Grain Orientation Parameters

A reconstructed grain is represented by the orientations measured at all of its EBSD points. Grain orientation parameters reduce that orientation cloud to three questions: where is its centre, how wide is it, and does it have a preferred direction of spread?

The last label is used inconsistently. This page does not average misorientation axes to obtain GAX. It fits a fibre to the orientation cloud and calls the result a dispersion axis.

The example is a deformed ferrite map. It assumes familiarity with grain reconstruction and misorientations. See Selecting Grains for the relation between a grain and its EBSD points.

close all;

% use the map frame shown throughout this example
plottingConvention.default('y↓→x');

% import the data
mtexdata ferrite silent

% reconstruct grains and store one grain id per measurement
[grains,ebsd] = calcGrains(ebsd,'angle',7.5*degree,'minPixel',5);
ebsd = ebsd.project2FundamentalRegion;

% smooth only the outlines used in the figures
grains = smoothBoundary(grains,5);

% use one colour key for the measurement and grain-mean maps
ipfKey = ipfColorKey(ebsd.CS);
ebsdColor = ipfKey.orientation2color(ebsd.orientations);

% plot the measured orientations and reconstructed grain boundaries
plot(ebsd,ebsdColor)
hold on
plot(grains.boundary,'lineWidth',2)
hold off

The colour changes inside the black outlines are the intragranular orientation variations measured below. Boundary smoothing changes the drawn outlines, but not which measurement belongs to which grain.

The mean orientation

Select one grain by its map coordinates. Indexing the EBSD map with that grain follows ebsd.grainId and returns all measurements assigned to it.

% select a grain by x and y coordinates
grainSel = grains(42,17);

% display all EBSD orientations within the grain
ori = ebsd(grainSel).orientations

ori = orientation (Ferrite → y↓→x)
  size: 1964 × 1

The summary reports how many orientations represent this grain. Their mean is the symmetry-aware centre of the cloud, not an arithmetic mean of Euler angles.

calcGrains has already stored this result as grains.meanOrientation. There is no reason to compute it again.

meanColor = ipfKey.orientation2color(grains.meanOrientation);
plot(grains,meanColor,'micronbar','off')

The map assigns one mean orientation colour to each grain. It therefore removes the within-grain colour gradients visible in the first map while preserving the orientation contrast between grains.

Deviation from the mean

With one reference orientation per grain, every measurement can be described by how far it departs from that reference. The resulting misorientation is the grain reference orientation deviation, GROD.

mis2meanSelected = inv(grainSel.meanOrientation) .* ori;

calcGROD performs the same calculation for the whole map and uses the appropriate mean orientation for each grain ID.

mis2mean = calcGROD(ebsd,grains);

Grain orientation spread

A misorientation has an angle and an axis. Averaging only the GROD angle over a grain gives the grain orientation spread, GOS. grainMean averages any per-measurement quantity over grains.

% average the GROD angles for each grain
GOS = ebsd.grainMean(mis2mean.angle,grains);

% plot the result in degrees
plot(grains,GOS./degree,'micronbar','off')
mtexColorbar('title','GOS in degree')

Grains that are internally uniform are dark. Grains with a larger orientation gradient stand out, but GOS alone does not reveal the shape or direction of that gradient.

The same value is available as grains.GOS because calcGrains computed it from the orientations used for reconstruction. If the orientations are subsequently denoised or otherwise changed, recompute GROD and GOS instead of treating the stored value as updated.

grainMean accepts a function handle as a further argument. Using @max gives the largest deviation in each grain rather than the average. A median, a quantile, or another statistic can be used in the same way.

% compute the maximum GROD angle for each grain
MGOS = ebsd.grainMean(mis2mean.angle,grains,@max);

% plot it
plot(grains,MGOS./degree,'micronbar','off')
mtexColorbar('title','maximum GROD in degree')

The maximum map emphasizes isolated peaks and sharp internal changes. The GOS map is less sensitive to one extreme measurement because it averages over the whole grain.

Grain average misorientation

GOS compares every measurement with the mean of its grain. It therefore grows with the total orientation change across a large grain, even when the change is gentle.

The grain average misorientation, GAM, asks a local question instead. It is the kernel average misorientation averaged over each grain. The default KAM compares first-order neighbours, applies no angle threshold, and does not cross reconstructed grain boundaries when ebsd.grainId is present.

kam = ebsd.KAM;
gam = ebsd.grainMean(kam,grains);

plot(grains,gam./degree,'micronbar','off')
mtexColorbar('title','GAM in degree')
setColorRange([0,3])

The GOS and GAM maps do not agree, and they are not meant to. GOS measures the total orientation change across a grain, whereas GAM measures the mean steepness of the local changes. A grain bent smoothly from one side to the other can have high GOS and low GAM. A subgrain boundary can instead produce high GAM. The two measures are regularly confused in the literature.

