International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 6-7
Section 1.1.2.4. Covariant coordinates – dual or reciprocal space^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Using the developments (1.1.2.1) and (1.1.2.5), the scalar products of a vector x and of the basis vectors can be written The n quantities are called covariant components, and we shall see the reason for this a little later. The relations (1.1.2.9) can be considered as a system of equations of which the components are the unknowns. One can solve it since (see the end of Section 1.1.2.2). It follows that with
The table of the 's is the inverse of the table of the 's. Let us now take up the development of x with respect to the basis :
Let us replace by the expression (1.1.2.10): and let us introduce the set of n vectors which span the space . This set of n vectors forms a basis since (1.1.2.12) can be written with the aid of (1.1.2.13) as
The 's are the components of x in the basis . This basis is called the dual basis. By using (1.1.2.11) and (1.1.2.13), one can show in the same way that
It can be shown that the basis vectors transform in a change of basis like the components of the physical space. They are therefore contravariant. In a similar way, the components of a vector x with respect to the basis transform in a change of basis like the basis vectors in direct space, ; they are therefore covariant:
Let us take the scalar products of a covariant vector and a contravariant vector : [using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13)].
The relation we obtain, , is identical to the relations defining the reciprocal lattice in crystallography; the reciprocal basis then is identical to the dual basis .
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.