International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 5.2, p. 87

Section 5.2.2.3. Metric matrix

H. Arnolda‡

a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.2.3. Metric matrix

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The metric matrix G of the unit cell in the direct lattice [{\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}\cr} = \pmatrix{aa \hfill &ab \cos \gamma \hfill &ac \cos \beta \hfill \cr ba \cos \gamma \hfill &bb \hfill &bc \cos \alpha \hfill \cr ca \cos \beta \hfill &cb \cos \alpha \hfill &cc \hfill \cr}] changes under a linear transformation, but G is invariant under a symmetry operation of the lattice. The volume of the unit cell V is obtained by [V^{2} = \det ({\bi G}).] The same considerations apply to the metric matrix [{\bi G}^{*}] of the unit cell in the reciprocal lattice and the volume [V^{*}] of the reciprocal-lattice unit cell. Thus, there are two invariants under an affine transformation, the product [VV^{*} = 1] and the product [{\bi G}{\bi G}^{*} = {\bi I}.]








































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