Abstract

A lattice-dynamical calculation applicable to crystals of any structural complexity and any symmetry (that permits optical activity) reveals three mechanisms causing optical activity in crystals: electric-dipole–magnetic-dipole interference, electric-dipole–electric-quadrupole interference, and first-order wave-vector dispersion of the bonding forces. Only the last mechanism was found by Born and Huang from lattice dynamics, while quantum-mechanical derivations have produced only the other two. Thus the present derivation removes a discrepancy between these two approaches, which, since they deal with a long-wavelength phenomenon, should produce closely comparable formulas. The first two mechanisms give terms in the rotatory power proportional to ω2/(ωi2ω2), whereas the third mechanism gives terms proportional to ω2/(ωi2ω2) (ωj2ω2). Both types of term have been shown to be necessary to fit frequency dispersion in particular crystals. Thus the present theory produces all the known mechanisms along with the needed dispersion from a general, unified, basic approach.

© 1989 Optical Society of America

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References

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  1. V. Vysin and V. Janku, Opt. Commun. 3, 305 (1971).
    [Crossref]
  2. Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].
  3. M. Born, Proc. R. Soc. London Ser. A 150, 84 (1935).
    [Crossref]
  4. E. U. Condon, Rev. Mod. Phys. 9, 432 (1937).
    [Crossref]
  5. M. Born, Phys. Z. 16, 251 (1915).
  6. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.
  7. L. Rosenfeld, Z. Phys. 28, 161 (1928).
  8. V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
    [Crossref]
  9. A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988 (1971).
    [Crossref]
  10. W. P. Healy, J. Phys. B 7, 1633 (1974).
    [Crossref]
  11. V. Schlagheck, Z. Phys. 266, 313 (1974).
    [Crossref]
  12. C. Duprez-Quesnoit, Mol. Phys. 49, 561 (1983).
    [Crossref]
  13. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, 1982).
  14. K. Natori, J. Phys. Soc. Jpn. 39, 1013 (1975).
    [Crossref]
  15. D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979). (Although this book is no longer in print, a limited number of paperback copies are available from the author.)
  16. E. Charney, The Molecular Basis of Optical Activity (Wiley, New York, 1979).
  17. A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr. 19, 108 (1986).
    [Crossref]
  18. T. M. Lowry, Optical Rotatory Power (Longmans, Green, London, 1935; reprinted by Dover, New York, 1964).
  19. J.-P. Mathieu, Les Théories Molécularies du Pouvoir Rotatoire Naturel (Centre National de la Recherche Scientifique, Paris, 1946).
  20. J. R. Partington, An Advanced Treatise on Physical Chemistry, Physico-Chemical Optics (Longmans, Green, London, 1953), Vol. IV.
  21. C. Djerassi, Optical Rotatory Dispersion (McGraw-Hill, New York, 1960).
  22. L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism (Academic, New York, 1965).
  23. P. Crabbé, Optical Rotatory Dispersion and Circular Dichroism in Organic Chemistry (Holden-Day, London, 1965).
  24. D. J. Caldwell and H. Eyring, The Theory of Optical Activity (Wiley-Interscience, New York, 1971).
  25. S. F. MasonMolecular Optical Activity and Chiral Discrimination (Cambridge U. Press, Cambridge, 1982).
  26. V. M. Agranovich and V. L. Ginsburg, Spatial Disperson in Crystal Optics and the Theory of Excitons, 2nd ed. (Springer-Verlag, Berlin, 1984).
    [Crossref]
  27. P. Drude, The Theory of Optics (Longmans, Green, London, 1902).
  28. W. Kuhn, Z. Phys. Chem. (Leipzig) B 20, 325 (1933).
  29. M. Born, Ann. Phys. (Leipzig) 55, 177 (1918).
  30. C. W. Oseen, Ann. Phys. (Leipzig) 48, 1 (1915).
  31. W. Kuhn, Z. Phys. Chem. (Leipzig) B4, 14 (1929); Trans. Faraday Soc. 26, 293 (1930).
  32. E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
    [Crossref]
  33. M. Born and M. Göppert-Mayer, in Handbuch der Physik, H. Geiger and K. Scheel, eds. (Springer-Verlag, Berlin, 1933), Vol. 24/2, p. 623.
  34. V. Vysin, Proc. Phys. Soc. London 87, 55 (1966).
    [Crossref]
  35. S. Chandrasekhar, Proc. R. Soc. London Ser. A 259, 531 (1961).
    [Crossref]
  36. J. W. Gibbs, Am. J. Sci., Ser. 3,  23, 460 (1882).
    [Crossref]
  37. Ref. 15, p. 77.
  38. Ref. 15, Chap. 6, p. 105.
  39. Ref. 15, p. 125.
  40. Ref. 15, p. 80.
  41. Ref. 15, p. 204.
  42. D. F. Nelson, Phys. Rev. Lett. 60, 608 (1988).
    [Crossref] [PubMed]
  43. Ref. 15, pp. 25, 80.
  44. Ref. 15, pp. 73–76,
  45. Ref. 15, p. 17.
  46. Ref. 6, App. IX, p. 408.
  47. R. M. Peterson, Am. J. Phys. 43, 969 (1975).
    [Crossref]

