Axis or line of symmetry: It is an imaginary line (or axis) about which the crystal may be rotated so that it presents the same appearance more than once in a complete rotation through 360°. In a cubic crystal the axis of symmetry may be of three types depending upon the number of times the identical appearance occurs during the course of a complete rotation of 360°.
(i) Axes of four-fold symmetry- Imagine a line passing through the centres of two opposite faces of a cube. On rotating the cube about this line as axis, the self-coincidence (identical appearance) of the cube occurs four times during the course of rotation through 360°, i.e. the original appearance is repeated as a result of rotation through 90°. Such an axis is said to be an axis of four-fold symmetry or a tetrad axis. Now since there are six faces or three pairs of opposite faces in a cubic crystal, 3 such axes at right angles to each other are possible.
(ii) Axes of three-fold symmetry- Imagine a line passing through the opposite corners of a cube along the body diagonal. On rotating the cube about this line as axis, the identical appearance of the cube occurs three times during the course of rotation through 360°, i.e. the original appearance is repeated as a result of rotation through 120°. Such as axis is said to be an axis of three-fold symmetry or a triad axis. Now since there are eight corners or four such body diagonally opposite pairs of corners in a cubic crystal, 4 such axes are possible.
(iii) Axis of two-fold symmetry- Imagine a line passing through the centres of the two diagonally opposite edges of a cube. On rotating the cube about this line as axis, the identical appearance of the cube occurs two times during the course of rotation through 360°, i.e. the original appearance is repeated as a result of rotation through 180°. Such an axis is said to be an axis of two-fold symmetry or a diad axis. Since there are twelve edges or six such diagonally opposite pairs of edges in a cubic crystal, 6 such axes are possible.
Thus in a cubic crystal, there are 3 axes of four-fold symmetry, 4 axes of three-fold symmetry and 6 axes of two-fold symmetry. Thus in all a cubic crystal has 3 + 4 + 6 = 13 axes of symmetry.
(iv) Axis of six-fold symmetry- This axis of symmetry is possible in hexagonal crystals and not in cubic crystals. Imagine a line passing through the centres of two opposite hexagonal faces of a hexagonal crystal. On rotating the crystal (hexagonal) about this line as axis, the identical appearance of the hexagon occurs six times in the course of one complete rotation of 360°, ie. the original appearance is repeated as a result of rotation through 60°. Such an axis is, therefore, known as an axis of six-fold symmetry or a hexad axis.
