Theory of Misorientations edit page

Misorientation describe the relative orientation of two crystal with respect to each other. Those crystal may be of the same phase or of different phases. Misorientation are used to describe

Grain Exchange Symmetry

Misorientation describes the relative orientation of two grains with respect to each other. Important concepts are twinnings and CSL (coincidence site lattice) misorientations. To illustrate this concept at a practical example let us first import some Magnesium EBSD data.

mtexdata twins silent

% use only proper symmetry operations
ebsd('M').CS = ebsd('M').CS.properGroup;

% compute grains
grains = calcGrains(ebsd('indexed'),'threshold',5*degree);
CS = grains.CS; % extract crystal symmetry

Next we plot the grains together with their mean orientation and highlight grain 74 and grain 85

plot(grains,grains.meanOrientation,'micronbar','off')

hold on
plot(grains([74,85]).boundary,'edgecolor','w','linewidth',2)
hold off

text(grains([74,85]),{'1','2'})

After extracting the mean orientation of grain 74 and 85

ori1 = grains(74).meanOrientation;
ori2 = grains(85).meanOrientation;

we may compute the misorientation angle between both orientations by

angle(ori1, ori2) ./ degree
ans =
   85.6996

Note that the misorientation angle is computed by default modulo crystal symmetry, i.e., the angle is always the smallest angles between all possible pairs of symmetrically equivalent orientations. In our example this means that symmetrisation of one orientation has no impact on the angle

angle(ori1, ori2.symmetrise) ./ degree
ans =
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996
   85.6996

The misorientation angle neglecting crystal symmetry can be computed by

angle(ori1, ori2.symmetrise,'noSymmetry')./ degree
ans =
  107.9490
  100.9100
   94.3022
  136.6144
  107.7191
  179.5950
  140.0710
  137.3683
  179.8142
  101.4342
  140.4132
   85.6996

We see that the smallest angle indeed coincides with the angle computed before.

Misorientations

Remember that both orientations ori1 and ori2 map crystal coordinates onto specimen coordinates. Hence, the product of an inverse orientation with another orientation transfers crystal coordinates from one crystal reference frame into crystal coordinates with respect to another crystal reference frame. This transformation is called misorientation

mori = inv(ori1) * ori2
mori = misorientation (Magnesium → Magnesium)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2    Inv.
  149.587 94.3019  150.14       0

In the present case the misorientation describes the coordinate transform from the reference frame of grain 85 into the reference frame of crystal 74. Take as an example the plane {11-20} with respect to the grain 85. Then the plane in grain 74 which alignes parallel to this plane can be computed by

round(mori * Miller(1,1,-2,0,CS))
ans = Miller (Magnesium)
  h  k  i  l
  2 -1 -1  0

Conversely, the inverse of mori is the coordinate transform from crystal 74 to grain 85.

round(inv(mori) * Miller(2,-1,-1,0,CS))
ans = Miller (Magnesium)
  h  k  i  l
  1  1 -2  0

Coincident lattice planes

The coincidence between major lattice planes may suggest that the misorientation is a twinning misorientation. Lets analyse whether there are some more alignments between major lattice planes.

%m = Miller({1,-1,0,0},{1,1,-2,0},{1,-1,0,1},{0,0,0,1},CS);
m = Miller({1,-1,0,0},{1,1,-2,0},{-1,0,1,1},{0,0,0,1},CS);

% cycle through all major lattice planes
close all
for im = 1:length(m)
  % plot the lattice planes of grains 85 with respect to the
  % reference frame of grain 74
  plot(mori * m(im).symmetrise,'MarkerSize',10,...
    'DisplayName',char(m(im)),'figSize','large','noLabel','upper')
  hold all
end
hold off

% mark the corresponding lattice planes in the twin
mm = round(unique(mori*m.symmetrise,'noSymmetry'),'maxHKL',6);
annotate(mm,'labeled','MarkerSize',5,'figSize','large','textAboveMarker')

% show legend
legend({},'location','SouthEast','FontSize',13);

we observe an almost perfect match for the lattice planes {11-20} to {-2110} and {1-101} to {-1101} and good coincidences for the lattice plane {1-100} to {0001} and {0001} to {0-661}. Lets compute the angles explicitly

angle(mori * Miller(1,1,-2,0,CS),Miller(2,-1,-1,0,CS)) / degree
angle(mori * Miller(1,0,-1,-1,CS),Miller(1,-1,0,1,CS)) / degree
angle(mori * Miller(0,0,0,1,CS) ,Miller(1,0,-1,0,CS),'noSymmetry') / degree
angle(mori * Miller(1,1,-2,2,CS),Miller(1,0,-1,0,CS)) / degree
angle(mori * Miller(1,0,-1,0,CS),Miller(1,1,-2,2,CS)) / degree
ans =
    0.4456
ans =
    0.2015
ans =
   59.6819
ans =
    2.6310
ans =
    2.5651

