Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section Sublattice relations for reciprocal lattices

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 Sublattice relations for reciprocal lattices

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Let us now consider the two reciprocal lattices [\Lambda_{\bf A}^{*}] and [\Lambda_{\bf B}^{*}]. Their period matrices [({\bf A}^{-1})^{T}] and [({\bf B}^{-1})^{T}] are related by: [({\bf B}^{-1})^{T} = ({\bf A}^{-1})^{T} {\bf N}^{T}], where [{\bf N}^{T}] is an integer matrix; or equivalently by [({\bf B}^{-1})^{T} = {\bf D}^{T} ({\bf A}^{-1})^{T}]. This shows that the roles are reversed in that [\Lambda_{\bf B}^{*}] is a sublattice of [\Lambda_{\bf A}^{*}], which we may write:

  • [\displaylines{\quad({\rm i})^*\hfill \Lambda_{\bf B}^{*} = {\bf D}^{T} \Lambda_{\bf A}^{*}\hfill}]

  • [\displaylines{\quad({\rm ii})^*\hfill\Lambda_{\bf A}^{*} = \bigcup_{{\boldell}^{*} \in \Lambda_{\bf A}^{*} / \Lambda_{\bf B}^{*}} ({\boldell}^{*} + \Lambda_{\bf B}^{*}).\hfill}]The residual lattice [\Lambda_{\bf A}^{*} / \Lambda_{\bf B}^{*}] is finite, with [[\Lambda_{\bf A}^{*}: \Lambda_{\bf B}^{*}] =] [ \det {\bf D} = \det {\bf N} = [\Lambda_{\bf B}: \Lambda_{\bf A}]], and we may again combine [(\hbox{i})^{*}] [link] and [(\hbox{ii})^{*}] [link] into

  • [\displaylines{\quad({\rm iii})^*\hfill\Lambda_{\bf A}^{*} = \bigcup_{{\boldell}^{*} \in \Lambda_{\bf A}^{*} / \Lambda_{\bf B}^{*}} ({\boldell}^{*} + {\bf D}^{T} \Lambda_{\bf A}^{*}).\hfill}]

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