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    The evaluation of grain orientation data
    Pole Figures and ODF

    The Orientation Density Function (ODF) and pole figures can be constructed directly from a data set of individual grain orientations by associating the orientations or the volume fractions of grains having these orientations with points or discrete cells of finite size in the Euler orientation space, on pole figures (stereographic projection) or on inverse pole figures (standard triangle of the crystal symmetry group). With increasing number of data points such discrete representations are difficult to survey. So they are often normalized, smoothed, filtered and converted to equal density line representations, or the densities are scaled to color intervals in the plots of ODF sections, pole figures or inverse pole figures. This method of discrete representation appears easy and straightforward, but has drawbacks even if the Euler plots and Pole Figure Plots are smoothed and normalized to equal densities. First, the distortions of the Euler space for small Φ angles are usually not taken into account (nor the distortions of the stereographic projection) since for simplicity the Euler space is subdivided into cells of constant angular increments. Second, the number of measured orientations is often not high enough to guarantee statistical relevance.

     An elegant approach to smooth and to condense the data set into a continuous representation is the calculation of the ODF, f(g), by series expansion into generalized spherical harmonics (T functions) after Bunge [1]:

      The Orientation Density Function

     The expansion coefficients (termed "C coefficients") are then expressed by

      The expension coefficients C

    Vs stands for the volume fraction of grain s, and Ks(l) for the convolution kernel of the expansion. Common convolution kernels are a Dirac δ function in case of a large number S of orientations, or Gauss type distributions in Euler space with a half width Ψ0s at 1/e maximum of the Gauss peak at the orientation point gs [2, 3]

     Gauss-type concolution kernel

    The C coefficients are a highly concise and convenient description of crystal texture. They enable the calculation of normal and inverse pole figures, important elastic and plastic materials properties, and tensorial properties in general [1]. ODFs calculated from EBSD and from X-ray pole figure measurements usually agree very well if the data have been acquired on the same specimen area.

    N.B.: In X-ray diffraction, the reflections and hence pole figures are indexed according to their diffracting lattice planes. By tradition, inverse pole figures referred to the specimen surface are also indexed according to the in-surface lattice planes rather than to the crystallographic directions perpendicular to the specimen surface. This inconsequent definition is of no relevance in case of cubic crystal symmetry, but has to be kept in mind when studying materials of lower symmetry.
    ______________

    [1] H.-J. Bunge: Texture analysis in materials science – Mathematical methods. Butterworths, London 1982   (ISBN 0-408-10642-5)
          Paperback reprint: Cuvillier-Verlag Göttingen 1993    (ISBN 3-928815-81-4))

    [2] F. Wagner: Texture determination by individual orientation measurements. In: Experimental Techniques of Texture Analysis, H.J. Bunge (ed.). DGM-Informations-Ges., Oberursel 1986, 115-123    (ISBN 3-88355-101-5)

    [3] J. Pospiech: The smoothing of the orientation distribution density introduced by calculation methods. Textures and Microstructures 26-27 (1996) 83-91


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