T J T J sin 2 ψ ,
( σ j k ) T J T J = ( ω - ω T J , T J ) 4 c 4 ( 2 J + 1 ) × M M ( C k j ) T J M T J M 2 .
σ 12 = σ z z cos 2 ψ + σ z x sin 2 ψ ,
ρ = σ z x / σ z z .
σ 12 = σ z z [ ( 1 - ρ ) cos 2 ψ + ρ ] .
I 2 = I 1 ( σ 12 / R 2 )
( σ z z ) T J T J = 4 ω 4 2 c 4 ( 2 J + 1 ) M { T J ω T J , T J ω 2 T J , T J - ω 2 × T J M D z T J M 2 } 2 ,
( σ z x ) T J T J = ω 4 2 c 4 ( 2 J + 1 ) × M M | T J { T J M D x T J M T J M D z T J M ω T J , T J - ω + T J M D z T J M T J M D x T J M ω T J , T J + ω } | 2 .
f T J , T J = 2 m e 2 ω T J , T J ( 2 J + 1 ) × M T J M D z T J M 2 .
( σ 12 ) T 0 T 0 = ω 4 ( e 2 M c 2 ) 2 × { T J f T 0 , T J ω 2 T J , T 0 - ω } 2 cos 2 ψ .
n 2 - 1 4 π N = e 2 m T J f T 0 , T J ω 2 T J , T 0 - ω 2 ,
( σ 12 ) T 0 T 0 = ω 4 c 4 ( n 2 - 1 4 π N ) 2 cos 2 ψ .
T J M D z T J M = ( - 1 ) J - M T J D T J ( J 1 M - M 0 M ) ,
T J M D x T J M = ( - 1 ) J - M + 1 2 / 2 T J D T J × [ ( J 1 J - M 1 M ) - ( J 1 J - M - 1 M ) ] .
( J 1 J - M 0 M )
( σ z z ) T J T J = 4 ω 2 2 c 4 ( 2 J + 1 ) M { T J ω T J , T J ω 2 T J , T J - ω 2 T J D T J 2 ( J 1 J - M 0 M ) 2 } 2 ,
( σ z x ) T J T J = ω 4 2 c 4 ( 2 J + 1 ) M { T J T J D T J 2 × ( ( J 1 J - M 1 M - 1 ) ( J 1 J - M 0 M ) ω T J , T J - ω + ( J 1 J - M + 1 0 M - 1 ) ( J 1 J - M + 1 - 1 M ) ω T J , T J + ω ) } 2 ,
f T J , T J = 2 m ω T J , T J 3 e 2 ( 2 J + 1 ) T J D T J 2 .
( σ z z ) T J T J = 9 ( 2 J + 1 ) ω 4 ( e 2 m c 2 ) 2 M { f T J , T J ω 2 T J , T J - ω 2 ( J 1 J - M 0 M ) 2 } 2
( σ z x ) T J T J = 9 4 ( 2 J + 1 ) ω 4 ( e 2 m c 2 ) 2 M { T J f T J , T J ω T J , T J × ( ( J 1 J - M 1 M - 1 ) ( J 1 J - M 0 M ) ω T J , T J - ω + ( J 1 J - M + 1 0 M - 1 ) ( J 1 J - M + 1 - 1 M ) ω T J , T J + ω ) } 2 .
( σ z z ) T J T J = 9 4 ( 2 J + 1 ) ( e 2 m c 2 ) 2 × ( ω - ω T J , T J ) 4 ω T J , T J ω T J , T J ( ω T J , T J - ω ) 2 × f T J , T J f T J , T J × M ( J 1 J - M 0 M ) 2 ( J 1 J - M 0 M ) 2 ,
( σ z x ) T J T J = 9 4 ( 2 J + 1 ) ( e 2 m c 2 ) 2 × ( ω - ω T J , T J ) 4 ω T J , T J ω T J , T J ( ω T J , T J - ω ) 2 × f T J , T J f T J , T J × M ( J 1 J - M 1 M - 1 ) 2 ( J 1 J - M 0 M ) 2 .
n 2 - 1 4 π N = 2 i g P i ω g i ω g i 2 - ω 2 g D z i 2 ,
n 2 - 1 4 π N = 2 3 1 2 J + 1 T J ω T J , T J ω 2 T J , T J - ω 2 × T J D T J 2 ,
( C 0 ) T J M T J M = 1 3 j = x , y , z ( C j j ) T J M T J M ,
( C k j s ) T J M T J M = 1 2 [ ( C k j ) T J M T J M + ( C j k ) T J M T J M ] - ( C 0 ) T J M T J M δ k j ,
( C k j a ) T J M T J M = 1 2 [ ( C k j ) T J M T J M - ( C j k ) T J M T J M ] .
( C k j ) T J M T J M = ( C 0 ) T J M T J M δ k j + ( C k j s ) T J M T J M + ( C k j a ) T J M T J M .
( σ j k ) T J T J = ( σ j k 0 ) T J T J + ( σ j k s ) T J T J + ( σ j k a ) T J T J
( σ j k s ) T J T J = ( ω - ω T J , T J ) 4 C 4 ( 2 J + 1 ) × M M ( C k j s ) T J M T J M 2 .
( σ z z ) T J T J = ( σ z z 0 ) T J T J + ( σ z z s ) T J T J ,
( σ z x ) T J T J = ( σ z x s ) T J T J + ( σ z x a ) T J T J .
