Elements of symmetry

The crystals of the various substances possess the following different types of symmetries:

(i)    Centre of symmetry

It is defined as an imaginary point within the crystal such that any line passing through this point intersects the opposite faces of the crystal at equal distances. The centre of symmetry is also called the centre of inversion because if the crystal or the molecule is inverted through the centre of symmetry it gives results equivalent * to the original and hence indistinguishable form the original one. For example, the centre of in ersion of a few molecules is given below (represented by solid dots):


                                       Centers of inversion of a few molecules.

(ii)    Plane of symmetry.

It is defined as an imaginary plane passing through, the crystal such that it divides the crystal into two parts in such a way that one part is the mirror image of the other. The planes of symmetry are, therefore, also called mirror planes.

(iii)    Rotation axis of symmetry.

The rotation axis of symmetry or simply called the axis of symmetry is defined as the imaginary line passing through the crystal such that when the crystal is related about this line, exactly similar appearance occurs more than once in one complete . Revolution, i.e., in a rotation through 360⁰. It similar appearance occurs twice in one complete revolution (i.e. after every 180⁰, the axis is called two fold axis of symmetry or a died axis. Similarly the other possibilities found for similar appearance in one complete revolution are thrice, four times of six times (i.e., after rotation through 120⁰, 90⁰, or 60⁰. The corresponding axes are called three-fold (triad), four-fold (tetrad) and six fold (hexed) axes of symmetry.

The above type of rotation which leads to equivalent configurations, which are congruent as well, is called proper rotation and the axis is called proper rotation axis. Further, it meaty be pointed out the we have only 2=fold, 3-fold, 4-fold, and 6-fold axes of rotation and we do not have 5-fold, 7-fold, 8-fold or higher-fold axes of rotation, as will be explained later in section 5.12.

(iv)Rotation-reflection axis of symmetry.

When the rotation of a molecule or a crystal about an n-fold axis of proper rotation is followed by reflection through a plane perpendicular to the axis, the axis is called an n-fold rotation-reflection axis The new configuration obtained is not congruent but is enantiomorphous. For example, rotation followed by reflection f an asymmetric object like the numeral ‘7’ is shown in Fig.

(v)Rotation-inversion axis of symmetry.

When the rotation of a molecule or a crystal is followed by inversion through the centre of symmetry the axis is called an n-fold rotation-inversion axis. The new configuration obtained is not congruent but is enantiomorphous, as in the above case.