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Euler angles and crystal symmetry
Author(s) -
Nolze Gert
Publication year - 2015
Publication title -
crystal research and technology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.377
H-Index - 64
eISSN - 1521-4079
pISSN - 0232-1300
DOI - 10.1002/crat.201400427
Subject(s) - euler angles , orientation (vector space) , electron backscatter diffraction , symmetry (geometry) , euler's formula , rotation (mathematics) , geometry , diffraction , crystallography , mathematics , physics , optics , mathematical analysis , chemistry
For the description of (single) crystal orientations, e.g. as measured by electron backscatter diffraction (EBSD) & X‐ray diffraction (XRD), Euler angles are still generally used to import and export data. However, because of the lack of standard definitions for the unit cell reference settings and specimen axes, several transformation descriptions exist which produce different sets of Euler angles for the same orientation. There is also no recommended region within the minimal Euler orientation space into which orientations should be placed. This is the reason why different sets of Euler angles for the same orientation are generated by the available software packages for indexing EBSD patterns. These issues are reviewed and addressed. The influence of crystal symmetry in form of chiral (enantiomorphic) groups is discussed, as well as how multiple, but symmetry‐equivalent sets of Euler angles can be reduced in order to deliver a unique orientation description. The Euler coloring algorithms applied to EBSD map data is critically discussed. The specific case of cubic symmetry, especially the effect of the three‐fold rotation on the Euler space is investigated in more detail for the highest‐symmetric chiral group 432. Recommendations for standard settings of the unit cell to orthogonal coordinate system transformation are given which exploit inherent symmetry.