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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 6-7

Section 1.1.2.4.3. Properties of the metric tensor

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

1.1.2.4.3. Properties of the metric tensor

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In a change of basis, following (1.1.2.3) and (1.1.2.5), the 's transform according to Let us now consider the scalar products, , of two contravariant basis vectors. Using (1.1.2.11) and (1.1.2.13), it can be shown that

In a change of basis, following (1.1.2.16), the 's transform according to The volumes V ′ and V of the cells built on the basis vectors and , respectively, are given by the triple scalar products of these two sets of basis vectors and are related by where is the determinant associated with the transformation matrix between the two bases. From (1.1.2.17) and (1.1.2.20), we can write

If the basis is orthonormal, and V are equal to one, is equal to the volume V ′ of the cell built on the basis vectors and This relation is actually general and one can remove the prime index:

In the same way, we have for the corresponding reciprocal basiswhere is the volume of the reciprocal cell. Since the tables of the 's and of the '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.