1988 (1)

D. F. Nelson, Phys. Rev. Lett. 60, 608 (1988).
[Crossref] [PubMed]

1986 (1)

A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr. 19, 108 (1986).
[Crossref]

1983 (1)

C. Duprez-Quesnoit, Mol. Phys. 49, 561 (1983).
[Crossref]

1975 (2)

K. Natori, J. Phys. Soc. Jpn. 39, 1013 (1975).
[Crossref]

R. M. Peterson, Am. J. Phys. 43, 969 (1975).
[Crossref]

1974 (3)

W. P. Healy, J. Phys. B 7, 1633 (1974).
[Crossref]

V. Schlagheck, Z. Phys. 266, 313 (1974).
[Crossref]

V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
[Crossref]

1973 (1)

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

1971 (2)

A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988 (1971).
[Crossref]

V. Vysin and V. Janku, Opt. Commun. 3, 305 (1971).
[Crossref]

1966 (1)

V. Vysin, Proc. Phys. Soc. London 87, 55 (1966).
[Crossref]

1961 (1)

S. Chandrasekhar, Proc. R. Soc. London Ser. A 259, 531 (1961).
[Crossref]

1937 (2)

E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
[Crossref]

E. U. Condon, Rev. Mod. Phys. 9, 432 (1937).
[Crossref]

1935 (1)

M. Born, Proc. R. Soc. London Ser. A 150, 84 (1935).
[Crossref]

1933 (1)

W. Kuhn, Z. Phys. Chem. (Leipzig) B 20, 325 (1933).

1929 (1)

W. Kuhn, Z. Phys. Chem. (Leipzig) B4, 14 (1929); Trans. Faraday Soc. 26, 293 (1930).

1928 (1)

L. Rosenfeld, Z. Phys. 28, 161 (1928).

1918 (1)

M. Born, Ann. Phys. (Leipzig) 55, 177 (1918).

1915 (2)

C. W. Oseen, Ann. Phys. (Leipzig) 48, 1 (1915).

M. Born, Phys. Z. 16, 251 (1915).

1882 (1)

J. W. Gibbs, Am. J. Sci., Ser. 3,  23, 460 (1882).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginsburg, Spatial Disperson in Crystal Optics and the Theory of Excitons, 2nd ed. (Springer-Verlag, Berlin, 1984).
[Crossref]

Altar, W.

E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
[Crossref]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, 1982).

Born, M.

M. Born, Proc. R. Soc. London Ser. A 150, 84 (1935).
[Crossref]

M. Born, Ann. Phys. (Leipzig) 55, 177 (1918).

M. Born, Phys. Z. 16, 251 (1915).

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.

M. Born and M. Göppert-Mayer, in Handbuch der Physik, H. Geiger and K. Scheel, eds. (Springer-Verlag, Berlin, 1933), Vol. 24/2, p. 623.

