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

International Tables for Crystallography (2006). Vol. A, ch. 5.2, pp. 86-87

Section 5.2.2. Invariants

H. Arnolda

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.2. Invariants

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A crystal structure and its physical properties are independent of the choice of the unit cell. This implies that invariants occur, i.e. quantities which have the same values before and after the transformation. Only some important invariants are considered in this section. Invariants of higher order (tensors) are treated by Altmann & Herzig (1994[link]), second cumulant tensors, i.e. anisotropic temperature factors, are given in International Tables for Crystallography (2004[link]), Vol. C.

The orthogonality of the basis vectors a, b, c of direct space and the basis vectors [{\bf a}^{*}], [{\bf b}^{*}], [{\bf c}^{*}] of reciprocal space, [\pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr} ({\bf a},{\bf b},{\bf c}) = \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} = {\bi I},] is invariant under a general (affine) transformation. Since both sets of basis vectors are transformed, [{\bf a}^{*}] is always perpendicular to the plane defined by b and c and [{\bf a}^{*}{'}] perpendicular to b′ and cetc. Position vector

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The position vector r in direct space, [{\bf r} = ({\bf a},{\bf b},{\bf c}) \pmatrix{x\cr y\cr z\cr} = x{\bf a} + y{\bf b} + z{\bf c},] is invariant if the origin of the coordinate system is not changed in the transformation (see example in Section 5.1.3[link] ). Modulus of position vector

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The modulus r of the position vector r gives the distance of the point xyz from the origin. Its square is obtained by the scalar product [\eqalignno{{\bf r}^{\;{\bi t}} \cdot {\bf r} = r^{2} &= (x,y,z) \pmatrix{{\bf a}\cr {\bf b}\cr {\bf c}\cr} ({\bf a},{\bf b},{\bf c}) \pmatrix{x\cr y\cr z\cr}\cr &= (x,y,z) {\bi G} \pmatrix{x\cr y\cr z\cr}\cr &= x^{2}a^{2} + y^{2}b^{2} + z^{2}c^{2} + 2yzbc \cos \alpha &\cr &\quad + 2xzac \cos \beta + 2xyab \cos \gamma,}] with [{\bf r}^{\;\bi t}] the transposed representation of r; a, b, c the moduli of the basis vectors a, b, c (lattice parameters); G the metric matrix of direct space; and α, β, γ the angles of the unit cell.

The same considerations apply to the vector [{\bf r}^{*}] in reciprocal space and its modulus [r^{*}]. Here, [{\bi G}^{*}] is applied. Note that [{\bf r}^{*}] and [r^{*}] are independent of the choice of the origin in direct space. 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}.] Scalar product

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The scalar product [{\bf r}^{*} \cdot{\bf r} = hx + ky + lz] of the vector [{\bf r}^{*}] in reciprocal space with the vector r in direct space is invariant under a linear transformation but not under a shift of origin in direct space.

A vector r in direct space can also be represented as a product of augmented matrices: [{\bf r} = ({\bf a},{\bf b},{\bf c},0) \pmatrix{x\cr y\cr z\cr 1\cr} = x{\bf a} + y{\bf b} + z{\bf c}.] As stated above, the basis vectors are transformed only by the linear part, even in the case of a general affine transformation. Thus, the transformed position vector [{\bf r}'] is obtained by [{\bf r}' = ({\bf a},{\bf b},{\bf c},0) \pmatrix{{\bi P} &{\bi o}^{{\bi t}}\cr {\bi o} &1\cr} \pmatrix{{\bi Q} &{\bi q}\cr {\bi o} &1\cr} \pmatrix{x\cr y\cr z\cr 1\cr}.] The shift p is set to zero. The shift of origin is contained in the matrix [\specialfonts\bbsf{Q}] only.

Similarly, a vector in reciprocal space can be represented by [{\bf r}^{*} = (h,k,l,1) \pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr 0\cr} = h{\bf a}^{*} + k{\bf b}^{*} + l{\bf c}^{*}.] The coordinates h, k, l in reciprocal space transform also only linearly. Thus, [{\bf r}^{*}{'} = (h,k,l,1)\pmatrix{{\bi P} &{\bi o}^{{\bi t}}\cr {\bi o} &{1}\cr} \pmatrix{{\bi Q} &{\bi q}\cr {\bi o} &{1}\cr} \pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr 0\cr}.] The reader can see immediately that the scalar product [{\bf r}^{*} \cdot{\bf r}] transforms correctly.


International Tables for Crystallography (2004). Vol. C, edited by E. Prince, Table Dordrecht: Kluwer Academic Publishers.
Altmann, S. L. & Herzig, P. (1994). Point-Group Theory Tables. Oxford Science Publications.

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