( σ 12 ) T J T J = [ ( σ z z 0 ) T J T J + ( σ z z s ) T J T J ] × [ ( 1 - ρ T J T J ) cos 2 ψ + ρ T J T J ] ,
ρ T J T J = ( σ z x s ) T J T J + ( σ z x a ) T J T J ( σ z z 0 ) T J T J + ( σ z z s ) T J T J .
( σ z x s ) T J T J = 3 4 ( σ z z s ) T J T J .
( σ 12 ) T J T J = [ ( σ z z 0 ) T J T J - 4 3 ( σ z x a ) T J T J ] × ( 3 / 3 - 4 ρ T J T J ) × [ ( 1 - ρ T J T J ) cos 2 ψ + ρ T J T J ] .
( σ z z 0 ) T J T J = ( ω / c ) 4 [ ( n 2 - 1 ) / 4 π N ] 2 .
( σ 12 ) T J T J = [ ( ω c ) 4 ( n 2 - 1 4 π N ) 2 - 4 3 ( σ z x a ) T J T J ] × 3 / ( 3 - 4 ρ T J T J ) [ ( 1 - ρ T J T J ) cos 2 ψ + ρ T J T J ] .
σ 12 = ( ω c ) 4 ( n 2 - 1 4 π N ) 2 ( 3 3 - 4 ρ ) [ ( 1 - ρ ) cos 2 ψ + ρ ] .
( σ 12 ) T J T J = ( ω c ) 4 ( n 2 - 1 4 π N ) 2 × [ ( 1 - ρ T J T J ) cos 2 ψ + ρ T J T J ] ,
( σ z z 0 ) T J T J = ( ω - ω T J , T J ) 4 9 2 c 4 ( 2 J + 1 ) 2 δ J J | T J ( Q 1 + R 1 ) | 2 ,
( σ z z s ) T J T J = ( ω - ω T J , T J ) 4 2 c 4 ( 2 J + 1 ) M = - J J | T 1 J 1 ( Q 1 + R 1 ) [ ( J 1 1 J - M 0 M ) ( J 1 1 J - M 0 M ) - δ J J 3 ( 2 J + 1 ) ] | 2 ,
( σ z x s ) T J T J = ( ω - ω T J , T J ) 4 4 2 c 4 ( 2 J + 1 ) M | T 1 J 1 ( Q 1 + R 1 ) × [ ( J 1 1 J - M 1 M - 1 ) ( J 1 1 J - M 0 M ) - ( J 1 1 J - M + 1 0 M - 1 ) ( J 1 1 J - M + 1 - 1 M ) ] | 2 ,
( σ z x a ) T J T J = ( ω - ω T J , T J ) 4 4 2 c 4 ( 2 J + 1 ) M | T 1 J 1 ( Q 1 - R 1 ) × [ ( J 1 1 J - M 1 M - 1 ) ( J 1 1 J - M 0 M ) + ( J 1 1 J - M + 1 0 M - 1 ) ( J 1 1 J - M + 1 - 1 M ) ] | 2 ,
Q 1 = T 1 J 1 D T J * T 1 J 1 D T J ω T 1 J 1 , T J - ω ,
R 1 = T 1 J 1 D T J * T 1 J 1 D T J ω T 1 J 1 , T J + ω .
{ j 1 j 2 j 3 l 1 l 2 l 3 }
m 3 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ( l 1 l 2 j 3 n 1 n 2 - m 3 ) = S M ( - 1 ) j 3 + S + m 1 + n 1 ( 2 S + 1 ) { j 1 j 2 j 3 l 1 l 2 S } × ( l 1 j 2 S n 1 m 2 M ) ( j 1 l 2 S m 1 n 2 - M ) ,
( σ z z s ) T J T J = ( ω - ω T J , T J ) 4 2 c 4 ( 2 J + 1 ) ( - 1 ) J - J × T 1 J 1 T 2 J 2 ( Q 1 + R 1 ) ( Q 2 * + R 2 * ) S = 0 2 ( 2 S + 1 ) × { J 1 1 J 1 J 2 S } { J 1 1 J 1 J 2 S } [ ( 1 1 S 0 0 0 ) 2 - 1 9 ] ,
( σ z x s ) T J T J = ( ω - ω T J , T J ) 4 2 2 c 4 ( 2 J + 1 ) ( - 1 ) J - J + 1 × T 1 J 1 T 2 J 2 ( Q 1 + R 1 ) ( Q 2 * + R 2 * ) S = 0 2 ( 2 S + 1 ) × { J 1 1 J 1 J s S } { J 1 1 J 1 J 2 S } [ ( 1 1 S - 1 1 0 ) ( 1 1 S 0 0 0 ) - ( 1 1 S 0 1 - 1 ) ( 1 1 S 1 0 - 1 ) ] .
( 1 1 S - 1 1 0 ) ( 1 1 S 0 0 0 ) - ( 1 1 S 1 0 - 1 ) ( 1 1 S 0 1 - 1 ) ] = - 3 2 [ ( 1 1 S 0 0 0 ) 2 - 1 9 ]
( σ z x s ) T J T J = 3 4 ( σ z z s ) T J T J .
σ 12 = [ σ z z 0 - 4 3 σ z x a ] [ 3 / ( 3 - 4 ρ ) ] × [ ( 1 - ρ ) cos 2 ψ + ρ ]
1 2 J + 1 T J ω T J , T J ω 2 T J , T J - ω T J D T J 2
σ z z 0 = ( ω / c ) 4 [ ( n 2 - 1 ) / 4 π N ] 2 .
σ 12 = ( ω c ) 4 ( n 2 - 1 4 π N ) ( 3 3 - 4 ρ ) [ ( 1 - ρ ) cos 2 ψ + ρ ] ,