Buckingham, A. D.

A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988 (1971).
[Crossref]

Burkov, V. I.

V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
[Crossref]

Caldwell, D. J.

D. J. Caldwell and H. Eyring, The Theory of Optical Activity (Wiley-Interscience, New York, 1971).

Chandrasekhar, S.

S. Chandrasekhar, Proc. R. Soc. London Ser. A 259, 531 (1961).
[Crossref]

Charney, E.

E. Charney, The Molecular Basis of Optical Activity (Wiley, New York, 1979).

Condon, E. U.

E. U. Condon, Rev. Mod. Phys. 9, 432 (1937).
[Crossref]

E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
[Crossref]

Crabbé, P.

P. Crabbé, Optical Rotatory Dispersion and Circular Dichroism in Organic Chemistry (Holden-Day, London, 1965).

Djerassi, C.

C. Djerassi, Optical Rotatory Dispersion (McGraw-Hill, New York, 1960).

Dobrzhanskii, G. F.

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

Drude, P.

P. Drude, The Theory of Optics (Longmans, Green, London, 1902).

Dunn, M. B.

A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988 (1971).
[Crossref]

Duprez-Quesnoit, C.

C. Duprez-Quesnoit, Mol. Phys. 49, 561 (1983).
[Crossref]

Eyring, H.

E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
[Crossref]

D. J. Caldwell and H. Eyring, The Theory of Optical Activity (Wiley-Interscience, New York, 1971).

Gibbs, J. W.

J. W. Gibbs, Am. J. Sci., Ser. 3,  23, 460 (1882).
[Crossref]

Ginsburg, V. L.

V. M. Agranovich and V. L. Ginsburg, Spatial Disperson in Crystal Optics and the Theory of Excitons, 2nd ed. (Springer-Verlag, Berlin, 1984).
[Crossref]

Glazer, A. M.

A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr. 19, 108 (1986).
[Crossref]

Göppert-Mayer, M.

M. Born and M. Göppert-Mayer, in Handbuch der Physik, H. Geiger and K. Scheel, eds. (Springer-Verlag, Berlin, 1933), Vol. 24/2, p. 623.

Grosjean, M.

L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism (Academic, New York, 1965).

Healy, W. P.

W. P. Healy, J. Phys. B 7, 1633 (1974).
[Crossref]

Huang, K.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.

Janku, V.

V. Vysin and V. Janku, Opt. Commun. 3, 305 (1971).
[Crossref]

Kizel’, V. A.

V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
[Crossref]

Krasilov, Yu. I.

V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
[Crossref]

Kuhn, W.

W. Kuhn, Z. Phys. Chem. (Leipzig) B 20, 325 (1933).

W. Kuhn, Z. Phys. Chem. (Leipzig) B4, 14 (1929); Trans. Faraday Soc. 26, 293 (1930).

Legrand, M.

L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism (Academic, New York, 1965).

Lowry, T. M.

T. M. Lowry, Optical Rotatory Power (Longmans, Green, London, 1935; reprinted by Dover, New York, 1964).

Mason, S. F.

S. F. MasonMolecular Optical Activity and Chiral Discrimination (Cambridge U. Press, Cambridge, 1982).

Mathieu, J.-P.

J.-P. Mathieu, Les Théories Molécularies du Pouvoir Rotatoire Naturel (Centre National de la Recherche Scientifique, Paris, 1946).

Natori, K.

K. Natori, J. Phys. Soc. Jpn. 39, 1013 (1975).
[Crossref]

Nelson, D. F.

D. F. Nelson, Phys. Rev. Lett. 60, 608 (1988).
[Crossref] [PubMed]

D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979). (Although this book is no longer in print, a limited number of paperback copies are available from the author.)

Oseen, C. W.

C. W. Oseen, Ann. Phys. (Leipzig) 48, 1 (1915).

Partington, J. R.

J. R. Partington, An Advanced Treatise on Physical Chemistry, Physico-Chemical Optics (Longmans, Green, London, 1953), Vol. IV.

Perekalina, Z. B.

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

Peterson, R. M.

