Alignment of the Crystal Axes edit page 
plottingConvention.default("y↑→x");

A crystal lattice is described by its lattice parameters \(a\), \(b\), \(c\) and the angles \(\alpha\), \(\beta\), \(\gamma\) between the axes. These fix the lattice, but they do not say how it is placed inside the Cartesian coordinate system that MTEX computes in - and for anything but the cubic case there is more than one common choice.

Unless told otherwise, MTEX uses

  • \(\vec z \parallel \vec c\) - the \(\vec c\) axis, i.e. the axis of highest symmetry, points along \(\vec z\)
  • \(\vec x \parallel \vec a^*\) - the \(\vec x\) direction is aligned with the reciprocal axis \(\vec a^*\)

The reference frame is printed with every crystal symmetry

cs = crystalSymmetry('12/m1',[4 5 6],[90 100 90]*degree,'mineral','test')
cs = crystalSymmetry
 
  mineral           : test             
  symmetry          : 12/m1            
  elements          : 4                
  a, b, c           : 4, 5, 6          
  alpha, beta, gamma: 90°, 100°, 90°   
  reference frame   : X||a*, Y||b, Z||c

Note that \(\vec a^*\), not \(\vec a\), is the direction placed along \(\vec x\). For a monoclinic lattice with \(\beta = 100^\circ\) the two differ by exactly the deviation of \(\beta\) from a right angle

[angle(cs.aAxis,vector3d.X), angle(cs.aAxisRec,vector3d.X)] ./ degree
ans =
   10.0000         0

For orthogonal lattices - orthorhombic, tetragonal and cubic - the direct and the reciprocal axes coincide and the distinction is immaterial. It matters for triclinic, monoclinic, trigonal and hexagonal symmetries, which is exactly where the competing conventions live.

A different alignment is requested by naming it in the constructor, e.g.

cs2 = crystalSymmetry('12/m1',[4 5 6],[90 100 90]*degree,'X||a','mineral','test')
cs2 = crystalSymmetry
 
  mineral           : test             
  symmetry          : 12/m1            
  elements          : 4                
  a, b, c           : 4, 5, 6          
  alpha, beta, gamma: 90°, 100°, 90°   
  reference frame   : X||a, Y||b, Z||c*

The two describe the same crystal, but Miller indices, Euler angles and tensor components refer to different Cartesian frames and are therefore not interchangeable between them.

Switching between different Alignment Options

Since, especially for lower symmetry groups, different conventions for aligning the crystal axes are used it might be necessary to transform data, e.g, orientations or tensors, from one convention into another. This can be done using the command transformReferenceFrame as it illustrated below.

First we import the stiffness tensor Forsterite with respect to the axes alignment

cs = crystalSymmetry('mmm',[4.7646 10.2296 5.9942],'mineral','Olivin');

% import some stiffness tensor
fname = fullfile(mtexDataPath,'tensor','Olivine1997PC.GPa');
C = stiffnessTensor.load(fname,cs)

plot(C)
C = stiffnessTensor (Olivin)
  unit: GPa              
  rank: 4 (3 x 3 x 3 x 3)
 
  tensor in Voigt matrix representation:
 320.5  68.2  71.6     0     0     0
  68.2 196.5  76.8     0     0     0
  71.6  76.8 233.5     0     0     0
     0     0     0    64     0     0
     0     0     0     0    77     0
     0     0     0     0     0  78.7

Let us now consider a different setup of the Forsterite symmetry, where the \(\vec a\) axis is the longest and the \(\vec c\)-axis is the shortest.

cs_new = crystalSymmetry('mmm',[10.2296 5.9942 4.7646],'mineral','Olivin')
cs_new = crystalSymmetry
 
  mineral : Olivin    
  symmetry: mmm       
  elements: 8         
  a, b, c : 10, 6, 4.8

In order to represent the stiffness tensor C with respect to this setup we use the command transformReferenceFrame.

C_new = C.transformReferenceFrame(cs_new)

nextAxis
plot(C_new)
C_new = stiffnessTensor (Olivin)
  unit: GPa              
  rank: 4 (3 x 3 x 3 x 3)
 
  tensor in Voigt matrix representation:
 196.5  76.8  68.2     0     0     0
  76.8 233.5  71.6     0     0     0
  68.2  71.6 320.5     0     0     0
     0     0     0    77     0     0
     0     0     0     0  78.7     0
     0     0     0     0     0    64