Reciprocal Space and Irreducible Representations of Space Groups
Many aspects of the physics and chemistry of crystalline solids require the use of group theory, for example, the study of band theory, of phonon theory and infra-red and Raman active modes, of soft modes and of the Landau theory of phase transitions are all based fundamentally upon the theory of space groups and their irreducible representations. The set of pure translational symmetry operations (ε/Ti) is a subgroup of the space group of a three dimensional crystalline solid, and this leads naturally to an initial search for the irr. reps, of this subgroup. A representation is a set of matrices which are associated with the symmetry operations (in this case the pure transitions) of the group and which multiply like the operations, i.e., if Ψ1 is a matrix associated with T1 and Ψ2 is associated with T2 then Ψ2Ψ1 associated with T1 + T2. It is not necessary that Ψ1 ≠ Ψ2.
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