International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 23-24
Section 1.3.2.2. Metric properties^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands |
In the three-dimensional vector space , the norm or length of a vector is (due to Pythagoras' theorem) given by From this, the scalar product is derived, which allows one to express angles by
The definition of a norm function for the vectors turns into a Euclidean space. A lattice that is contained in inherits the metric properties of this space. But for the lattice, these properties are most conveniently expressed with respect to a lattice basis. It is customary to choose basis vectors a, b, c which define a right-handed coordinate system, i.e. such that the matrix with columns a, b, c has a positive determinant.
Definition
For a lattice with lattice basis the metric tensor of is the 3 × 3 matrix If is the 3 × 3 matrix with the vectors as its columns, then the metric tensor is obtained as the matrix product . It follows immediately that the metric tensor is a symmetric matrix, i.e. .
Example
Letbe the basis of a lattice . Then the metric tensor of (with respect to the given basis) is
With the help of the metric tensor the scalar products of arbitrary vectors, given as linear combinations of the lattice basis, can be computed from their coordinate columns as follows: If and , then
From this it follows how the metric tensor transforms under a basis transformation . If , then the metric tensor of with respect to the new basis is given by
An alternative way to specify the geometry of a lattice in is using the cell parameters, which are the lengths of the lattice basis vectors and the angles between them.
Definition
For a lattice in with lattice basis the cell parameters (also called lattice parameters, lattice constants or metric parameters) are given by the lengths of the basis vectors and by the interaxial angles
Owing to the relation for the scalar product of two vectors, one can immediately write down the metric tensor in terms of the cell parameters: