The influence of crystal and sample symmetries on the orientation distribution function of the crystallites in polycrystalline materials

In the most general case the orientation distribution of crystals in a polycrystalline sample is to be described by a function of orthogonal transformations which splits up into two functions of rotations corresponding to right- and left-handed crystals. The properties of these functions are influenced by crystal and sample symmetry. The rotational subgroup of crystal symmetry leads to symmetry relations which may be written in the form of selection rules. Elements of the second kind of the crystal symmetry give rise to a determinability condition, according to which the texture function may be split into a part (g) which can be determined from polycrystal diffraction experiments and a part (g) which cannot. The determinability condition may take on three different forms according to whether the crystal symmetry contains a centre of inversion, a mirror plane or a inversion axis. In the case of normal scattering the Laue symmetry is to be considered instead of the true crystal symmetry. The sample symmetry is to be described by a black-white or Shubnikov group containing four kinds of elements which give rise to four kinds of symmetry conditions in the function f(g). The sample symmetry may be a conventional one consisting of one-to-one relationships between crystal orientations. It may, however, also be a non-conventional one defined by an integral relation between an infinite number of crystal orientations.