(ii) 4-fold axis parallel to c-axis and passing through the center of the unit cell.
There is only one such axis.
(iii) Seven mirror planes, 3 bisecting the parallel faces, and 4 connecting diagonal
edges. All of them are passing through the center of the unit cell.
(iv) One center of symmetry at the center of the unit cell.
Thus, there are:
4-diads + 1-tetrad + 7 surface planes + 1 center of symmetry = 13
Therefore, the total number of symmetry elements in a tetragonal unit cell = 13.
Example 9 Find the total number of symmetry elements that exist in a hexagonal
unit cell.
Solution: Looking at Fig. 6.12, we find that there exist:
(i) 2-fold axes parallel to the face diagonal and passing through the center of the
unit cell. There are six such axes.
(ii) 6-fold axis parallel to c-axis and passing through the center of the unit cell.
There is only one such axis.
(iii) Seven mirror planes, 3 parallel to the opposite (vertical) faces, 3 parallel to
opposite vertical edges, all of them are passing through the center of the unit
cell and parallel to the principal axis. 1 horizontal plane passing through the
center of the unit cell.
(iv) One center of symmetry at the center of the unit cell.
Thus, there are:
6-diads + 1-hexad + 7 surface planes + 1 center of symmetry = 15
Therefore, the total number of symmetry elements in a hexagonal unit cell = 15.
Fig. 6.12 Simple hexagonal
unit cell
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6 Unit Cell Symmeteries and Their Representations