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The texture of a polycrysta11ine material is quantitatively defined by the orientation distribution function of the crystallites. In addition, higher-order textural quantities may aiso be considered such as the mis-orientation distribution function which describes the pair distribution of neighbouring crystals, texture fields, and multi-phase textures, thus approaching a comprehensive description of the statistical crysta110graphy of polycrystalline aggregates. Te xtures of materials can be measured mainly by X-ray diffraction but also using neutron and electron diffraction. The original textural data are pole density distribution functions from which the orientation distribution function can be ca\culated. Te xtures have an influence on the properties of materials and they originate from a11 kinds of anisotrop solid-state processes. This establishes the interest in texture in materials science as we11 as in the earth sciences as iIlustrated by some examples.

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density distribution functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-(hkl) superposition, the solution is mathematically not unique. Rather it contains a range of possible solutions (kernel) which is only limited by the positivity condition of the distribution function. The complete distributio 9 functionf(g) can be split into two parts f(g) and f(g) expressed by even-and odd-order terms of the series expansions. For the calculation of the even part f(g), the positivity condition for all pole figures contributes essentially to an 'economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time.

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