Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 46-47   | 1 | 2 |

Section Relation between lattice distributions

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Relation between lattice distributions

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The above relations between lattices may be rewritten in terms of the corresponding lattice distributions as follows: [\displaylines{\quad (\hbox{i}) \hfill R_{\bf A} = {1 \over |\det {\bf D}|} {\bf D}^{\#} R_{\bf B}^{*} \;\hfill\cr \quad (\hbox{ii}) \hfill R_{\bf B} = T_{{\bf B} / {\bf A}} * R_{\bf A}\qquad \hfill\cr \quad (\hbox{i})^{*} \hfill \;\;R_{\bf B}^{*} = {1 \over |\det {\bf D}|} ({\bf D}^{T})^{\#} R_{\bf A}^{*} \hfill\cr \quad (\hbox{ii})^{*} \hfill R_{\bf A}^{*} =T_{{\bf A} / {\bf B}}^{*} * R_{\bf B}^{*} \qquad\;\;\hfill}] where [T_{{\bf B} / {\bf A}} = {\textstyle\sum\limits_{{\boldell} \in \Lambda_{\bf B} / \Lambda_{\bf A}}} \delta_{({\boldell})}] and [T_{{\bf A}/{\bf B}}^{*} = {\textstyle\sum\limits_{{\boldell}^{*} \in \Lambda_{\bf A}^{*} / \Lambda_{\bf B}^{*}}} \delta_{({\boldell}^{*})}] are (finite) residual-lattice distributions. We may incorporate the factor [1/|\det {\bf D}|] in (i) and [(\hbox{i})^{*}] into these distributions and define [S_{{\bf B}/{\bf A}} = {1 \over |\det {\bf D}|} T_{{\bf B}/{\bf A}},\quad S_{{\bf A}/{\bf B}}^{*} = {1 \over |\det {\bf D}|} T_{{\bf A}/{\bf B}}^{*}.]

Since [|\det {\bf D}| = [\Lambda_{\bf B}: \Lambda_{\bf A}] = [\Lambda_{\bf A}^{*}: \Lambda_{\bf B}^{*}]], convolution with [S_{{\bf B}/{\bf A}}] and [S_{{\bf A}/{\bf B}}^{*}] has the effect of averaging the translates of a distribution under the elements (or `cosets') of the residual lattices [\Lambda_{\bf B}/\Lambda_{\bf A}] and [\Lambda_{\bf A}^{*}/\Lambda_{\bf B}^{*}], respectively. This process will be called `coset averaging'. Eliminating [R_{\bf A}] and [R_{\bf B}] between (i) and (ii), and [R_{\bf A}^{*}] and [R_{\bf B}^{*}] between [(\hbox{i})^{*}] and [(\hbox{ii})^{*}], we may write: [\displaylines{\quad (\hbox{i}')\hfill \! R_{\bf A} = {\bf D}^{\#} (S_{{\bf B}/{\bf A}} * R_{\bf A})\;\;\;\hfill\cr \quad (\hbox{ii}')\hfill \! R_{\bf B} = S_{{\bf B}/{\bf A}} * ({\bf D}^{\#} R_{\bf B})\;\;\;\;\hfill\cr \quad (\hbox{i}')^{*}\hfill R_{\bf B}^{*} = ({\bf D}^{T})^{\#} (S_{{\bf A}/{\bf B}}^{*} * R_{\bf B}^{*}) \hfill\cr \quad (\hbox{ii}')^{*}\hfill R_{\bf A}^{*} = S_{{\bf A}/{\bf B}}^{*} * [({\bf D}^{T})^{\#} R_{\bf A}^{*}]. \;\hfill}] These identities show that period subdivision by convolution with [S_{{\bf B}/{\bf A}}] (respectively [S_{{\bf A}/{\bf B}}^{*}]) on the one hand, and period decimation by `dilation' by [{\bf D}^{\#}] on the other hand, are mutually inverse operations on [R_{\bf A}] and [R_{\bf B}] (respectively [R_{\bf A}^{*}] and [R_{\bf B}^{*}]).

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