Theory of Crystal Space Groups and Infra-Red and Raman Lattice Processes of Insulating Crystals

  • Joseph L. Birman
Part of the Encyclopedia of Physics / Handbuch der Physik book series (HDBPHYS, volume 5 / 25 / 2 / 2b)

Abstract

This article was written with the objective of developing, and illustrating, the principles and practice of the application of group theory methods to the analysis of the infra-red and Raman lattice optical processes in insulating crystals. The group theory methods have proven to be very powerful tools in the interpretation and prediction of optical processes. It is also our objective to make these methods as accessible, and transparent, and hence as widely used as possible.

Keywords

Anisotropy Radium Perovskite Bors Clarification 

References

  1. 1.
    M. Born and T. von Karman: Physik. Z. 14, 15 (1913).MATHGoogle Scholar
  2. This will be fully discussed in Sects. 126 et seq. 1 This will be discussed in Sects. 126 et seq.Google Scholar
  3. 1.
    F. Bloch: Z. Physik 52, 555 (1928).ADSMATHCrossRefGoogle Scholar
  4. 1.
    G.F. Koster: Phys. Rev. 109, 227 (1958).MathSciNetADSCrossRefGoogle Scholar
  5. 1.
    P.P. Ewald: Z. Physik 2, 232 (1920).CrossRefGoogle Scholar
  6. 1.
    G.F. Koster, in: Solid State Physics (eds. F. Seitz and D. Turnbull), Vol. 5. New York: Academic Press 1957.Google Scholar
  7. 2.
    E. Wigner and F. Seitz: Phys. Rev. 43, 804 (1933).ADSCrossRefGoogle Scholar
  8. 3.
    L. Bouckaert, R. Smoluchowski, and E. Wigner: Phys. Rev. 50, 58 (1936).ADSMATHCrossRefGoogle Scholar
  9. 1.
    For the mathematical aspects of the work we follow A.H. Clifford: Ann. of Math. 38, 533 (1937). Also see [12]. MathSciNetMATHCrossRefGoogle Scholar
  10. 2.
    Other references in the physics literature are: F. Sfjtz: Ann. of Math. 37, 17 (1936)MathSciNetCrossRefGoogle Scholar
  11. 2a.
    L. Bouckaert, R. Smoluchowski, and E. Wigner: Phys. Rev. 50, 58 (1936)ADSMATHCrossRefGoogle Scholar
  12. 2b.
    C. Herring: J.Franklin Inst. 233, 525 (1942).MathSciNetMATHCrossRefGoogle Scholar
  13. 2c.
    G.F. Koster, in: Solid State Physics (eds. F. Seitz and D. Turnbull), Vol.5. New York: Academic Press 1957.Google Scholar
  14. 2d.
    R.J. Elliott: Phys. Rev. 96, 130 (1954).CrossRefGoogle Scholar
  15. 1.
    C. Herring: J. Franklin Inst. 233, 525 (1942).MathSciNetMATHCrossRefGoogle Scholar
  16. 1.
    R.J. Elliott and R. Loudon: J. Phys. Chem. Solids 15, 146 (1960).MathSciNetADSCrossRefGoogle Scholar
  17. 1.
    W. Döring: Z. Naturforsch. 14a, 343 (1959).ADSGoogle Scholar
  18. 1a.
    A. Kitz: Phys. stat. solidi 8, 813 (1965).ADSCrossRefGoogle Scholar
  19. 1b.
    See A.C. Hurley: Phil. Trans. Roy. Soc. London, Ser. A260, 1 (1966) for other references.MathSciNetADSMATHCrossRefGoogle Scholar
  20. 1.
    Hurley: op. cit., Sect. 41, footnote 1.Google Scholar
  21. 1.
    Hurley: op. cit., Sect. 41, footnote 1.Google Scholar
  22. 1.
    W. Döring and V. Zehler: Ann. Physik 13, 214 (1953).MATHCrossRefGoogle Scholar
  23. 1.
    J. Zak: J. Math. Phys. 1, 165 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  24. 2.
    L.J. Klauder Jr., and J.G. Gay: J. Math. Phys. 9, 1488 (1968).ADSMATHCrossRefGoogle Scholar
  25. 1.
    L. Bouckaert, R. Smoluchowski, and E. Wigner: Phys. Rev. 50, 58 (1936).ADSMATHCrossRefGoogle Scholar
  26. 1.
    J.L. Birman: Phys. Rev. 127, 1093 (1962)ADSMATHCrossRefGoogle Scholar
  27. 1a.
    J.L. Birman: Phys. Rev. 131, 1489 (1963).ADSCrossRefGoogle Scholar
  28. 2.
    J. Zak: J. Math. Phys. 3, 1278 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
  29. 3.
    R.J. Elliott and R. Loudon: J. Phys. Chem. Solids 15, 146 (1960).MathSciNetADSCrossRefGoogle Scholar
  30. 4.
    M. Lax and J.J. Hopfield: Phys. Rev. 124, 115 (1961).ADSMATHCrossRefGoogle Scholar
  31. 5.
    J. Zak: Phys. Rev. 151, 464 (1966).ADSCrossRefGoogle Scholar
  32. 6.
    H. Poulet: J. de Physique (Paris) 26, 684 (1965).CrossRefGoogle Scholar
  33. 1.
    See Sect. 117 infra. Google Scholar
  34. 2.
    See Sect 117 infra. Google Scholar
  35. A considerable portion of this section was written by Dr. Rhoda Berenson. See Sects. 18 and 134.Google Scholar
  36. 1.
    I. Itzkan: The Clebsch-Gordan Coefficients for the Crystallographic Space Groups. Ph. D. thesis, Physics Department, New York University, June 1969.Google Scholar
  37. 1a.
