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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 6-7

Section Properties of the metric tensor

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Properties of the metric tensor

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In a change of basis, following ([link] and ([link], the [g_{ij}]'s transform according to [\left. \matrix{g_{ij} \ = A_{i}^{k} A_{j}^{m} g'_{km}\cr g'_{ij} \ = B_{i}^{k} B_{j}^{m} g_{km}.\cr}\right\} \eqno(]Let us now consider the scalar products, [{\bf e}^{i} \cdot {\bf e}\hskip 1pt^{j}], of two contravariant basis vectors. Using ([link] and ([link], it can be shown that [{\bf e}^{i} \cdot {\bf e}\hskip 1pt^{j} = g^{ij}. \eqno(]

In a change of basis, following ([link], the [g^{ij}]'s transform according to [\left. \matrix{g^{ij} \ = B\hskip1pt_{k}^{i} B\hskip1pt_{m}^{j} g'^{km}\cr g'^{ij} \ = A_{k}^{i} A\hskip1pt_{m}^{j} g^{km}.\cr}\right\} \eqno(] The volumes V ′ and V of the cells built on the basis vectors [{\bf e}'_{i}] and [{\bf e}_{i}], respectively, are given by the triple scalar products of these two sets of basis vectors and are related by [\eqalignno{V' &= ({\bf e}'_{1}, {\bf e}'_{2}, {\bf e}'_{3}) &\cr &= \Delta (B_{j}^{i}) ({\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}) &\cr & = \Delta (B_{j}^{i}) V, &(\cr}]where [\Delta (B_{j}^{i})] is the determinant associated with the transformation matrix between the two bases. From ([link] and ([link], we can write [\Delta (g'_{ij}) = \Delta (B_{i}^{k}) \Delta (B_{j}^{m}) \Delta(g_{km}).]

If the basis [{\bf e}_{i}] is orthonormal, [\Delta (g_{km})] and V are equal to one, [\Delta (B_{j})] is equal to the volume V ′ of the cell built on the basis vectors [{\bf e}'_{i}] and [\Delta (g'_{ij}) = V'^{2}.]This relation is actually general and one can remove the prime index: [\Delta (g_{ij}) = V^{2}. \eqno(]

In the same way, we have for the corresponding reciprocal basis[\Delta (g^{ij}) = V^{*2},]where [V^{*}] is the volume of the reciprocal cell. Since the tables of the [g_{ij}]'s and of the [g^{ij}]'s are inverse, so are their determinants, and therefore the volumes of the unit cells of the direct and reciprocal spaces are also inverse, which is a very well known result in crystallography.

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