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Next: 5. Comparison with IT A83 Up: Definition of Symmetry Elements in Space Previous: 3. Partial symmetry elements

4. Some examples

According to the above definition, each symmetry operation belongs to one and only one symmetry element (not counting partial elements). The latter is determined by the geometric element of the operation. For example, we can `decompose' any site symmetry group into symmetry elements.
Example 1. A site symmetry with point group $\bar{6}$ consists of:

$\bar{6}^1$ and $\bar{6}^5$ , forming a rotoinversion axis $E\bar{6}$

$\bar{6}^2$ and $\bar{6}^4$ , belonging to a rotation axis E3

$\bar{6}^3$ , belonging to a mirror plane Em.

This `decomposition' is always exhaustive (the identity operation is not listed in these examples since it occurs in every point group). Strictly speaking, it is not a true decomposition, because the mirror plane for instance has in its element set not only the reflection but also the infinite number of coplanar equivalents which are of course not part of the site symmetry.

Example 2. Site symmetry 4/m consists of:

4¹, 4² and 4³, belonging to a rotation axis E4

$\bar{4}^1$ and $\bar{4}^3$ , forming a rotoinversion axis $E\bar{4}$

m¹, belonging to a mirror plane Em

$\bar{1}^1$ , forming a center $E\bar{1}$ .

Example 3. Site symmetry 2/ $m\, \bar{3}$ consists of:

$4 \times (3^1$ and 3²), belonging to four rotation axes E3

$4 \times (\bar{3}^1$ and $\bar{3}^5$ ), forming four rotoinversion axes $E\bar{3}$

$3 \times 2^1$ , belonging to three rotation axes E2

$3 \times m^1$ , belonging to three mirror planes Em

$\bar{1}^1$ , forming a center $E\bar{1}$ .

Next: 5. Comparison with IT A83 Up: Definition of Symmetry Elements in Space Previous: 3. Partial symmetry elements

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Example 1. A site symmetry with point group $\bar{6}$ consists of:
$\bar{6}^1$ and $\bar{6}^5$ ,	forming a rotoinversion axis $E\bar{6}$
$\bar{6}^2$ and $\bar{6}^4$ ,	belonging to a rotation axis E3
$\bar{6}^3$ ,	belonging to a mirror plane Em.

Example 2. Site symmetry 4/m consists of:
4¹, 4² and 4³,	belonging to a rotation axis E4
$\bar{4}^1$ and $\bar{4}^3$ ,	forming a rotoinversion axis $E\bar{4}$
m¹,	belonging to a mirror plane Em
$\bar{1}^1$ ,	forming a center $E\bar{1}$ .

Example 3. Site symmetry 2/ $m\, \bar{3}$ consists of:
$4 \times (3^1$ and 3²),	belonging to four rotation axes E3
$4 \times (\bar{3}^1$ and $\bar{3}^5$ ),	forming four rotoinversion axes $E\bar{3}$
$3 \times 2^1$ ,	belonging to three rotation axes E2
$3 \times m^1$ ,	belonging to three mirror planes Em
$\bar{1}^1$ ,	forming a center $E\bar{1}$ .