Pole Figures of an ODF

A pole density function is a two-dimensional projection of an ODF. Fix a crystal direction \(\vec h\) and ask how much material points it along each specimen direction \(\vec r\).

This page assumes the ODF and mrd normalization introduced in ODF Theory. Crystal directions and their Miller indices are introduced in Miller Indices.

Formally, the pole density function is the ODF integrated over every orientation that maps \(\vec h\) onto \(\vec r\),

\[ P_{\vec h}(\vec r) = \int_{g \vec h = \vec r} f(g)\, \mathrm{d}g. \]

This integral is the spherical Radon transform of the ODF. It collapses a whole orientation fibre to one value, so a pole figure is a projection rather than a section through orientation space. One pole figure cannot determine a three-dimensional ODF, and different ODFs can produce the same pole figure.

Like the ODF itself, the result is a density in multiples of a random distribution (mrd). A value of 2 mrd means twice the random density at that specimen direction, not two percent of the material.

After background, defocusing, and structure-factor corrections, this density is proportional to what a diffraction experiment measures. Pole figures therefore connect diffraction measurements to texture; see Reconstructing an ODF for the inverse problem.

A plotting convention states how a reference frame is laid out on screen. The following convention draws specimen Y upward and specimen X to the right. It does not rotate the specimen or change the ODF.

plottingConvention.default('y↑→x');

A Model Texture to Look At

The example combines two single-orientation components with one fibre component. Trigonal symmetry makes the difference between a direction and its opposite visible in the third pole figure.

cs = crystalSymmetry('32');
mod1 = orientation.byEuler(90*degree,40*degree,110*degree,'ZYZ',cs);
mod2 = orientation.byEuler(50*degree,30*degree,-30*degree,'ZYZ',cs);

odf = 0.2*unimodalODF(mod1) ...
  + 0.3*unimodalODF(mod2) ...
  + 0.5*fibreODF(Miller(0,0,1,cs),vector3d.X,...
  'halfwidth',10*degree);

% select three crystal directions in four-index notation
h = Miller({1,0,-1,0},{0,0,0,1},{1,1,-2,1},cs);

plotPDF fixes each crystal direction in h and draws its pole density over specimen directions. The panels use the default spherical projection.

plotPDF(odf,h);
mtexColorMap LaboTeX;

The two single-orientation components make localized spots. The fibre component makes a ring because its remaining rotational freedom becomes a curve in the pole figure.

MTEX draws only the upper hemisphere when the lower hemisphere contains no independent information. That is why the first two pole figures use one disc while the third uses two.

When the Two Hemispheres Differ

The 'complete' flag displays both hemispheres for every direction. It changes the displayed region; it does not symmetrize the pole density or restore information lost by the projection.

plotPDF(odf,h,'complete');
mtexColorMap LaboTeX;

The upper and lower hemispheres agree in exactly three situations:

Dynamical or resonant diffraction may distinguish the two sides. The antipodal assumption therefore belongs to the experiment, not to the definition of an ODF.

The 'antipodal' flag averages opposite directions. With 'complete' retained below, both equal halves remain visible for comparison.

plotPDF(odf,h(3),'antipodal','complete');
mtexColorMap LaboTeX;

Opposite points in the two discs now have the same colour. Drawing both discs does not make them independent measurements.

Values Rather Than Pictures

calcPDF with only a crystal direction returns a spherical function. Passing a specimen direction as well evaluates that function at one point.

pdf = calcPDF(odf,Miller(1,0,0,cs))
densityAtX = calcPDF(odf,Miller(1,0,0,cs),vector3d.X)

pdf = S2FunHarmonicSym (y↑→x)
  bandwidth: 25
  antipodal: true
 
densityAtX =
    0.1982

The printed summary identifies pdf as a spherical function. Its value along specimen X is 0.1982 mrd, so this texture avoids that direction.

Averaged over the whole sphere, every normalized pole density function is 1 mrd, just as every normalized ODF has mean 1 mrd.

meanDensity = mean(pdf)

meanDensity =
    1.0000

Following a Pole-Figure Maximum into Orientation Space

A pole-figure maximum still represents a whole orientation fibre. Find the strongest specimen direction for the third pole figure, then inspect the ODF along the corresponding fibre with plotFibre.

pdfPeak = calcPDF(odf,h(3));
[~,rPeak] = max(pdfPeak);
f = fibre(h(3),rPeak);

close all;
plotFibre(odf,f,'LineWidth',2,'figSize','small');

The line is the ODF density itself, not another projection. Its peaks locate the components that this fibre passes through, while their average along the curve is the pole-figure value at rPeak. This is the degree of freedom that the pole figure hides.

Further Reading

Next

The other projection fixes a specimen direction instead of a crystal direction; continue with Inverse Pole Figures. Going the other way, from measured pole figures to an ODF, is Reconstructing an ODF.