# Coincidence-site lattices

Coincidence-site lattices The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined. Let T be the 3 x 3 matrix that maps a basis of lattice 1 onto a basis of lattice 2 and let ||T|| be the absolute value of its determinant. It may be assumed that ||T|| 1. There is a CSL if, and only if, T is rational. The main result is that the density ratio, 2, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T-1 are integral matrices. A basis for the CSL can be determined quickly if lattices 1 and 2 are related by a rotation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography International Union of Crystallography

# Coincidence-site lattices

, Volume 32 (5): 783 – Sep 1, 1976

## Coincidence-site lattices

, Volume 32 (5): 783 – Sep 1, 1976

### Abstract

The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined. Let T be the 3 x 3 matrix that maps a basis of lattice 1 onto a basis of lattice 2 and let ||T|| be the absolute value of its determinant. It may be assumed that ||T|| 1. There is a CSL if, and only if, T is rational. The main result is that the density ratio, 2, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T-1 are integral matrices. A basis for the CSL can be determined quickly if lattices 1 and 2 are related by a rotation.  /lp/international-union-of-crystallography/coincidence-site-lattices-qnT9yzhwzb
Publisher
International Union of Crystallography
Copyright (c) 1976 International Union of Crystallography
ISSN
0567-7394
DOI
10.1107/S056773947601231X
Publisher site
See Article on Publisher Site

### Abstract

The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined. Let T be the 3 x 3 matrix that maps a basis of lattice 1 onto a basis of lattice 2 and let ||T|| be the absolute value of its determinant. It may be assumed that ||T|| 1. There is a CSL if, and only if, T is rational. The main result is that the density ratio, 2, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T-1 are integral matrices. A basis for the CSL can be determined quickly if lattices 1 and 2 are related by a rotation.

### Journal

Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General CrystallographyInternational Union of Crystallography

Published: Sep 1, 1976