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

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

Section 1.3.2.2. Metric properties

B. Souvigniera*

aRadboud 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 }. ]








































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