    R. Berenson: Theory of Crystal Clebsch-Gordan Coefficients. Ph. D. Thesis, New York University, February 1974. -R.Berenson, I. Itzkan and J.L. Birman: J. Math. Phys. (to be published). Available from University Microfilms Inc., Ann Harbor, Mich.Google Scholar
  38. 2.
    D.B. Litvin and J. Zak: J. Math. Phys. 9, 212 (1969).MathSciNetADSCrossRefGoogle Scholar
  39. 3.
    B.K. Novosadov, L.K. Saulevich, D.T. Sviridov and Y.F. Smirnov: Soviet Phys. Doklady 14, 50 (1969).MathSciNetADSGoogle Scholar
  40. 3a.
    L.K. Saulevich, D.T. Svirdov, and Y.F. Smirnov: Soviet Phys. Cryst. 15, 351, 355 (1970).MATHGoogle Scholar
  41. 3b.
    J.F. Cornwell: phys. stat. solidi 37, 225 (1970).MathSciNetADSCrossRefGoogle Scholar
  42. 4.
    T Sakata- T Math Phys (in press) (1973)Google Scholar
  43. 5.
    I. Itzkan: Ph.D. Thesis. New York University 1969.-R. Berenson: Ph.D. Thesis. New York University 1974. See footnote 1.Google Scholar
  44. 1.
    R.J. Elliott and R. Loudon: J. Phys. Chem. Solids 15, 146 (1960).MathSciNetADSCrossRefGoogle Scholar
  45. 1.
    E.P. Wigner: Nachr. Akad. Wiss. Göttingen, Math-Physik. Kl. 133 (1930).Google Scholar
  46. 2.
    S. Yanagawa: Progr. Theoret. Phys. (Kyoto) 10, 83 (1953).ADSMATHCrossRefGoogle Scholar
  47. 3.
    I. V. V. Raghavacharyulu: Can. J. Phys. 39, 1704 (1961).MathSciNetADSMATHCrossRefGoogle Scholar
  48. 4.
    H.W. Streitwolf: phys. stat. solidi 5, 383 (1964).ADSCrossRefGoogle Scholar
  49. 5.
    S.H. Chen: Ph. D. Thesis (1964) cited in [47], p. 138.Google Scholar
  50. 6.
    A.A. Maradudin and S.H. Vosko: Rev. Mod. Phys. 40, 1 (1968).ADSCrossRefGoogle Scholar
  51. 1.
    Single dot scalar product is sum on (k / α).Google Scholar
  52. 2.
    Double dot scalar product is sum on (jp).Google Scholar
  53. 1.
    Originally discussed by E. P. Wigner, Nachr. Akad. Wiss. Göttingen, Math.-Physik. Kl. 546 (1932).Google Scholar
  54. 1.
    C. Herring: Phys. Rev. 52, 361 (1937).ADSCrossRefGoogle Scholar
  55. 1.
    V. Frei: Czech. J. Phys. 16, 207 (1966).ADSCrossRefGoogle Scholar
  56. 1.
    A. W. Clifford: Ann. Math. 38, 533 (1937).MathSciNetMATHCrossRefGoogle Scholar
  57. 2.
    J. Dimmock: J. Math. Phys. 4, 1307 (1963).MathSciNetADSMATHCrossRefGoogle Scholar
  58. 1.
    The use of ray corepresentations for electronic energy bands was discussed by N. V. Kudryavtseva, Soviet Phys. Solid State 7, 803 (1965) [orig. FTT 7, 998 (1965)], and references cited therein.Google Scholar
  59. 2.
    See reference 1, p. 196, of this section, and references to work of Karavaev, Kudryartseva, Kovalev, and Chaldyshev cited therein.Google Scholar
  60. 1.
    See [S], Eq. (38.33).Google Scholar
  61. 2.
    See [8], Eq. (38.38).Google Scholar
  62. 1.
    This approach was apparently first used in crystals by K. Huang, Z. Physik 171, 213 (1963).ADSCrossRefGoogle Scholar
  63. 1a.
    See also B. Klein: Ph. D. thesis, New York University (1969). Available from University microfilms, Ann Arbor, Mich.Google Scholar
  64. B. Klein in [45]; and in [12]. Google Scholar
  65. 2.
    D.B. Litvin and J. Zak: J. Math. Phys. 9, 212 (1968).MathSciNetADSMATHCrossRefGoogle Scholar
  66. 3.
    I. Itzkan: Clebsch-Gordon coefficients for the crystallographic space groups. Ph. D. thesis, Department of Physics, New York University, June 1969.Google Scholar
  67. 3a.
    R. Berenson: Theory of Crystal Clebsch-Gordan Coefficients, Ph. D. Thesis, Physics Department, New York University, February 1974. Available from University Microfilms, Ann. Arbor, Mich.-R. Berenson, I. Itzkan and J.L. Birman: J. Math. Phys. (to be published).Google Scholar
  68. 3b.
    J.F. Cornwall: phys. stat. solidi 37, 225 (1970).ADSCrossRefGoogle Scholar
  69. 2.
    A. Aviran and J. Zak: J. Math. Phys. 9, 2138 (1968).ADSMATHCrossRefGoogle Scholar
  70. 3.
    A. Aviran and J. Zak: J. Math. Phys. 9, 2138 (1968).ADSMATHCrossRefGoogle Scholar
  71. 1.
    J.C. Phillips: Phys. Rev. 104, 1263 (1956).MathSciNetADSCrossRefGoogle Scholar
  72. 2.
    L. van Hove: Phys. Rev. 89, 1189 (1953).ADSMATHCrossRefGoogle Scholar
  73. 3.
    J.C. Phillips: Phys. Rev. 104, 1263 (1956).MathSciNetADSCrossRefGoogle Scholar
  74. 4.
    N.D. Kudryavtseva: Soviet Phys. Solid State 9, 1850 (1968).Google Scholar
  75. 1.
