Abstract

A unified treatment of the Rayleigh, Einstein-Smoluchowski, and Cabannes-Kingformulas.—Schuster has shown that the well-known Rayleigh formula for molecular scattering of light can be derived from general wave theory without special postulates (customarily introduced) which are peculiar to the elastic-solid, the electromagnetic, or the electron theory. The present report—the content of which is theoretical and mathematical—discusses the restrictions of the Rayleigh formula; and extends Schuster’s demonstration to the two generalizations which have been developed, respectively, by Smoluchowski and Einstein, and by Cabannes and King. The former applies to dense media, and the latter to scattering by anisotropic molecules.

Extensions of the Rayleigh formula.—The same method leads to two new generalizations. The first theoretically applies to resonance scattering, and, in a special case, is equivalent to a formula derived by Lamb. The second applies to scattering by a gas which contains active molecules of more than one type.

The relation of scattering to opacity is briefly discussed. A partial bibliography is included, of Rayleigh scattering and of opacity in general. The standpoint throughout is that of classical theory; the quantum theory of scattering is not dealt with in this report.

© 1925 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. L. Nichols, Physical Review,  26, p. 497; 1908.
  2. E. Brücke, Poggendorf’s Annalen,  88, p. 363; 1853.
    [Crossref]
  3. J. Tyndall, Proc. Roy. Soc.,  17, p. 223; 1868.Phil. Mag.,  37, (4), p. 384; 1869.
    [Crossref]
  4. Rayleigh, Phil. Mag.,  41, (4), pp. 107; 274, and 447; 1871;Phil. Mag. 12, (5), p. 81; 1881;Phil. Mag. 44, (5), p. 28; 1897;Phil. Mag. 47, (5), p. 375, 1899;Phil. Mag. 35, (6), p. 373; 1918. Reprinted in Scientific Papers.
    [Crossref]
  5. J. Cabannes, Comptes Rendus,  160, p. 62; 1915;Comptes Rendus 171, p. 852; 1920.Annales de Physique,  15, p. 5; 1921; (an inclusive treatise).
  6. R. J. Strutt, Roy. Soc. Proc.,  94 A, p. 453; 1918;Roy. Soc. Proc. 95 A, pp. 155 and 476; 1919;Roy. Soc. Proc. 97 A, p. 435; 1920.
    [Crossref]
  7. J. Kunz, Phil. Mag.,  39, p. 416; 1920.
    [Crossref]
  8. L. Natanson, Phil. Mag.,  38, p. 269; 1919;A. Somerfeld, “Atomic Structure and Spectral Lines,” p. 533, (note 2), London, 1923.
    [Crossref]
  9. A. Schuster, “The Theory of Optics,” 2nd edition, p. 325, 1909;“The Theory of Optics,” 3rd edition, p. 320, 1924.
  10. M. v. Smoluchowski, Annalen der Physik,  25, p. 205; 1908.
    [Crossref]
  11. A. Einstein, Annalen der Physik,  33, p. 1275; 1910.
    [Crossref]
  12. L. V. King, Proc. Roy. Soc.,  104 A, p. 333; 1923.Phil. Trans.,  212 A, p. 375; 1922.
    [Crossref]
  13. H. Lamb, Cambridge Phil. Soc. Trans.,  18, p. 348; 1899. (Stokes Commemoration.)
  14. H. A. Lorentz, Proc. Amsterdam Academy,  8, p. 591, 1905;Proc. Amsterdam Academy 18, p. 34; 1914.L. Page, Astrophysical Journal,  52, p. 65; 1920.J. Q. Stewart, Astrophysical Journal,  59, p, 30, 1924.
    [Crossref]
  15. F. E. Fowle, Astroph. Journ.,  38, p. 392; 1913;Astroph. Journ. 40, p. 435; 1914.H. Dember, Ann. der Physik,  49, p. 549; 1916.
    [Crossref]
  16. C. V. Raman and K. R. Ramanathan, Phil. Mag.,  45, pp. 113, and 213; 1923.K. R. Ramanathan, Proc. Roy. Soc., London,  102 A, p. 151; 1922.
    [Crossref]
  17. E. A. Milne, Monthly Notices of R. A. S.,  85, p. 111; 1924.J. C. Slater, Amer. Phys. Soc., Abstracts of 130th meeting, no. 55, 1924;Physical Review; 1925.
  18. R. C. Tolman, Phys. Rev.,  23, p. 693; 1924.
    [Crossref]
  19. R. W. Wood, Phil Mag.,  23, p.689; 1912; etc.
    [Crossref]
  20. C. Füchtbauer, P. Z.,  12, p. 722; 1911;P. Z.,  21, pp. 322 and 635, 1920; Füchtbauer and Hofmann, P. Z.,  14, p. 1169; 1913;Ann. der Physik,  43, p. 96; 1924; Füchtbauer and Schell, P. Z.,  14, p. 1164; 1913;H. Bartels, Ann. der Physik,  65, p. 143; 1921; Füchtbauer, Joos, and Dinkelacker, Ann. der Physik,  71, p. 204; 1913.
    [Crossref]
  21. A. Schuster, Astrophysical Journal,  21, p. 1; 1905.
    [Crossref]
  22. M. Planck, “The Theory of Heat Radiation.”
  23. J. Q. Stewart, Monthly Notices of R.A.S., June, 1925.
  24. J. Q. Stewart, Phys. Rev.,  22, p. 324; 1923.
    [Crossref]
  25. A. S. Eddington, Monthly Notices of R.A.S.,  83, pp. 32 and 98, 1922;Monthly Notices of R.A.S. 83, p. 431, 1923.
  26. A. H. Compton, Phil. Mag.,  46, p. 807; 1923, etc.
    [Crossref]

