Pole Figures

A pole figure answers one question: given a crystal direction, where in the specimen does it point? A pole is a direction fixed in the lattice, commonly the normal to a lattice plane. The plot therefore turns a population of orientations into a two dimensional view in the specimen reference frame.

This page assumes the coordinate map from Theory, the equivalent directions from Symmetry, and the Miller notation from Crystal Directions. The choice of hemisphere and projection is developed in Spherical Projections. MTEX uses an equal-area projection by default.

A reference frame is the coordinate system in which data are expressed. The plotting convention below lays specimen Y upward and specimen X to the right. It changes only the screen layout; it does not rotate the specimen or re-express the data.

plottingConvention.default('y↑→x');

cs = crystalSymmetry('321');

ori = orientation.rand(cs);

Building One by Hand

Fix the normal to the \((100)\) lattice plane in the crystal frame. Parentheses denote plane-normal hkl notation; square brackets would denote a lattice direction in uvw notation.

h = Miller({1,0,0},cs);

symmetrise applies the crystal point group to the pole. The orientation then maps every returned pole into the specimen frame.

r = ori * h.symmetrise;

poleCount = length(r)

poleCount =
     6

The count is six. Point group 321 has six operations, and all six produce distinct directions for this pole. A pole on a symmetry axis can have fewer distinct directions because several operations may coincide.

Plot the specimen directions in a spherical projection.

plot(r);

Notice three points on each hemisphere. Together they are the six crystallographically equivalent positions of the \((100)\) pole for one orientation; they are not six different orientations.

The Shortcut

plotPDF performs the symmetrisation, coordinate map, and projection for several poles at once.

plotPDF(ori,Miller({1,0,-1,0},{0,0,0,1},{1,1,-2,1},ori.CS));

The first two pole figures need only one hemisphere, while the third needs both. MTEX makes that decision from symmetry. It uses one hemisphere when h and -h are crystallographically equivalent, because the other half then repeats the same information.

In 321 this holds for \((10\bar{1}0)\) and \((0001)\). The twofold axes in the basal plane turn the c-axis pole into its opposite. It does not hold for \((11\bar{2}1)\), so that pole figure retains both hemispheres.

The 'antipodal' flag identifies opposite poles even when crystal symmetry does not. This is a modelling choice, not another point-group operation. It is conventional for kinematic diffraction under Friedel's law, but resonant scattering can distinguish a Friedel pair. See Axes and Antipodal Symmetry.

This orientation has the default identity specimen symmetry. If an orientation carries a nontrivial specimen symmetry, plotPDF also repeats the poles by that symmetry in the specimen frame. Crystal symmetry acts before the orientation map; specimen symmetry acts after it. See Specimen Symmetry.

What One Pole Figure Leaves Out

A pole position fixes where one crystal direction points, but not the rotation of the crystal about that direction. All orientations that put h at one specimen direction form an orientation fibre. One pole figure therefore cannot determine a complete orientation or ODF by itself.

Combining pole figures for several lattice planes constrains the missing information. That is the inverse problem treated in Pole Figure Analysis.

Contour Plots

The option 'contourf' replaces the discrete markers with a kernel density estimate on the sphere.

plotPDF(ori,Miller({1,0,-1,0},{0,0,0,1},{1,1,-2,1},ori.CS),...
  'contourf');
mtexColorbar;

For this single orientation the contours merely spread each discrete pole into a small spot; the first family still represents only six poles. For a population of measured or simulated orientations, the contours instead show where poles concentrate.

The colour scale is in multiples of a random distribution (m.r.d.). A value of 1 is the pole density of an untextured population, while 2 means twice that density. Pole densities computed directly from an ODF are developed in Pole Figures of an ODF.

The Maths Behind a Pole Figure

Let \(\mathbf{O}\) map crystal coordinates to specimen coordinates. Let \(\mathbf{C}\) be a crystal-symmetry operation and \(\mathbf{P}\) a specimen-symmetry operation. Every pole drawn by plotPDF has the form

\[ \mathbf{r} = \mathbf{P}\,\mathbf{O}\,\mathbf{C}\,\mathbf{h}, \qquad \mathbf{C} \in \mathrm{S}_{\mathrm{c}}, \quad \mathbf{P} \in \mathrm{S}_{\mathrm{s}}. \]

In this example the specimen group contains only the identity, so the hand construction reduces to \(\mathbf{r}=\mathbf{O}\mathbf{C}\mathbf{h}\).

References

Next

The opposite question - given a specimen direction, which crystal direction points along it - is the Inverse Pole Figure. Both are projections and both discard information. The full orientation space is shown in 3D Plots and Section Plots.