Period lattice, reciprocal lattice and structure factors

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 59-60 | 1 | 2 |

Section 1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors

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Let ρ be the distribution of electrons in a crystal. Then, by definition of a crystal, ρ is Λ-periodic for some period lattice Λ (Section 1.3.2.6.5) so that there exists a motif distribution $[\rho^{0}]$ with compact support such that $[\rho = R * \rho^{0},]$ where $[R = {\textstyle\sum_{{\bf x}\in \Lambda}} \delta_{({\bf X})}]$ . The lattice Λ is usually taken to be the finest for which the above representation holds.

Let Λ have a basis $[({\bf a}_{1}, {\bf a}_{2}, {\bf a}_{3})]$ over the integers, these basis vectors being expressed in terms of a standard orthonormal basis $[({\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3})]$ as $[{\bf a}_{k} = {\textstyle\sum\limits_{j = 1}^{3}} a_{jk} {\bf e}_{j}.]$ Then the matrix $[{\bf A} = \pmatrix{a_{11} &a_{12} &a_{13}\cr a_{21} &a_{22} &a_{23}\cr a_{31} &a_{32} &a_{33}\cr}]$ is the period matrix of Λ (Section 1.3.2.6.5) with respect to the unit lattice with basis $[({\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3})]$ , and the volume V of the unit cell is given by $[V = |\det {\bf A}|]$ .

By Fourier transformation $[\bar{\scr F}[\rho] = R^{*} \times \bar{\scr F}[\rho^{0}],]$ where $[R^{*} = {\textstyle\sum_{{\bf H}\in \Lambda^{*}}} \delta_{({\bf H})}]$ is the lattice distribution associated to the reciprocal lattice $[\Lambda^{*}]$ . The basis vectors $[({\bf a}_{1}^{*}, {\bf a}_{2}^{*}, {\bf a}_{3}^{*})]$ have coordinates in $[({\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3})]$ given by the columns of $[({\bf A}^{-1})^{T}]$ , whose expression in terms of the cofactors of A (see Section 1.3.2.6.5) gives the familiar formulae involving the cross product of vectors for [n = 3] . The H-distribution F of scattered amplitudes may be written $[F = \bar{\scr F}[\rho]_{{\bf H}} = {\textstyle\sum\limits_{{\bf H}\in \Lambda^{*}}} \bar{\scr F}[\rho^{0}]({\bf H})\delta_{({\bf H})} = {\textstyle\sum\limits_{{\bf H}\in \Lambda^{*}}} F_{{\bf H}}\delta_{({\bf H})}]$ and is thus a weighted reciprocal-lattice distribution, the weight $[F_{{\bf H}}]$ attached to each node $[{\bf H} \in \Lambda^{*}]$ being the value at H of the transform $[\bar{\scr F}[\rho^{0}]]$ of the motif $[\rho^{0}]$ . Taken in conjunction with the assumption that the scattering is elastic, i.e. that H only changes the direction but not the magnitude of the incident wavevector $[{\bf K}_{0}]$ , this result yields the usual forms (Laue or Bragg) of the diffraction conditions: $[{\bf H} \in \Lambda^{*}]$ , and simultaneously H lies on the Ewald sphere.

By the reciprocity theorem , $[\rho^{0}]$ can be recovered if F is known for all $[{\bf H} \in \Lambda^{*}]$ as follows [Section 1.3.2.6.5, e.g. (iv)]: $[\rho_{\bf x} = {1 \over V} {\sum\limits_{{\bf H}\in \Lambda^{*}}} F_{{\bf H}} \exp (-2\pi i{\bf H} \cdot {\bf X}).]$

These relations may be rewritten in terms of standard, or `fractional crystallographic', coordinates by putting $[{\bf X} = {\bf Ax}, \quad {\bf H} = ({\bf A}^{-1})^{T}{\bf h},]$ so that a unit cell of the crystal corresponds to $[{\bf x} \in {\bb R}^{3}/{\bb Z}^{3}]$ , and that $[{\bf h} \in {\bb Z}^{3}]$ . Defining $[\rho\llap{$-\!$}]$ and $[\rho\llap{$-\!$}^{0}]$ by $[\rho = {1 \over V} A^{\#} \rho\llap{$-\!$}, \quad \rho^{0} = {1 \over V} A^{\#} \rho\llap{$-\!$}^{0}]$ so that $[\rho ({\bf X}) \;\hbox{d}^{3}{\bf X} = \rho\llap{$-\!$} ({\bf x}) \;\hbox{d}^{3}{\bf x}, \quad \rho^{0} ({\bf X}) \;\hbox{d}^{3}{\bf X} = \rho\llap{$-\!$}^{0} ({\bf x}) \;\hbox{d}^{3}{\bf x},]$ we have $[\eqalign{\bar{\scr F}[\rho\llap{$-\!$}]_{{\bf h}} &= {\textstyle\sum\limits_{{\bf h}\in {\bb Z}^{3}}} F({\bf h})\delta_{({\bf h})},\cr F({\bf h}) &= \langle \rho\llap{$-\!$}_{\bf x}^{0}, \exp (2\pi i{\bf h} \cdot {\bf x})\rangle\cr &= {\textstyle\int\limits_{{\bb R}^{3}/{\bb Z}^{3}}} \rho\llap{$-\!$}^{0} ({\bf x}) \exp (2\pi i{\bf h} \cdot {\bf x}) \;\hbox{d}^{3}{\bf x} \quad \hbox{if } \rho\llap{$-\!$}^{0} \in L_{\rm loc}^{1} ({\bb R}^{3}/{\bb Z}^{3}),\cr \rho\llap{$-\!$}_{\bf x} &= {\textstyle\sum\limits_{{\bf h}\in {\bb Z}^{3}}} F({\bf h}) \exp (-2\pi i{\bf h} \cdot {\bf x}).}]$ These formulae are valid for an arbitrary motif distribution $[\rho\llap{$-\!$}^{0}]$ , provided the convergence of the Fourier series for $[\rho\llap{$-\!$}]$ is considered from the viewpoint of distribution theory (Section 1.3.2.6.10.3).

The experienced crystallographer may notice the absence of the familiar factor [1/V] from the expression for $[\rho\llap{$-\!$}]$ just given. This is because we use the (mathematically) natural unit for $[\rho\llap{$-\!$}]$ , the electron per unit cell, which matches the dimensionless nature of the crystallographic coordinates x and of the associated volume element $[\hbox{d}^{3}{\bf x}]$ . The traditional factor [1/V] was the result of the somewhat inconsistent use of x as an argument but of $[\hbox{d}^{3}{\bf X}]$ as a volume element to obtain ρ in electrons per unit volume (e.g. Å³). A fortunate consequence of the present convention is that nuisance factors of V or [1/V] , which used to abound in convolution or scalar product formulae, are now absent.

It should be noted at this point that the crystallographic terminology regarding $[{\scr F}]$ and $[\bar{\scr F}]$ differs from the standard mathematical terminology introduced in Section 1.3.2.4.1 and applied to periodic distributions in Section 1.3.2.6.4: F is the inverse Fourier transform of ρ rather than its Fourier transform, and the calculation of ρ is called a Fourier synthesis in crystallography even though it is mathematically a Fourier analysis. The origin of this discrepancy may be traced to the fact that the mathematical theory of the Fourier transformation originated with the study of temporal periodicity, while crystallography deals with spatial periodicity; since the expression for the phase factor of a plane wave is $[\exp [2 \pi i(\nu t - {\bf K} \cdot {\bf X})]]$ , the difference in sign between the contributions from time versus spatial displacements makes this conflict unavoidable.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 59-60