1925 (1)

J. Q. Stewart, Monthly Notices of R.A.S., June, 1925.

1924 (2)

E. A. Milne, Monthly Notices of R. A. S.,  85, p. 111; 1924.J. C. Slater, Amer. Phys. Soc., Abstracts of 130th meeting, no. 55, 1924;Physical Review; 1925.

R. C. Tolman, Phys. Rev.,  23, p. 693; 1924.
[Crossref]

1923 (4)

L. V. King, Proc. Roy. Soc.,  104 A, p. 333; 1923.Phil. Trans.,  212 A, p. 375; 1922.
[Crossref]

J. Q. Stewart, Phys. Rev.,  22, p. 324; 1923.
[Crossref]

C. V. Raman and K. R. Ramanathan, Phil. Mag.,  45, pp. 113, and 213; 1923.K. R. Ramanathan, Proc. Roy. Soc., London,  102 A, p. 151; 1922.
[Crossref]

A. H. Compton, Phil. Mag.,  46, p. 807; 1923, etc.
[Crossref]

1922 (1)

A. S. Eddington, Monthly Notices of R.A.S.,  83, pp. 32 and 98, 1922;Monthly Notices of R.A.S. 83, p. 431, 1923.

1920 (1)

J. Kunz, Phil. Mag.,  39, p. 416; 1920.
[Crossref]

1919 (1)

L. Natanson, Phil. Mag.,  38, p. 269; 1919;A. Somerfeld, “Atomic Structure and Spectral Lines,” p. 533, (note 2), London, 1923.
[Crossref]

1918 (1)

R. J. Strutt, Roy. Soc. Proc.,  94 A, p. 453; 1918;Roy. Soc. Proc. 95 A, pp. 155 and 476; 1919;Roy. Soc. Proc. 97 A, p. 435; 1920.
[Crossref]

1915 (1)

J. Cabannes, Comptes Rendus,  160, p. 62; 1915;Comptes Rendus 171, p. 852; 1920.Annales de Physique,  15, p. 5; 1921; (an inclusive treatise).

