the orientation distribution function

Pole figures and inverse pole figures are handy 2D snapshots of texture, but each throws some information away. The orientation distribution function (ODF) is the complete, honest bookkeeping: it states, for every possible grain orientation, how many grains have that orientation. If texture is which way the grains point, the ODF is the full histogram over all possible pointings.

A grain's orientation needs three numbers to specify — a rotation has three degrees of freedom, usually given as three Euler angles (phi1, Phi, phi2). The ODF, f(g), is the density of grains at each orientation g in that three-dimensional orientation space, normalized so that a texture-free (random) sample has f = 1 everywhere. Values are read in multiples of random density (m.r.d.): f = 8 at some orientation means grains are eight times more common there than in a random sample; f near 0 means that orientation is avoided. Because a single pole figure is a 2D projection of this 3D function, several measured pole figures are mathematically inverted to reconstruct the ODF, or it is built directly from thousands of individual EBSD orientation measurements.

The ODF is the reference description of texture because everything else follows from it: any pole figure or inverse pole figure can be computed from the ODF, and direction-dependent (anisotropic) properties of the polycrystal — stiffness, yield, magnetic behaviour — can be predicted by averaging single-crystal properties weighted by f(g). Honest caveats: reconstructing an ODF from a few pole figures is an inversion that can be non-unique (the classic ghost artefacts), so it needs care; and orientation space has its own symmetry (from the crystal's point group), so the ODF is only plotted over a reduced, non-redundant region.