Brief notes on the symmetry elements of crystals

Examination of a crystal indicates the existence of regularity in the arrangement of like faces, edges, solid angles etc.

This regularity constitutes the symmetry of the crystal and depends on the internal atomic structure of the mineral. The symmetry can be defined with reference to three criteria viz. axis of symmetry, plane of symmetry and centre of symmetry generated by rotation, reflection and inversion operations respectively.

Axis of symmetry:

Symmetry operations like rotation and rotation combined with inversion give rise to axes of symmetry. An axis of symmetry is an imaginary line, which passes through the centre of the crystal.

The rotational axis, commonly known as axis of symmetry, involves rotation of the crystal about the concerned axis by certain angle as a result of which the crystal occupies the immediately next similar position in space.

The angle between two consecutive similar positions in space is known as the elementary angle of rotation.

The axes of symmetry are one-fold, two-fold, three-fold, four-fold and six-fold with elementary angles of rotation 360°, 180°, 120°, 90° and 60° respectively. These are represented by numbers 1, 2, 3, 4 and 6 respectively. The axes of 2-fold, 3-fold, 4-fold and 6-fold are also known as that, triad, tetrad and hexed axes respectively.

The degree of the axis indicates the number of times the crystal occupies the same position in space in a complete rotation of 360° about the concerned axis. Axis of one-fold rotation (1) is universally present in all the objects whatever the shape may be. Hence, it is not regarded as a symmetry element by many.

However, in the true sense, it indicates the lack of symmetry and is used in classification of crystals into 32 classes. A pinewood is shown in. It consists of two faces, which interchange their positions twice in a complete rotation of 360°. In the trifocal, tetragonal and hexagonal pyramids possess rotational axes of 3-, 4- and 6-fold respectively. Even, a single form may have axes of different degrees.

For example, a cube has six axes of 2-fold symmetry, which join the mid points of opposite edges, four axes of 3-fold symmetry, which pass through the opposite corners (solid angles) and three axes of 4-fold symmetry, which emerge at the mid points of opposite faces. Axis of 5-fold symmetry is not possible in crystallography.

It can be proved both by experiment and mathematical inference. According to the rule, there should not be any empty space in crystal structure of stable compounds i.e. the available trace should be completely filled in. Any amount of two-dimensional empty space can be completely filled in by suitable size paper-cuttings of rectangular, equilateral triangular, square and regular hexagonal shapes, which are the two-dimensional horizontal cross sections of three-dimensional solid objects having 2-, 3-, 4- and 6- fold axes of symmetry respectively.

The horizontal cross-section of a three-dimensional object having an axis of 5-fold symmetry is regular pentagons, whose sides are equal and subtend 72° at the centre. If attempt is made to fill in any specified space by small paper cuttings of regular pentagons, small empty spaces (holes) will remain.

Existence of empty spaces (holes) is not permitted within stable and ordered crystal structure. This leads to the impossibility of existence of axis of 5-fold symmetry. Similar arguments are also true for 7-fold and other higher degree symmetry axes. The mathematical treatment of this problem will be addressed in higher classes.

The rot inversion axes involve rotation by the elementary angle of rotation followed by inversion through a point. Corresponding to five] elementary angles stated above, there are five rot inversion axes designated as \, 2, 3, 4 and 5.

These are read as bar one etc. equivalent to a centre of symmetry (or centre of inversion, i), 2 is equivalent to a mirror plane (m), 3 is equivalent to a 3-fold rotation axis plus centre of symmetry, 4 is unique and q is equivalent to a 3-fold rotation axis with a perpendicular mirror plane. These will be discussed at length in higher classes.

Plane of symmetry:

A plane of symmetry is an imaginary plane that passes through the centre and divides a figure or a crystal into two similar and similarly placed halves, which are mirror images of each other, appears as if one half of the figure or the crystal is generated by reflection of the other half on a plane mirror surface.

The plane of symmetry is also known as mirror plane or mirror and is denoted by the letter’s’ (some authors denote it by ‘P’). The presence of the plane of symmetry can be well visualized from different two-dimensional figures shown in. The dashed lines represent planes of symmetry.

In an irregular figure is shown which is not symmetrical with respect to any plane and thus lacks a plane of symmetry. The isosceles triangle, rectangle, equilateral triangle, square and hexagon shown in (b), (c), (d), (e) and (0 have 1, 2, 3, 4 and 6 numbers of planes of symmetry respectively.

The circle has infinite number of planes of symmetry (only a few are shown) each coincident with a diameter of the circle. Two three- dimensional crystal forms, a prism and a cube are shown in and 3.8 respectively.

The upper two faces of the prism are related with the lower two faces by a horizontal plane of symmetry. Similarly, a cube has nine planes of symmetry indicated by numbers 1 to 9. A sphere has infinite number of planes of symmetry.

Centre of symmetry:

A crystal is said to have a centre of symmetry if like faces, edges, solid angles etc. are arranged in pairs in corresponding positions on opposite sides of a central point. The centre of symmetry is an imaginary point located at the centre of the crystal. If a straight line is drawn through it, similar faces, edges and solid angles are encountered on either side of the point at equal distances. It is observed that when a crystal having centre of symmetry is made to stand on a certain face, an identify face remains to the top of the crystal. A crystal having an axis of even-of perpendicular to a mirror plane generally possesses a centre of symmetry.