1913 (1)

F. E. Fowle, Astroph. Journ.,  38, p. 392; 1913;Astroph. Journ. 40, p. 435; 1914.H. Dember, Ann. der Physik,  49, p. 549; 1916.
[Crossref]

1912 (1)

R. W. Wood, Phil Mag.,  23, p.689; 1912; etc.
[Crossref]

1911 (1)

C. Füchtbauer, P. Z.,  12, p. 722; 1911;P. Z.,  21, pp. 322 and 635, 1920; Füchtbauer and Hofmann, P. Z.,  14, p. 1169; 1913;Ann. der Physik,  43, p. 96; 1924; Füchtbauer and Schell, P. Z.,  14, p. 1164; 1913;H. Bartels, Ann. der Physik,  65, p. 143; 1921; Füchtbauer, Joos, and Dinkelacker, Ann. der Physik,  71, p. 204; 1913.
[Crossref]

1910 (1)

A. Einstein, Annalen der Physik,  33, p. 1275; 1910.
[Crossref]

1908 (2)

M. v. Smoluchowski, Annalen der Physik,  25, p. 205; 1908.
[Crossref]

E. L. Nichols, Physical Review,  26, p. 497; 1908.

1905 (2)

H. A. Lorentz, Proc. Amsterdam Academy,  8, p. 591, 1905;Proc. Amsterdam Academy 18, p. 34; 1914.L. Page, Astrophysical Journal,  52, p. 65; 1920.J. Q. Stewart, Astrophysical Journal,  59, p, 30, 1924.
[Crossref]

A. Schuster, Astrophysical Journal,  21, p. 1; 1905.
[Crossref]

1899 (1)

H. Lamb, Cambridge Phil. Soc. Trans.,  18, p. 348; 1899. (Stokes Commemoration.)

1871 (1)

Rayleigh, Phil. Mag.,  41, (4), pp. 107; 274, and 447; 1871;Phil. Mag. 12, (5), p. 81; 1881;Phil. Mag. 44, (5), p. 28; 1897;Phil. Mag. 47, (5), p. 375, 1899;Phil. Mag. 35, (6), p. 373; 1918. Reprinted in Scientific Papers.
[Crossref]

1868 (1)

J. Tyndall, Proc. Roy. Soc.,  17, p. 223; 1868.Phil. Mag.,  37, (4), p. 384; 1869.
[Crossref]

1853 (1)

E. Brücke, Poggendorf’s Annalen,  88, p. 363; 1853.
[Crossref]

Brücke, E.

E. Brücke, Poggendorf’s Annalen,  88, p. 363; 1853.
[Crossref]

Cabannes, J.

J. Cabannes, Comptes Rendus,  160, p. 62; 1915;Comptes Rendus 171, p. 852; 1920.Annales de Physique,  15, p. 5; 1921; (an inclusive treatise).

Compton, A. H.

A. H. Compton, Phil. Mag.,  46, p. 807; 1923, etc.
[Crossref]

Eddington, A. S.

A. S. Eddington, Monthly Notices of R.A.S.,  83, pp. 32 and 98, 1922;Monthly Notices of R.A.S. 83, p. 431, 1923.

Einstein, A.

A. Einstein, Annalen der Physik,  33, p. 1275; 1910.
[Crossref]

Fowle, F. E.

F. E. Fowle, Astroph. Journ.,  38, p. 392; 1913;Astroph. Journ. 40, p. 435; 1914.H. Dember, Ann. der Physik,  49, p. 549; 1916.
[Crossref]

Füchtbauer, C.

C. Füchtbauer, P. Z.,  12, p. 722; 1911;P. Z.,  21, pp. 322 and 635, 1920; Füchtbauer and Hofmann, P. Z.,  14, p. 1169; 1913;Ann. der Physik,  43, p. 96; 1924; Füchtbauer and Schell, P. Z.,  14, p. 1164; 1913;H. Bartels, Ann. der Physik,  65, p. 143; 1921; Füchtbauer, Joos, and Dinkelacker, Ann. der Physik,  71, p. 204; 1913.
[Crossref]

King, L. V.

L. V. King, Proc. Roy. Soc.,  104 A, p. 333; 1923.Phil. Trans.,  212 A, p. 375; 1922.
[Crossref]

Kunz, J.

J. Kunz, Phil. Mag.,  39, p. 416; 1920.
[Crossref]

Lamb, H.

H. Lamb, Cambridge Phil. Soc. Trans.,  18, p. 348; 1899. (Stokes Commemoration.)

Lorentz, H. A.

H. A. Lorentz, Proc. Amsterdam Academy,  8, p. 591, 1905;Proc. Amsterdam Academy 18, p. 34; 1914.L. Page, Astrophysical Journal,  52, p. 65; 1920.J. Q. Stewart, Astrophysical Journal,  59, p, 30, 1924.
[Crossref]

Milne, E. A.

