Theory of crystal space groups and lattice dynamics : infra-red and Raman optical processes of insulating crystals

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Bibliographic Information

Theory of crystal space groups and lattice dynamics : infra-red and Raman optical processes of insulating crystals

Joseph L. Birman

Springer-Verlag, 1984, c1974

  • : Germany
  • : U.S.

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Note

"Reissue of Encyclopedia of physics/Handbuch der Physik, vol. xxv/2b"--Pref

Bibliography: p. [507]-509

Includes indexes

Volume editor: L. Genzel, chief editor: S. Flügge

Description and Table of Contents

Description

Reissue of Encyclopedia of Physics/Handbuch der Physik, Vol. XXV/2b I am very pleased that my book is now to be reprinted and rebound in a new format which should make it accessible at a modest price to students and active researchers in condensed matter physics. In writing this book I had in mind an audience of physicists and chemists with no previous deep exposure to symmetry analysis of crystalline matter, non to the use of symmetry in simplifying and refining predictions of the results of optical experiments. Hence the book was written to explain and illustrate in all necessary detail how to: 1) describe the space group symmetry in terms of space group symmetry operations; 2) obtain irreducible representations and selection rules for optical infra-red and Raman and other transition processes. On the physical side I redeveloped the traditional theory of classical and quantum lattice dynamics, illustrating how space-time symmetry designations in the equations of motion can: 1) simplify and rationalize calculations of the classical eigenvectors of the dynamical equation; 2) permit classification of the eigenstates of the quantum lattice-dynamic pro blem; 3) give specific selection rules for optical infra-red and Raman lattice processes, and thus make "go, no-go" predictions including polarization of absorbed or scattered radiation; and 4) simplify the modern many-body theories of optical processes.

