Alignment of the Crystal Axes

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

The reference frame is printed with every crystal symmetry

cs = crystalSymmetry('12/m1',[4 5 6],[90 100 90]*degree,'mineral','test')

cs = crystalSymmetry (⊙c*→b)
 
  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 - ten degrees here, which is how far the direct axis \(\vec a\) sits from \(\vec x\) while \(\vec a^*\) sits on it.

[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. Separately, different sources may rename or permute lattice axes even when those axes are orthogonal; the olivine example below shows that case.

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 (⊙c*→b)
 
  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 Alignments

Data published under one convention has to be transformed before it can be used under another, and transformReferenceFrame is what does it - for tensors as for orientations. Take a published stiffness tensor of olivine, given for the standard setup

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 × 3 × 3 × 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

The plot is the directional stiffness: the crystal is stiffest along the short \(\vec a\) axis. Another setup of the same mineral names the axes the other way round, with \(\vec a\) the longest and \(\vec c\) the shortest.

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

cs_new = crystalSymmetry (⊙c→a)
 
  mineral : Olivin    
  symmetry: mmm       
  elements: 8         
  a, b, c : 10, 6, 4.8

Expressing the same tensor in that setup permutes its components.

C_new = C.transformReferenceFrame(cs_new)

nextAxis
plot(C_new)

C_new = stiffnessTensor (Olivin)
  unit: GPa              
  rank: 4 (3 × 3 × 3 × 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

The two plots show the same physical crystal - the stiff direction has not moved in the material, only the axis it is called. What would be wrong is to use the published numbers unchanged with cs_new: that describes a crystal whose stiff direction points somewhere else.

Next

How the Cartesian frame is inscribed into the crystal axes, and what depends on it, is The Crystal Reference System. Directions written in the crystal frame are Miller Indices.