Space-Time Symmetry of Transport Coefficients

The symmetry properties of linear transport coefficients are derived treating time inversion and spatial transformations on the same footing. The possible presence of a uniform external magnetic field is taken into account. The method used can be applied more generally—to nonlinear transport coefficients, for example. It is shown that the usual Onsager reciprocity relations do not in general apply in practice to magnetic crystals; appropriate generalized Onsager relations are given. The 1651 3-dimensional space groups which exist when time inversion is taken into account fall into three categories: (a) 230 which contain time inversion as an element, (b) 230 which do not involve time inversion, and (c) 1191 which contain time inversion only in combination with spatial transformations; (a) refers to nonmagnetic crystals and (b) and (c) refer to magnetic crystals. Onsager's relations are shown to apply in their usual form to crystals in category (a), not at all to crystals in category (b), and in general only in a modified form to crystals in category (c). As an application, the equations derived which determine the symmetry restrictions are used to obtain symmetry-restricted matrices for the thermogalvanomagnetic coefficients for each of the 1651 space-group symmetries, and the results are tabulated.