Table of Contents

  • Theory of Crystal Space Groups and Infra-Red and Raman Lattice Processes of Insulating Crystals.- A. Scope and plan of the article.- 1. General survey.- 2. Plan of the article: An overview.- B. The crystal space group.- 3. Crystal symmetry - Introduction.- 4. The translation subgroup of a crystal.- ?) Translation operators {? | RL}.- ?) The translation group T.- ?) Structure of T.- ?) Born-Karman boundary conditions.- ?) A property of the {? | t].- 5. Rotational symmetry elements: The crystal point group.- ?) Rotational operators {?|0}.- ?) The point group P.- 6. General symmetry element in a crystal: Space group G.- ?) The operator {?|t(?)}.- ?) Group property of the set {?|t(?)}.- ?) Compatibility of rotation and translation.- ?) The operator {?|t} in non-Cartesian axes.- ?) Order of the space group G.- ?) Normality of translation subgroup T.- ?) Factor group.- ?) Site symmetry.- 7. The space group G as a central extension of T by P.- 8. Symmorphic space groups.- 9. Non-symmorphic space groups.- 10. Some subgroups of a space group.- C. Irreducible representations and vector spaces for finite groups.- 11. Introduction.- 12. Transformation operators on functions.- 13. Group of transformation operators on functions.- 14. Functions and representations.- 15. Irreducible representations and spaces.- 16. Idempotent transformation operators.- 17. Direct products.- ?) Direct products of representations.- ?) Reduction coefficients.- ?) Irreducible representations of direct product groups.- 18. Clebsch-Gordan coefficients.- D. Irreducible representations of the crystal translation group T.- 19. Introduction.- 20. Irreducible representations of T.- 21. The reciprocal lattice.- 22. Irreducible representations of T = T1 ? T2 ? T3.- 23. Wave vector: First Brillouin zone.- 24. Completeness and orthonormality for D(k).- 25. Irreducible vector spaces for T: Bloch vectors.- 26. Direct products in T.- E. Irreducible representations and vector spaces of space groups.- 27. Introduction.- 28. Irreducible representation D(?k)(m) of G.- 29. Representation of T subduced by D(?k)(m) of G.- 30. Transformation of Bloch vectors by rotation operators.- 31. Conjugate representations of T.- 32. Characterization of the subduced representation.- 33. Block structure of D(?k)(m) of G.- 34. Group of the canonical k: G(k).- 35. Irreducibility of the acceptable representations $${D^{\left( {<!-- -->{k_1}} \right)\left( m \right)}}$$ of G(k1).- 36. D(?k)(m) of G induced from D(kl) (m) of G(k1).- 37. Characters of D(?k)(m) of G
  • induced characters.- 38. Allowable irreducible D(k)(m) : General star with G(k1)= T.- 39. Allowable irreducible D(k)(m): Special star: Little group technique.- 40. Non-allowable irreducible D(k)(?): Little group technique.- 41. Allowable irreducible D(k)(m) as ray representations.- 42. Ray representations of P(k): The covering group P?(k).- 43. Gauge transformations of ray representations.- 44. Relationship between little group and ray representation methods.- 45. Full D(?k)(m) for symmorphic groups: Illustration.- 46. Full D(?k)(m) for non-symmorphic groups.- 47. Complete set of all D(?k)(m) for a space group.- 48. Verification of completeness of D(?k)(m).- 49. Verification of orthonormality relations for D(?k)(m).- 50. Induction of D(k)(m) from sub-space groups.- 51. Compatibility relations for D(?k)(m) and subduction.- F. Reduction coefficients for space groups: Full group methods.- 52. Introduction.- 53. Direct product D(?k)(m) ? D(?k')(m').- 54. Symmetrized powers [D(?k)(m)](p).- ?) Ordinary Kronecker powers.- ?) Symmetrized Kronecker powers.- 55. Definition of reduction coefficients.- 56. Wave vector selection rules.- ?) Star reduction coefficients for the ordinary product.- ?) Star reduction coefficients for the symmetrized product.- 57. Determination of reduction coefficients: Method of linear algebraic equations...- 58. Determination of reduction coefficients: Method of the reduction group.- 59. Determination of reduction coefficients: Use of basis functions.- 60. Theory of Clebsch-Gordan coefficients for space groups.- G. Reduction coefficients for space groups: Subgroup methods.- 61. Introduction.- 62. Complete subgroup character system.- 63. Subgroup reduction coefficients.- 64. Comparison of full group and subgroup methods.- 65. Reduction coefficients: A little group technique.- H. Space group theory and classical lattice dynamics.- 66. Introduction.