A Classification of the Symmetries of Uniform Discrete Defective Crystals
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Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as ‘inelastic’ symmetries.
KeywordsCrystals Defects Lie groups
Mathematics Subject Classification74A20 74E25
The author acknowledges the support of the UK Engineering and Physical Sciences Research Council through grant EP/G047162/1. The author wishes to thank Gareth Parry for many useful discussions and his advice concerning this paper.
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