misorientation angle

[crecentluo]

• Hello，Here I met two problems: Q1 and Q2
• Q1:  ori = orientation('Miller',[h k l],[u v w],cs,ss);
  ------------------------------
  % win7, matlab R2010a, mtex3.3.1
  clear;
  cs=symmetry('cubic');
  ori1=orientation('Miller',[1 0 0],[0 1 1],cs);
  ------------------------------
  % result
  oril=  crystal symmetry: cubic;    sample symmetry : triclinic
               Bunge Euler angles in degree
               phi1  Phi phi2
                  45    0    0
  ------------------------------
  To my understanding, ori1 is to rotating [100]||[100],and [010]||[011],respectively. but, if  rot=rotation('euler',45*degree,0,0), rotating the crystal coordinates to [100]||[110], [010]||[-110], and [001]||[110], respectively.
  The calculated results cannot match, so what is wrong here?
• Q2:
  here what I want to get is the smallest misorientation angle of two orientation. I found that there were something wrong here!
• If I only change the orientation of g1,there would be a different smallest angles. In fact,if the rotation matrix donot change which means that the misorientations are equivalent,so there should be the same smallest misorientation angle through different g1.
  So, anthing wrong here? 
  --------------------------------------------------
  % in window7, matlab R2010a, mtex3.3.1
  clear;
  cs=symmetry('m-3m'); % cubic crystal
  l1=Miller(1,1,1,cs,'uvw'); % [111] crystal direction
  rot=rotation('axis',l1,'angle',180*degree); % rotating around axis [111] by 180 degree
  g1=orientation('euler',10*degree,20*degree,40*degree,cs);
  g2=rot*g1; % g1 rotates around [111] by 180 degree
  a=angle(g1,g2)/degree;
  --------------------------------------------------
  output:  a = 45.4308 % in degree --------------------------------------------------

[Ralf Hielscher]

Question 1

To my understanding Miller indice notation

ori1 = orientation('Miller',[1 0 0],[0 1 1],cs)

should give the orientation that maps [1 0 0]||(001) and [001] || (100). Do you have any reference, that supports your definition? 

Anyway, the main point is that an orientation corresponds to a class of crystallographic equivalent rotations. In fact, the the rotation you are looking for is within the same class as the rotation returned by MTEX. You may use 

symmetrise(ori1)

to see this. If you are not interested in classes of rotations you shouldn't work with orientations but with rotations. 

Question 2

The point here is that g1 is not symmetrically equivalent to g2. The reason is that 

r = g1 * h

maps crystal coordinates to specimen coordinates. Thus applying a rotation rot to the left of g1 does correspond to a rotation of the specimen coordinate system. What you want to do is a change of the crystal coordinate system. Therefore, the rotation has to be applied from the right. Furthermore, since it transforms the crystal coordinates into crystal coordinates it is actually a misorientation. A misorientation is defined by

mori = orientation('axis',Miller(1,1,1,cs),'angle',180*degree,cs,cs)

Applying to the right gives

g2 = g1 * mori

which is again an orientation. Now

angle(g1,g2)

is zero whenever g1 and g2 are symmetrically equivalent. This is the case when mori is within the symmetry group cs. Here this is not the case as [1,1,1] is a threefold symmetry axis but not a twofold one. Note also that always

angle(g1,g2) == angle(mori)

[crecentluo]

Thanks a lot.
In Q2:  I know where is my mistake. I confuse the crystal direction with the specimen direction.   The rotating axis is defined by Miller indices which is a crystal coordinates (which is local coordinate). When I apply a rotation to the g1, the rotation axis l1 should in the same coordinate with g1.

 But I also cannot understand why specimen symmetry(ss) is equal to cs in the following equation? Beside,  I try to ignore the second cs, but it shows symmetry mismatch ( g2 and g1),  why?

[Ralf Hielscher]

This is mainly a problem in the documentation

orientation(....,cs,ss)

defines an orientation with crystal symmetry cs and specimen symmetry ss

orientation(....,cs1,cs2)

defines a misorientation between an orientation 1 (crystal 1) with crystal symmetry cs1 and an orientation 2 (crystal 2) with crystal symmetry cs2. This misorientation describes the rotation to bring the crystal coordinate system of crystal 1 into coincidence with the crystal coordinate system of crystal 2.