- 67. Equations of motion in the harmonic approximation.- 68. Translation symmetry and particle displacements.- 69. Translation symmetry and force matrix.- 70. General symmetry and particle displacements.- 71. General symmetry and force matrix.- 72. Solution of the equations of motion: Eigenvectors [ej].- 73. Real normal coordinates qj.- 74. Crystal symmetry and the eigenvectors [ej] of [D].- 75. Necessary degeneracy of the eigenvectors [ej].- 76. Crystal symmetry and the transformation of normal coordinates qj.- 77. Fourier transformations.- 78. Fourier transformed displacements and force matrix: The dynamical matrix [D(k)].- 79. Eigenvectors of the dynamical matrix [D(k)].- 80. Complex normal coordinates.- 81. Crystal symmetry and the dynamical matrix [D(k)] and its eigenvectors.- 82. Eigenvectors of [D(k)] as bases for representation D(k)(e) of G(k).- 83. Eigenvectors of [D(k)] as bases for representations D(k)(j) of G(k).- 84. Equivalence of D(k)(e) and D(k)(j).- 85. Necessary degeneracy under G(k) and the eigenvectors of [D(k)].- 86. Complex normal coordinates $$Q\left( {\begin{array}{*{20}{c}} k \\ {<!-- -->{j_\mu }} \end{array}} \right)$$ as bases for the representation D(k)(j) of G(k).- I. Space-time symmetry and classical lattice dynamics.- 87. Introduction.- 88. The antilinear, antiunitary transformation operator K and time reversal.- 89. The complete space-time symmetry group G.- 90. Eigenvectors $$e\left( {\left| {\begin{array}{*{20}{c}} k \\ {<!-- -->{j_\lambda }} \end{array}} \right.} \right)$$ and normal coordinates $$Q\left( {\begin{array}{*{20}{c}} k \\ {<!-- -->{j_\lambda }} \end{array}} \right)$$ as bases for representation of G.- 91. Necessary degeneracy under the full space-time crystal symmetry group G.- 92. Test for reality of D(?k)(j) of G.- 93. Simplification of the reality test of D(?k)(m).- 94. Classification of D(?k)(m) according to reality by use of a new test.- 95. Physically irreducible representations of G as corepresentations of G.- 96. Structure of corepresentations of G: The costar, co ?k.- 97. Corepresentations of G: Class III costar.- 98. Corepresentations of G: Class II costar and general theory.- 99. Corepresentations of G: Class I costar.- 100. Acceptable irreducible corepresentations of G(k) as irreducible ray corepresentations.- 101. Complex normal coordinates as bases for irreducible corepresentations of G.- 102. Eigenvectors of D(k) as bases for irreducible corepresentations of G.- 103. Determination of actual normal mode symmetry in a crystal.- 104. Determination of eigenvectors $$e\left( {\left| {\begin{array}{*{20}{c}} k \\ j \end{array}} \right.} \right)$$ by symmetry: Factorization of the dynamical matrix.- J. Applications of results on symmetry adapted eigenvectors in classical lattice dynamics.- 105. Introduction.- 106. Tensor calculus for lattice dynamics.- ?) Effect of unitary elements.- ?) Effect of antiunitary elements.- 107. Critical points.- ?) Representation theory for the "symmetry set".- ?) Determination of potential critical points by point symmetry.- 108. Compatibility or connectivity theory for representations.- 109. Construction crystal invariants.- ?) The crystal Hamiltonian: Harmonic and anharmonic.- ?) Force constant coupling parameters.- ?) Anharmonic terms in the potential.- 110. Construction of crystal co variants: Electric moment and polarizability.- K. Space-time symmetry and quantum lattice dynamics.- 111. Introduction.- 112. The many-body electron-ion Hamiltonian.- 113. Born-Oppenheimer adiabatic approximation.- 114. Normal coordinates and quantization.- 115. Lattice eigenfunctions in harmonic adiabatic approximation.- 116. Symmetry of harmonic lattice eigenfunctions: Introduction.- 117. Transformations of products of Hermite polynomials: Symmetrized Kronecker product.- 118. Transformation of the lattice eigenfunction: Summary and generalities.- L. Interaction of radiation and matter: Infra-red absorption and Raman scattering by phonons.- 119. Introduction.- 120. Infra-red absorption by phonons.- ?) Semi-classical radiation theory for ions and electrons.- ?) Transition rate.- ?) Analysis of the transition matrix element for infra-red lattice absorption.- ?) Symmetry of the matrix element for infra-red absorption.- ?) One phonon and multiphonon processes.- 121. Raman scattering by phonons: Generalized Placzek theory.- ?) Hamiltonian.- ?) Transition rate for scattering.- ?) Simplification of the scattering matrix elements for an insulator.