A pseudo symmetry is a rotation that is not a symmetry of the crystal but nevertheless maps the diffraction pattern almost onto itself. The indexing algorithm has then no reliable criterion to decide between the true orientation and the orientation rotated by the pseudo symmetry and picks essentially at random. As a consequence single grains appear in the orientation map as two or more grains that are separated by a boundary with exactly the pseudo symmetric misorientation.
Such boundaries can be told apart from real grain boundaries by their shape. A real grain boundary is a physical interface and hence a reasonably smooth curve. A boundary that only separates the two indexing solutions follows the noise of the indexing and is therefore ragged. This is measured by the tortuosity
\[ \tau = \frac{\textrm{length of the boundary}}{\textrm{distance between its end points}} \]
which is close to one for a straight boundary and large for a ragged one. The command cleanUpPseudoSym uses exactly this criterion: it merges all grains that are separated by a tortuous boundary with a pseudo symmetric misorientation and rotates the orientations of the smaller part by the pseudo symmetry.
A Data Set With Pseudo Symmetry
Olivine is a classical example. Its oxygen sublattice is almost hexagonally close packed with the stacking axis along [100], i.e., rotations about [100] by multiples of 60 degree map the oxygen positions nearly onto themselves although the true symmetry mmm contains only the 180 degree rotation. Let us import a Forsterite data set and reconstruct the grains.
plottingConvention.default('y↑→x');
mtexdata forsterite
% consider only indexed data
ebsd = ebsd('indexed');
% reconstruct the grain structure
[grains,ebsd] = calcGrains(ebsd,'angle',10*degree,'minPixel',5);
% plot the orientation map of the Forsterite phase
plot(ebsd('Fo'),ebsd('Fo').orientations)
hold on
plot(grains.boundary,'linewidth',1.5)
hold offebsd = EBSDsquare (y↑→x)
Phase Orientations Mineral Color Symmetry Crystal reference frame
0 58485 (24%) notIndexed none
1 152345 (62%) Forsterite LightSkyBlue mmm
2 26058 (11%) Enstatite DarkSeaGreen mmm
3 9064 (3.7%) Diopside Goldenrod 12/m1 X||a*, Y||b, Z||c
Properties: bands, bc, bs, error, mad, oldId
Scan unit : um
X x Y x Z : [0 → 36550] x [0 → 16750] x [0 → 0]
Normal vector: (0,0,1)
Square grid :336 x 732
Note that we did not smooth the grain boundaries. Since cleanUpPseudoSym decides by the raggedness of the boundaries, smoothing has to be applied after the correction and not before.
Detecting the Pseudo Symmetry
If we do not know the pseudo symmetry beforehand we may look for it in the distribution of the boundary misorientations. Since we expect a rotation about 60 degree we restrict ourselves to those boundaries and plot the distribution of the corresponding rotational axes.
% the misorientations at all Forsterite - Forsterite boundaries
mori = grains.boundary('Fo','Fo').misorientation;
% restrict to those with a rotational angle close to 60 degree
mori = mori(mori.angle > 55*degree & mori.angle < 65*degree);
% and plot the distribution of the rotational axes
plot(mori.axis,'contourf','fundamentalRegion','halfwidth',5*degree)
mtexColorbar
The sharp maximum at [100] confirms our expectation. Accordingly we define the pseudo symmetry as the rotation about [100] by 60 degree.
cs = ebsd('Fo').CS;
psSym = orientation.byAxisAngle(Miller(1,0,0,cs,'uvw'),60*degree)psSym = misorientation (Forsterite → Forsterite)
Bunge Euler angles in degree
phi1 Phi phi2
0 60 0Modulo the true symmetry mmm the rotations by 60 and by 120 degree are two different operations - the latter being the inverse of the former - and both occur in the map, since the indexing may have picked either solution. It is nevertheless enough to pass one of them. A measurement is a fixed representative of an orientation, so the alternative solution is not simply ori * psSym but ori * s * psSym for some symmetry element s, and cleanUpPseudoSym generates all these operators itself. Passing 60 degree, 120 degree or both gives the same result.
