Symmetricaly Equivalent Orientations

Symmetricaly Equivalent Orientations edit page

A crystal orientation always appears as a class of symmetrically equivalent rotations which all transform the crystal reference frame into the specimen reference frame and are physicaly not distinguishable.

Lets start by defining some random orientation

% trigonal crystal symmetry
cs = crystalSymmetry('6')

% monoclinic specimen symmetry with respect to the x-axis
ss = specimenSymmetry('112')

% a random orientation
ori = orientation.rand(cs,ss)

cs = crystalSymmetry
 
  symmetry       : 6                 
  elements       : 6                 
  a, b, c        : 1, 1, 1           
  reference frame: X||a*, Y||b, Z||c*
 
 
ss = monoclinic specimenSymmetry  
 
 
ori = orientation (6 → xyz (112))
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  105.628 103.798 48.7838

Since orientations transform crystal coordinates into specimen coordinates crystal symmetries will act from the right and specimen symmetries from the left

% symmetrically equivalent orientations with respect to crystal symmetry
ori * cs

ans = orientation (6 → xyz (112))
  size: 1 x 6
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  105.628 103.798 48.7838
  105.628 103.798 108.784
  105.628 103.798 168.784
  105.628 103.798 228.784
  105.628 103.798 288.784
  105.628 103.798 348.784

We observe that only the third Euler angle phi2 changes as this Euler angle applies first to the crystal coordinates.

% symmetrically equivalent orientations with respect to specimen symmetry
ss * ori

ans = orientation (6 → xyz (112))
  size: 2 x 1
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  105.628 103.798 48.7838
  285.628 103.798 48.7838

Combining crystal and specimen symmetry we obtain 6 crystallographically equivalent orientations to ori

ss * ori * cs

ans = orientation (6 → xyz (112))
  size: 2 x 6
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  105.628 103.798 48.7838
  285.628 103.798 48.7838
  105.628 103.798 108.784
  285.628 103.798 108.784
  105.628 103.798 168.784
  285.628 103.798 168.784
  105.628 103.798 228.784
  285.628 103.798 228.784
  105.628 103.798 288.784
  285.628 103.798 288.784
  105.628 103.798 348.784
  285.628 103.798 348.784

A shortcut for this operation is the command symmetrise

symmetrise(ori)

ans = orientation (6 → xyz (112))
  size: 12 x 1
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  105.628 103.798 48.7838
  285.628 103.798 48.7838
  105.628 103.798 108.784
  285.628 103.798 108.784
  105.628 103.798 168.784
  285.628 103.798 168.784
  105.628 103.798 228.784
  285.628 103.798 228.784
  105.628 103.798 288.784
  285.628 103.798 288.784
  105.628 103.798 348.784
  285.628 103.798 348.784

For specific orientations, e.g. for the cube orientations, symmetrisation leads to multiple identical orientations. This can be prevented by passing the option unique to the command symmetrise

symmetrise(orientation.id(cs,ss),'unique')

ans = orientation (6 → xyz (112))
  size: 6 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2
   180    0    0
   240    0    0
   120    0    0
   300    0    0
    60    0    0
     0    0    0

Crystal symmetries in computations

Note that all operation on orientations are preformed taking all symmetrically equivalent orientations into account. As an example consider the angle between a random orientation and all orientations symmetricall equivalent to the goss orientation

ori = orientation.rand(cs);
angle(ori,symmetrise(orientation.goss(cs))) ./ degree

The value is the same for all orientations and equal to the smallest angle to one of the symmetrally equivalent orientations. This can be verified by computing the rotational angle ignoring symmetry.

angle(ori,symmetrise(orientation.goss(cs)),'noSymmetry') ./ degree

Functions that respect crystal symmetry but allow to switch it off using the flag noSymmetry include dot, unique, calcCluster.