    See, for example, the article by Leibfried and Ludwig [4], pp. 286–288 et seq.Google Scholar
  76. 2.
    See [13] Eq. (39.2).Google Scholar
  77. 1.
    See [13], Eqs. (39.11)—(39.18) for discussion of restrictions due to translational symmetry alone.Google Scholar
  78. 2.
    Near resonance, antisymmetric components of P may become important in Raman scattering (see Sect. 124). Also morphic effects in certain crystal classes may involve those components.Google Scholar
  79. 1.
    M. Born and J.R. Oppenheimer: Ann. d. Phys. 84, 457 (1927).ADSMATHCrossRefGoogle Scholar
  80. 2.
    R.J. Elliott, R.T. Harley, W. Hayes, and S.R.P. Smith: Proc. Roy. Soc. (London), Ser. A328, 217 (1972).ADSCrossRefGoogle Scholar
  81. 1.
    M. Born and J.R. Oppenheimer: Ann. d. Phys. 84, 457 (1927).ADSMATHCrossRefGoogle Scholar
  82. 2.
    Ref. [13], pp. 166–173.Google Scholar
  83. 3.
    Ref. [13] Appendix VIII.Google Scholar
  84. 4.
    J.W. Garland: Phys. Rev. 153, 460 (1967).ADSCrossRefGoogle Scholar
  85. 4a.
    P.N. Keating: Phys. Rev. 175, 1169 (1968)ADSCrossRefGoogle Scholar
  86. 4b.
    P.N. Keating: Phys. Rev.187, 1190(1969).ADSCrossRefGoogle Scholar
  87. 5.
    G. Baym: Ann. Phys. 14, 1 (1961).MathSciNetADSMATHCrossRefGoogle Scholar
  88. 6.
    R.M. Pick, M.H. Cohen, and R.M. Martin: Phys. Rev. 1B, 910 (1970).ADSCrossRefGoogle Scholar
  89. 6a.
    R.M. Martin: Phys. Rev. Lett. 21, 536 (1968)ADSCrossRefGoogle Scholar
  90. 6b.
    R.M. Martin: Phys. Rev. 186, 871 (1969).ADSCrossRefGoogle Scholar
  91. 7.
    F.A. Johnson: Proc. Roy. Soc. (London), Ser.A310, 79, 89, 101, 111 (1969).ADSCrossRefGoogle Scholar
  92. 8.
    G. Gliss and H. Bilz: Phys. Rev. Lett. 21, 884 (1968).ADSCrossRefGoogle Scholar
  93. 9.
    A.K. Rajagopal and M.H. Cohen: Collective Phenomena 1, 9 (1972).MathSciNetGoogle Scholar
  94. 10.
    H. Bilz and E. Zybell: Unpublished.Google Scholar
  95. 11.
    J. Prince: Ph. D. Thesis. New York: New York University 1967. Unpublished: available from University Microfilms Ann Arbor, Mich.Google Scholar
  96. 12.
    See especially papers of H. Bilz, W. Weber and F. A. Johnson in [59]. Google Scholar
  97. 1.
    G. Baker: Phys. Rev. 103, 1119 (1956).MathSciNetADSMATHCrossRefGoogle Scholar
  98. 1.
    See [16] p. 12 et seq.Google Scholar
  99. 2.
    We follow, with modifications L. Tisza: Z. Physik 82, 48 (1933).ADSCrossRefGoogle Scholar
  100. 3.
    The above argument was pointed out by Dr. Y.C. Cho.Google Scholar
  101. 4.
    C. J. Bradley and B.L. Davies: J. Math. Phys. 11, 1536 (1970).MathSciNetADSMATHCrossRefGoogle Scholar
  102. 1.
    L. Biedenharn: J. Math. Phys. 4, 436 (1963).MathSciNetADSMATHCrossRefGoogle Scholar
  103. 1a.
    L. Weber: Z. Physik 190, 25 (1966).MathSciNetADSCrossRefGoogle Scholar
  104. 2.
    J.M. Jauch and R. Hill: Phys. Rev. 57, 641 (1940).MathSciNetADSMATHCrossRefGoogle Scholar
  105. 2a.
    G. Baker: Phys. Rev. 103, 1119 (1956). See also [61] and references cited therein.MathSciNetADSMATHCrossRefGoogle Scholar
  106. 1.
    M. Lax and E. Burstein: Phys. Rev. 97, 42 (1955).ADSGoogle Scholar
  107. 2.
    Ref. [32], p. 265.Google Scholar
  108. 3.
    See H. Bilz: Infra-red lattice vibration spectra of perfect crystals, in [79].Google Scholar
  109. 4.
    I. P. Ipatova, A.A. Maradudin, and R.F. Wallis: Soviet Phys. Solid State 8, 850 (1966)Google Scholar
  110. 4a.
    I. P. Ipatova, A.A. Maradudin, and R.F. Wallis: Phys. Rev. 155, 882 (1967).ADSCrossRefGoogle Scholar
  111. 5.
    B. Szigeti: Proc. Roy. Soc. (London), Ser. A 258, 377 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  112. 6.
    V.S. Vinogradov: Soviet Phys. Solid State 4, 519 (1962).Google Scholar
  113. 7.
    R. Wehner: phys. stat. solidi 15, 725 (1966).ADSCrossRefGoogle Scholar
  114. 8.
    M. Sparks and L.J. Sham: Solid State Comm. 11, 1451 (1972)ADSCrossRefGoogle Scholar
  115. 8a.
    M. Sparks and L.J. Sham: Phys. Rev. B8, 3037 (1973)ADSCrossRefGoogle Scholar
  116. 8b.
    M. Sparks and L.J. Sham: Phys. Rev. Lett 31, 718 (1973).ADSCrossRefGoogle Scholar
  117. 9.
