Metric properties

Souvignier, B.

doi:10.1107/97809553602060000921

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 23-24

Section 1.3.2.2. Metric properties

B. Souvignier^a ^*

^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
Correspondence e-mail: souvi@math.ru.nl

1.3.2.2. Metric properties

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In the three-dimensional vector space $[{\bb V}^3]$ , the norm or length of a vector $[{\bf v} = \pmatrix{ v_x \cr v_y \cr v_z }]$ is (due to Pythagoras' theorem) given by $[ |{\bf v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}. ]$ From this, the scalar product $[ {\bf v} \cdot {\bf w} = v_x w_x + v_y w_y + v_z w_z \ {\rm for }\ {\bf v} = \pmatrix{ v_x \cr v_y \cr v_z }, {\bf w} = \pmatrix{ w_x \cr w_y \cr w_z } ]$ is derived, which allows one to express angles by $[ \cos \angle ({\bf v}, {\bf w}) = {{{\bf v} \cdot {\bf w}}\over{| {\bf v} | \, | {\bf w} |}}. ]$

The definition of a norm function for the vectors turns $[{\bb V}^3]$ into a Euclidean space. A lattice $[{\bf L}]$ that is contained in $[{\bb V}^3]$ 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 $[{\bf L} \subseteq {\bb V}^3]$ with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ the metric tensor of $[{\bf L}]$ is the 3 × 3 matrix $[ {\bi G} = \pmatrix{ {\bf a} \cdot {\bf a} & {\bf a} \cdot {\bf b} & {\bf a} \cdot {\bf c} \cr {\bf b} \cdot {\bf a} & {\bf b} \cdot {\bf b} & {\bf b} \cdot {\bf c} \cr {\bf c} \cdot {\bf a} & {\bf c} \cdot {\bf b} & {\bf c} \cdot {\bf c} }. ]$ If $[{\bi A}]$ is the 3 × 3 matrix with the vectors $[{\bf a}, {\bf b}, {\bf c} ]$ as its columns, then the metric tensor is obtained as the matrix product $[{\bi G} = {\bi A}^{\rm T}\cdot {\bi A}]$ . It follows immediately that the metric tensor is a symmetric matrix, i.e. $[{\bi G}^{\rm T} = {\bi G} ]$ .

Example

Let $[{\bf a} = \pmatrix{ 1 \cr 1 \cr 1 },\quad {\bf b} = \pmatrix{ 1 \cr 1 \cr 0 },\quad {\bf c} = \pmatrix{ 1 \cr -1 \cr 0 } ]$ be the basis of a lattice $[{\bf L}]$ . Then the metric tensor of $[{\bf L}]$ (with respect to the given basis) is $[{\bi G} = \pmatrix{ 3 & 2 & 0 \cr 2 & 2 & 0 \cr 0 & 0 & 2 }.]$

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 $[{\bf v} = x_1 {\bf a} + y_1 {\bf b} + z_1 {\bf c} ]$ and $[{\bf w} = x_2 {\bf a} + y_2 {\bf b} + z_2 {\bf c}]$ , then $[ {\bf v} \cdot {\bf w} = (x_1 \, y_1 \, z_1) \cdot {\bi G} \cdot \pmatrix{ x_2 \cr y_2 \cr z_2 }. ]$

From this it follows how the metric tensor transforms under a basis transformation $[{\bi P}]$ . If $[({\bf a}', {\bf b}', {\bf c}') = ({\bf a}, {\bf b}, {\bf c}) {\bi P} ]$ , then the metric tensor $[{\bi G}']$ of $[{\bf L}]$ with respect to the new basis $[{\bf a}', {\bf b}', {\bf c}']$ is given by $[ {\bi G}' = {\bi P}^{\rm T} \cdot {\bi G} \cdot {\bi P}. ]$

An alternative way to specify the geometry of a lattice in $[{\bb V}^3 ]$ is using the cell parameters, which are the lengths of the lattice basis vectors and the angles between them.

Definition

For a lattice $[{\bf L}]$ in $[{\bb V}^3]$ with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ the cell parameters (also called lattice parameters, lattice constants or metric parameters) are given by the lengths $[ a = | {\bf a} | = \sqrt{ {\bf a} \cdot {\bf a} }, \quad b = | {\bf b} | = \sqrt{ {\bf b} \cdot {\bf b} }, \quad c = | {\bf c} | = \sqrt{ {\bf c} \cdot {\bf c} } ]$ of the basis vectors and by the interaxial angles $[ \alpha = \angle ({\bf b}, {\bf c}), \quad \beta = \angle ({\bf c}, {\bf a}), \quad \gamma = \angle ({\bf a}, {\bf b}). ]$

Owing to the relation $[{\bf v} \cdot {\bf w} = | {\bf v} | \, | {\bf w} | \, \cos \angle ({\bf v}, {\bf w}) ]$ for the scalar product of two vectors, one can immediately write down the metric tensor in terms of the cell parameters: $[ {\bi G} = \pmatrix{ a^2 & a b \cos \gamma & a c \cos \beta \cr a b \cos \gamma & b^2 & b c \cos \alpha \cr a c \cos \beta & b c \cos \alpha & c^2 }. ]$

References

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 23-24