Abstract
Abstract Let L L be a lattice of rank n n in an n n -dimensional Euclidean space. We show that the coincidence isometry group of L L is generated by coincidence reflections if and only if L L contains an orthogonal subset of order n n .
Highlights
The theory of coincidence site lattice (CSL) gives partial answers to some questions that appeared in the physics of interfaces and grain boundaries, see [1,2]
We focus on the structure of the coincidence isometry group of a lattice in a finitedimensional Euclidean space
The coincidence problem was completely solved in the planar case. They found that coincidence reflections play an important role and conjectured that if L = n is the lattice spanned by the canonical basis in n, any coincidence isometry is a product of coincidence reflections
Summary
The theory of coincidence site lattice (CSL) gives partial answers to some questions that appeared in the physics of interfaces and grain boundaries, see [1,2]. The CSL theory mainly studies the coincidence problem between two lattices in a finite-dimensional Euclidean space. The coincidence problem was completely solved in the planar case They found that coincidence reflections play an important role and conjectured that if L = n is the lattice spanned by the canonical basis in n, any coincidence isometry is a product of coincidence reflections. Zou in [14] showed that if the reflection defined by an arbitrary nonzero vector of L is a coincidence isometry of L, any coincidence isometry of L is a product of coincidence reflections defined by the vectors of L This result includes the conjecture as a special case and an algorithm to decompose a coincidence isometry into coincidence reflections was obtained.
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