Recasting Classical Expansion of Orientation Distribution Function as Tensorial Fourier Expansion

In classical texture analysis the orientation distribution function (ODF) is expanded as an infinite series of Wigner D -functions D-mn(l) (.), with the classical texture coefficients c(mn)(l) serving as undetermined expansion parameters. Herein we present a systematic procedure by which the classical expansion of an ODF (with arbitrary crystallite and texture symmetry) truncated at the order / = L can be directly rewritten as a tensorial Fourier expansion truncated at the same order, where the same set of classical texture coefficients c(mn)(l) appear as undetermined parameters. The crucial step of this procedure is as follows: Let H-1 be the complexified space of l-th order harmonic tensors and .,. the Hermitian inner product onz. Let R be a rotation and R-circle times l : H-1 -> H(1)be the l-th order tensor product of R with itself. For each 1 <= 1 <= L, determine an orthonormal irreducible tensor basis B-m(l) (-1 <= m <= 1) in H-1 so that D-mn(l) (R) = B-m(l), R-circle times l B-n(l)) for each rotation R. We have determined explicit sets of irreducible basis tensors B-m(l) in H-1 for 1 <= 1 <= 8, which are displayed in tables in this paper. In addition to examples which illustrate the general procedure, we provide, for orthorhombic aggregates of cubic and hexagonal crystallites, explicit expressions of tensorial Fourier expansions of the ODF up to terms with 1 = 8, in which the tensorial texture coefficients are given in terms of the classical texture coefficients and the basis tensors B-m(l) .

Automated summary

In this paper, a systematic procedure for rewriting the classical expansion of orientation distribution function (ODF) as a tensorial Fourier expansion is presented. The researchers also provide explicit expressions for specific crystallite structures and show the irreducible basis tensors used in the process.