R. M. Peterson, Am. J. Phys. 43, 969 (1975).
[Crossref]

Rosenfeld, L.

L. Rosenfeld, Z. Phys. 28, 161 (1928).

Schlagheck, V.

V. Schlagheck, Z. Phys. 266, 313 (1974).
[Crossref]

Shpil’ko, I. A.

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

Smirnova, N. L.

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

Stadnicka, K.

A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr. 19, 108 (1986).
[Crossref]

Velluz, L.

L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism (Academic, New York, 1965).

Vysin, V.

V. Vysin and V. Janku, Opt. Commun. 3, 305 (1971).
[Crossref]

V. Vysin, Proc. Phys. Soc. London 87, 55 (1966).
[Crossref]

Am. J. Phys. (1)

R. M. Peterson, Am. J. Phys. 43, 969 (1975).
[Crossref]

Am. J. Sci. (1)

J. W. Gibbs, Am. J. Sci., Ser. 3,  23, 460 (1882).
[Crossref]

Ann. Phys. (Leipzig) (2)

M. Born, Ann. Phys. (Leipzig) 55, 177 (1918).

C. W. Oseen, Ann. Phys. (Leipzig) 48, 1 (1915).

J. Appl. Crystallogr. (1)

A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr. 19, 108 (1986).
[Crossref]

J. Chem. Phys. (1)

E. U. Condon, W. Altar, and H. Eyring, J. Chem. Phys. 5, 752 (1937).
[Crossref]

J. Chem. Soc. A (1)

A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988 (1971).
[Crossref]

J. Phys. B (1)

W. P. Healy, J. Phys. B 7, 1633 (1974).
[Crossref]

J. Phys. Soc. Jpn. (1)

K. Natori, J. Phys. Soc. Jpn. 39, 1013 (1975).
[Crossref]

Kristallogr. (1)

Z. B. Perekalina, N. L. Smirnova, G. F. Dobrzhanskii, and I. A. Shpil’ko, Kristallogr. 18, 852 (1973) [Sov. Phys. Crystallogr. 18, 538 (1974)].

Mol. Phys. (1)

C. Duprez-Quesnoit, Mol. Phys. 49, 561 (1983).
[Crossref]

Opt. Commun. (1)

V. Vysin and V. Janku, Opt. Commun. 3, 305 (1971).
[Crossref]

Phys. Rev. Lett. (1)

D. F. Nelson, Phys. Rev. Lett. 60, 608 (1988).
[Crossref] [PubMed]

Phys. Z. (1)

M. Born, Phys. Z. 16, 251 (1915).

Proc. Phys. Soc. London (1)

V. Vysin, Proc. Phys. Soc. London 87, 55 (1966).
[Crossref]

Proc. R. Soc. London Ser. A (2)

S. Chandrasekhar, Proc. R. Soc. London Ser. A 259, 531 (1961).
[Crossref]

M. Born, Proc. R. Soc. London Ser. A 150, 84 (1935).
[Crossref]

Rev. Mod. Phys. (1)

E. U. Condon, Rev. Mod. Phys. 9, 432 (1937).
[Crossref]

Usp. Fiz. Nauk (1)

V. A. Kizel’, Yu. I. Krasilov, and V. I. Burkov, Usp. Fiz. Nauk 114, 295 (1974) [Sov. Phys. Usp. 17, 745 (1975)].
[Crossref]

Z. Phys. (2)

L. Rosenfeld, Z. Phys. 28, 161 (1928).

V. Schlagheck, Z. Phys. 266, 313 (1974).
[Crossref]

Z. Phys. Chem. (Leipzig) (1)

W. Kuhn, Z. Phys. Chem. (Leipzig) B4, 14 (1929); Trans. Faraday Soc. 26, 293 (1930).

Z. Phys. Chem. (Leipzig) B (1)

W. Kuhn, Z. Phys. Chem. (Leipzig) B 20, 325 (1933).

Other (24)

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, 1982).

D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979). (Although this book is no longer in print, a limited number of paperback copies are available from the author.)

