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 |

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**.

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

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'); plot(cs)cs = crystalSymmetry('-3m',[1.7,1.7,1.4],'Y||a'); plot(cs)## Complete Function list

Laue | return the corresponding Laue group |

LaueName | get Laue name |

alignment | return alignment of the reference frame as string, e.g. x||a, y||b* |

calcAngleDistribution | compute the angle distribution of a uniform ODF for a crystal symmetry |

calcAxisDistribution | compute the axis distribution of an uniform ODF or MDF |

calcQuat | calculate quaternions for Laue groups |

check | symmetry |

disjoint | returns the disjoint of two symmetry groups |

ensureCS | ensures that an obj has the right crystal symmetry |

eq | check S1 == S2 |

factor | izes s1 and s2 into l, d, r such that s1 = l * d and s2 = d * r |

fundamentalRegion | fundamental region in orientation space for a (pair) of symmetries |

fundamentalRegionEuler | get the fundamental region in Euler angles |

fundamentalSector | get the fundamental sector for a symmetry in the inverse pole figure |

length | number of symmetry elements |

maxAngle | get the maximum angle of the fundamental region |

multiplicityPerpZ | maximum angle rho |

multiplicityZ | maximum angle rho |

nfold | maximal nfold of symmetry axes |

plot | visualize symmetry elements according to international table |

properGroup | return the corresponding Laue group |

properSubGroup | return the corresponding Laue group |

rotation_special | returns symmetry elements different from rotation about caxis |

subsref | overloads subsref |

symmetry | Supported Symmetries |

union | returns the union of two symmetry groups |

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