E. A. Milne, Monthly Notices of R. A. S.,  85, p. 111; 1924.J. C. Slater, Amer. Phys. Soc., Abstracts of 130th meeting, no. 55, 1924;Physical Review; 1925.

Natanson, L.

L. Natanson, Phil. Mag.,  38, p. 269; 1919;A. Somerfeld, “Atomic Structure and Spectral Lines,” p. 533, (note 2), London, 1923.
[Crossref]

Nichols, E. L.

E. L. Nichols, Physical Review,  26, p. 497; 1908.

Planck, M.

M. Planck, “The Theory of Heat Radiation.”

Raman, C. V.

C. V. Raman and K. R. Ramanathan, Phil. Mag.,  45, pp. 113, and 213; 1923.K. R. Ramanathan, Proc. Roy. Soc., London,  102 A, p. 151; 1922.
[Crossref]

Ramanathan, K. R.

C. V. Raman and K. R. Ramanathan, Phil. Mag.,  45, pp. 113, and 213; 1923.K. R. Ramanathan, Proc. Roy. Soc., London,  102 A, p. 151; 1922.
[Crossref]

Rayleigh,

Rayleigh, Phil. Mag.,  41, (4), pp. 107; 274, and 447; 1871;Phil. Mag. 12, (5), p. 81; 1881;Phil. Mag. 44, (5), p. 28; 1897;Phil. Mag. 47, (5), p. 375, 1899;Phil. Mag. 35, (6), p. 373; 1918. Reprinted in Scientific Papers.
[Crossref]

Schuster, A.

A. Schuster, Astrophysical Journal,  21, p. 1; 1905.
[Crossref]

A. Schuster, “The Theory of Optics,” 2nd edition, p. 325, 1909;“The Theory of Optics,” 3rd edition, p. 320, 1924.

Smoluchowski, M. v.

M. v. Smoluchowski, Annalen der Physik,  25, p. 205; 1908.
[Crossref]

Stewart, J. Q.

J. Q. Stewart, Monthly Notices of R.A.S., June, 1925.

J. Q. Stewart, Phys. Rev.,  22, p. 324; 1923.
[Crossref]

Strutt, R. J.

R. J. Strutt, Roy. Soc. Proc.,  94 A, p. 453; 1918;Roy. Soc. Proc. 95 A, pp. 155 and 476; 1919;Roy. Soc. Proc. 97 A, p. 435; 1920.
[Crossref]

Tolman, R. C.

R. C. Tolman, Phys. Rev.,  23, p. 693; 1924.
[Crossref]

Tyndall, J.

J. Tyndall, Proc. Roy. Soc.,  17, p. 223; 1868.Phil. Mag.,  37, (4), p. 384; 1869.
[Crossref]

Wood, R. W.

R. W. Wood, Phil Mag.,  23, p.689; 1912; etc.
[Crossref]

Annalen der Physik (2)

M. v. Smoluchowski, Annalen der Physik,  25, p. 205; 1908.
[Crossref]

A. Einstein, Annalen der Physik,  33, p. 1275; 1910.
[Crossref]

Astroph. Journ. (1)

F. E. Fowle, Astroph. Journ.,  38, p. 392; 1913;Astroph. Journ. 40, p. 435; 1914.H. Dember, Ann. der Physik,  49, p. 549; 1916.
[Crossref]

Astrophysical Journal (1)

A. Schuster, Astrophysical Journal,  21, p. 1; 1905.
[Crossref]

Cambridge Phil. Soc. Trans. (1)

H. Lamb, Cambridge Phil. Soc. Trans.,  18, p. 348; 1899. (Stokes Commemoration.)

Comptes Rendus (1)

J. Cabannes, Comptes Rendus,  160, p. 62; 1915;Comptes Rendus 171, p. 852; 1920.Annales de Physique,  15, p. 5; 1921; (an inclusive treatise).

Monthly Notices of R. A. S. (1)

E. A. Milne, Monthly Notices of R. A. S.,  85, p. 111; 1924.J. C. Slater, Amer. Phys. Soc., Abstracts of 130th meeting, no. 55, 1924;Physical Review; 1925.