E. Charney, The Molecular Basis of Optical Activity (Wiley, New York, 1979).

M. Born and M. Göppert-Mayer, in Handbuch der Physik, H. Geiger and K. Scheel, eds. (Springer-Verlag, Berlin, 1933), Vol. 24/2, p. 623.

Ref. 15, p. 77.

Ref. 15, Chap. 6, p. 105.

Ref. 15, p. 125.

Ref. 15, p. 80.

Ref. 15, p. 204.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.

T. M. Lowry, Optical Rotatory Power (Longmans, Green, London, 1935; reprinted by Dover, New York, 1964).

J.-P. Mathieu, Les Théories Molécularies du Pouvoir Rotatoire Naturel (Centre National de la Recherche Scientifique, Paris, 1946).

J. R. Partington, An Advanced Treatise on Physical Chemistry, Physico-Chemical Optics (Longmans, Green, London, 1953), Vol. IV.

C. Djerassi, Optical Rotatory Dispersion (McGraw-Hill, New York, 1960).

L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism (Academic, New York, 1965).

P. Crabbé, Optical Rotatory Dispersion and Circular Dichroism in Organic Chemistry (Holden-Day, London, 1965).

D. J. Caldwell and H. Eyring, The Theory of Optical Activity (Wiley-Interscience, New York, 1971).

S. F. MasonMolecular Optical Activity and Chiral Discrimination (Cambridge U. Press, Cambridge, 1982).

V. M. Agranovich and V. L. Ginsburg, Spatial Disperson in Crystal Optics and the Theory of Excitons, 2nd ed. (Springer-Verlag, Berlin, 1984).
[Crossref]

P. Drude, The Theory of Optics (Longmans, Green, London, 1902).

Ref. 15, pp. 25, 80.

Ref. 15, pp. 73–76,

Ref. 15, p. 17.

Ref. 6, App. IX, p. 408.

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Equations (61)

