Description of the texture by distribution functions on the space of orthogonal transformations. Implications on the inversion centre

2014 In the most general case the orientation distribution function (texture) is a function on the space of orthogonal transformations which splits into a couple of functions on the space of rotations. In the case of non-enantiomorphic crystal symmetry groups only one of these functions is independent. Sample symmetry (especially centrosymmetry) can be achieved in a trivial and a non-trivial way. The first one leads to relations between individual functional values the latter one to integral relations. J. Physique LETTRES 41 (1980) L-543 L-545 Classification Physics Abstracts 81.40E 15 NOVEMBRE 1980, 1. Crystal orientation. The crystallographic texture of a polycrystalline material is the orientation statistic of its crystallites [1]. The orientation of a crystallite may be described by means of the crystalfixed reference frame Kb (consisting of crystallographically equivalent directions in each crystal). The reference frame Kb will be referred to a sample fixed reference frame Ka by the orthogonal transformation 0 which brings Ka into coincidence with Kb where 0 is an element of the group 0(3) of orthogonal transformations in three dimensions. Because of crystal symmetry there may be other reference frames K’ b related to Kb by an element t)b of the crystal symmetry group Ob which are indistinguishable from Kb where =means equivalent or indistinguishable. Because of the statistical sample symmetry the reference frame Ka will be equivalent with other frames Ka. This is the case if the orientation statistic is the same from the viewpoints of Ka and Ka. The two reference systems are related by an operation 0~ of the sample symmetry group Oa Hence, crystal orientation may be described by the double coset of all equivalent orthogonal transformations induced by crystal and sample symmetry. 2. The texture function. The texture function /(0) is defined by the volume fraction d~(0)/~ of crystals in orientation 0, i.e. a function which images the space 0(3) of orthogonal transformations onto the space R+ of real positive numbers which can be symbolically expressed by Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019800041022054300 L-544 JOURNAL DE PHYSI~JUE LETTRES in such a way that the element f(O) of R+ corresponds to the element 0 of 0(3). Because of the symmetry relation equation (4) the texture function must fulfil the invariance conditions An orthogonal transformation (9 is either a rotation g (determinant + 1) or the product of a rotation (including the identity E) by the inversion centre I (determinant 1). These two kinds of elements cannot be transformed continuously one into the other. Hence, the space 0(3) is said to consist of two connexe components containing the elements with determinant + 1 and 1 respectively. The group 0(3) of orthogonal transformations is the cartesian product of the group of rotations SO(3) in three dimensions and the group of the elements E (the identity) and I (the inversion centre). The texture function f is thus decomposed into a couple of functions f and fl 1 defined on the space of rotation SO(3) each