Tensor properties and rotational symmetry of crystals. II. Groups with 1-, 2- and 4-fold principal symmetry and trigonal and hexagonal groups different from group 3

Abstract

The first part of the paper emphasizes that the problem of the effect of the rotational symmetry of crystals on their tensor properties is completely solved for the groups of 1-, 2- and 4-fold principal symmetry since simple general formulas can be given which provide the schemes of a (polar or axial) general tensor of any rank in these groups, thus yielding a closed-form solution. These formulas are derived both by the new method of vector representatives [introduced in paper I: Fumi & Ripamonti (1980). Acta Cryst. A36, 535-551] and by the direct-inspection method. In the second part, it is emphasized that simple general rules can be given to obtain the schemes of a (general or particular, polar or axial) tensor of any rank in the trigonal and hexagonal groups other than group 3 from the corresponding scheme in group 3(3z). These rules are given directly by the formulas obtained in the first part for the groups (or generators) of order 2. These compact formulas and rules are applied to two specific tensor properties discussed in recent literature, pointing out errors in some of the reported schemes. Brief discussions are finally given of various techniques to obtain the tensor schemes in the cylindrical and spherical groups, in particular of the new methods introduced in paper I.