Note also that the pseudo symmetry is a misorientation, i.e., it has the same crystal symmetry on both sides. This is required by cleanUpPseudoSym and at the same time tells the command which phase it should correct.
Pseudo Symmetric Grain Boundaries
Let us select all boundary segments whose misorientation is the pseudo symmetry and have a closer look at one of the affected regions. A grain boundary has no direction - its misorientation carries the grain exchange symmetry - so this catches the segments of both solutions at once.
gB = grains.boundary('Fo','Fo');
gB = gB(any(angle(gB.misorientation,psSym) < 2*degree,2))gB = grainBoundary (y↑→x)
Segments length mineral 1 mineral 2
143 7534 µm Forsterite Forsterite% a region containing one of the pseudo symmetric grains
region = [27300 1900 1000 700];
% plot the orientation map of this region
ebsdS = ebsd(inpolygon(ebsd,region));
plot(ebsdS('Fo'),ebsdS('Fo').orientations)
% and on top the grain boundaries
hold on
plot(grains.boundary,'linewidth',1.5)
plot(gB,'linewidth',3,'linecolor','r')
hold off
The blue region has been indexed with the second solution. Note how the red boundary meanders around single pixels instead of following a smooth curve - this is the signature we are after.
The Tortuosity Criterion
Internally cleanUpPseudoSym splits the selected boundary segments into connected components and computes for each component its total length and the distance between its extreme points.
% the connected components of the pseudo symmetric boundaries
compId = gB.componentId;
% the length of each component
len = accumarray(compId,gB.segLength);
% the extent of each component
xmin = accumarray(compId,gB.midPoint.x,[],@min);
xmax = accumarray(compId,gB.midPoint.x,[],@max);
ymin = accumarray(compId,gB.midPoint.y,[],@min);
ymax = accumarray(compId,gB.midPoint.y,[],@max);
dist = sqrt((xmax-xmin).^2 + (ymax-ymin).^2);
% the tortuosity and the number of segments of each component
tortuosity = len ./ dist;
numSeg = accumarray(compId,1);
% components above the threshold are considered as pseudo symmetric
plot(numSeg,tortuosity,'o','MarkerFaceColor','b')
hold on
plot(xlim,[1.5 1.5],'r--')
hold off
xlabel('number of segments'), ylabel('tortuosity')
Very short components are unreliable - a component consisting of a single segment has zero extent and hence infinite tortuosity, which is why two of the components are missing in the plot above. For this reason only components with more than four segments are taken into account.
Correcting the Map
All this is done by a single call to cleanUpPseudoSym. It returns the corrected EBSD data, the grains after merging and the number of pixels that have been rotated.
[ebsdC,grainsC,numChanged] = cleanUpPseudoSym(ebsd,grains,psSym);
numChangednumChanged =
275Plotting the same region again we see that the two indexing solutions have been unified into a single grain.
ebsdS = ebsdC(inpolygon(ebsdC,region));
plot(ebsdS('Fo'),ebsdS('Fo').orientations)
hold on
plot(grainsC.boundary,'linewidth',1.5)
hold off
Only now, with the pseudo symmetric boundaries removed, it makes sense to smooth the grain boundaries
grainsC = smoothBoundary(grainsC,5);Options
The tolerance within which a boundary misorientation has to match the pseudo symmetry is controlled by the option 'delta', the tortuosity above which a component is considered as pseudo symmetric by the option 'threshold'
[ebsdC,grainsC] = cleanUpPseudoSym(ebsd,grains,psSym,...
'delta',2*degree,'threshold',1.5)Lowering the threshold corrects more boundaries but eventually starts to remove real grain boundaries as well. It is therefore a good idea to look at the tortuosity plot above before choosing it.
Since the pseudo symmetry determines the phase that is corrected, only a single phase is treated per call. In order to clean up several phases simply call the command once for every phase, each time with the pseudo symmetries of that phase.