MTEX vs. Bunge Convention

For historical reasons MTEX defines orientations in a slightly different way than they have been defined by Bunge. As explained in topic orientations MTEX defines them as coordinate transformations from the crystal reference frame into the specimen reference frame. In contrast to this Bunge orientations are coordinate transformations from the specimen reference frame into the crystal reference frame. Lets demonstrate this by a simple example:

% consider cubic symmetry
cs = crystalSymmetry('cubic')

% and a random orientation
ori = orientation.rand(cs)

cs = crystalSymmetry
 
  symmetry: m-3m   
  elements: 48     
  a, b, c : 1, 1, 1
 
 
ori = orientation (m-3m → xyz)
 
  Bunge Euler angles in degree
    phi1     Phi    phi2
  185.43 150.883 267.377

This is now an MTEX orientation and can be used to translate crystal coordinates, i.e., Miller indices into specimen coordinates,

% either by multiplying from the left
r = ori * Miller({1,0,0},cs)

% or using the command rotate
rotate(Miller({1,0,0},cs),ori)

r = vector3d
         x         y         z
  0.128138 -0.864465 -0.486088
 
ans = vector3d
         x         y         z
  0.128138 -0.864465 -0.486088

A Bunge orientation is exactly the inverse of an MTEX orientation, i.e.,

ori_Bunge = inv(ori)

ori_Bunge = orientation (xyz → m-3m)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  272.623 150.883  354.57

and translates specimen coordinates into Miller indices

ori_Bunge * r

ans = Miller (m-3m)
  h k l
  1 0 0

Euler angles

Since the Euler angles are the most common way to describe orientations MTEX implements them such that the Euler angles of an MTEX orientation coincide with the Euler angles of a Bunge orientation. Thus the Euler angles of orientations in MTEX agree with the Euler angles reported by all common EBSD devices, simulation software, text books and paper.

Matrix notation

Due to the above explained inverse relationship of orientations defined in MTEX and in Bunge convention, a matrix generated from an orientation in MTEX is the inverse, or equivallently, the transpose of the matrix in Bunge notation.

ori.matrix
ori_Bunge.matrix^(-1)
ori_Bunge.matrix'

ans =
    0.1281   -0.9907   -0.0460
   -0.8645   -0.1343    0.4844
   -0.4861   -0.0223   -0.8736
ans =
    0.1281   -0.9907   -0.0460
   -0.8645   -0.1343    0.4844
   -0.4861   -0.0223   -0.8736
ans =
    0.1281   -0.9907   -0.0460
   -0.8645   -0.1343    0.4844
   -0.4861   -0.0223   -0.8736

Misorientations

Since, MTEX orientations translates crystal to specimen coordinates misorientations are defined by the formula

ori1 = orientation.rand(cs);
ori2 = orientation.rand(cs);

mori = inv(ori1) * ori2

mori = misorientation (m-3m → m-3m)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  330.123  133.96  281.89

as they are commonly defined coordinate transformations from crystal to crystal coordinates. This formula is different to the misorientation formula for Bunge orientations

ori1_Bunge = inv(ori1);
ori2_Bunge = inv(ori2);

mori = ori1_Bunge * inv(ori2_Bunge)

mori = misorientation (m-3m → m-3m)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  330.123  133.96  281.89

However, both formula result in exactly the same misorientation.

Summary

This list summarizes the differences between MTEX orientations and Bunge orientations.