Monthly Notices of R.A.S. (2)

J. Q. Stewart, Monthly Notices of R.A.S., June, 1925.

A. S. Eddington, Monthly Notices of R.A.S.,  83, pp. 32 and 98, 1922;Monthly Notices of R.A.S. 83, p. 431, 1923.

P. Z. (1)

C. Füchtbauer, P. Z.,  12, p. 722; 1911;P. Z.,  21, pp. 322 and 635, 1920; Füchtbauer and Hofmann, P. Z.,  14, p. 1169; 1913;Ann. der Physik,  43, p. 96; 1924; Füchtbauer and Schell, P. Z.,  14, p. 1164; 1913;H. Bartels, Ann. der Physik,  65, p. 143; 1921; Füchtbauer, Joos, and Dinkelacker, Ann. der Physik,  71, p. 204; 1913.
[Crossref]

Phil Mag. (1)

R. W. Wood, Phil Mag.,  23, p.689; 1912; etc.
[Crossref]

Phil. Mag. (5)

Rayleigh, Phil. Mag.,  41, (4), pp. 107; 274, and 447; 1871;Phil. Mag. 12, (5), p. 81; 1881;Phil. Mag. 44, (5), p. 28; 1897;Phil. Mag. 47, (5), p. 375, 1899;Phil. Mag. 35, (6), p. 373; 1918. Reprinted in Scientific Papers.
[Crossref]

C. V. Raman and K. R. Ramanathan, Phil. Mag.,  45, pp. 113, and 213; 1923.K. R. Ramanathan, Proc. Roy. Soc., London,  102 A, p. 151; 1922.
[Crossref]

J. Kunz, Phil. Mag.,  39, p. 416; 1920.
[Crossref]

L. Natanson, Phil. Mag.,  38, p. 269; 1919;A. Somerfeld, “Atomic Structure and Spectral Lines,” p. 533, (note 2), London, 1923.
[Crossref]

A. H. Compton, Phil. Mag.,  46, p. 807; 1923, etc.
[Crossref]

Phys. Rev. (2)

J. Q. Stewart, Phys. Rev.,  22, p. 324; 1923.
[Crossref]

R. C. Tolman, Phys. Rev.,  23, p. 693; 1924.
[Crossref]

Physical Review (1)

E. L. Nichols, Physical Review,  26, p. 497; 1908.

Poggendorf’s Annalen (1)

E. Brücke, Poggendorf’s Annalen,  88, p. 363; 1853.
[Crossref]

Proc. Amsterdam Academy (1)

H. A. Lorentz, Proc. Amsterdam Academy,  8, p. 591, 1905;Proc. Amsterdam Academy 18, p. 34; 1914.L. Page, Astrophysical Journal,  52, p. 65; 1920.J. Q. Stewart, Astrophysical Journal,  59, p, 30, 1924.
[Crossref]

Proc. Roy. Soc. (2)

J. Tyndall, Proc. Roy. Soc.,  17, p. 223; 1868.Phil. Mag.,  37, (4), p. 384; 1869.
[Crossref]

L. V. King, Proc. Roy. Soc.,  104 A, p. 333; 1923.Phil. Trans.,  212 A, p. 375; 1922.
[Crossref]

Roy. Soc. Proc. (1)

R. J. Strutt, Roy. Soc. Proc.,  94 A, p. 453; 1918;Roy. Soc. Proc. 95 A, pp. 155 and 476; 1919;Roy. Soc. Proc. 97 A, p. 435; 1920.
[Crossref]

Other (2)

A. Schuster, “The Theory of Optics,” 2nd edition, p. 325, 1909;“The Theory of Optics,” 3rd edition, p. 320, 1924.

M. Planck, “The Theory of Heat Radiation.”

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (57)