Neither quantity is a direct strain or dislocation-density measurement. Segmentation, grain size, scan step, angular precision, denoising, and the KAM neighbourhood all affect the result. Report those choices whenever GOS or GAM is compared between maps.

The crystal dispersion axis

Slip on one dominant system can spread the orientations of a grain along a curve rather than equally in every direction. Such a curve is an orientation fibre. Its axis may constrain the slip system that produced the orientation gradient.

fibre.fit finds the fibre that best fits a list of orientations. It always returns a fibre, even for a round cloud, so the fit must be checked before its axis is interpreted.

% visualize the selected grain orientations in a pole figure
figure(2)
h = Miller({1,0,0},ebsd.CS);
plotPDF(ori,h,'MarkerSize',2,'all')

% fit a fibre to those orientations
[f,lambda,fit] = fibre.fit(ori,'local');

% add the fibre to the pole figure
hold on
plotPDF(f.symmetrise,h,'lineColor','orange','lineWidth',2)
hold off

The orientations lie in short streaks rather than round blobs, and the orange fibre follows them. The fitted axis is f.r in the specimen frame and f.h in the crystal frame.

axisSpecimen = f.r
axisCrystal = f.h

axisSpecimen = vector3d (y↓→x)
      x       y       z
  0.282 -0.0471  -0.958
 
axisCrystal = Miller (Ferrite)
       h       k       l
  1.6922 -0.1497 -2.3132

Rotation about an axis leaves that axis itself fixed. The crystal direction f.h should therefore scatter less than the other directions in the pole figure.

hold on
plot(ori.*f.h,'MarkerSize',2,'all','MarkerFaceColor','k',...
  'antipodal','micronbar','off')
hold off

The black points form a compact cluster where the other poles form streaks. This is a direct visual check on the fitted crystal axis.

The second output of fibre.fit contains four eigenvalues of the orientation tensor in ascending order. The largest describes the overall concentration. The second largest, lambda(3), describes spread along the fitted fibre. The two smallest describe scatter away from it; lambda(2) is the larger of those two. The third output is the mean angular distance from the orientations to the fibre.

lambda
fitDegrees = fit./degree

lambda =
    0.0001    0.0001    0.0004    0.9994
fitDegrees =
    1.2233

Fitting every sufficiently large grain

A separate fibre must be fitted to each grain. Very small grains contain too few orientations to define the shape of an orientation cloud, so the example keeps grains with more than 50 measurements.

grainsLarge = grains(grains.isIndexed & grains.numPixel > 50);
nLarge = length(grainsLarge);
lambda = nan(nLarge,4);
fit = nan(nLarge,1);
GAX_C = Miller.nan(nLarge,1,ebsd.CS);
GAX_S = vector3d.nan(nLarge,1);

% fit one fibre per grain and store its axes in both frames
for k = 1:nLarge

  [f,lambda(k,:),fit(k)] = ...
    fibre.fit(ebsd(grainsLarge(k)).orientations,'local');
  GAX_C(k) = f.h;
  GAX_S(k) = f.r;

end

The ratio lambda(3)/lambda(2) compares spread along the fibre with scatter away from it. A ratio near 1 describes a blob rather than a line. The threshold 3 used below is a screening choice, not a universal material constant.

linearity = lambda(:,3)./lambda(:,2);
isFibre = linearity > 3;

corrFitLambda2 = corrcoef(fit,lambda(:,2));
corrFitLambda2 = corrFitLambda2(1,2);
corrFitLambda3 = corrcoef(fit,lambda(:,3));
corrFitLambda3 = corrFitLambda3(1,2);

fitStatistics = table(nLarge,nnz(isFibre),...
  corrFitLambda2,corrFitLambda3,...
  min(fit./degree),max(fit./degree),...
  'VariableNames',{'largeGrains','acceptedAxes','corrFitLambda2',...
  'corrFitLambda3','minFitDegree','maxFitDegree'})

fitStatistics =
  1×6 table
    largeGrains    acceptedAxes    corrFitLambda2    corrFitLambda3    minFitDegree    maxFitDegree
    ___________    ____________    ______________    ______________    ____________    ____________
        161            103            0.92233           0.60797          0.25359          2.1785

The table makes the screening reproducible. Of the 161 large grains, 103 pass the ratio threshold. Fit and lambda(2) correlate at 0.92, whereas fit and lambda(3) correlate at 0.61. The fit angles range from 0.25 to 2.18 degrees.

