Rhoda Berenson

Clebsch−Gordan coefficients for crystal space groups

A practical method for calculating Clebsch−Gordan coefficients for crystal space groups is presented. It is based on properties of the space group irreducible representations as induced from ray representations of subgroups. Using this method, we obtain all Clebsch−Gordan coefficients for a family of representations in a single calculation: For space groups, for a given triangle of stars *k, *k′, *k″, where *k ⊗ *k′ ‐ *k″, the coefficients for all allowable little group representations l, l′, l″ are obtained. In the following paper this is applied to rocksalt O5h−Fm3m and diamond O7h−Fd3m space groups.

Clebsch−Gordan coefficients for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5n−Fm3m

By using the method described in the previous paper, based on properties of space group irreducible representations as induced from ray representations of subgroups, Clebsch−Gordan coefficients are calculated for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5h−Fm3m structures. Tables of coefficients for these stars are presented. An example of explicit calculation of the coefficients is given for these symmorphic and nonsymmorphic groups with multiplicity included in the former.

Molien generating functions, invariants and covariants of magnetic point groups

The general properties of the Molien generating function and invariant and covariant tensors for the corepresentations of nununitary point groups are presented. The generating functions and (D1,Dm) (invariant) and (Dr,Dm) (covariant) tensors are obtained for the 32 grey point groups and the 58 black‐white point groups.

Tensor fields in crystals and colour groups

The theory of permutational colour groups is used to construct tables of the k = 0 irreducible representations whose basis functions are linear combinations of the components of tensor fields defined on the atoms of a crystal. As examples of their use, these tables are shown to be applicable in determining the k = 0 vibrational modes of a crystal and in applying the Tensor Field Criterion in the Landau theory of continuous phase transitions.

Invariants and covariants of corepresentations of magnetic point groups

Invariants and covariants have been obtained for the 32 grey point groups and the 58 black-white point groups.