Crystallographic Texture and Group Representations

This exposition consists of three parts. Part I is an introduction to classical texture analysis. The harmonic method and the approach initiated by Roe, where the orientation distribution function (ODF) is always defined on the rotation group SO(3), is emphasized and given a systematic treatment. Basic concepts (e.g., the orientation density function) are made precise through their mathematical definition. The active view of rotations is implemented throughout. A conscientious effort is made to use machinery already available in mathematics and physics. TheWigner D-functions, whose properties are familiar in physics, are used instead of Bunge's and Roe's versions of generalized spherical harmonics. By including three mathematical appendices, it is hoped that engineering students would find Part I readable. The objectives of Parts II and III are threefold, namely: (i) To delve deeper into the mathematical foundations of the harmonic method. The Weyl method is used to prove that the Wigner D-functions are the matrix elements of a complete set of pairwise-inequivalent, continuous, irreducible unitary representation of SO(3). General formulas of the Wigner D-functions, valid for any parametrization of SO(3), are derived. An elementary proof (attributed to Wigner) of the Peter-Weyl theorem is presented. (ii) To provide mathematical prerequisites in group representations for research on representation theorems that delineate the effects of crystallographic texture on material properties defined by tensors or pseudotensors. (iii) To introduce tensorial Fourier expansion of the ODF and the tensorial texture coefficients. The classical ODF expansion in Wigner D-functions is recast as a special tensorial Fourier series. The relation between the tensorial and classical texture coefficients in this context is derived.

Automated summary

This article introduces the basic concepts and mathematical foundations of classical texture analysis, focusing on the harmonic method and the approach initiated by Roe. The orientation distribution function (ODF) is defined on the rotation group SO(3), and the Wigner D-functions are used instead of generalized spherical harmonics. The article also discusses tensorial Fourier expansion of the ODF and tensorial texture coefficients.