Screw axes

A rototranslational axis is a symmetry axis of order n if all the properties of the space remain unchanged after a   rotation around the axis and a translation t along the axis. We will consider here the restrictions imposed by the periodic nature of crystals on the translational component t.

Suppose that the rototranslational axis lies along a lattice row with period T.   The rotational component must correspond to n=1,2,3,4,6 (see the preceding section). If we apply the symmetry axis n times the overall rotational component will be   and the final translational displacement will be nt .

In order to maintain the periodicity of the crystal we must have

nt = pT, with integer p

or

t = (p/n) T             (1)

For example, for a screw axis of order 4 , the allowed translational component t will be

(0/4)T , (1/4)T , (2/4)T , (3/4)T , (4/4)T , (5/4)T,...

Because of the periodical nature of the crystals only the first four will be distinct. It follows that:

a) in (1) p can be restricted within the interval (0 , n);

b) the n-fold axis may be thought as a special screw axis with t=0 ;

c) the nature of a screw is completely defined by the symbol

n_p

where n defines the order of the rotation and p/n defines the translational part. The allowed screw axes are then :

2 , 2₁
3 , 3₁ , 3₂
4 , 4₁ , 4₂ , 4₃
6 , 6₁ , 6₂ , 6₃ , 6₄ , 6₅