Question

It is common to use rotation-inversion axes (rather than rotation-reflection axes) to classify the symmetry of crystals. Any Sn axis is equivalent to a rotation-inversion axis (symbolized by p) whose order p may differ from n. A rotation-inversion operation consists of rotation by 2Ï€/p radians followed by inversion. Show that

Thus we have the following correspondence:

Give the next three pairs of entries in this table.

Answer 1

Figure 12.6 shows that 2

 

Since two rotations about the z axis clearly commute with each other, we have

 

For n = 1, the relation

 

(Eq. 1) becomes

 

So an S1 axis is a 2 axis. For n = 2, Eq. 1 becomes

 

So an S2 axis is a 1 axis. For n = 3, Eq. 1 becomes

 

So an S3 axis is a 6 axis. For n = 4, Eq. 1 becomes

 

So an S4 axis is a 4 axis. For n = 5

 

And

 

So an S5 axis is a 10 axis.

 

For

 

And an S6 axis is a 3 axis. For n = 7,

 

 

And an S7 axis is a 14 axis.