Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 43-44   | 1 | 2 |

Section The case of nonstandard period 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 The case of nonstandard period lattices

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Let Λ denote the nonstandard lattice consisting of all vectors of the form [{\textstyle\sum_{j=1}} m_{j} {\bf a}_{j}], where the [m_{j}] are rational integers and [{\bf a}_{1}, \ldots, {\bf a}_{n}] are n linearly independent vectors in [{\bb R}^{n}]. Let R be the corresponding lattice distribution: [R = {\textstyle\sum_{{ x} \in \Lambda}} \delta_{({\bf x})}].

Let A be the nonsingular [n \times n] matrix whose successive columns are the coordinates of vectors [{\bf a}_{1}, \ldots, {\bf a}_{n}] in the standard basis of [{\bb R}^{n}]; A will be called the period matrix of Λ, and the mapping [{\bf x} \,\longmapsto\, {\bf Ax}] will be denoted by A. According to Section[link] we have[\langle R, \varphi \rangle = {\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} \varphi ({\bf Am}) = \langle r, (A^{-1})^{\#} \varphi \rangle = |\!\det {\bf A}|^{-1} \langle A^{\#} r, \varphi \rangle]for any [\varphi \in {\scr S}], and hence [R = |\!\det {\bf A}|^{-1} A^{\#} r]. By Fourier transformation, according to Section[link],[{\scr F}[R] = |\!\det {\bf A}|^{-1} {\scr F}[A^{\#} r] = [({\bf A}^{-1})^{T}]^{\#} {\scr F}[r] = [({\bf A}^{-1})^{T}]^{\#} r,]which we write:[{\scr F}[R] = |\!\det {\bf A}|^{-1} R^{*}]with[R^{*} = |\det {\bf A}| [({\bf A}^{-1})^{T}]^{\#} r.]

[R^{*}] is a lattice distribution:[R^{*} = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} \delta_{[({\bf A}^{-1})^{T} {\boldmu}]} = {\textstyle\sum\limits_{{\boldxi} \in \Lambda^{*}}} \delta_{({\boldxi})}]associated with the reciprocal lattice [\Lambda^{*}] whose basis vectors [{\bf a}_{1}^{*}, \ldots, {\bf a}_{n}^{*}] are the columns of [({\bf A}^{-1})^{T}]. Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of co-factors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section[link] for the crystallographic case [n = 3]).

A distribution T will be called Λ-periodic if [\tau_{\boldxi} T = T] for all [{\boldxi} \in \Lambda]; as previously, T may be written [R * T^{0}] for some motif distribution [T^{0}] with compact support. By Fourier transformation,[\eqalignno{{\scr F}[T] &= |\!\det {\bf A}|^{-1} R^{*} \cdot {\scr F}[T^{0}]\cr &= |\!\det {\bf A}|^{-1} {\textstyle\sum\limits_{{\boldxi} \in \Lambda^{*}}} {\scr F}[T^{0}] ({\boldxi}) \delta_{({\boldxi})}\cr &= |\!\det {\bf A}|^{-1} {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} {\scr F}[T^{0}] [{({\bf A}^{-1})^{T}}{\boldmu}] \delta_{{[({\bf A}^{-1})^{T}} {\boldmu}]}}]so that [{\scr F}[T]] is a weighted reciprocal-lattice distribution, the weight attached to node [{\boldxi} \in \Lambda^{*}] being [|\!\det {\bf A}|^{-1}] times the value [{\scr F}[T^{0}](\boldxi)] of the Fourier transform of the motif [T^{0}].

This result may be further simplified if T and its motif [T^{0}] are referred to the standard period lattice [{\bb Z}^{n}] by defining t and [t^{0}] so that [T = A^{\#} t], [T^{0} = A^{\#} t^{0}], [t = r * t^{0}]. Then[{\scr F}[T^{0}] ({\boldxi}) = |\!\det {\bf A}| {\scr F}[t^{0}] ({\bf A}^{T} {\boldxi}),]hence[{\scr F}[T^{0}] [{({\bf A}^{-1})^{T}}{\boldmu}] = |\!\det {\bf A}| {\scr F}[t^{0}] ({\boldmu}),]so that[{\scr F}[T] = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} {\scr F}[t^{0}] ({\boldmu}) \delta_{[({\bf A}^{-1})^{T} {\boldmu}]}]in nonstandard coordinates, while[{\scr F}[t] = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} {\scr F}[t^{0}] ({\boldmu}) \delta_{({\boldmu})}]in standard coordinates.

The reciprocity theorem may then be written:[\displaylines{\quad (\hbox{iii}) \hfill W_{\boldxi} = |\!\det {\bf A}|^{-1} \langle T_{\bf x}^{0}, \exp (-2 \pi i {\boldxi} \cdot {\bf x})\rangle, \quad {\boldxi} \in \boldLambda^{*} \hfill\cr \quad (\hbox{iv}) \hfill T_{\bf x} = {\textstyle\sum\limits_{{\boldxi} \in \Lambda^{*}}} W_{\boldxi} \exp (+2 \pi i {\boldxi} \cdot {\bf x})\qquad\qquad\qquad\quad\hfill}]in nonstandard coordinates, or equivalently:[\displaylines{\quad (\hbox{v}) \hfill w_{\boldmu} = \langle t_{\bf x}^{0}, \exp (-2 \pi i {\boldmu} \cdot {\bf x})\rangle, \quad {\boldmu} \in {\bb Z}^{n} \hfill\cr \quad (\hbox{vi}) \hfill t_{\bf x} = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} w_{\boldmu} \exp (+2 \pi i {\boldmu} \cdot {\bf x}) \quad\qquad\hfill}]in standard coordinates. It gives an n-dimensional Fourier series representation for any periodic distribution over [{\bb R}^{n}]. The convergence of such series in [{\scr S}\,' ({\bb R}^{n})] will be examined in Section[link].

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