Geometric description of sublattices

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 46 | 1 | 2 |

Section 1.3.2.7.2.1. Geometric description of sublattices

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.7.2.1. Geometric description of sublattices

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Let $[\Lambda_{\bf A}]$ be a period lattice in $[{\bb R}^{n}]$ with matrix A, and let $[\Lambda_{\bf A}^{*}]$ be the lattice reciprocal to $[\Lambda_{\bf A}]$ , with period matrix $[(A^{-1})^{T}]$ . Let $[\Lambda_{\bf B}, {\bf B}, \Lambda_{\bf B}^{*}]$ be defined similarly, and let us suppose that $[\Lambda_{\bf A}]$ is a sublattice of $[\Lambda_{\bf B}]$ , i.e. that $[\Lambda_{\bf B} \supset \Lambda_{\bf A}]$ as a set.

The relation between $[\Lambda_{\bf A}]$ and $[\Lambda_{\bf B}]$ may be described in two different fashions: (i) multiplicatively, and (ii) additively.

(i) We may write $[{\bf A} = {\bf BN}]$ for some non-singular matrix N with integer entries. N may be viewed as the period matrix of the coarser lattice $[\Lambda_{\bf A}]$ with respect to the period basis of the finer lattice $[\Lambda_{\bf B}]$ . It will be more convenient to write $[{\bf A} = {\bf DB}]$ , where $[{\bf D} = {\bf BNB}^{-1}]$ is a rational matrix (with integer determinant since det $[{\bf D} = \det {\bf N}]$ ) in terms of which the two lattices are related by $[\Lambda_{\bf A} = {\bf D} \Lambda_{\bf B}.]$
(ii) Call two vectors in $[\Lambda_{\bf B}]$ congruent modulo $[\Lambda_{\bf A}]$ if their difference lies in $[\Lambda_{\bf A}]$ . Denote the set of congruence classes (or `cosets') by $[\Lambda_{\bf B} / \Lambda_{\bf A}]$ , and the number of these classes by $[[\Lambda_{\bf B} : \Lambda_{\bf A}]]$ . The `coset decomposition' $[\Lambda_{\bf B} = \bigcup_{{\boldell} \in \Lambda_{\bf B} / \Lambda_{\bf A}} ({\boldell} + \Lambda_{\bf A})]$ represents $[\Lambda_{\bf B}]$ as the disjoint union of $[[\Lambda_{\bf B} : \Lambda_{\bf A}]]$ translates of $[\Lambda_{\bf A} .\; \Lambda_{\bf B} / \Lambda_{\bf A}]$ is a finite lattice with $[[\Lambda_{\bf B} : \Lambda_{\bf A}]]$ elements, called the residual lattice of $[\Lambda_{\bf B}]$ modulo $[\Lambda_{\bf A}]$ .

The two descriptions are connected by the relation $[[\Lambda_{\bf B} : \Lambda_{\bf A}] = \det {\bf D} = \det {\bf N}]$ , which follows from a volume calculation. We may also combine (i) and (ii) into
$[\displaylines{\quad({\rm iii})\hfill \Lambda_{\bf B} = \bigcup_{{\boldell} \in \Lambda_{\bf B} / \Lambda_{\bf A}} ({\boldell} + {\bf D} \Lambda_{\bf B})\hfill}]$ which may be viewed as the n-dimensional equivalent of the Euclidean algorithm for integer division: $[\boldell]$ is the `remainder' of the division by $[\Lambda_{\bf A}]$ of a vector in $[\Lambda_{\bf B}]$ , the quotient being the matrix D.