Some basic concepts of texture analysis and comparison of three methods to calculate orientation distributions from pole figures

Several methods have been developed which derive the orientation distribution (ODF) in a polycrystalline sample from pole figures measured by X-ray or neutron diffraction techniques. The theoretical backgrounds of the conventional harmonic method, of the vector method and of the method of Williams [J. Appl. Phys. (1968). 39, 4329-4335], Imhof [Textures Microstruct. (1982). 5, 73-86] and Matthies & Vinel [Phys. Status Solidi B. (1982). 112, K111-K120] (WIMV) are reviewed. A quantitative comparison is then made using the same input data and the same computer to evaluate resolution, errors and efficiency. The input data consist of standard functions, Taylor predictions and measured pole figures covering a realistic range of possibilities for both cubic and trigonal crystal symmetries. Comprehensive error criteria are introduced, and it is proposed to use both integral errors (RP) and difference pole figures to assess the quality of the pole-figure inversion. The harmonic method and WIMV are able to reproduce the original pole figures from the ODF within computer roundoff errors. Resolution of the vector method, particularly for low crystal symmetry, is considerably worse owing to the large-volume cells in orientation space. Computing time is optimal for the conventional harmonic method (for medium termination L), slightly worse for WIMV and about an order of magnitude higher for the vector method. Whereas the conventional harmonic method only reproduces the ghost-afflicted part (g) of the ODF, the vector method satisfies automatically the non-negativity criterion; however, only WIMV provides a general (conditional) ghost correction. In the examples chosen the ghost-corrected ODF f(g) closely coincides with the starting model (model with standard functions or Taylor prediction) supporting its physical relevance. An attractive feature of WIMV is that it leads to results of satisfactory quality using fewer pole figures than the harmonic method. This is particularly important for low crystal symmetries. Furthermore, the treatment of incomplete pole figures is straightforward.