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

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#### 1.3.2.7.2.1. Geometric description of sublattices

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Let be a period lattice in with matrix A, and let be the lattice reciprocal to , with period matrix . Let be defined similarly, and let us suppose that is a sublattice of , i.e. that as a set.

The relation between and may be described in two different fashions: (i) multiplicatively, and (ii) additively.

 (i) We may write for some nonsingular matrix N with integer entries. N may be viewed as the period matrix of the coarser lattice with respect to the period basis of the finer lattice . It will be more convenient to write , where is a rational matrix (with integer determinant since det ) in terms of which the two lattices are related by (ii) Call two vectors in congruent modulo if their difference lies in . Denote the set of congruence classes (or cosets') by , and the number of these classes by . The coset decomposition'represents as the disjoint union of translates of is a finite lattice with elements, called the residual lattice of modulo . The two descriptions are connected by the relation , which follows from a volume calculation. We may also combine (i) and (ii) into which may be viewed as the n-dimensional equivalent of the Euclidean algorithm for integer division: is the `remainder' of the division by of a vector in , the quotient being the matrix D.