Entropy optimization in quantitative texture analysis. II. Application to pole-to-orientation density inversion

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Abstract
Anisotropic behavior of single‐phase polycrystalline material is controlled by its constituent crystal grains and their spatial orientation within the specimen. More specifically, a macroscopic physical property in a given direction is the mean value of the corresponding property of the individual crystallites with respect to the statistical distribution of their orientations. Thus, the statistical orientation distribution is a mathematical approach of describing and quantifying anisotropy. Unfortunately, the orientation density function (ODF) can generally not be measured directly. Therefore, it is common practice to measure pole density functions (PDFs) of several distinct reflections in x‐ray‐ or neutron‐diffraction experiments with a texture goniometer. Recovering an ODF from its corresponding PDFs is then the crucial prerequisite of quantitative texture analysis. This mathematical problem of texture goniometry is essentially a projection problem because the measured PDFs represent integral properties of the specimen along given lines; it may also be addressed as a tomographic problem specified by the crystal and statistical specimen symmetries and the properties of the diffraction experiment itself. Mathematically, it reads as a Fredholm integral equation of the first kind and was conventionally tackled by transform methods. Because of the specifics of the problem, these are unable to recover the part of the ODF represented by the odd terms of the (infinite) series expansion. In this situation, more sophisticated iterative methods, especially finite‐series‐expansion methods were developed within which the required non‐negativity of the ODF to be recovered plays the prominent role. An efficient way to guarantee non‐negativity is to employ entropy optimization.