
EBSD and BKD 
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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 expansion coefficients (termed "C coefficients") are then expressed by
V_{s} stands for the volume fraction of grain s, and K_{s}(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 Ψ_{0s} at 1/e maximum of the Gauss peak at the orientation point g_{s} [2, 3]
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 Xray pole figure measurements usually agree very well if the data have been acquired on the same specimen area.
N.B.: In Xray 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 insurface
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 0408106425) [2] F. Wagner: Texture determination by individual orientation measurements. In: Experimental Techniques of Texture Analysis, H.J. Bunge (ed.). DGMInformationsGes., Oberursel 1986, 115123 (ISBN 3883551015) [3] J. Pospiech: The smoothing of the orientation distribution density introduced by calculation methods. Textures and Microstructures 2627 (1996) 8391 