2. Symmetry elements

Two or more different symmetry operations of the same space group or crystal structure may have identical geometric elements, even when they belong to different categories as specified by Table 1. An example is provided by the powers of a rotation (same category) or by a reflection and a glide reflection in the same plane (different categories). In a given space group, the complete set of symmetry operations which have the same point or line etc. as their common geometric element will be called the element set of that geometric element.

The combination of a geometric element and its element set is indicated by the term `symmetry element' (Table 2). This allows such statements as `This point lies on a rotation axis', and also `The operations belonging to a glide plane' to be made.


 
Table 2: Symmetry elements in point groups and space groups
Name of symmetry element Symbol Geometric element Defining operation (d.o.) Operations in element set
Mirror plane Em Plane A Reflection in A D.o. and its coplanar equivalents*
Glide plane Eg Plane A Glide reflection in A, $2\nu$ (not $\nu$) a lattice translation D.o. and its coplanar equivalents*
Rotation axis En Line b Rotation about b, angle $2{\pi}/n,n=2,3,4$ or 6 1st$\dots(n-1)$th powers of d.o., and their coaxial equivalents$\dagger$
Screw axis Enj Line b Screw rotation about b, angle $2{\pi}/n,u=j/n$ times shortest lattice translation along b, right-hand screw; n= 2,3,4 or 6, $j=1,\dots,(n-1)$ 1st,$\dots,(n-1)$th powers of d.o., and their coaxial equivalents$\dagger$
Rotoinversion axis $E\bar{n}$ Line b and point P on b Rotoinversion: rotation about b, angle $2{\pi}/n$, and inversion through P; n=3, 4 or 6 D.o. and its inverse
Center $E\bar{1}$ Point P Inversion through P D.o. only

*That is, all glide reflections with the same reflection plane, with glide vectors differing from that of d.o. (taken to be zero for a reflection) by a lattice translation vector.

$\dagger$ That is, all rotations and screw rotations with the same axis, b, the same angle and sense of rotation, and the same screw vector u (zero for a rotation) up to a lattice translation vector.


The first column of Table 2 lists the name of the symmetry element. Of the four different kinds of geometric elements (Table 1), `point' and `point plus line' each yield one type of symmetry element. Two types arise from both `plane' and `line', depending on the presence or absence of a pure reflection and a pure rotation in the element set (cf. $\S$5). The symbols in the second column all begin with E, thereby indicating symmetry elements as defined in this Report.

The fourth column, `defining operation', states what to look for in order to identify a symmetry element, for instance in a structure model. The defining operation alone (for which the simplest is selected when there is a choice) suffices. However, for a rotoinversion axis $E\bar{3}$ or $E\bar{6}$, it will be easier to verify the presence of both its square and its cube, cf. $\S$3. The last column explicitly describes the full element set.

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Next: 3. Partial symmetry elements Up: Definition of Symmetry Elements in Space Previous: 1. Geometric elements

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