Recovering the Orientation Distribution Function with Maximum Entropy from Experimental Pole Distribution Functions

Abstract

Anisotropic behavior of singlephase 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 orientation distribution is a mathematical approach of describing and quantifying anisotropy. Unfortunately, the orientation distribution function (odf) can generally not be measured directly. Therefore it is common practice to measure pole distribution functions (pdfs) of several distinct reflexions in X-ray or neutron diffraction experiments with a texture goniometer. Recovering the 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 tomographical problem specified by the crystal and statistical specimen symmetries and the properties of the diffraction experiment. Mathematically it reads as a Fredholm integral equation of the first kind and was conventionally tackled by transform methods. Due to 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, finite series expansion, or discrete, methods have been developed primarily because they are capable of incorporating additional information, e.g. the non-negativity of the odf to be recovered, as truly constitutive elements of an inversion procedure. This approach leads to a system of linear equations which is large, sparse, structureless, highly (column) rank deficient, and often inconsistent. Its entropy maximizing solution is proposed here because it yields the least presumptive odf with the smallest information content consistent with the experimental data thus avoiding artificial components, especially “ghosts” caused by the specific properties of the diffraction experiment. With respect to the Backus-Gilbert formalism it provides the most safely balanced solution. Therefore, it allows interpretation in geoscientific terms.