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

International Tables for Crystallography (2015). Vol. A, ch. 1.5, p. 84

Section 1.5.2.2. Metric tensors of direct and reciprocal lattices

H. Wondratscheka and M. I. Aroyob

1.5.2.2. Metric tensors of direct and reciprocal lattices

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The metric tensor of a crystal lattice with a basis [{\bf a},{\bf b},{\bf c}] 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}},]which can formally be described as[{\bi G}=({\bf a},{\bf b},{\bf c})^T\cdot({\bf a},{\bf b},{\bf c})=\pmatrix{{\bf a}\cr {\bf b}\cr {\bf c}}\cdot({\bf a},{\bf b},{\bf c})](cf. Section 1.3.2[link] ). The transformation of the metric tensor under the coordinate transformation [({\bi P}, {\bi p})] follows directly from its definition:[\eqalignno{{\bi G}'&=({\bf a}',{\bf b}',{\bf c}')^T\cdot({\bf a}',{\bf b}',{\bf c}')=[({\bf a},{\bf b},{\bf c}){\bi P}]^T\cdot ({\bf a},{\bf b},{\bf c}){\bi P} &\cr&= {\bi P}^T({\bf a},{\bf b},{\bf c})^T\cdot({\bf a},{\bf b},{\bf c}){\bi P}={\bi P}^T{\bi G}{\bi P},&(1.5.2.4)}]where [{\bi P}^T] is the transposed matrix of P. The transformation behaviour of G under [({\bi P}, {\bi p})] is determined by the matrix P, i.e. G is not affected by an origin shift p.

The volume V of the unit cell defined by the basis vectors [{\bf a},{\bf b},{\bf c}] can be obtained from the determinant of the metric tensor, [V^2=\det({\bi G})]. The transformation behaviour of V under a coordinate transformation follows from the transformation behaviour of the metric tensor [note that [\det({\bi P})=\det({\bi P}^T)]]: [(V')^2] = [\det({\bi G}')] = [\det({\bi P}^T{\bi G}{\bi P})] = [\det({\bi P})\det({\bi P}^T)\det({\bi G})] = [\det({\bi P})^2V^2], i.e.[V'=|\!\det({\bi P})|V,\eqno(1.5.2.5)]which is reduced to [V'=\det({\bi P})V] if [\det({\bi P})\,\gt\,0].

Similarly, the metric tensor [{\bi G}^*] of the reciprocal lattice and the volume [V^*] of the unit cell defined by the basis vectors [{\bf a}^{*},{\bf b}^{*},{\bf c}^{*}] transform as[{\bi G}^{*}{'}={\bi Q}{\bi G}^{*}{\bi Q}^T,\eqno(1.5.2.6)][\eqalignno{&{V}^{*}{'}\!=\!|\!\det({\bi Q})|{V}^* \, {\rm or }\, {V}^{*}{'}\!=\!\det({\bi Q}){V}^*\!=\! [1/\det({\bi P})]{V}^* \, {\rm if }\, \det({\bi Q})\,\gt\,0.&\cr&&(1.5.2.7)}]Again, it is only the linear part [{\bi Q}={\bi P}^{-1}] that determines the transformation behaviour of [{\bi G}^*] and [V^*] under coordinate transformations.








































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