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Crystallographic site-symmetry operations

In theoretical and practical work one frequently needs to know the symmetry around a position in a molecule or in a crystal structure. The symmetry of the surroundings of an atom or of the center of gravity of a (more or less complex) group of atoms (ion, molecule, etc.) is determined, among others, by chemical bonds. The surroundings of such a constituent strongly influence the physical and chemical properties of a substance. A striking example is the pair `graphite and diamond', which both are chemically carbon but display different surroundings of the carbon atoms and thus extremely different chemical and physical properties.

The symmetry of the surroundings of a point $P$, called the site symmetry or point symmetry of $P$, is determined by the symmetry of the whole molecule or crystal and by the locus of $P$ in the molecule or crystal. Here, we are interested in crystallographic site symmetries only, i.e. the local symmetries around points $P$ in a crystal (better, in a crystal pattern). Strictly, one defines:

Definition (D 3.2.2) The set $\mbox{$\mathcal{R}$}$ of all symmetry operations of a crystal pattern is called the space group of the crystal pattern. The set of all elements of $\mbox{$\mathcal{R}$}$, i.e. of the space group, which leave a given point $P$ fixed, is called the site symmetry, site-symmetry group, point-symmetry group, or point group $\mbox{$\mathcal{S}$}$ of $P$ with regard to the space group $\mbox{$\mathcal{R}$}$.

In this manuscript the term site-symmetry group or, for short, site symmetry, is preferred for reasons which will become clear in Section 3.4.

Because of its periodicity each crystal has an infinite number of translations as symmetry operations, i.e. $\mbox{$\mathcal{R}$}$ is an infinite set. However, a translation can not be an element of a site-symmetry group because a translation has no fixed point at all. The same holds for screw rotations and glide reflections.

For the description of the crystallographic symmetry operations, it is convenient to have available the notion of the `order of an isometry'.

Definition (D 3.2.2) An isometry W has the (or: is of) order $k$, if $\mathsf{W}^k=\mathsf{I}$ holds, where $\mathsf{I}$ is the identity operation, and $k>0$ is the smallest number, for which this equation is fulfilled.

Remark. The $k$ different isometries $\mathsf{W}^j$, $j$ = 1, ..., $k$, form a group with $k$ elements. See also the definition (D 3.4.2) of the group order.

The following types of isometries may be elements of crystallographic site symmetries:

  1. Identity. The identity I is a member of any crystallographic site-symmetry group because it leaves any point fixed. It is the only operation whose order is 1, its Hermann-Mauguin symbol (HM symbol) is also 1. [The $HM$ symbols have been introduced by CARL HERMANN and CHARLES MAUGUIN around 1930. There are $HM$ symbols for point-group and for space-group operations, as well as for site-symmetry, space, and point groups. In IT A the $HM$ symbols form the standard nomenclature].

  2. Inversion. If the inversion is a member of the site-symmetry group, then the point $P$ is the center of inversion. The order of the inversion is 2, its $HM$ symbol is $\bar{1}$.
  3. Rotations. The point $P$ is placed on the rotation axis. Due to the periodicity of the crystals, the rotation angles of crystallographic rotations are restricted to multiples of $60^{\circ}$ and $90^{\circ}$, i.e. to $60^{\circ}$, $120^{\circ}$, $180^{\circ}$, $240^{\circ}$, $300^{\circ}$, $90^{\circ}$, and $270^{\circ}$. All these angles are of the form $j\,360^{\circ}/N$, where $N$=2, 3, 4, or 6, and $j$ is an integer which is relative prime to $N$. (This restriction does not hold for the symmetry of molecules which may display, e.g., a rotation angle of $360^{\circ}/5=72^{\circ}$ and its multiples.) Moreover, the angles between different axes of crystallographic rotations are limited to a small number of values only.

    A rotation with the rotation angle $j\,360^{\circ}/N$ is called an $N$-fold rotation. Its $HM$ symbol is $N^j$. The $HM$ symbols of the crystallographic rotations are $2^1$ = 2$(180^{\circ})$, $3^1$=3$(120^{\circ})$, $3^2$=$(240^{\circ})$, $4^1$=4$(90^{\circ})$, $4^3$=$(270^{\circ})$, $6^1$=$6$$(60^{\circ})$, $6^5$=$(300^{\circ})$, (and 1 $(0^{\circ}
\mbox{\,or\,}360^{\circ}$) for the identity). The order of the rotation $N^j$ is $k=N$.

  4. Rotoinversions. The point $P$ is placed in the inversion point on the rotoinversion axis. The restrictions on the angles $\Phi$ of the rotational parts are the same as for rotations. If $\Phi=j\,360^{\circ}/N$, the rotoinversion is called an $N$-fold rotoinversion. The $HM$ symbol for such a rotoinversion is $\bar{N}^j$. In crystals can occur:

    ($\bar{1}$ inversion), $\bar{3}^1=\bar{3}$, $\bar{3}^5$, $\bar{4}^1=\bar{4}$, $\bar{4}^3$, $\bar{6}^1=\bar{6}$, and $\bar{6}^5$. The rotoinversion $\bar{2}$ is identical with a reflection, see next item.

    Question. Which isometry is $\bar{3}^3$, $\bar{4}^2$, $\bar{6}^2$, and $\bar{6}^3$ ? The answer to this question is found at the end of this chapter.

  5. Reflections. The point $P$ is situated on the mirror plane. There are only a few possible angles between the normals of different mirror planes belonging to the reflections of a site-symmetry group: $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $90^{\circ}$. The $HM$ symbol of a reflection is $m$ (mirror, miroir) instead of $\bar{2}$. The order of a reflection is 2, because its 2-fold application yields the identity operation.

The combination of these types of symmetry operations to site-symmetry or point groups will be discussed in Section 3.4.

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