## Abstract

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further 'grain exchange symmetry', which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D _{2}(222), D_{4}(422) and D_{6}(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.

Original language | English (US) |
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Pages (from-to) | 1489-1508 |

Number of pages | 20 |

Journal | Philosophical Magazine |

Volume | 91 |

Issue number | 10 |

DOIs | |

State | Published - Apr 1 2011 |

## Keywords

- crystal symmetry
- crystal topology
- crystalline interface
- crystallography
- grain boundary

## ASJC Scopus subject areas

- Condensed Matter Physics