Matter tensors in the crystallographic groups of cartesian symmetry

Abstract

I. Many recent publications in crystals physics give the components of matter tensors in the various crystallographic groups. We think it is useful to complete the known results with a table of the components of general matter tensors of polar and axial nature, of second, third and fourth order, for the 20 Cartesian groups: triclinic, monocl~ic, orthorhombic, tetragonal and cubic. Our results are derived by means of the method of direct inspection, which has been explained elsewhere 1), i). We shall give here a very simple example. In Cartesian frames, under coordinate transformations of the first kind, tensor components transform as (non-commutative) products of coordinates: under coordinate transformations of the second kind, the components of polar tensors transform also as coordinate products, however, the transformation equations for components of axial tensors differ from those for products of coordinates by a factor -1. Let us consider two second-order tensors, respectively of polar and axial nature, in the point group m. We choose a Cartesian reference frame with the y axes parallel to the inversion diad: the symmetry transformation which corresponds to the inversion diad is then