Space Group Operations

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(a) purely translational operations expressed by the Bravais lattice (denoted by a capital letter in the space group symbol), and

(b) operations of point symmetry elements, glide planes and/or screw axes, as listed in the following table:


Symmetry element	Point symmetry elements	Symmetry elements involving translations

of the first kind	rotation axes	screw axes
(proper)	1 2 3 4 6	2₁ 3₁ 3₂ 4₁ 4₂ 4₃ etc

of the second kind	rotary inversion	glide planes
(improper)	axes	a b c n d
	$\overline{1}$ $\overline{2}$ $\overline{3}$ $\overline{4}$ $\overline{6}$

These symmetry operations can be depicted as shown in Figs. 2 and 3 (if necessary, see also Fig. 4 for explanation of the diagrams). Symmetry operations of the second kind, unlike those of the first kind (or proper operations), transform a right-handed into a left-handed object and vice versa. Two enantiomeric objects can be related only by a symmetry operation of the second kind (or an improper operation). This distinction of the two kinds of symmetry elements is also of considerable practical importance, as it will soon be evident, in recognizing the space group of a given pattern.

**Figure 2:** Symmetry operations of the first kind (examples).
$\begin{figure} \includegraphics {fig2.ps} \end{figure}$

**Figure 3:** Symmetry operations of the second kind.
$\begin{figure} \includegraphics {fig3.ps} \end{figure}$

**Figure 4:** Scheme used in space group patterns (worksheets (i) to (xv)).
$\begin{figure} \includegraphics {fig4.ps} \end{figure}$

Some point symmetry operations (which leave at least one point in space unaffected) are shown on the left hand side of Figs. 2 and 3. Examples of operations containing translational components (which are incompatible with a finite array of objects) are presented on the right hand side in these figures.

More details on crystallographic symmetry elements can be readily obtained from most textbooks on crystallography.⁴ For information on symbols and notation it is best to consult the International Tables .⁵