Graphically oriented method for obtaining atomic displacement fields in crystals from irreducible representations of space groups

A method for obtaining distortion fields in a crystal from a given irreducible representation of the underlying space group is described. The method, based on projection operators of group theory, is graphically oriented and therefore calculation-free. For an example (space group P4¯2₁m), complete sets of representation matrices are analytically calculated for all irreducible representations which correspond to all wave vectors of the form k=(q,q,0). All 96 linear independent atomic displacement modes in the (3×3×1) supercell, which are induced by the two irreducible representations with k=(13,13,0), are explicitly determined: the obtained atomic displacement fields are plane waves with wave vector k.