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

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

Section 1.3.2.6.7. The Poisson summation formula

G. Bricognea

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1.3.2.6.7. The Poisson summation formula

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Let [\varphi \in {\scr S}], so that [{\scr F}[\varphi] \in {\scr S}]. Let R be the lattice distribution associated to lattice Λ, with period matrix A, and let [R^{*}] be associated to the reciprocal lattice [\Lambda^{*}]. Then we may write:[\eqalignno{\langle R, \varphi \rangle &= \langle R, \bar{\scr F}[{\scr F}[\varphi]]\rangle\cr &= \langle \bar{\scr F}[R], {\scr F}[\varphi]\rangle\cr &= |\!\det {\bf A}|^{-1} \langle R^{*}, {\scr F}[\varphi]\rangle}]i.e.[{\textstyle\sum\limits_{{\bf x} \in \Lambda}} \varphi ({\bf x}) = |\!\det {\bf A}|^{-1} {\textstyle\sum\limits_{{\boldxi} \in \Lambda^{*}}} {\scr F}[\varphi] ({\boldxi}).]

This identity, which also holds for [\bar{\scr F}], is called the Poisson summation formula. Its usefulness follows from the fact that the speed of decrease at infinity of ϕ and [{\scr F}[\varphi]] are inversely related (Section 1.3.2.4.4.3[link]), so that if one of the series (say, the left-hand side) is slowly convergent, the other (say, the right-hand side) will be rapidly convergent. This procedure has been used by Ewald (1921)[link] [see also Bertaut (1952)[link], Born & Huang (1954)[link]] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4[link] in this volume on convergence acceleration techniques for crystallographic lattice sums and Chapter 3.5[link] on modern extensions of the Ewald summation method).

When ϕ is a multivariate Gaussian[\varphi ({\bf x}) = G_{\bf B} ({\bf x}) = \exp (-\textstyle{{1 \over 2}} {\bf x}^{T} {\bf Bx}),]then[{\scr F}[\varphi] (\boldxi) = |\!\det (2 \pi {\bf B}^{-1})|^{1/2} G_{{\bf B}^{-1}} (\boldxi),]and Poisson's summation formula for a lattice with period matrix A reads:[\eqalignno{{\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} G_{\bf B} ({\bf Am}) &= |\!\det {\bf A}|^{-1}| \!\det (2 \pi {\bf B}^{-1})|^{1/2}\cr &\quad \times \textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}} G_{4 \pi^{2}{\bf B}^{-1}} [({\bf A}^{-1})^{T} {\boldmu}]}]or equivalently[{\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} G_{\bf C} ({\bf m}) = |\!\det (2 \pi {\bf C}^{-1})|^{1/2} {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} G_{4 \pi^{2}}{{_{{\bf C}^{-1}}}} ({\boldmu})]with [{\bf C} = {\bf A}^{T} {\bf BA}.]

References

Bertaut, E. F. (1952). L'énergie électrostatique de réseaux ioniques. J. Phys. Radium, 13, 499–505.
Born, M. & Huang, K. (1954). Dynamical Theory of Crystal Lattices. Oxford University Press.
Ewald, P. P. (1921). Die Berechnung optischer und electrostatischer Gitterpotentiale. Ann. Phys. Leipzig, 64, 253–287.








































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