    D.L. Mills and A.A. Maradudin: Phys. Rev. B8, 1617 (1973).ADSCrossRefGoogle Scholar
  118. 10.
    B. Bendow, S.C. Ying, and S.P. Yukon: Phys. Rev. B8, 1679 (1973).ADSCrossRefGoogle Scholar
  119. 11.
    H. Rosenstock: Phys. Rev. B9, 1963 (1974).ADSCrossRefGoogle Scholar
  120. 1.
    In this section we shall follow the general approach originally due to G. Placzek, Handbuch der Radiologie (ed. E. Marx), Vol. VI, 2, pp. 209–374 (1934). Leipzig: Akademische Verlagsgesellschaft 1934. See also Ref. [4], Born-Huang, Sects. 19–21, and also Chap. VII.Google Scholar
  121. 2.
    W. Heitler: Quantum Theory of Radiation. Oxford: University Press (1954).MATHGoogle Scholar
  122. 3.
    P. P. Platzman and N. Tzoar: Phys. Rev. 182, 510 (1969).ADSCrossRefGoogle Scholar
  123. 1.
    J.L. Birman: Phys. Rev. 127, 1093 (1962)ADSMATHCrossRefGoogle Scholar
  124. 1a.
    J.L. Birman: Phys. Rev. 131, 1489 (1963).ADSCrossRefGoogle Scholar
  125. 2.
    M. Lax: Comment on paper by J.L. Birman, in: [32], p. 712.Google Scholar
  126. 3.
    R. Loudon: Phys. Rev. 137, 1784 (1965).MathSciNetADSCrossRefGoogle Scholar
  127. 4.
    R. Berenson and J.L. Birman: Phys. Rev. B9, 4512 (1974)MathSciNetADSCrossRefGoogle Scholar
  128. 4a.
    J.L. Birman: Phys. Rev. B9, 4518 (1974).MathSciNetADSCrossRefGoogle Scholar
  129. 1.
    H. Poulet: Ann. Phys. (Paris) 10, 908 (1955).Google Scholar
  130. 2.
    J.L. Birman and R. Berenson: Phys. Rev. B9, 4512 (1974).MathSciNetADSCrossRefGoogle Scholar
  131. 2a.
    J.L. Birman: Phys. Rev. B9 4518 (1974).MathSciNetADSCrossRefGoogle Scholar
  132. 3.
    R. Loudon: Proc. Roy. Soc. A 275, 218 (1963).ADSCrossRefGoogle Scholar
  133. 3a.
    R. Loudon: Adv. Phys. 13, 423 (1964).ADSCrossRefGoogle Scholar
  134. 4.
    M. H. Cohen and J. Ruvalds: Phys. Rev. Letters 23, 1378 (1969)ADSCrossRefGoogle Scholar
  135. 4a.
    J. Ruvalds and A. Zawadewski: Phys. Rev. B2, 1172(1970).ADSCrossRefGoogle Scholar
  136. 1.
    K. Huang: E.R.A. Report L/T 239 (1950); — Proc. Roy. Soc. (London), Ser. A208, 352 (1951); and private communication to the author (May 1973).ADSMATHCrossRefGoogle Scholar
  137. 2.
    W. Cochran and R. Cowley, in [20]. -H. Bilz and R.K. Wehner, in [22]. Google Scholar
  138. 3.
    R.A.Cowley, in [19] See also A.D. Bruce and R.A. Cowley: J. Phys. C. (Solid State Physics) 5, 595 (1972).ADSCrossRefGoogle Scholar
  139. 4.
    A.A. Maradudin and A.E. Fein: Phys. Rev. 128, 2589 (1962).ADSCrossRefGoogle Scholar
  140. 5.
    M. Born and K. Huang, in [4], Eq. (38.38).Google Scholar
  141. 6.
    W. Cochran and R. Cowley, in [20].-H. Bilz and R.K. Wehner, in [22]. Google Scholar
  142. 7.
    R.A. Cowley: Proc. Phys. Soc. (London) 84, 281 (1964)J. Phys. Radium 26, 659 (1965).-In [20] ADSCrossRefGoogle Scholar
  143. 8.
    R. A. Cowley: J. Phys. Radium 26, 659 (1965), Eq. (3.9).Google Scholar
  144. 9.
    R.F. Wallis, I.P. Ipatova, and A.A. Maradudin: Soviet Phys. Solid State 8, 850 (1966).Google Scholar
  145. 9a.
    L.P. Ipatova, A.A. Maradudin, and R.F. Wallis: Phys. Rev. 155, 882 (1967).ADSCrossRefGoogle Scholar
  146. 9b.
    L.E. Gurevich and I.P. Ipatova: Soviet Phys. Solid State 4, 1513 (1963)Google Scholar
  147. 9c.
    L.E. Gurevich and I.P. Ipatova: Soviet Phys. JETP 18, 162 (1964).Google Scholar
  148. 10.
    A.A. Abrikosov: Soviet Phys. JETP 16, 765 (1963).ADSGoogle Scholar
  149. 11.
    A.A. Maradudin and I.P. Ipatova: Tech. Report U. of California, Irvine, March 1966;-J. Math. Phys. 9, 525 (1968).ADSCrossRefGoogle Scholar
  150. 12.
    R.A. Cowley: J. di Physique (Paris) 26, 659 (1965).CrossRefGoogle Scholar
  151. 13.
    W. Cochran and R. Cowley, in [20], see also A.D. Bruce: J. Phys. C. (Solid State Physics) 6, 174 (1973).ADSCrossRefGoogle Scholar
  152. 14.
    H. Bilz and R.K. Wehner, in [22]. 15H.Bilz, in [29].Google Scholar
  153. 16.
    R.F. Wallis et al in Ref. [12] Google Scholar
  154. 17.
    R.K. Wehner: phys. stat. solidi 15, 725 (1966).ADSCrossRefGoogle Scholar
  155. 18.
