Theory of Crystal Space Groups and Lattice Dynamics

Infra-Red and Raman Optical Processes of Insulating Crystals

InhaltsangabeTheory 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