Detecting sample symmetry

Explains how to detect orthotropic symmetry in an ODF.

On this page ...
A synthetic example
Reconstruct an ODF from simulated EBSD data
Detect the sample symmetry axis in the reconstructed ODF
Sample symmetry in an ODF computed from pole figure data

A synthetic example

We start by modeling a orthotropic ODF with cubic crystal symmetry.

CS = crystalSymmetry('cubic');
SS = specimenSymmetry('222');

% some component center
ori = [orientation('euler',135*degree,45*degree,120*degree,CS,SS) ...
  orientation('euler', 60*degree, 54.73*degree, 45*degree,CS,SS) ...
  orientation('euler',70*degree,90*degree,45*degree,CS,SS)...
  orientation('euler',0*degree,0*degree,0*degree,CS,SS)];

% with corresponding weights
c = [.4,.13,.4,.07];

% the model odf
odf = unimodalODF(ori(:),'weights',c,'halfwidth',12*degree)

% lets plot some pole figures
h = [Miller(1,1,1,CS),Miller(2,0,0,CS),Miller(2,2,0,CS)];
plotPDF(odf,h,'antipodal','silent','complete')
 
odf = ODF  
  crystal symmetry : m-3m
  specimen symmetry: 222
 
  Radially symmetric portion:
    kernel: de la Vallee Poussin, halfwidth 12°
    center: Rotations: 4x1
    weight: 1
 

Reconstruct an ODF from simulated EBSD data

Next we simulated some EBSD data, rotate them and estimate an ODF from the individual orientations.

% define a sample rotation
%rot = rotation('euler',0*degree,0*degree,1*degree);
rot = rotation('euler',15*degree,12*degree,-5*degree);


% Simulate individual orientations and rotate them.
% Note that we loose the sample symmetry by rotating the orientations
ori = rot * calcOrientations(odf,1000)

% estimate an ODF from the individual orientations
odf_est = calcODF(ori,'halfwidth',10*degree)

% and visualize it
figure, plotPDF(odf_est,h,'antipodal',8,'silent');
 
ori = orientation  
  size: 1 x 1000
  crystal symmetry : m-3m
  specimen symmetry: 1
 
 
odf_est = ODF  
  crystal symmetry : m-3m
  specimen symmetry: 1
 
  Harmonic portion:
    degree: 28
    weight: 1
 

Detect the sample symmetry axis in the reconstructed ODF

We observe that the reconstructed ODF has almost orthotropic symmetry, but with respect to axed different from x, y, z. With the following command we can determine an rotation such that the rotated ODF has almost orthotropic symmetry with respect to x, y, z. The second argument is some starting direction where MTEX locks for a symmetry axis.

[odf_corrected,rot_inv] = centerSpecimen(odf_est);

figure
plotPDF(odf_corrected,h,'antipodal',8,'silent')

% the difference between the applied rotation and the estimate rotation
angle(rot,inv(rot_inv)) / degree
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ans =
    0.8195

Sample symmetry in an ODF computed from pole figure data

In the next example we apply the function centerSpecimen to an ODF estimated from pole figure data. Lets start by importing them

fname = fullfile(mtexDataPath,'PoleFigure','aachen_exp.EXP');
pf = loadPoleFigure(fname);

plot(pf,'silent')

In a second step we compute an ODF from the pole figure data

odf = calcODF(pf,'silent')

plotPDF(odf,h,'antipodal','silent')
 
odf = ODF  
  crystal symmetry : 432
  specimen symmetry: 1
 
  Radially symmetric portion:
    kernel: de la Vallee Poussin, halfwidth 5°
    center: 4790 orientations, resolution: 5°
    weight: 1
 

Finally, we detect the orthotropic symmetry axes a1, a2, a3 by

[~,~,a1,a2] = centerSpecimen(odf,yvector)
a3 = cross(a1,a2)

annotate([a1,a2,a3],'label',{'RD','TD','ND'},'backgroundcolor','w','MarkerSize',8)
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a1 = vector3d  
 size: 1 x 1
 resolution: 0.71°
          x          y          z
  0.0499876   0.998745 0.00325886
 
a2 = vector3d  
 size: 1 x 1
         x         y         z
  -0.99875 0.0499879         0
 
a3 = vector3d  
 size: 1 x 1
 resolution: 0.71°
             x            y            z
  -0.000162903  -0.00325478     0.999995