    R. Loudon: Proc. Roy. Soc. (London), Ser. A275, 218 (1963).ADSCrossRefGoogle Scholar
  156. 19.
    A.K. Ganguly and J.L. Birman: Phys. Rev. 162, 806 (1967).ADSCrossRefGoogle Scholar
  157. 20.
    Suggested by Prof. H. Bilz. At the time of writing (Sept. 1974) such tests are apparently under way in several laboratories.Google Scholar
  158. 21.
    D.L. Mills and E. Burstein: Phys. Rev. 188, 1465 (1969).ADSCrossRefGoogle Scholar
  159. 21.
    Polaritons were apparently first discussed in the context of phonons by Huang [13], Sect. 7, and references therein. Huang (private communication to the author May (1973)), also discussed exciton polaritons about that time (1950) but they were not named by him at that time. Their popularity seems to derive from the work of U. Fano, Phys. Rev. 103, 1202 (1956) and J. J. Hopfield, Phys. Rev. 112, 1555 (1958).ADSCrossRefGoogle Scholar
  160. 22.
    L.N. Ovander: Soviet Phys. Usp. 8, 337 (1965).ADSCrossRefGoogle Scholar
  161. 22a.
    J.J. Hopfield: Phys. Rev. 182, 945 (1969).ADSCrossRefGoogle Scholar
  162. 23.
    B. Bendow and J.L. Birman: Phys. Rev. B, 1, 1678 (1970) et seq.ADSCrossRefGoogle Scholar
  163. 24.
    D.L. Mills and E. Burstein: Phys. Rev. 188, 1465 (1969).ADSCrossRefGoogle Scholar
  164. 25.
    See the recent review by A. S. Barker Jr., and R. Loudon: Rev. Mod. Phys. 44, 18 (1972) and citations therein, especially to the work of Mills, Mills and Benton, Loudon and Barker, Tait, Mavroyannis and Martin. Another review is J. A. Deverin and C. Mavroyannis: Helv. Phys. Acta. 45, 1005 (1972).ADSCrossRefGoogle Scholar
  165. 26.
    R. Zeyher, C.S. Ting, and J.L. Birman: Phys. Rev. B10, 1725 (1974).ADSCrossRefGoogle Scholar
  166. 27.
    J J. Hopfield: Phys. Rev. 182, 945 (1969). But see also reference 26.ADSCrossRefGoogle Scholar
  167. 28.
    B. Bendow, J.L. Birman, A.K. Ganguly, T.C. Damen, R.C.C. Leite, and J.F. Scott: Optics Comm. 1, 267 (1970).ADSCrossRefGoogle Scholar
  168. 28a.
    R. Callender, S.S. Sussman, M. Selders, and R.K. Chang: Phys. Rev. B7, 388 (1973) and references cited in the latter.Google Scholar
  169. 29.
    A. Compaan and H.Z. Cummins: Phys. Rev. Letters 31, 41 (1973).ADSCrossRefGoogle Scholar
  170. 30.
    J.L. Birman and R. Berenson: Phys. Rev. B9, 4512 (1974).MathSciNetADSCrossRefGoogle Scholar
  171. 30a.
    J. L. Birman: Phys. Rev. B9, 4518 (1974).MathSciNetADSCrossRefGoogle Scholar
  172. 31.
    R.M. Martin in [48] and also: Phys. Rev. B4, 3676 (1971).ADSCrossRefGoogle Scholar
  173. 1.
    L. Bouckaert, R. Smoluchowski, and E. Wigner: Phys. Rev. 50, 58 (1936).ADSMATHCrossRefGoogle Scholar
  174. 2.
    L.C. Chen, R. Berenson, and J.L. Birman: Phys. Rev. 170, 639 (1968).ADSCrossRefGoogle Scholar
  175. In Appendix C a treatment by ray representation theory is given.Google Scholar
  176. 1.
    Adapted from R. Loudon and R.J. Elliott: J. Phys. Chem. Solids 15, 146 (1960).MathSciNetADSCrossRefGoogle Scholar
  177. These numbers refer to Appendix B.Google Scholar
  178. This section was largely written by Dr. Rhoda Berenson.Google Scholar
  179. 2.
    R. Berenson: Theory of Crystal Clebsch-Gordan Coefficients. Ph. D. Thesis, New York University, February 1974 (unpublished). Available from University Microfilms, Ann. Arbor, Mich. Also cf. references in Sect. 60.Google Scholar
  180. See p. 395 infra. Google Scholar
  181. 1.
    M. Lax: Proceedings of the Exeter Conference on Physics of Semiconductors (ed. A.C. Stickland), p. 395. Published by Institute of Physics and the Physical Society London SW. 7 (1962).Google Scholar
  182. 2.
    M. Hulin: phys. stat. solidi 21, 607 (1967).ADSCrossRefGoogle Scholar
  183. 3.
    A. Aviron and J. Zak: J. Math. Phys. 9, 2138 (1968).ADSCrossRefGoogle Scholar
  184. 4.
    G.F. Karavaev: Fiz. Tverd. Tela 6, 3676 (1964).MathSciNetGoogle Scholar
  185. 1.
    C. Herring: J. Franklin Inst. 233, 525 (1942) discussed these questions for the non-symmorphic groups diamond math and hexagonal close-packed math.MathSciNetMATHCrossRefGoogle Scholar
  186. 2.
    D.L. Rode: phys. stat. solidi (b), 53, 245 (1972).ADSCrossRefGoogle Scholar
  187. 3.
    H.W. Streitwolf: phys. stat. solidi 37, K47 (1970), has discussed this case.ADSCrossRefGoogle Scholar
  188. 3a.
    He reached the correct conclusion although he mistakenly claimed that earlier work by M. Lax and J. J. Hopfield: Phys. Rev. 124, 115 (1961)ADSMATHCrossRefGoogle Scholar
  189. 3b.
