Plotting Pole Figures

A measured pole figure stores an intensity at every sampled specimen direction for one family of crystal-plane normals. plot places those samples on a projection of the specimen hemisphere. It does not turn the measurements into a continuous function.

This page assumes the pole-figure idea introduced in Pole Figures and a PoleFigure object imported as in Import. Review Miller Indices if the plane labels are unfamiliar.

plottingConvention.default('y↑→x');
mtexdata ptx

pf = PoleFigure (y↑→x)
  crystal symmetry : mmm
 
  h = (104), r = 72 × 17 points
  h = (110), r = 72 × 17 points
  h = (202), r = 72 × 17 points

The display reports three measured pole figures, for (104), (110), and (202). Each one contains a regular \(72 \times 17\) grid of specimen directions. The convention above draws y upwards and x to the right. It makes the layout explicit instead of inheriting the session default. See Axes Alignment for how that convention relates to the specimen frame; it does not rescale the intensities.

Measured directions as markers

With no plot-type option, each measured direction becomes a circle coloured by its stored intensity.

plot(pf)
mtexColorMap parula
mtexColorbar('title','intensity')

The circles expose the sampling grid as well as the intensity pattern. Each panel has its own colour range, so the same colour does not yet mean the same value in all three panels.

MTEX estimates a marker size from the angular spacing and the figure size. The 'MarkerSize' option replaces that estimate when it hides gaps or makes neighbouring circles overlap.

plot(pf,'MarkerSize',2)
mtexColorMap parula
mtexColorbar('title','intensity')

The smaller circles separate neighbouring samples more clearly. Their size has no physical meaning, and the intensity pattern is unchanged.

Contour plots

Contours need values between the measured directions. On a regular grid, 'contourf' lets the plotting routine interpolate the stored intensities. This is useful for a quick visual summary.

plot(pf,'contourf')
mtexColorMap parula
mtexColorbar('title','intensity')

The isolated spots now appear as continuous patches. That continuity is interpolation, not additional measurement, and it can be misleading on an irregular grid. When a smooth pole-density function is required, reconstruct an ODF and recalculate the pole figure as shown below.

One colour range for all

Comparing pole figures by eye only works when they share a colour range. setColorRange with 'equal' gives all axes of the figure the union of their ranges. A single colorbar then applies to every panel.

mtexColorbar % remove colorbars
setColorRange('equal');
mtexColorbar('title','intensity') % add a single colorbar

The colours can now be compared directly. The (202) figure reaches 15.8, whereas (104) stops at 9.8. The same orientation population projects differently for different crystal-plane normals, which is why several pole figures provide more information than one.

The colourbar shows the values stored in pf. Do not call them multiples of a random distribution (mrd), or compare maxima as material fractions, until the pole figures have been corrected and normalized.

Some stored intensities are negative, down to -1.8. Diffracted intensity cannot be negative, so the background correction has subtracted too much. Data Correction covers that problem and the normalization step.

Recalculated pole figures

An orientation distribution function (ODF) is a density over crystal orientations. At one point of a recalculated pole figure, MTEX integrates that density over every orientation that carries the chosen crystal-plane normal into the corresponding specimen direction.

The result can be calculated for any crystal-plane normal, including one that was never measured. Reconstructing the ODF from measured pole figures is the subject of ODF Estimation.

odf = calcODF(pf,'silent');

plotPDF recalculates the same three pole figures. Diffraction identifies a plane normal with its opposite, so 'antipodal' folds the two directions into the same hemisphere.

plotPDF(odf,pf.h,'antipodal')
mtexColorMap parula
setColorRange('equal');
mtexColorbar('title','mrd')

These are smooth because the ODF is, not because the data were. Comparing them with the measured figures is the standard first check on a reconstruction, and what to compare is where the maxima sit.

Their heights need not agree. Read the two colour bars here: the measured peaks are two to three times the recalculated ones, because the finite width of the reconstruction kernel spreads a sharp measured maximum out.

A displaced or missing maximum points to a reconstruction or data problem. Continue with ODF Estimation for quantitative error measures and Data Correction for systematic defects. A visual match does not make the reconstructed ODF unique; the ambiguity is explained in The Ghost Effect.

Further reading