- ?) Symmetry of the Raman scattering matrix element.- ?) One phonon and multiphonon processes.- ?) The A * A term in scattering Hamiltonian.- 122. A mutual exclusion selection rule for certain two phonon overtones in infra-red and Raman processes in crystals with an inversion.- 123. Polarization effects in infra-red and Raman lattice processes.- ?) Raman tensor and Clebsch-Gordan coefficients.- ?) Polarization effects in Raman scattering: The Raman tensor in a cubic crystal with inversion.- ?) Polarization effects due to macroscopic electric field.- i) Cubic crystals with inversion symmetry.- ii) Cubic crystals without inversion.- ?) Polarization effects and two phonon bound states.- ?) Polarization in infra-red absorption due to anisotropy.- 124. Aspects of modern quantum theories of lattice Raman scattering and infra-red absorption.- ?) Many-body polarizability theory of Raman scattering.- ?) Many-body theory of infra-red absorption.- ?) Group theory and the thermal phonon Green functions.- ?)Microscopic theory of Raman scattering: Bloch picture.- ?) Microscopic theory of Raman scattering: Exciton picture.- ?) Microscopic theory of Raman scattering: Polariton picture.- ?) Resonance Raman scattering and symmetry breaking.- M. Group theory of diamond and rocksalt space groups.- 125. Introduction.- 126. Geometry of the rocksalt and diamond space groups.- 127. Irreducible representations in rocksalt.- 128. Some wave vector selection rules in rocksalt.- 129. Reduction of ?X(4?) ? ?X(5?) in rocksalt.- 130. Reduction of ?L(3?) ? ?L(3+) in rocksalt.- 131. Additional reduction coefficients in rocksalt.- 132. Irreducible representations D(?)(m), D(?X)(m)D(?L)(m) in diamond.- 133. Reduction coefficients.- ?) Products D(?)(m), D(?X)(m)D(?L)(m) in diamond.- ?) Additional reduction coefficients in diamond.- 134. Clebsch-Gordan coefficients in diamond structure for D(?X)(m) ? D(?X)(m').- 135. Test of effect of time reversal symmetry in diamond and rocksalt structure.- 136. Connectivity and labelling of irreducible representations in diamond and rocksalt structures: Consequences for selection rules.- N. Phonon symmetry, infra-red absorption and Raman scattering in diamond and rocksalt space groups.- 137. Introduction.- 138. Phonon symmetry in rocksalt and diamond.- 139. Compatibility and phonon symmetry in diamond and rocksalt.- 140. Critical points for phonons in diamond structure: Germanium, silicon and diamond.- Diamond structure: Point ?.- Diamond structure: Point X1.- Point L1.- Point W1.- Line ?.- Line Q.- 141. Two phonon density of states and critical points in diamond structure.- 142. Interpretation of lattice Raman and infra-red spectra in crystals of the diamond structure.- Diamond.- Silicon.- Germanium.- C, Si, Ge.- 143. Symmetry set of critical points in rocksalt structure.- 144. Two phonon density of states and critical points in rocksalt-NaCl.- 145. Interpretation of lattice Raman and infra-red spectra in some rocksalt structure crystals.- NaCl.- NaF.- Other alkali halides.- 146. Polarization effects in two phonon Raman scattering in rocksalt and diamond structures.- ?) Rocksalt.- ?) Diamond.- O. Some aspects of the optical properties of crystals with broken symmetry: Point imperfections and external stresses.- 147. Introduction.- 148. Symmetry group of the imperfect crystal with a point defect.- 149. "Band" phonons in imperfect diamond and rocksalt crystals.- 150. Local phonons in imperfect diamond and rocksalt crystals.- 151. Dynamical aspects of perturbed crystal vibrations.- 152. Infra-red absorption in the perturbed system.- 153. Raman scattering in the perturbed system.- 154. Symmetry breaking and induced lattice absorption and scattering.- ?) Symmetry breaking.- ?) Morphic Effects.- P. Respice, adspice, prospice.- Q. Acknowledgements.- Appendix A: Complete tables of reduction coefficients-selection rules for rocksalt structure Of Oh5 (Tables A.1 to A.11).- Appendix B: Complete tables of reduction coefficients-selection rules for the diamond space group Oh7 (Tables B.1 to B.10).- Appendix C: Illustration of ray representation method: Point X in diamond (Table C.1).- Appendix D: Tables for the zincblende structure: $$F\bar 43m$$
  • Td2 (Tables D.1 to D.10).- References.- Index of key equations.- Index of tables.- Index of figures.- Sachverzeichnis (Deutsch-Englisch).- Subject Index (English-German).

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