    J.L. Birman: Phys. Rev. 127, 1093 (1962) was in disagreement.ADSMATHCrossRefGoogle Scholar
  190. 3c.
    See M. Lax and J.L. Birman: phys. stat. solidi 49, K153 (1972). a Note Ganesan et al incorrectly give all parities at *L as even in J. Phys. 26, 640 (1966), in contrast to Burstein, Johnson, and Loudon, Phys. Rev. 139 A, 1240 (1965).ADSCrossRefGoogle Scholar
  191. 1.
    A. Karo and J.R. Hardy: Phys. Rev. 141, 701 (1966).ADSCrossRefGoogle Scholar
  192. 2.
    R. A. Cowley: Phys. Rev. 131, 1030 (1963).ADSCrossRefGoogle Scholar
  193. 3.
    J.L. Warren, J.L. Yarnell, G. Dolling and R.A. Cowley: Phys. Rev. 158, 805 (1967).ADSCrossRefGoogle Scholar
  194. 1.
    H.B. Rosenstock: J. Phys. Chem. Solids 2, 44 (1957).MathSciNetADSCrossRefGoogle Scholar
  195. 1a.
    J.C. Phillips and H.B. Rosenstock: J. Phys. Chem. Solids 5, 288 (1958).ADSCrossRefGoogle Scholar
  196. 2.
    F.A. Johnson and R. Loudon: Proc. Roy. Soc. (London), Ser. A281, 274 (1964), especially Fig. 2.ADSCrossRefGoogle Scholar
  197. 3.
    J.C. Phillips: Phys. Rev. 113, 147 (1959), Table VI.ADSCrossRefGoogle Scholar
  198. 4.
  199. 5.
    See W. Cochran and R. A. Cowley, in [20], p. 97, Fig. 16.Google Scholar
  200. 6.
    ibid., p. 98, Fig. 17.Google Scholar
  201. 7.
    See G. Dolling and R. A. Cowley: Proc. Phys. Soc. (London) 88, 463 (1966) Fig. 4.-J.L. Warren, J.L. Yarnell, G. Dolling, and R.A. Cowley: Phys. Rev. 158, 805 (1967).ADSCrossRefGoogle Scholar
  202. 8.
    F. A. Johnson and R. Loudon: Proc. Roy. Soc. (London), Ser. A281, 274 (1964), Fig. 1.ADSCrossRefGoogle Scholar
  203. 9.
    F.A. Johnson and R. Loudon: op. cit.-J.C. Phillips: op. cit. Google Scholar
  204. 10.
    F.A. Johnson and R. Loudon: ibid., Table I.Google Scholar
  205. 11.
    H. Bilz, R. Geick, and K.F. Renk, in: Proceedings of the Conference on Lattice Dynamics, Copenhagen 1963 (ed. R. F. Wallis). New York: Pergamon Press 1965.Google Scholar
  206. 1.
    F. A. Johnson and R. Loudon: Proc. Roy. Soc. (London), Ser. A281, 274 (1964).ADSCrossRefGoogle Scholar
  207. Recall Sect. 122.Google Scholar
  208. 1.
    J.L. Warren, J.L. Yarnell, G. Dolling, and R.A. Cowley: Phys. Rev. 158, 805 (1967).ADSCrossRefGoogle Scholar
  209. 2.
    J.L. Warren: Reference 1 this section.Google Scholar
  210. 3.
    J.R. Hardy and S.D. Smith: Phil. Mag. 6, 1163 (1961).ADSCrossRefGoogle Scholar
  211. 4.
    S.A. Solin and A.K. Ramdas: Phys. Rev. B1, 1687 (1970), earlier work was reported by R.S. Krishnan: Proc. Indian Acad. Sci. Sect. A26, 399 (1947).ADSCrossRefGoogle Scholar
  212. 5.
    F.A. Johnson: Proc. Phys. Soc. (London) 73, 265 (1959).ADSCrossRefGoogle Scholar
  213. 6.
    F.A. Johnson and R. Loudon: loc. cit. Google Scholar
  214. 7.
    P.A. Temple and CE. Hathaway: Phys. Rev. B7, 3685 (1973).ADSCrossRefGoogle Scholar
  215. 8.
    B. A. Weinstein and M. Cardona: Solid State Comm. 10, 961 (1972).ADSCrossRefGoogle Scholar
  216. 9.
    H. Bilz, R. Geick, and K. Renk, in: Proceedings of the International Conference on Lattice Dynamics, Copenhagen 1963 (ed. R.F. Wallis), p. 355. New York: Pergamon Press 1965.Google Scholar
  217. 10.
    R.J. Collins and H.Y. Fan: Phys. Rev. 93, 674 (1954).ADSCrossRefGoogle Scholar
  218. 11.
    S. J. Fray, F. A. Johnson, and J. E. Quarrington : Proc. Phys. Soc. (London) 85, 153 (1965).ADSCrossRefGoogle Scholar
  219. 12.
    B. A. Weinstein and M. Cardona: Phys. Rev. B7, 2545 (1973).ADSCrossRefGoogle Scholar
  220. 13.
    G. Nilsson and G. Nelin: Phys. Rev. B3, 364 (1971).ADSGoogle Scholar
  221. 13a.
    G. Nelin and G. Nilsson: Phys. Rev. B5, 3151 (1972).ADSCrossRefGoogle Scholar
  222. 14.
    H. Bilz, R. Geick, and K. Renk: loc. cit. [32]. Google Scholar
  223. 15.
    M. Lax: in [32].Google Scholar
  224. 16.
    M.J.M. Musgrave and J.A. Pople: Proc. Roy. Soc. (London), Ser. A268, 474 (1962); a pioneer paper is M.Born: Ann. Phys. 4, 44 (1914).ADSCrossRefGoogle Scholar
  225. 16a.
