Defining Orientations

An orientation answers one question: how is this crystal placed in this specimen? In MTEX it is a rotation that maps coordinates from the crystal reference frame into the specimen reference frame. It also carries the symmetry attached to each frame.

This page assumes the three-dimensional directions introduced in Defining Three-Dimensional Vectors, the plane and direction notation from Miller Indices, and basic matrix algebra. The constructors are the same as on Defining Rotations, with a crystalSymmetry supplied as an extra argument.

What this mapping means is developed in Theory and compared with other conventions in MTEX vs. Bunge Convention. This page concentrates on building orientations.

plottingConvention.default('y↑→x');

% load the crystal symmetry and reference frame from a CIF file
cs = crystalSymmetry.load('Cu-Copper.cif');

Euler Angles

Euler angles are the most common input and the one most easily misinterpreted. Their axes, order, and mapping direction belong to the convention. Equal angle triplets in different conventions need not describe the same orientation.

MTEX uses the Bunge convention by default. Naming it explicitly keeps a reusable script independent of the current session preference. Angles are in radians, so values stated in degrees are multiplied by degree.

ori = orientation.byEuler(30*degree,50*degree,10*degree,'Bunge',cs)

ori = orientation (Copper → y↑→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
    30   50   10

The display gives the three Bunge angles and names the copper crystal symmetry alongside them. This attached crystal frame and symmetry are what distinguish an orientation from a bare rotation.

Rotation Matrix

A \(3 \times 3\) matrix can define the same mapping. Its convention must be checked when it comes from another program: this matrix maps crystal-frame coordinates into specimen-frame coordinates.

M = eye(3);

ori = orientation.byMatrix(M,cs)

ori = orientation (Copper → y↑→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
     0    0    0

The identity matrix gives the orientation in which the Cartesian crystal frame is aligned with the specimen frame. It is the reference setting from which the Euler angles of every other orientation are counted.

The point group does not by itself determine how the Cartesian crystal frame is inscribed into the lattice axes. A statement such as X || a*, Z || c belongs to the crystal reference frame, not to the symmetry. Changing that alignment changes the coordinate description without moving the crystal; see The Crystal Reference System.

Miller Indices

Metallurgy often names an orientation by two crystal quantities: the lattice plane facing the specimen Z axis and the lattice direction pointing along specimen X. The inputs must describe an orthogonal plane normal and direction. That is what orientation.byMiller takes, here for the Goss orientation \((011)[100]\).

ori = orientation.byMiller([0 1 1],[1 0 0],cs)

ori = orientation (Copper → y↑→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
     0   45    0

Apply the orientation to the plane normal and the lattice direction to check where they point in the specimen frame.

rPlane = ori * Miller(0,1,1,cs,'hkl');
rDirection = ori * Miller(1,0,0,cs,'uvw');

plot([rPlane,rDirection],'upper','grid','MarkerSize',10,...
  'label',{'(011)','[100]'},'backgroundColor','w','nolabel')

Notice that the \((011)\) pole is at the centre, the specimen Z direction, while \([100]\) is on the specimen X axis at the rim.

Goss and the other named texture components are predefined. The zero angular difference confirms that this result is also orientation.goss(cs); see Standard Orientations.

angle(ori,orientation.goss(cs)) ./ degree

ans =
     0

Random Orientations

As for rotations, rand generates uniformly distributed orientations and needs the crystal symmetry as well. MTEX stores the 100 results in one vectorized orientation array.

ori = orientation.rand(100,cs);

length(ori)

ans =
   100

Symmetrically Equivalent Orientations

A crystal cannot distinguish its symmetrically equivalent settings, so an orientation represents a whole class of rotations. symmetrise lists that class.

ori = orientation.byEuler(30*degree,50*degree,10*degree,'Bunge',cs);

length(ori.symmetrise)

ans =
    48

Copper has point group m-3m with 48 elements. Only the 24 proper elements describe settings into which the crystal can be physically turned.

length(ori.symmetrise('proper'))

ans =
    24

The other 24 are improper lattice symmetries. They still matter when a calculation treats opposite plane normals as equivalent, as conventional diffraction does under Friedel's law.

This equivalence is why the angle between two orientations is the smallest angle over all equivalent pairs. The dedicated Symmetry page develops this rule and explains when to use the 'noSymmetry' option.

Specimen Symmetry

A specimen may have symmetry of its own. A rolled sheet, for example, is commonly modelled with orthorhombic symmetry: three mutually perpendicular twofold axes, or equivalently three mirror planes in the full point group. It is represented by a specimenSymmetry and passed alongside the crystal symmetry.

ss = specimenSymmetry('orthorhombic');

ori = orientation.byEuler(30*degree,50*degree,10*degree,'Bunge',cs,ss)

ori = orientation (Copper → y↑→x (mmm))
 
  Bunge Euler angles in degree
  phi1  Phi phi2
    30   50   10

Crystal symmetry acts in the crystal frame and specimen symmetry in the specimen frame. With the full point groups, the class contains the 48 copper elements times the 8 orthorhombic elements.

length(ori.symmetrise)

ans =
   384

Restricting both groups to proper operations leaves the 24 crystal rotations times the 4 specimen rotations.

length(ori.symmetrise('proper'))

ans =
    96

Specimen symmetry is a statement about the sample, not about the measurement. Its axes must match the physical specimen frame, and imposing a symmetry that is not present hides real texture components. Specimen Symmetry explains when to use it.

References

Next

Theory explains how an orientation maps coordinates, which is the definition the rest of MTEX rests on. Pole Figures and Inverse Pole Figures are the two ways of looking at one. Existing orientation files are handled by Import.