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. 5-7
Section 1.1.2. Basic properties of vector spaces^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
[The reader may also refer to Section 1.1.4 of Volume B of International Tables for Crystallography (2001).]
Let us consider a vector space spanned by the set of n basis vectors , , . The decomposition of a vector using this basis is written using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:
Let us now consider a second basis, . The vector x is independent of the choice of basis and it can be decomposed also in the second basis:
If and are the transformation matrices between the bases and , the following relations hold between the two bases: (summations over j and i, respectively). The matrices and are inverse matrices: (Kronecker symbol: if if ).
Important Remark. The behaviour of the basis vectors and of the components of the vectors in a transformation are different. The roles of the matrices and are opposite in each case. The components are said to be contravariant. Everything that transforms like a basis vector is covariant and is characterized by an inferior index. Everything that transforms like a component is contravariant and is characterized by a superior index. The property describing the way a mathematical body transforms under a change of basis is called variance.
We shall limit ourselves to a Euclidean space for which we have defined the scalar product. The analytical expression of the scalar product of two vectors and is Let us put The nine components are called the components of the metric tensor. Its tensor nature will be shown in Section 1.1.3.6.1. Owing to the commutativity of the scalar product, we have
The table of the components is therefore symmetrical. One of the definition properties of the scalar product is that if for all x, then . This is translated as
In order that only the trivial solution exists, it is necessary that the determinant constructed from the 's is different from zero: This important property will be used in Section 1.1.2.4.1.
An orthonormal coordinate frame is characterized by the fact that One deduces from this that the scalar product is written simply as
Let us consider a change of basis between two orthonormal systems of coordinates: Multiplying the two sides of this relation by , it follows that which can also be written, if one notes that variance is not apparent in an orthonormal frame of coordinates and that the position of indices is therefore not important, as
The matrix coefficients, , are the direction cosines of with respect to the basis vectors. Similarly, we have so that where ^{T} indicates transpose. It follows that so that The matrices A and B are unitary matrices or matrices of rotation and
One can write for the coefficients giving six relations between the nine coefficients . There are thus three independent coefficients of the matrix A.
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.
References
International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.