    Recently this type of model was used by R. Tubino, L. Piseri, and G. Zerbi: J. Chem. Phys. 56, 1022 (1972).ADSCrossRefGoogle Scholar
  226. 17.
    G. Dolling and R.A. Cowley: Proc. Phys. Soc. (London) 88, 463 (1964).ADSCrossRefGoogle Scholar
  227. 17a.
    G. Dolling, R.A. Cowley, and A.D. B. Woods: Can. J. Phys. 43, 1397 (1965).ADSCrossRefGoogle Scholar
  228. 18.
    R.A. Cowley: J. Phys. (Paris) 26, 659 (1965);CrossRefGoogle Scholar
  229. 18a.
    R.A. Cowley: Advan. Phys. 12, 421 (1963)ADSCrossRefGoogle Scholar
  230. 18b.
    R.A. Cowley: Proc. Phys. Soc. (London) 84, 281 (1964).ADSCrossRefGoogle Scholar
  231. 1.
    A. M. Karo and J. R. Hardy : Phys. Rev. 141, 701 (1966).ADSCrossRefGoogle Scholar
  232. 2.
    J.R. Hardy and A.M. Karo, p. 99 in [31]Google Scholar
  233. 1.
    J.R. Hardy and A.M. Karo: Phys. Rev. 141, 703 (1966).Google Scholar
  234. 2.
    E. Burstein, F. A. Johnson, and R. Loudon: Phys. Rev. 139, A 1240 (1965).ADSCrossRefGoogle Scholar
  235. 1.
    H.L. Welsh, M.F. Crawford, and W.J. Staple: Nature 164, 737 (1949).ADSCrossRefGoogle Scholar
  236. 2.
    J.R. Hardy and A.M. Karo: Phys. Rev. 141, 703 (1966).Google Scholar
  237. 3.
    E. Burstein, F.A. Johnson, and R. Loudon: Phys. Rev. 139, A, 1240 (1965).ADSCrossRefGoogle Scholar
  238. 4.
    C. Smart, G.R. Wilkinson, A.M. Karo, and J.R. Hardy, p. 387 in [32j.Google Scholar
  239. 5.
    H. Bilz, L. Genzel, and H. Happ: Z. Physik 160, 535 (1960).ADSCrossRefGoogle Scholar
  240. 5a.
    H. Bilz and L. Genzel: Z. Physik 169, 53 (1962).ADSCrossRefGoogle Scholar
  241. 6.
    M. Krauzman, p. 109 in [32]. Also M. Krauzman: Thèse de Doctorat D’État ès Sciences Physiques, Fac. des Sciences, Paris, 10 Mars, 1969.Google Scholar
  242. 7.
    J.R. Hardy, A.M. Karo, I. Morrison, CT. Sennett, and J.P. Russell: Phys. Rev. 179, 837 (1969). See esp. p. 838 and the footnote therein.ADSCrossRefGoogle Scholar
  243. 8.
    J.M. Worlock: Private communication.Google Scholar
  244. 9.
    Also the reported work by A. I. Stekhanov and A. P. Corolkov, p. 119 in [32] for which only an abstract exists, seems to agree with the other work reported above.Google Scholar
  245. 10.
    J.R. Hardy, A.M. Karo, I. Morrison, C.T. Sennett, and J. P. Russell: Phys. Rev. 179, 837 (1969).ADSCrossRefGoogle Scholar
  246. 11.
    A.M. Karo and J.R. Hardy: Phys. Rev. 181, 1272 (1969).ADSCrossRefGoogle Scholar
  247. 12.
    E. Burstein, F. A. Johnson, and R. Loudon: Phys. Rev. 139A, 1240 (1965).ADSCrossRefGoogle Scholar
  248. 13.
    M. Krauzman: Thèse (footnote 6, p 424).Google Scholar
  249. 14.
    A.D. Bruce and R.A. Cowley: J. Phys. C. Solid state Phys. 5, 595 (1972)ADSCrossRefGoogle Scholar
  250. 14a.
    A.D. Bruce, ibid, 5, 2909 (1972);- ibid, 6, 174 (1973).Google Scholar
  251. 1.
    M. Krauzman: Thèse (footnote 6, p. 424).Google Scholar
  252. 2.
    L. Kleinman: Solid State Communications 3, 47 (1967) has given some examples of calculation of depolarization of Raman scattering. Kleinman’s results are based upon obtaining partial reduction coefficients by what amounts to the method based on use of a partial linear vector space. But in certain cases his results contradict those obtainable using the full basis (full group method). It seems that Kleinman did not verify the completeness of his partial vector space, and so his work is in error. Recall here the caution discussed on p. 358, Eq. (129.51), et seq. Google Scholar
  253. 3.
    P.A. Temple and CE. Hathaway: Phys. Rev. B7, 3685 (1973).ADSCrossRefGoogle Scholar
  254. 4.
    B. A. Weinstein and M. Cardona: Solid State Comm. 10, 961 (1972).ADSCrossRefGoogle Scholar
  255. 5.
    B.A. Weinstein and M. Cardona: Phys. Rev.B7, 2545 (1973).ADSCrossRefGoogle Scholar
  256. 1.
    H. Winston and R.H. Halford: J. Chem. Phys. 17, 607 (1950) (Site group).ADSCrossRefGoogle Scholar
  257. 1.
    R. Loudon: Proc. Phys. Soc. (London) 84, 379 (1964).ADSCrossRefGoogle Scholar
  258. 1.
    A.A. Maradudin: Theoretical and experimental aspects of the effects of point defects and disorder as the vibrations of crystals, in: Solid State Physics (eds. F. Seitz and D. Turnbull), Part 1, Vol. 18, Part 2, Vol. 19. New York: Academic Press 1966.Google Scholar
  259. 2.
