General connections for the form of property tensors in the 122 Shubnikov point groups

The 122 Shubnikov point groups (SPGs) are obtained from the 32 ordinary crystallographic point groups (OPGs) by taking time inversion into account. Like the OPGs, the SPGs can be grouped into 11 Laue classes. Tensors can be invariant under space inversion (s tensors), time inversion (t tensors), space-time inversion (u tensors) or all three inversions (i tensors). The restrictions imposed on the form of a property tensor by the SPG of the material under consideration depend, for i tensors, only on the Laue class of the SPG. If these restrictions are known for an i tensor, the corresponding restrictions for s, t and u tensors of the same rank and internal symmetry can be written down immediately for all the 122 SPGs and for all orientations in which the SPG under consideration appears in the corresponding holohedry. These connections provide tests for the forms of tensors given in the literature. A number of corrections and of possible simplifications are pointed out. The results are illustrated by showing how the form of the i tensor describing linear electrogyration determines the form of the piezoelectric t tensor and the piezomagnetic s tensor for all 122 SPGs. Similarly, the form of the t tensor describing quadratic electrogyration is derived explicitly from the i tensor describing the piezooptic effect.