Pole Figures and the Orientation Distribution Function

From 'there is a texture' to 'how strong, and pointing where?'

Guide 4 in this rung made the case that a real engineering metal almost never has a random spread of grain orientations. Roll a sheet, draw a wire, evaporate a film, and the grains come out with a preferred orientation — a statistical clustering of their crystallographic orientations around a few favoured settings. That is a genuine, measurable feature of the microstructure, and it is the reason a rolled sheet can be noticeably stronger along one direction than another. But 'the grains cluster near a rolling texture' is a sentence, not a measurement. To predict how much anisotropy a texture will cause, we need to say precisely how many grains sit near which orientation. This guide is about the three drawings and functions crystallographers invented to do exactly that.

The obstacle is dimensional. A full orientation is a three-dimensional object — it takes three numbers (three rotation angles) to say exactly how one grain's crystal axes are turned relative to the sample. But every diffraction experiment measures directions, and a direction is only a point on a sphere, a two-dimensional thing. So we cannot photograph the full orientation directly; we can only capture flattened, two-dimensional shadows of it. The whole art of texture analysis is choosing clever shadows and then reasoning back to the solid object that cast them. The drawing tool for every one of these shadows is the stereographic projection you met back in the symmetry rung: pierce a unit sphere with the direction of interest, then project that piercing point onto a flat disk.

One more piece of vocabulary from earlier makes everything click. A pole is the point where a plane's normal pierces that unit sphere — the plane (111) is represented not by a big flat sheet but by the single dot where its perpendicular stabs the sphere. Representing a whole plane by one point is exactly what lets us plot thousands of grains on one small disk. When we then colour that disk by how densely the poles pile up, we have turned an abstract statistical statement — 'the orientations cluster' — into a picture you can read at a glance.

The pole figure: a density map of one plane-normal

A pole figure is built by one firm rule: fix a crystal plane family, then plot where its normals land in the SAMPLE's frame. Pick, say, the {100} cube faces. Every grain has three of them, so every grain contributes {100} poles somewhere on the disk. Crucially, the disk itself is glued to the sample, not to any crystal: its centre is the sheet normal (ND), and its rim carries the rolling direction (RD) and transverse direction (TD). Now overlay all the grains. If the polycrystal is random, the poles smear out evenly and the disk is a featureless grey. If there is a texture, the poles pile into hotspots — and the position of each hotspot tells you which sample direction that crystal plane prefers to face.

{100} POLE FIGURE : where each grain's cube-face normals land,
projected onto a disk carrying the SAMPLE axes RD / TD / ND.

     RANDOM polycrystal                SHARP CUBE texture {100}<001>
     (~1x random everywhere)           (poles pile up; >> 1x random)

              RD                                RD
            .......                           .:####:.
          ...........                       ..:(8x) :..
         .............                     ...        ...
    TD ......  +  ...... TD          TD   ####   +   ####   TD
         .............                     ...        ...
          ...........                       ..:(8x) :..
            .......                           ':####:'

  + = ND (disk centre, out of the page)
  read-out:  1 = random ;  value > 1 = a surplus of that normal there
  a cube-textured sheet points cube faces along RD, TD and ND at once

How is a pole figure actually measured? The classic way is an X-ray texture goniometer using the Schulz reflection method: mount the flat sheet, lock the detector on one strong (hkl) reflection so that only planes of that spacing diffract, then systematically tilt and rotate the sample through every orientation while recording how the diffracted intensity rises and falls. High intensity at a given tilt means many grains have that plane facing the beam there — that is the pole density. Neutrons do the same job for bulk samples centimetres thick because they penetrate so deeply, while grain-by-grain electron backscatter diffraction (EBSD) in an SEM builds pole figures the modern way, one measured grain orientation at a time.

The inverse pole figure: flip the question around

A pole figure fixes a CRYSTAL feature (say {100}) and asks where it points in the SAMPLE. The inverse pole figure swaps the two roles: it fixes a SAMPLE direction and asks which CRYSTAL directions line up with it. So instead of a disk carrying RD, TD and ND, you draw the crystal's own standard stereographic triangle — the little wedge with corners [001], [011] and [111] that captures every distinct direction of a cubic crystal by symmetry — and you shade it by how often each crystal direction happens to fall along your chosen sample axis.

This flip is a perfect fit when only one sample direction really matters — a fibre texture, where grains share a common axis but tumble freely about it. A drawn wire is the archetype: it has an axis of rotational symmetry, so a full pole figure is redundant, but a single inverse pole figure of the wire axis says everything. Two concrete cases worth memorizing: a cold-drawn FCC wire (copper, aluminium) develops a double fibre with both <111> and <100> lying along the wire axis, so its inverse pole figure lights up at the [111] and [001] corners; a cold-drawn BCC wire (iron, tungsten) develops a single <110> fibre, so its inverse pole figure has one strong spot at the [011] corner. Read the corner, and you know the fibre.

