Reciprocal Space and Irreducible Representations of Space Groups

Abstract

The set of pure translational symmetry operations { ε |≠_i} is a subgroup of the space group of a three dimensional crystalline solid. It is therefore meaningful to seek irreducible representations (irr. reps) and basis functions for this pure translational subgroup, and these play an important role in the theory of crystalline solids. For the benefit of the reader unfamiliar with group theory, a set of basis functions for a representation is made up of functions which transform into each other, or linear combinations thereof, under symmetry operations of the group. The irr. rep. is then the set of matrices which transform the functions under the symmetry operations (a representation) when the set cannot be reduced in the sense that a coordinate transformation results in splitting the basis functions into two or more sets of new functions each of which is a set of basis functions for a representation.