Twinning misorientations

Lets define a misorientation that makes a perfect fit between the {11-20} lattice planes and between the {10-11} lattice planes

mori = orientation.map(Miller(1,1,-2,0,CS),Miller(2,-1,-1,0,CS),...
  Miller(-1,0,1,1,CS),Miller(-1,1,0,1,CS))


% the rotational axis
round(mori.axis)

% the rotational angle
mori.angle / degree
mori = misorientation (Magnesium → Magnesium)
 
 (01-10) || (10-10)   [0001] || [0001]
 
 
ans = Miller (Magnesium)
  h  k  i  l
  1  0 -1  0
ans =
     0

and plot the same figure as before with the exact twinning misorientation.

% cycle through all major lattice planes
close all
for im = 1:length(m)
  % plot the lattice planes of grains 85 with respect to the
  % reference frame of grain 74
  plot(mori * m(im).symmetrise,'MarkerSize',10,...
    'DisplayName',char(m(im)),'figSize','large','noLabel','upper')
  hold all
end
hold off

% mark the corresponding lattice planes in the twin
mm = round(unique(mori*m.symmetrise,'noSymmetry'),'maxHKL',6);
annotate(mm,'labeled','MarkerSize',5,'figSize','large')

% show legend
legend({},'location','NorthWest','FontSize',13);

Highlight twinning boundaries

It turns out that in the previous EBSD map many grain boundaries have a misorientation close to the twinning misorientation we just defined. Lets Lets highlight those twinning boundaries

% consider only Magnesium to Magnesium grain boundaries
gB = grains.boundary('Mag','Mag');

% check for small deviation from the twinning misorientation
isTwinning = angle(gB.misorientation,mori) < 5*degree;

% plot the grains and highlight the twinning boundaries
plot(grains,grains.meanOrientation,'micronbar','off')
hold on
plot(gB(isTwinning),'edgecolor','w','linewidth',2)
hold off

From this picture we see that large fraction of grain boudaries are twinning boundaries. To make this observation more evident we may plot the boundary misorientation angle distribution function. This is simply the angle distribution of all boundary misorientations and can be displayed with

close all
plotAngleDistribution(gB.misorientation)

From this we observe that we have about 50 percent twinning boundaries. Analogously we may also plot the axis distribution

plotAxisDistribution(gB.misorientation,'contour')

which emphasises a strong portion of rotations about the (-12-10) axis.

Phase transitions

Misorientations may not only be defined between crystal frames of the same phase. Lets consider the phases Magnetite and Hematite.

CS_Mag = loadCIF('Magnetite')
CS_Hem = loadCIF('Hematite')
CS_Mag = crystalSymmetry
 
  mineral : Magnetite    
  symmetry: m-3m         
  elements: 48           
  a, b, c : 8.4, 8.4, 8.4
 
 
CS_Hem = crystalSymmetry
 
  mineral        : Hematite          
  symmetry       : -3m1              
  elements       : 12                
  a, b, c        : 5, 5, 14          
  reference frame: X||a*, Y||b, Z||c*

The phase transition from Magnetite to Hematite is described in literature by {111}_m parallel {0001}_h and {-101}_m parallel {10-10}_h The corresponding misorientation is defined in MTEX by

Mag2Hem = orientation.map(...
  Miller(1,1,1,CS_Mag),Miller(0,0,0,1,CS_Hem),...
  Miller(-1,0,1,CS_Mag),Miller(1,0,-1,0,CS_Hem))
Mag2Hem = misorientation (Magnetite → Hematite)
 
 (01-1) || (0-110)   [111] || [0001]

Assume a Magnetite grain with orientation

ori_Mag = orientation.byEuler(0,0,0,CS_Mag)
ori_Mag = orientation (Magnetite → xyz)
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0    0    0    0

Then we can compute all variants of the phase transition by

symmetrise(ori_Mag) * inv(Mag2Hem)
ans = orientation (Hematite → xyz)
  size: 48 x 1

and the corresponding pole figures by

plotPDF(symmetrise(ori_Mag) * inv(Mag2Hem),...
  Miller({1,0,-1,0},{1,1,-2,0},{0,0,0,1},CS_Hem))