The Orientation Distribution Function

A texture is a population of orientations. The orientation distribution function (ODF) describes that population as a density over orientation space, rather than as a list of measured orientations.

This page assumes the crystal-to-specimen map introduced in Orientation Theory and the symmetry-equivalent representatives from Orientation Symmetry.

Density is the word to hold on to. No finite fraction of a continuous distribution sits at one exact orientation. An ODF instead says how much material occurs per unit of orientation space near that orientation. Values are reported in multiples of a random distribution, mrd. A uniform texture is 1 mrd everywhere; 9 mrd means nine times the random density near that orientation, not nine percent of the material.

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

plottingConvention.default('y↑→x');

From a Map to Orientations

The example is a titanium alloy measured by EBSD on a hexagonal grid. An orientation is stored at every indexed measurement point.

% import the data without printing its EBSD summary
mtexdata titanium silent

The map below uses an inverse pole figure key for the specimen Z direction. The overlaid hexagonal prisms make the crystal orientation behind selected colours explicit.

% define the habitus of titanium as a sample hexagonal prism
cS = crystalShape.hex(ebsd.CS);

% keep a regular spatial subsample for the crystal and orientation plots
ebsdPlot = reduce(ebsd,4);

% plot coloured orientations
plot(ebsd,ebsd.orientations,'ipfDirection',zvector,'micronbar','off','figSize','large')

% overlay the orientations as rotated hexagonal prisms
hold on
plot(ebsdPlot,40*cS)
hold off

Neighbouring points with similar colours have similar orientations, and the prisms show the corresponding lattice directions. This spatial arrangement is useful for EBSD analysis, but an ODF deliberately discards it.

Keep only the orientations and draw the regular subsample in Bunge Euler angle space.

oriPlot = ebsdPlot('indexed').orientations;
plot(oriPlot,'Euler')

The same measurements now form a point cloud. Dense clusters and sparse regions reveal the texture, while their former positions in the map are no longer present.

Orientation space is curved, so no drawing of it is canonical. The alternatives are compared in 3D Plots.

From Orientations to a Density

calcDensity places a kernel at every orientation and adds the kernels. The full indexed pixel list is used here, so equal-area pixels have equal statistical weight.

oriData = ebsd('indexed').orientations;
odf = calcDensity(oriData)

odf = SO3FunHarmonic (Titanium (Alpha) → y↑→x)
  bandwidth: 25
  weight: 1

The printed summary identifies the harmonic representation and its bandwidth. Other ODF representations share the same SO3Fun interface.

This conversion is an estimate, not a unique rewriting of the points. Its most important choice is the kernel halfwidth. A small halfwidth keeps sample-scale peaks. A large one can merge real components; see Density Estimation.

The weighting also answers a physical question. Pixel orientations describe mapped area, whereas equally weighted grain mean orientations describe the fraction of grains. The 'weights' option can make grain areas supply the weights when area fraction is wanted instead.

Reading Density and Volume

An ODF is a function, so it can be evaluated at an orientation that was never measured. Here the orientation has zero Bunge Euler angles.

ori0 = orientation.byEuler(0,0,0,ebsd.CS);
valueAtOri0 = odf.eval(ori0)

valueAtOri0 =
    0.8166

The value is 0.8166 mrd, slightly less common than the random density of 1 mrd. The strongest density in this texture is about 9.0003 mrd.

[maxValue,oriMax] = max(odf);
maxValue

maxValue =
    9.0003

The mean is 1 mrd by construction, whatever the texture. This is why an isolated ODF value only has meaning relative to the random density.

meanValue = mean(odf)

meanValue =
    1.0000

A volume fraction comes from integrating over a region. The region below contains every orientation within \(10^\circ\) of the strongest one.

volumeFraction = volume(odf,oriMax,10*degree)

volumeFraction =
    0.0235

The result is 0.0235, or about 2.35 percent, not nine percent. The 9 mrd value is a density, and the orientation region around the maximum is small.

Looking at an ODF

An ODF is a function on a three-dimensional curved space. A 3-D plot can show how the measured orientations relate to its high-density regions.

plot3d(odf,'Euler')
hold on
plot(oriPlot,'Euler','MarkerEdgeColor','k')
hold off

The black point clusters pass through the coloured density lobes. This is the visual connection between the discrete measurements and their ODF.

Equal-looking boxes in Euler angle space do not represent equal volumes of orientation space. Use this plot to locate components, not to estimate their volume by eye.

For hexagonal crystals, sigma sections often give a more direct view.

plotSection(odf,'sigma')
mtexColorMap LaboTeX

Position within a section fixes the c-axis direction, while the section angle records the remaining rotation about it. The panels separate the components. They do not invite a volume estimate from distorted boxes.

Both plots evaluate the ODF itself. A pole density function instead integrates the ODF along orientation fibres and therefore loses information.

The Maths Behind the Normalization

For a region \(A\) of orientation space, its material volume fraction is

\[\frac{V(A)}{V} = \int_A \mathrm{odf}(g)\,\mathrm{d}g.\]

In differential form this is

\[\mathrm{odf}(g) = \frac{1}{V}\frac{\mathrm{d}V(g)}{\mathrm{d}g}.\]

Here \(\mathrm{d}g\) is the normalized volume measure on the symmetry-reduced orientation space. Integrating 1 over the whole space gives 1, so the uniform ODF is \(\mathrm{odf}(g)=1\) everywhere.

Crystal and specimen symmetry identify equivalent numerical rotations. MTEX carries both symmetries with the ODF. Equivalent copies are not counted as distinct material during evaluation or integration.

References

Next

Continue with Density Estimation for kernels and halfwidth selection. Modeling builds known ODFs, and Properties extracts texture index, entropy and volume fractions. Diffraction projections begin with Pole Figures. The next chapter continues with pole-figure reconstruction.