Symmetry operations

We can represent every symmetry operation by a matrix A:

$$A = \left[\begin{array} {ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32 }&a_{33}\end{array}\right]; (8)$$ (8)

the value of the elements of this matrix is dependent on the kind and orientation of the corresponding symmetry element with respect to the base system, and on the choice of the latter. In fact, in direct space a symmetry operation transforms a given vector r into the vector r$^{\prime}$; in matrix notation we can write:

$$r^{\prime} = Ar$$ (9)

where r and $r^{\prime}$ are the two column matrices whose elements are given by the components of the two vectors.

If the base system is given by the three vectors ${\tau}_1$, ${\tau}_2$,${\tau}_3$ of a primitive lattice, the elements a_ij of the A matrix are necessarily integers. In fact relation (9) must hold true for every vector r of the lattice; A transforms r in another vector r$^{\prime}$: in this case the components of r and r$^{\prime}$ are integers, and since relation (9) holds for every group of these integers relative to r, the elements of A must be integers.

We will now examine other restrictions on A which allow us to define the single elements a_ij as a function of the metric tensor. A symmetry operation obviously must not change the length of a vector or the angle between vectors. Therefore we have:

$$\textbf{r}^{\prime} \cdot \textbf{r}^{\prime} = \textbf{r} \cdot \textbf{r}$$

from which follows, applying relation (4):

$$r^{{\prime}t}Gr^{\prime} = r^tGr$$

and from (9):

r^tA^tGAr = r^tGr

and finally, since the previous relation must hold for any value of r:

G = A^tGA

i.e.:

$$\left[\begin{array} {ccc} g_{11}&g_{12}&g_{13}\\ g_{12}&g_{2... ...\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array}\right]$$ (11)

This identity is the matrix expression of the scalar product conservation on the crystallographic base system. All the matrices satisfying relation (10) are symmetry operations on the base system defined by G (see the example in the Appendix).

From relation (10), using matrix and determinant properties, we obtain:

$$\vert G\vert = \vert A^t\vert \cdot \vert G\vert \cdot \vert A\vert$$

from which, keeping in mind that |A^t| = |A|, follows that the determinant associated with the A matrix must be equal to $\pm$ 1. If the determinant is equal to +1 the symmetry operation is said to belong to the type I and it is defined as a rotation; if the determinant is equal to -1 the symmetry operation is of type II and defined as a rotoinversion.

Copyright © 1980, 1998 International Union of Crystallography