Description of Crystallite Orientation in Polycrystalline Materials. III. General Solution to Pole Figure Inversion

Abstract

A method is presented here by which orientation distribution of crystallites in anisotropic polycrystalline samples can be derived from a set of plane‐normal distributions obtained by x‐ray diffraction measurements. It is the generalization of the similar procedure proposed previously for analysis of samples having fiber texture. It thus represents a completely general solution to the problem of pole figure inversion, applicable to samples having any arbitrary symmetry elements. The plane‐normal distribution function is expanded in a series of spherical harmonics, the coefficients of which, Q_lmⁱ, can be determined by numerical integration of experimental diffraction data. The crystallite distribution function is expanded in a series of generalized spherical harmonics which appear as solutions to the Schrödinger wave equation of a symmetric top. The coefficients of the crystallite distribution function, W_lmn, are then obtained as linear combinations of Q_lmⁱ. Symmetry properties of W_lmn arising from crystallographic or statistical symmetry elements existing in the sample are examined. The methods of estimating the series truncation errors and of minimizing the experimental error by a least‐squares method, previously proposed in connection with fiber texture analysis, are still applicable here with appropriate generalizations. In addition it is shown that the effect of diffraction line broadening due to finite size or imperfection of crystallites can also be allowed for at least approximately.