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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 48   | 1 | 2 |

Section 1.3.2.7.2.1. Geometric description of sublattices

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, 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 nonsingular 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)[link] and (ii)[link] 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.








































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