Grain Reference Orientation Deviation (GROD)

The grain reference orientation deviation asks how far each measurement has turned away from a chosen orientation for its own grain. The grain mean is the usual reference.

KAM compares a point with its immediate neighbours and asks how sharply the lattice is bent. GROD instead compares the point with the grain reference. It reveals orientation variation across the whole grain. GROD records lattice rotation relative to that reference. It is not a strain measurement or a measure of accumulated deformation.

Both the angle and the axis of the deviation carry information. The angle shows its magnitude, while the axis can distinguish rotation patterns that the angle alone hides.

The example is a deformed ferrite specimen. The misorientation axes are very sensitive to noise, so the orientations are denoised first. See Denoising.

The page assumes that the map has already been segmented into grains. See Grain Reconstruction for that step. Misorientation angle and axis should also be familiar; see Misorientations.

close all;

% use the rolling-direction frame carried by the ferrite specimen
plottingConvention.default('y↑→x');

mtexdata ferrite silent

highAngle = 10*degree;
lowAngle = 1*degree;
[grains,ebsd] = calcGrains(ebsd,'angle',[highAngle lowAngle],'minPixel',3);

% smooth grain boundaries
grains = smoothBoundary(grains,5);

% denoise the orientations
F = halfQuadraticFilter;
ebsd = smooth(ebsd,F,grains,'fill');

ipfKey = ipfColorKey(ebsd.CS);
plot(ebsd,ipfKey.orientation2color(ebsd.orientations))
hold on
plot(grains.boundary,'lineWidth',2)
hold off

The colour changes within the larger grains are the orientation gradients that GROD will quantify. The overlaid outlines show which grain supplies the reference orientation for each measurement.

The deviation angle

calcGROD computes the deviation as a misorientation, one per measurement. It needs the reconstructed grains as a second argument. It also needs ebsd.grainId, which calcGrains set.

% compute the grain reference orientation deviation
grod = ebsd.calcGROD(grains);

The first plot maps its angle. Subgrain boundaries are drawn on top and faded in proportion to their own misorientation.

% plot the misorientation angle of the GROD
plot(ebsd,grod.angle./degree,'micronbar','off')
mtexColorbar('title',{'misorientation angle in degree'})
mtexColorMap LaboTeX

% overlay grain and sub-grain boundaries
hold on
plot(grains.boundary,'lineWidth',1.5)
plot(grains.innerBoundary,'edgeAlpha',grains.innerBoundary.misorientation.angle / (5*degree))
hold off

grodAngle = angle(grod(:)) ./ degree;
fprintf('GROD angle: median %.2f degree; maximum %.1f degree\n',...
  median(grodAngle,'omitnan'),max(grodAngle,[],'omitnan'));

GROD angle: median 1.55 degree; maximum 18.9 degree

Half the measurements are within 1.6° of their grain mean and the worst reach 18.9°. The large values are not scattered. They fill extended parts of grains and often end at the inner boundaries.

Those boundaries were detected from local neighbour misorientations, not from GROD. A subgrain boundary can coincide with a jump in GROD, but a GROD jump is not its definition.

Grain orientation spread

Averaging the deviation angle over each grain gives the grain orientation spread. grainMean performs that averaging.

Although calcGrains stored grains.GOS, that value belongs to the raw orientations. The calculation below deliberately uses the denoised data.

GOS = grainMean(ebsd, grod.angle, grains);

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

fprintf('GOS for %d grains: median %.2f degree; maximum %.1f degree\n',...
  length(grains),median(GOS./degree,'omitnan'),...
  max(GOS./degree,[],'omitnan'));

GOS for 377 grains: median 0.69 degree; maximum 6.6 degree

Of the 377 grains the median spread is 0.69° and the largest 6.6°. The map is not uniform. Some grains have much more internal orientation variation than their neighbours, producing an order-of-magnitude spread.

GOS depends on segmentation, denoising, reference orientation, and grain size. Comparisons require the same choices.

The misorientation axis in crystal coordinates

Every deviation also has an axis, and it matters in which reference frame that axis is expressed. The crystal frame is attached to the lattice of the phase. In this frame the axis is an \((hk\ell)\) crystal direction, obtained with axis.

axCrystal = grod.axis;

plot(axCrystal,'contourf','fundamentalRegion','antipodal','figSize','small')
mtexColorbar('title','mrd')

crystalDensityRange = clim;
fprintf('Crystal-frame axis density: %.2f to %.2f mrd\n',...
  crystalDensityRange);

Crystal-frame axis density: 0.83 to 1.21 mrd

The range is 0.83 to 1.21 times uniform. In crystal coordinates the axes are therefore close to evenly distributed, with a slight preference for \([101]\). A colour key shows where those directions occur in the map.

colorKey = HSVDirectionKey(ebsd.CS,'antipodal');

plot(colorKey,'figSize','small')

Directions related by ferrite crystal symmetry have the same colour in this key. The deviation angle serves as transparency in the map below, so that points which have barely turned fade to white.