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

K = 32 π 3 ( μ 1 ) 2 3 N λ 4 ,
I = I 0 K x ,
Z = Z 0 cos 2 π λ ( c t x )
z = Z 0 r sin θ [ A cos 2 π λ ( c t x 1 r ) + B sin 2 π λ ( c t x 1 r ) ] ;
z = Z 0 r sin θ ( A 2 + B 2 ) cos [ 2 π λ ( c t x r ) + ϕ π 2 ] ,
cot ϕ = B / A .
S = I ( A 2 + B 2 ) sin 2 θ r 2 ,
z = Z 0 [ A sin 2 π λ ( c t p ) B cos 2 π λ ( c t p ) ] λ N d x .
Z = Z 0 cos 2 π λ ( c t p ) ;
( 1 k ) Z 0 cos 2 π λ ( c t p δ ) .
δ = ( μ 1 ) d x ;
k = N B λ d x K = 2 N B λ ;
δ = N A λ 2 d x / 2 π , A = 2 π ( μ 1 ) / N λ 2 .
S = 4 π 2 ( μ 1 ) 2 N λ 4 sin 2 θ r 2 I .
S = 32 π 3 ( μ 1 ) 2 3 N λ 4 I .
S = 4 π 2 ( μ 1 ) 2 N λ 4 ( 1 + cos 2 Φ ) 2 r 2 I .
K = 32 π 3 3 n υ 2 γ 2 λ 4 / μ 4 ,
δ A = p δ υ + 1 2 d p d υ ( δ υ ) 2 .
1 2 d p d υ ( δ υ ) 2 ¯ = 1 2 R T ,
μ 2 1 μ 2 + 2 = b / V
γ = ( μ 2 1 ) ( μ 2 + 2 ) 6 μ 2 δ υ υ .
K = 32 π 3 3 υ μ 4 γ 2 ¯ λ 4 ,
K = 8 π 3 27 β R T ( μ 2 1 ) 2 ( μ 2 + 2 ) 2 λ 4 ,
K = R T 6 π ( 2 π λ ) 4 V ( μ 2 V ) 2 2 ψ V 2 .
z = Z 0 a r sin χ cos 2 π λ ( c t r ) ;
S = 8 π 3 N a 2 ¯ I ,
a 1 = R i n i l i , a 2 = R i n i m i , a 3 = R ι n i 2 ,
a 1 ¯ = a 2 ¯ = 0 ,
a 2 ¯ = a 1 2 ¯ + a 2 2 ¯ + a 3 2 ¯ ,
a 1 2 ¯ = a 2 2 ¯ ,
3 ( a 3 ¯ ) 2 = 3 a 3 2 ¯ 4 a 1 2 ¯ .
a 3 ¯ = 2 π ( μ 1 ) / N λ 2 .
ρ ( π / 2 ) = ρ = 2 a 1 2 ¯ / ( a 1 2 ¯ + a 3 2 ¯ ) .
K = 32 π 3 3 ( μ 1 ) 2 N λ 4 6 + 3 ρ 6 7 ρ ;
ρ ( Φ ) = ρ + ( 1 ρ ) cos 2 Φ .
K = 8 π 3 N ( A 2 + B 2 ) .
B / A = cot ϕ = K λ / 4 π ( μ 1 ) .
K = 32 π 3 3 ( μ 1 ) 2 N λ 4 csc 2 ϕ ;
K λ / 4 π = α ± { α 2 ( μ 1 ) 2 } 1 2 ,
α = 3 16 π 2 N λ 3 .
K m = 3 2 π N λ 2 ,
K = 3 2 π N λ 2 cos 2 ϕ ;
μ 1 = 3 8 π 2 N λ 3 sin ϕ cos ϕ .
tan ϕ = λ 0 π a 2 β sinh β ;
λ = λ 0 β ,
K = 2 N λ ω l B i , ω i A i = 2 π ( μ 1 ) / N λ 2 ;
K I = 8 π 3 N I ω i ( A i 2 + B i 2 ) .
ω i = 1 .
K = K i ; μ = μ i .
cot ϕ i = B i / A i ; ζ = ω i cos 2 ϕ i = ω cos 2 ϕ ; η = ω i sin ϕ i cos ϕ i = ω sin ϕ cos ϕ ;
cot ϕ = ζ / η ;
ω = ( η 2 + ζ 2 ) / ζ .
K = 32 π 3 3 ( μ 1 ) 2 N ω λ 4 cos 2 ϕ .
cos ϕ i = 2 π a λ i 2 / λ 3 ;
ω = [ ω i λ i 2 ] 2 ω i λ i 4 ,
I 2 = I 1 1 + K x
d I d s = K I + E ,