Reconstructing the Dubna Quartz ODF

This page follows a three-file subset of the seven bundled Dubna pole figures from their files to a reconstructed orientation distribution function (ODF), then checks its fit. Florian Wobbe measured the quartz specimen at Dubna in 2005 using neutron diffraction, as recorded in the original MTEX Dubna example.

The example brings together the import, inspection, reconstruction, and validation steps developed earlier in this chapter. It assumes the pole-figure idea from Pole Figures, the import model from Import, and the inversion workflow from ODF Reconstruction. The origin of the ghost effect is explained in The Ghost Effect.

A plotting convention states how the specimen reference frame is drawn. This data set uses Y upward and X to the right. The convention does not rotate the specimen directions or change their intensities.

plottingConvention.default('y↑→x');

Import the three measurements

Quartz is trigonal. The lattice parameters below define its crystal frame as well as the metric used to interpret the four-index notation introduced in Miller Indices.

CS = crystalSymmetry('-3m',[4.9 4.9 5.4]);

fname = {...
  fullfile(mtexDataPath,'PoleFigure','dubna','Q(10-10)_amp.cnv'),...
  fullfile(mtexDataPath,'PoleFigure','dubna','Q(10-11)(01-11)_amp.cnv'),...
  fullfile(mtexDataPath,'PoleFigure','dubna','Q(11-22)_amp.cnv')};

% crystal-plane normals, one cell per measured file
h = {Miller(1,0,-1,0,CS),...
  [Miller(0,1,-1,1,CS),Miller(1,0,-1,1,CS)],...
  Miller(1,1,-2,2,CS)};

% relative structure coefficients, in the same order as h
c = {1,[0.52,1.23],1};

The second diffraction peak contains the unresolved \((01\bar{1}1)\) and \((10\bar{1}1)\) reflections. Its measured intensity is therefore a weighted superposition of two pole figures, not a fourth measurement. Passing c at import makes that same weighted sum part of the forward model used during reconstruction. See Import for how the structure coefficients are found.

pf = PoleFigure.load(fname,h,CS,'interface','dubna','superposition',c)

pf = PoleFigure (y↑→x)
  crystal symmetry : -3m1, X||a*, Y||b, Z||c
 
  h = (101̅0), r = 72 × 19 points
  h = (011̅1)(101̅1), r = 72 × 19 points
  h = (112̅2), r = 72 × 19 points

The summary reports three entries on identical \(72 \times 19\) direction grids. The double Miller label on the second entry confirms that its two reflections have not been mistaken for separate measurements.

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

The three panels contain sharp maxima in different specimen directions. The second panel has a much larger raw intensity range, which is why the solver estimates one scale factor per pole figure. These counts are not yet pole densities in multiples of a random distribution (mrd); see ODF Theory.

Inspect the stored data

The object exposes the measured intensities, crystal-plane normals, and specimen directions as ordinary arrays. Different pole figures can use different direction grids, so PoleFigure also provides the cell-valued properties allI, allH, and allR.

I = pf.intensities;
latticeDirections = pf.h;
specimenDirections = pf.r;

Minimum and maximum intensities give a first scale check. The rows follow the three entries in the object summary above.

poleFigureLabel = {'(10-10)';'(01-11)+(10-11)';'(11-22)'};
intensitySummary = table(min(pf).',max(pf).',...
  'VariableNames',{'minimum','maximum'},'RowNames',poleFigureLabel)

intensitySummary =
  3×2 table
                       minimum    maximum
                       _______    _______
    (10-10)               0        89.8  
    (01-11)+(10-11)       0        1360  
    (11-22)               0         962

isOutlier compares every measurement with its neighbourhood. Use it to create a mask for inspection:

outlierMask = isOutlier(pf);

A flag is a prompt to inspect the experiment, not permission to delete a value automatically. Background, defocusing, normalization, and an executable outlier example are in Data Correction.

Select a diagnostic band

High-tilt measurements are especially vulnerable to defocusing. The next condition demonstrates indexed selection by removing only the two rings from 70 through 75 degrees. It deliberately retains directions above 75 degrees, so it is not a recommended high-tilt correction.

keep = pf.r.theta < 70*degree | pf.r.theta > 75*degree;
pf_bandRemoved = pf(keep)

plot(pf_bandRemoved)
mtexColorbar('title','intensity')

pf_bandRemoved = PoleFigure (y↑→x)
  crystal symmetry : -3m1, X||a*, Y||b, Z||c
 
  h = (101̅0), r = 1 × 1224 points
  h = (011̅1)(101̅1), r = 1 × 1224 points
  h = (112̅2), r = 1 × 1224 points

The blank band is the direct visual consequence of the selection. Each entry now contains 1224 of its original 1368 specimen directions.

Rotate the measured directions

rotate actively moves every measured specimen direction while leaving its intensity unchanged. This is not a frame change and not a plotting convention. If import assigned the wrong specimen reference frame, correct the import whenever possible.

rot = rotation.byAxisAngle(xvector-yvector,25*degree);
pf_rotated = rotate(pf,rot);

plot(pf_rotated)
mtexColorbar('title','intensity')

All three intensity patterns turn together relative to the plotted axes. Their values and the number of sampled directions do not change.

Make a coarse reconstruction

A 10 degree orientation grid and at most six solver iterations provide a quick consistency check.

recCoarse = calcODF(pf,'resolution',10*degree,'iterMax',6)

recCoarse = SO3FunRBF (3̅m1 → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 10°
  center: 2472 orientations, resolution: 10°
  weight: 1

The first check is always the recalculated pole figures against the measured ones above. Recalculate exactly the three measured entities. The 'superposition' option combines the two unresolved reflections with their imported structure coefficients instead of drawing them as separate pole figures.

plotPDF(recCoarse,pf.allH,'antipodal','silent',...
  'superposition',pf.c)
mtexColorbar('title','mrd')

The broad maxima occupy the same regions as in the measured panels and have the same relative order. At this coarse resolution that agreement is only a screening result, not evidence that the recovered ODF is unique. ODF Reconstruction explains the cost of resolution and iteration count. Iterative ODF Reconstruction shows what successively narrowing the kernel can gain.

Reconstruct at the default resolution

The final reconstruction uses the default orientation grid and kernel. Its iteration trace is suppressed because ODF Reconstruction explains that output in detail.

rec = calcODF(pf,'silent');

calcError returns one regularised relative residual per measured pole figure. Label the values so the superposed measurement remains identifiable.

regularisedRelativeResidual = calcError(pf,rec,'silent').';
fitSummary = table(regularisedRelativeResidual,...
  'RowNames',poleFigureLabel)

fitSummary =
  3×1 table
                       regularisedRelativeResidual
                       ___________________________
    (10-10)                      0.42687          
    (01-11)+(10-11)              0.18203          
    (11-22)                      0.35339

The superposed measurement in the middle fits best, at 0.18, and the \((10\bar{1}0)\) measurement fits worst, at 0.43. A smaller residual means a better match to the measured projection. It does not prove that the ODF itself is unique or true.

Locate the remaining mismatch

plotDiff shows the same regularised relative residual at every measured direction.

plotDiff(pf,rec,'silent')
mtexColorbar('title','relative residual')

The mismatch is scattered rather than concentrated in one patch, which is consistent with measurement noise. It also grows towards the rim, where the high specimen tilt makes defocusing correction least reliable. Both patterns are diagnostic clues, not proof of a single cause.

Exercises

Working through these on the same data set covers the rest of the chapter:

Further reading

Next

Export shows how to write measured and recalculated pole figures. Continue to ODF Analysis to quantify the assessed ODF and derive texture characteristics from it.