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ρ = ω 2 i A i ω i 2 - ω 2 + ω 2 i j B i j ( ω i 2 - ω 2 ) ( ω j 2 - ω 2 2 ) ,
i j ω 2 B i j ( ω i 2 - ω 2 ) ( ω j 2 - ω 2 ) = i ω 2 B i i ( ω i 2 - ω 2 ) 2 + i j i j ( B i j + B j i ) ω i 2 ( ω j 2 - ω i 2 ) ( ω i 2 - ω 2 ) .
y T ν = Y ν + y ν             ( ν = 1 , 2 , , N - 1 ) ,
L M = 1 2 ν m ν y ˙ i ν y ˙ i ν - ν μ M i j ν μ y i ν y j μ - ν μ L i j k ν μ y i ν y j , k μ .
L F = 0 2 ( E 2 - c 2 B 2 ) .
E = Φ - A t ,
B = × A .
× E + B t = 0 ,
· B = 0 ,
L I = P · E + M · B - E · ( · Q ) ,
P = ν q ν y T ν ,
M = 1 2 μ ν q μ ν y T μ × y ˙ T ν ,
Q = 1 2 μ ν q μ ν y T μ y T ν ,
L = L M + L F + L I
d d t L ( A i / t ) = L A i - z j L A i , j ,
1 μ 0 × B - 0 E t = j D
j D = P t + × M - · Q t .
× ( × E ) + 1 c 2 2 E t 2 = - μ 0 j D t .
m ν y ¨ i ν + 1 2 μ q μ ν ( B ˙ × Y μ ) i = q ν E i + 1 2 μ q μ ν ( Y j μ E i j + Y j μ E j , i ) - 2 μ M i j ν μ y j μ μ ( L j i k μ ν - L i j k μ ν ) y j , k μ .
m ν y ¨ i ν = q ν E i - 2 μ M i j ν μ y j μ + μ q μ ν Y j μ E i , j + μ ( L j i k μ ν - L i j k μ ν ) y j , k μ .
η k = μ ( m μ ) 1 / 2 n k μ · y μ ,
2 μ n i k μ M i j μ ν ( m μ m ν ) 1 / 2 = Ω k 2 n j k ν
μ n k μ · n l μ = δ k l ,
k n i k μ n j k ν = δ i j δ μ ν ,
η ¨ k = c k · E - Ω k 2 η k ,
c k = μ q μ ( m μ ) 1 / 2 n k μ .
η k = c k · E Ω k 2 - ω 2 .
( Ω k 2 - ω 2 ) η k = c k · E + l q i j i k l N l E i , j + l ( L j l k - L j k l ) η , j l ,
q i j k l = ν μ n i k ν q ν μ n l l μ ( m ν m μ ) 1 / 2 = q j i l k ,
L p k l = ν μ n i k ν L i j p ν μ n j l μ ( m ν m μ ) 1 / 2 ,
η k = 1 Ω k 2 - ω 2 { c k · E + l [ q i j k l N l + ( L j l k - L j k l ) c i l Ω l 2 - ω 2 ] E i , j } .
- μ 0 j i D t = ω 2 0 c 2 ( ξ q ξ y i ξ - ξ ζ q ξ ζ Y j ξ y i , j ζ ) ,
- μ 0 j i D t = ω 2 0 c 2 ( k c i k η k - k l N k q j i k l η , j l ) .
- μ 0 j i D t = ω 2 c 2 [ χ i j ( ω ) E j + g i j k ( ω ) E j , k ] ,
χ i j ( ω ) = 1 0 a c i a c j a Ω a 2 - ω 2 ,
g i j k ( ω ) = 1 0 a b c i a q j k a b - c j a q i k a b Ω a 2 - ω 2 N b + 1 0 a b c i a c j b ( L k b a - L k a b ) ( Ω a 2 - ω 2 ) ( Ω b 2 - ω 2 ) .
g i j k ( ω ) = - g j i k ( ω ) .
a b ( c i a q j k a b - c j a q i k a b ) N b = 0 ,
L I = j · A - q Φ ,
q D = - · P + : Q ,
j D = P t + × ( P × x ˙ ) - t ( · Q ) + × M - × [ ( · Q + Q · ) × x ˙ ] ,
L I = P · ( E + x ˙ × B ) + M · B - E · ( · Q ) - B · [ ( · Q + Q · ) × x ˙ ] .
D = 0 E + P - · Q ,
H = 1 μ 0 B - M
× H - D t = 0.
0 · E = q D ,
· D = 0 ,
D i = 0 ( χ i j ( ω ) + δ i j ) E j + 0 g i j k e ( ω ) E k , j ,
H n = 1 μ 0 B n + i ω 0 n i j g i j k m ( ω ) E k ,
0 g i j k e ( ω ) = l m c i l c j m ( L k m l - L k l m ) ( Ω m 2 - ω 2 ) ( Ω l 2 - ω 2 ) + l m c i m q j k m l N l Ω m 2 - ω 2 - 1 2 l m c j m q i k m l N l Ω m 2 - ω 2 - 1 2 l m c j m q k i m l N l Ω m 2 - ω 2 ,
0 g i j k m ( ω ) = 1 2 l m c k m q j i m l N l Ω m 2 - ω 2 ,
j D = t ( D - 0 E ) + × ( H - 1 μ 0 B ) ,
g i j k ( ω ) = g i j k e ( ω ) + g i k j m ( ω ) - g k i j m ( ω ) .
r = ( i + i j ) / 2 ,
l = ( i - i j ) / 2 ,
n r = κ 1 / 2 - g 2 κ 1 / 2 ,
n l = κ 1 / 2 + g 2 κ 1 / 2 ,
g = κ 2 s k k l m g l m n ( ω ) s n = k s · G ( ω ) · s = s · G ( ω ) ,
G i j ( ω ) = ¼ [ i m n g m n j ( ω ) + j m n g m n i ( ω ) ] ,
G i ( ω ) = k 2 i l m g l m n ( ω ) s n .
ρ = π λ ( n l - n r ) = 2 π 2 λ 2 s · G ( ω ) · s = ω 2 2 c 2 s · G ( ω ) · s ,

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