    I.M. Lifschitz: Soviet Phys. Usp. 7, 549 (1965).ADSCrossRefGoogle Scholar
  260. 2a.
    From A.A. Maradudin: Reports. Prog. Phys. 28, 378 (1969), Table 2.Google Scholar
  261. It is assumed that the coordination number of the impurity site is 4, and that the impurity interacts with no more than its four nearest neighbours. The components of the position vectors {x(lk)} of the nearest neighbours to the impurity are given in units of one quarter of the lattice parameter for the diamond structure.Google Scholar
  262. 3.
    W. Ludwig, in: Ergeb. der exakten Naturwissenschaften (ed. S. Flügge and F. Trendelenburg), Bd. 35. Berlin-Heidelberg-New York: Springer 1964.Google Scholar
  263. 1.
    I. M. Lifschitz: Soviet Phys. Usp. 7, 549 (1965).ADSCrossRefGoogle Scholar
  264. 2.
    A.G. Dawber and R.J. Elliott: Proc. Roy. Soc. (London), Ser. A273, 222 (1963).ADSMATHCrossRefGoogle Scholar
  265. 3.
    A.A. Maradudin: Footnote l, Sect. 150Google Scholar
  266. 1.
    M. Balkanski and W. Nazariewicz: J. Phys. Chem. Solids 25, 437 (1964); 27, 671 (1966).ADSCrossRefGoogle Scholar
  267. 2.
    J.F. Angress, S.D.Smith, and K.F.Renk, in [32], p.467. — J.F. Angress, A.R. Goodwin, and S.D. Smith: Proc. Roy. Soc. (London), Ser. A287, 64 (1965).ADSCrossRefGoogle Scholar
  268. 3.
    L. Bellomante and M.H.L. Pryce: Proc. Phys. Soc. (London) 89, 967, 973 (1966); et seq. ADSCrossRefGoogle Scholar
  269. 4.
    R.J. Elliott and P. Pfeuty, in [34], p. 193.Google Scholar
  270. 5.
    Work on the U center is actively going on. See H. Bilz, R. Zeyher, and R.K. Wehner: phys. stat. solidi 20, K167 (1967). Also see [50]. 6 See also M.V. Klein, in [34].ADSCrossRefGoogle Scholar
  271. 1.
    N.X. Xinh (thesis unpublished); referred to in A. A. Maradudin, footnote 1, Sect. 150, and also Westinghouse Research Laboratories paper 65–9 F 5–442, p. 8.Google Scholar
  272. 2.
    A. Ganguly and J.L. Birman, in [33].Google Scholar
  273. 3.
    E. Mulazzi: Phys. Rev. Letters 25, 228 (1970).ADSCrossRefGoogle Scholar
  274. 4.
    R.S. Leigh and B. Szigetti, in [33], p. 477.Google Scholar
  275. 2.
    H. Hartmann: Phys. Rev. 147, 663 (1966).ADSCrossRefGoogle Scholar
  276. 3a.
    E. Anastassakis and E. Burstein: J. Phys. Chem. Solids 32, 313, 563; 33, 519 (1972).CrossRefGoogle Scholar
  277. 3b.
    E. Anastassakis, S. Iwasa, and E. Burstein: Phys. Rev. Letters 17, 1051 (1966).ADSCrossRefGoogle Scholar
  278. 4.
    J. M. Worlock and P. Fleury: Phys. Rev. Letters 19, 1176 (1967)ADSCrossRefGoogle Scholar
  279. 4a.
    J. M. Worlock and P. Fleury: Phys. Rev. Letters 18, 665 (1967). Also J. M. Worlock, in [33], p. 415.ADSCrossRefGoogle Scholar
  280. 5.
    V. Dvorak: Phys. Rev. 159, 652 (1967).ADSCrossRefGoogle Scholar
  281. 6.
    Such effects were reported in CdS on the magnetic field dependence of the Raman scattering by T.C. Daman and J. Shah: Bull. Am. Phys. Soc, Ser. 2, 16, 29 (1971), Abstract AK10.Google Scholar
  282. 7.
    F. Cerdeiri, C.J. Buchenauer, F.H. Pollak, and M. Cardona: Bull. Am. Phys. Soc, Ser. 2, 16, 29 (1971), Abstract AK9.Google Scholar
  283. 8.
    G.S. Hobson and E.G.S. Paige: Proc. Phys. Soc. (London) 88, 437 (1966).ADSCrossRefGoogle Scholar
  284. 9.
    L.B. Humphreys and A.A. Maradudin: Solid State Comm. 11, 1003 (1972). 10 See ref. 2, p. 371 and references to Sect. 60.ADSCrossRefGoogle Scholar
  285. 1.
    L.P. Bouckaert, R. Smoluchowski, and E.P. Wigner: Phys. Rev. 50, 58 (1936).ADSMATHCrossRefGoogle Scholar
  286. 2.
    F. Seitz: Ann. Math. 37, 17 (1936).MathSciNetCrossRefGoogle Scholar
  287. 3.
    E.P. Wigner: Göttinger Nachrichten, Math.-Phys. Klasse, 546 (1932).Google Scholar
  288. 4.
    R.B. Barnes, R. Brattein, and F. Seitz: Phys. Rev. 48, 582 (1935).ADSMATHCrossRefGoogle Scholar
  289. 5.
    J. Neubuser: Proc. Und Colloquium on Group Theory in Physics, Nijmegen, June 1973 (ed. A. Janner). Available from Prof. J anner, University of Nijmegen, Netherlands.Google Scholar
  290. 6.
    T.G. Worlton and J.L. Warren: Computer Phys. Comm. 3, 88 (1972).ADSCrossRefGoogle Scholar
  291. 6a.
    J.L. Warren: Rev. Mod. Phys. 40, 38 (1970).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Joseph L. Birman
    • 1
  1. 1.The City College of the CityUniversity of New YorkNew YorkUSA

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