One crystal, one pole figure

A pole figure answers a single question, over and over: for one chosen family of crystal planes, where do their normals point, in specimen coordinates? Take the {100} planes of one crystal: three planes, six normals, three of them pointing into the upper hemisphere. Project those three onto a disc and you have that crystal's {100} pole figure: three dots. That is the entire construction.

Drag the crystal and watch its dots move. Switch families and the same orientation draws a different constellation: four dots for {111}, six for {110}. The orientation is the object; each pole figure is one shadow of it.

The pole figure machine Drag the crystal; the dots follow

Drag to rotate the crystal

Euler φ₁

Euler Φ

Euler φ₂

Set all three sliders to zero: the cube orientation. The {100} poles land at the centre and on the rim at RD and TD: the crystal axes lie along the specimen axes, and the pole figure says so at a glance. Every named texture component is read this way.

pole figure of {hkl}: for each grain, plot n̂_specimen = g⁻¹ · n̂_{hkl} on the hemisphere the same g as everywhere in this series, used backwards: crystal → specimen. Projections: stereographic preserves angles (used above); equal-area preserves densities and is what texture work should use: modules 2 and 3 below do. Note the pole figure keeps only a direction, not the rotation about it: the same two lost-and-kept degrees of freedom as the IPF colours of part 2, which is exactly why one pole figure can never pin down a texture.

Build a texture, grain by grain

A polycrystal's pole figure is just module 1 repeated for every grain. Pick a texture and pour grains in; watch three dots at a time accumulate into the smeared, symmetric figures that papers print. Two pole figures are shown, because one is about to prove insufficient.

Texture builder Both pole figures show the same grains

Scatter about the ideal texture

Grains

Peak intensity

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× random (m.r.d.-style)

Signature

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Try the ⟨111⟩ fibre with a few hundred grains: the {111} figure grows a bright centre spot and an outer ring, a wire texture's fingerprint, while {100} shows only rings. Fibres are where single pole figures mislead soonest: a ring tells you an axis is shared, and nothing about the rotation around it.

Go deeper: reading the classic signatures

Cube {001}⟨100⟩: {100} poles at the centre and at four rim positions: the recrystallisation texture of many fcc metals, and the one grain-oriented electrical steel is grown for. Rolled fcc: the smeared "β-fibre" running Copper {112}⟨111⟩ → S → Brass {110}⟨112⟩, recognisable as the characteristic lobed {111} figure of any cold-rolled aluminium or copper sheet. ⟨111⟩ fibre: drawn wires and many thin films; an axis pinned along the wire or growth direction, all rotations about it equally represented, hence rings.

Real measured pole figures come with intensity in "multiples of random density" (m.r.d.), exactly the normalisation the peak-intensity readout above imitates: 1.0 everywhere means random; a cube component at 20 m.r.d. means the sheet will behave nothing like an isotropy handbook says. And note the symmetry: rolled figures inherit the orthorhombic symmetry of the rolling process itself; the sample's history is stamped into the figure's symmetry before you read any component off it.

The ODF: where components live

The orientation distribution function is nothing exotic: it is the histogram of the full three-parameter orientations: the object whose shadows the pole figures are. Euler space is its address system. Below is a slice through that space for whatever you built in module 2: drag the φ₂ slider and watch the named components light up at their textbook addresses.

ODF section viewer Sections of Euler space, φ1 × Φ, at fixed φ2

Section φ₂

Components detected

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Build the rolled texture in module 2, then step φ₂ from 0° to 90°: Brass appears in the 0° section, Copper near 45°, S near 65°: the β-fibre threading through Euler space section by section. Components that smear together in a pole figure sit at cleanly separated addresses here.

And now the point of the whole page. X-ray texture measurement records pole figures (shadows) and must reconstruct the ODF from several of them. That inversion is mathematically ill-posed: the even-order part of the ODF is all a pole figure retains, "ghost" components can appear or vanish, and a whole literature of regularisation exists to manage it. EBSD does not have this problem, and it is worth saying plainly: EBSD measures every grain's full orientation directly. The ODF is simply binned from the data. The pole figures a texture paper shows from EBSD are drawn from the ODF, not the other way round.

The thing to remember

Pole figure: where one crystal direction family points; a projection, two degrees of freedom, and different textures can share one. ODF: the full three-parameter distribution, the object itself. When a conclusion depends on texture, it should rest on the ODF (or on the orientations directly), with pole figures as the pictures that make it legible.

A trap worth naming

EBSD's ODF is only as good as its statistics and its sampling. A texture from 200 grains has grain- counting noise all over it (a few hundred grains is a minimum for bulk texture claims, and thousands are better), and a single map measures one plane of one region: a rolled sheet's surface and centre can carry genuinely different textures. X-ray's weakness is the inversion; EBSD's is the sample size. Neither one is "the true texture" by default.

Next · Part 5

Cleaning data honestly

Noise, pseudosymmetry, and what dilation and CI-standardisation actually do to your statistics, with a "how much cleanup is too much" slider.

Previous · Part 3

Misorientation, grain boundaries & CSL

The rotation between two grains, the threshold that defines a grain, and the Σ3 twins that light up the Mackenzie plot.

Sources & further reading

H.-J. Bunge, Texture Analysis in Materials Science, Butterworths (1982): Euler space, the ODF, and the series-expansion machinery.
V. Randle, O. Engler, Introduction to Texture Analysis: Macrotexture, Microtexture and Orientation Mapping, 2nd ed., CRC Press (2010): the accessible standard; the component addresses used above are its tables.
U. F. Kocks, C. N. Tomé, H.-R. Wenk, Texture and Anisotropy, Cambridge (1998): texture to properties, and the honest treatment of conventions.
S. Matthies, “On the reproducibility of the orientation distribution function of texture samples from pole figures (ghost phenomena),” Phys. Status Solidi B 92, K135 (1979): the ghost problem named.
H.-R. Wenk, P. Van Houtte, “Texture and anisotropy,” Rep. Prog. Phys. 67, 1367 (2004): the modern review.
R. Hielscher, H. Schaeben, “A novel pole figure inversion method: specification of the MTEX algorithm,” J. Appl. Cryst. 41, 1024 (2008): how open-source texture analysis actually does it; MTEX is the tool to learn.
S. I. Wright, M. M. Nowell, J. F. Bingert, “A comparison of textures measured using X-ray and electron backscatter diffraction,” Metall. Mater. Trans. A 38, 1845 (2007): the sampling-vs-inversion trade this page's last warning compresses.