International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 6

Section 1.1.2.4.2. Reciprocal space

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.2.4.2. Reciprocal space

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Let us take the scalar products of a covariant vector [{\bf e}_i] and a contravariant vector [{\bf e}\hskip 1pt^{j}]: [{\bf e}_{i} \cdot {\bf e}\hskip 1pt^{j} = {\bf e}_{i} \cdot g\hskip 1pt^{jk}{\bf e}_{k} = {\bf e}_{i} \cdot {\bf e}_{k} g\hskip 1pt^{jk} = g_{ik}g\hskip 1pt^{jk} = \delta\hskip1pt_{i}^{j}][using expressions (1.1.2.5)[link], (1.1.2.11)[link] and (1.1.2.13)[link]].

The relation we obtain, [{\bf e}_{i} \cdot {\bf e}\hskip1pt^{j} = \delta\hskip1pt_{i}^{j}], is identical to the relations defining the reciprocal lattice in crystallography; the reciprocal basis then is identical to the dual basis [{\bf e}^{i}].








































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