The inverse pole figure is also the secret behind the rainbow micrographs everyone recognizes from modern EBSD. In an 'IPF map' each grain is coloured by which crystal direction points along one chosen sample axis (usually ND): grains with <001> along ND go red, <101> green, <111> blue, and everything in between blends. That single false-colour image is a spatial inverse pole figure — it shows you both the orientation of every grain and, because it is a real map, where each grain physically sits, which a bare pole figure throws away.

The orientation distribution function: the whole truth

Here is the honest limitation that motivates everything else. A pole figure is a shadow, and shadows are many-to-one: several completely different full orientations can throw poles onto the very same spot, so no single pole figure can be uniquely un-projected back to the three-dimensional truth. It is the same information loss you met with the phase problem in diffraction — the measurement keeps some of the story and quietly discards the rest. To recover the solid object we need the thing the shadows are shadows OF: a function that gives the density of grains at every possible orientation.

That function is the orientation distribution function (ODF), written f(g). Its argument g is a full orientation, most often written as three Bunge Euler angles (phi1, PHI, phi2) — the three successive rotations that carry the sample frame onto the crystal frame. Those three angles span a three-dimensional box, orientation space, and f(g) is a density living inside that box, normalized on the same 'times random' scale so that f = 1 everywhere means a random polycrystal. A pole figure is then, mathematically, a two-dimensional projection of this three-dimensional f(g) — you get it by integrating the ODF along the line of orientations that all happen to put a chosen plane in a chosen sample direction. Measure two or three pole figures from different (hkl), invert them numerically, and you reconstruct f(g) itself.

Working metallurgists rarely stare at the raw box. Instead they name the peaks. A handful of ideal orientations recur so often that they have proper names: the cube {100}<001>, Goss {110}<001>, brass {110}<112>, copper {112}<111>, and S {123}<634> components dominate rolled FCC sheet, while BCC steels are described by fibres — the alpha fibre with <110> parallel to RD and the gamma fibre with <111> parallel to ND. A real texture is then reported as 'strong gamma fibre, weak cube', with each named component sitting as a peak of some height in f(g). Crystal symmetry and sample symmetry fold the full Euler box down to a small asymmetric wedge, exactly as the asymmetric unit did for a space group, so you only ever plot that reduced region.

From the ODF back to properties

The reason to go to all this trouble is the payoff at the end: the ODF lets you predict anisotropy. A single crystal is anisotropic — its stiffness, yield behaviour, magnetic and thermal response all depend on direction. A random polycrystal averages that away and behaves almost isotropically, because every orientation is equally represented. A textured polycrystal is the interesting in-between: you take each grain's single-crystal property tensor, rotate it into the sample frame, and average over all grains WEIGHTED by f(g). Because the weighting is lopsided, the average keeps a directional flavour — the polycrystal inherits a muted, blended version of the single crystal's anisotropy. This is the quantitative core of the structure-property relationship at the grain scale.

Two industrial stories make this vivid. First, deep-drawing a can: the plastic anisotropy of textured sheet is captured by the r-value, and its variation with direction decides whether the drawn cup comes out with a smooth rim or with scalloped 'ears' at 0/90 or 45 degrees. Sheet-makers tune the annealing texture specifically to flatten those ears and stop wasting metal in the trim. Second, transformer cores: grain-oriented silicon steel is deliberately given a razor-sharp Goss texture {110}<001> so that iron's easy magnetization axis <001> lies right along the rolling direction the flux will follow — the result is dramatically lower core loss, and it powers the world's electricity grid. In both cases the ODF is not academic bookkeeping; it is the thing the process engineer is paid to control.

And texture is not frozen — it is built and rebuilt by processing, which loops us back to the previous guides in this rung. Deformation rotates grains toward a deformation texture; annealing then nucleates new grains and grows them, and grain growth can wholly replace one texture with another as certain favoured orientations devour their neighbours. The extreme case, abnormal (secondary) grain growth, is precisely how the Goss texture in silicon steel is sharpened: a few Goss grains grow enormous at the expense of the rest, driven by the same boundary-energy accounting that drove ordinary coarsening. Texture, in other words, is a running record of everything that has happened to the metal — and the ODF is how we read that record with numbers.

1. Choose a strong (hkl) reflection and define the sample frame — mount the flat sheet with RD, TD and ND fixed on the texture goniometer.
2. Tilt and rotate through all orientations, recording the diffracted intensity; corrected and normalized to times-random, that intensity map IS the (hkl) pole figure.
3. Repeat for two or three independent (hkl) so the shadows come from genuinely different viewpoints.
4. Invert the several pole figures numerically to reconstruct the three-dimensional ODF f(g) in Euler space — or, with EBSD, skip inversion and just histogram the per-grain orientations directly.
5. Read off the named texture components as peaks, then integrate f(g) against the single-crystal property tensor to predict the polycrystal's anisotropy.