This is expected because fit measures scatter away from the fibre. A grain worth reading a dispersion axis from has large lambda(3), small lambda(2), and therefore a large ratio.

newMtexFigure('figSize','wide');
plot(grainsLarge,lambda(:,3),'micronbar','off')
mtexTitle('\(\lambda_3\)')

nextAxis(1,2)
plot(grainsLarge,lambda(:,2),'micronbar','off')
mtexTitle('\(\lambda_2\)')

nextAxis(1,3)
plot(grainsLarge,fit./degree,'micronbar','off')
mtexTitle('fit in degree')

The lambda(3) map shows where orientation clouds extend along a fibre. The lambda(2) and fit maps instead highlight scatter away from that fibre. Their visual similarity agrees with the correlations in the table.

Axes in crystal coordinates

The fitted axes are crystal directions. An HSVDirectionKey colours those directions, and it needs 'antipodal' because an axis and its negative are the same axis.

% define the colour key
cKey = HSVDirectionKey(ebsd.CS,'antipodal');

% plot the key and the accepted dispersion axes
plot(cKey)
hold on
plot(project2FundamentalRegion(GAX_C(isFibre),'antipodal'),...
  'MarkerFaceColor','black')
hold off

The black dots show which crystal directions occur as credible axes. A cluster would point to a preferred crystal direction and could constrain a common slip mechanism. The colours below show where grains with similar axes lie in the map.

% compute colours from the accepted crystal axes
color = cKey.direction2color(GAX_C(isFibre));

% plot the corresponding grains
plot(grainsLarge(isFibre),color,'micronbar','off')

Similar colours do not form one dominant spatial population in this map. The accepted axes also cover much of the fundamental sector rather than gathering at one crystal direction.

Axes in specimen coordinates

A continuous one-to-one colour key cannot cover a whole sphere without a discontinuity. MTEX instead draws specimen axes as compass needles: grey for an axis in the map plane, and split into black and white halves when its ends point out of and into the plane.

plot(grains,GOS./degree,'micronbar','off')
mtexColorbar('title','GOS in degree')

hold on
plot(grainsLarge(isFibre),GAX_S(isFibre),'micronbar','off')
hold off

The needles let orientation spread be compared with position and GOS. An axis is not trustworthy merely because a needle can be drawn; only the grains that passed the eigenvalue-ratio screen are shown.

Dispersion axes can relate to flow. Under simple shear they may align with the vorticity axis. Under pure-shear-dominated deformation they may form a girdle whose normal follows the shortening direction. These are kinematic interpretations of an aggregate, not identities that every deformed grain must obey.

plot(GAX_S(isFibre),'antipodal','MarkerSize',4)

The scatter plot is the direct evidence. Contours make a preferred population easier to judge, but only after axes without a fibre-like orientation cloud have been removed.

Do not weight these contours by fit. A larger fit is a larger distance from the fibre and therefore a worse fit, not a stronger axis.

axisDensity = calcDensity(GAX_S(isFibre),'antipodal',...
  'halfwidth',10*degree);
[axisDensityMaximum,axisDensityDirection] = max(axisDensity)

hold on
plot(axisDensity,'contour','antipodal','contours',[1 2 3],...
  'lineWidth',2)
hold off

axisDensityMaximum =
    4.8582
 
axisDensityDirection = vector3d (y↓→x)
 antipodal: true
      x      y      z
  0.994 0.0439 0.0999

The accepted axes still fill much of the projection, but the distribution is not uniform. Its maximum is 4.86 mrd near the specimen X axis. This supports a preferred population along X rather than one axis shared by every grain, and it does not form a girdle. A flow interpretation also requires the independently established specimen and kinematic frames. This example is ferritic steel rather than a shear-zone rock, so the geological kinematic patterns above are possibilities, not ground truth.

The definitions

Let \(o_{g,i}\) be measurement \(i\) in grain \(g\), and let \(\overline{o}_g\) be the symmetry-aware mean orientation of that grain. MTEX chooses the symmetrically equivalent misorientation with the smallest angle when it computes

\[ \mathrm{GROD}_{g,i} = \mathrm{inv}(\overline{o}_g)\,o_{g,i}. \]

For \(N_g\) measurements in the grain, the scalar summaries are

\[ \mathrm{GOS}_g = \frac{1}{N_g}\sum_i \angle(\mathrm{GROD}_{g,i}), \]

\[ \mathrm{GAM}_g = \frac{1}{N_g}\sum_i \mathrm{KAM}_{g,i}. \]

The neighbour definition inside KAM and the reference orientation inside GROD are therefore part of the definitions, not plotting options.

Further reading

Related pages

Grain Reference Orientation Deviation develops GROD angle and axis maps. Kernel Average Misorientation develops the neighbourhood and threshold choices behind GAM. Dispersion Axes gives the detailed single-grain validation workflow used before interpreting a fitted axis. The next page, Grain Neighbours, changes from properties inside one grain to relationships between adjacent grains.