Textures and Microstructures

Volume 25, Issue 2-4, Pages 71-108

http://dx.doi.org/10.1155/TSM.25.71

## Physical Versus Mathematical Aspects in Texture Analysis

Department of Physical Metallurgy, Technical University of Clausthal, Germany

Received 4 December 1995

Copyright © 1996 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Texture deals with the orientational aspects of the crystal lattice in polycrystalline aggregates. This comprises the classical orientation distribution function ODF as well as higher-order textural quantities. The quantitative treatment of these quantities requires a good deal of mathematical methods. This concerns particularly the representation of orientation data including all kinds of symmetries, the transformation of experimentally measured raw data into the required distribution functions, as well as mathematical models of texture formation by physical processes and of the texture-property relationship.

When physical facts are idealized in terms of a mathematical description or by mathematical models then the physical limits, within which these are valid, must be known. Such physical limits are, for instance, definition of crystal orientation by the crystal lattice which leads to an unsharpness relationship between location and orientation resolution as well as a relationship between statistical relevance and angular resolving power. Other physical limits are given by the “fuzzyness” of sample symmetry. A central problem is pole figure inversion i.e. the inversion of the projection equation. This problem may have a “physical” solution even if it does not have a mathematical one. Finally, in the problem of rationalizing orientation distribution functions in terms of a low number of “components”, mathematical aspects may be quite different from physical ones.

In all these problems it is thus necessary to keep the mathematical aspects apart from physical aspects.