Crystal and Specimen Symmetries (The Class symmetry)

This section describes the class symmetry and gives an overview how to deal with crystal symmetries in MTEX.

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Class Description
Defining a Crystal Symmetry
The Crystal Coordinate System
Complete Function list

Class Description

Crystal symmetries are sets of rotations and mirroring operations that leave the lattice of a crystal invariant. They form so-called groups since the concatenation of two symmetry operations is again a symmetry operation. Crystal symmetries are classified in various ways - either according to the corresponding space group or the corresponding point group, or the corresponding Laue group. In total, there are only 11 different Laue groups present in crystallography. All these 11 Laue groups are supported by MTEX. More precisely, in MTEX a Laue group is represented by a variable of the class symmetry.

Defining a Crystal Symmetry

MTEX supports the Schoenflies notation on Laue groups as well as the international notation. In the case of noncubic crystal symmetry the length of the crystal axis has to be specified as a second argument to the constructor symmtry and in the case of triclinic crystal symmetry the angles between the axes has to be passed as the third argument. Hence, valid definitions are:

Laue Group - international notation

cs = crystalSymmetry('m-3m')
cs = crystalSymmetry  
  symmetry: m-3m   
  a, b, c : 1, 1, 1

Laue Group - Schoenflies notation

cs = crystalSymmetry('O')
cs = crystalSymmetry  
  symmetry: 432    
  a, b, c : 1, 1, 1

Point Group or its Space Group

If not the name of a Laue group was specified but the name of a point group or a space group MTEX automatically determines the corresponding Laue group and assigns it to the variable.

cs = crystalSymmetry('Td')
cs = crystalSymmetry  
  symmetry: -43m   
  a, b, c : 1, 1, 1

CIF Files

Finally, MTEX allows defining a crystal symmetry by importing a crystallographic information file (*.cif).

cs = loadCIF('quartz')
cs = crystalSymmetry  
  mineral        : Quartz           
  symmetry       : 321              
  a, b, c        : 4.9, 4.9, 5.4    
  reference frame: X||a, Y||b*, Z||c

The Crystal Coordinate System

In the case of cubic crystal symmetry the crystal coordinate system is already well defined. However, especially in the case of low order crystal symmetry, the crystal coordinate system has to be specified by the length of the axis and the angle between the axis.

cs = crystalSymmetry('triclinic',[1,2.2,3.1],[80*degree,85*degree,95*degree]);

A and B Configurations

In the case of trigonal and hexagonal crystal symmetries different conventions are used. One distinguishes between the A and the B configuration depending whether the a-axis is aligned parallel to the x-axis or parallel to the y-axis. In order to specify the concrete

cs = crystalSymmetry('-3m',[1.7,1.7,1.4],'X||a');
cs = crystalSymmetry('-3m',[1.7,1.7,1.4],'Y||a');

Complete Function list

Lauereturn the corresponding Laue group
LaueNameget Laue name
alignmentreturn alignment of the reference frame as string, e.g. x||a, y||b*
calcAngleDistributioncompute the angle distribution of a uniform ODF for a crystal symmetry
calcAxisDistributioncompute the axis distribution of an uniform ODF or MDF
calcQuatcalculate quaternions for Laue groups
disjointreturns the disjoint of two symmetry groups
ensureCSensures that an obj has the right crystal symmetry
eqcheck S1 == S2
factorizes s1 and s2 into l, d, r such that s1 = l * d and s2 = d * r
fundamentalRegionfundamental region in orientation space for a (pair) of symmetries
fundamentalRegionEulerget the fundamental region in Euler angles
fundamentalSectorget the fundamental sector for a symmetry in the inverse pole figure
lengthnumber of symmetry elements
maxAngleget the maximum angle of the fundamental region
multiplicityPerpZmaximum angle rho
multiplicityZmaximum angle rho
nfoldmaximal nfold of symmetry axes
plotvisualize symmetry elements according to international table
properGroupreturn the corresponding Laue group
properSubGroupreturn the corresponding Laue group
rotation_specialreturns symmetry elements different from rotation about caxis
subsrefoverloads subsref
symmetrySupported Symmetries
unionreturns the union of two symmetry groups