A small-angle axis is poorly constrained by conventional EBSD. Whitening it prevents uncertain axes from dominating the picture, but it is a visualization choice rather than an uncertainty correction.

% compute the color from the misorientation axis
color = colorKey.direction2color(axCrystal);

% and set the transparency from the misorientation angle
alpha = min(grod.angle/degree/7.5,1);

% plot the data
plot(ebsd,color,'micronbar','off','faceAlpha',alpha,'figSize','large')

hold on
plot(grains.boundary,'lineWidth',2)
plot(grains.innerBoundary,'edgeAlpha',grains.innerBoundary.misorientation.angle / (5*degree))
hold off

Whole regions of a grain share one colour, that is one misorientation axis. This is what makes the crystal frame worth looking at.

A low-angle boundary formed by a single slip system has its axis fixed by the geometry of that system. A pure tilt wall turns about an axis in the slip plane and perpendicular to the Burgers vector. A pure twist wall turns about the slip plane normal. This rule applies directly to the misorientation across a boundary. GROD compares a point with its reference.

A coherent GROD-axis cluster can nevertheless constrain candidate slip systems when combined with the angle map and independent slip geometry. V. Tong, E. Wielewski and B. Britton use long-range rotations in Characterisation of Slip and Twinning in High-Rate-Deformed Zirconium with EBSD.

A cluster alone does not identify a unique active system. The slip system and dislocation system chapters describe how to set up candidates in MTEX. See Tilt and Twist Boundaries for the boundary geometry itself.

The misorientation axis in specimen coordinates

The specimen frame is the reference frame in which the map and sample are expressed. The same axis in this frame is the crystal axis carried over by the orientation of the measurement.

The option 'noSymmetry' is essential. The axis must be the representative that belongs to this orientation, not a symmetric equivalent.

axSpecimen = ebsd.orientations .* grod.axis('noSymmetry');

plot(axSpecimen,'contourf','halfwidth',2.5*degree)
mtexColorbar('title','distribution of misorientation axes in mrd')

specimenDensityRange = clim;
fprintf('Specimen-frame axis density: %.2f to %.2f mrd\n',...
  specimenDensityRange);

Specimen-frame axis density: 0.24 to 6.77 mrd

This distribution runs from 0.24 to 6.77 times uniform. The same axes that were spread evenly over the crystal are strongly clustered in the specimen frame. Loading is defined in this frame, which makes the clustering useful. Texture and a few highly deformed grains can also produce maxima; the map below distinguishes those possibilities.

In the specimen frame the axes have no crystal symmetry and are directional rather than antipodal. The colour key therefore uses the whole sphere.

colorKey = HSVDirectionKey;

plot(colorKey,'figSize','small')

Opposite specimen directions have different colours in this key. The spatial plot follows the same construction as the crystal-frame map.

% compute color and transparency
omega = min(grod.angle/degree/7.5,1);
color = colorKey.direction2color(axSpecimen);

% plot the data
plot(ebsd,color,'micronbar','off','FaceAlpha',omega,'figSize','large')

hold on
plot(grains.boundary,'lineWidth',2)
plot(grains.innerBoundary,'edgeAlpha',grains.innerBoundary.misorientation.angle / (5*degree))
hold off

Each deformed grain carries a single colour over large parts of itself, and neighbouring grains rarely share it. The maxima of the distribution above therefore come from individual grains, not from a pattern running across the map.

Only some of the 377 grains are strongly deformed. A larger map would be needed to read the loading geometry from this figure.

Choosing another reference orientation

The grain mean is convenient, but it removes a rigid rotation shared by the whole grain. calcGROD accepts a third argument: one reference rotation for every grain ID, or one scalar for all grains.

A registered map of the undeformed specimen can supply one initial orientation per grain. GROD then retains whole-grain rotation relative to that state. Such a comparison first requires spatial registration; see TrueEBSD Distortion Correction. Changing the reference changes the angle, axis, and GOS. State that choice in every reported method.

The definition

For the orientation \(o_{i,j}\) at position \((i,j)\) and the chosen reference orientation \(r_g\) of its grain,

\[ \mathrm{GROD}_{i,j} = \mathbf S_{i,j} \cdot \mathrm{inv}(r_g) \cdot o_{i,j} \]

where the symmetry element \(\mathbf S_{i,j}\) is the one that makes the misorientation angle as small as possible. With no third input to calcGROD, \(r_g\) is the grain mean orientation.

Further reading

Next

Continue with Line Profiles to turn a colour gradient into a curve along a chosen path. The broader family of grain-scale statistics is described in Grain Orientation Parameters.