Let us denote the operator  by O(q). The operators s_x, s_y and s_z are the Pauli matrices given on p.67. The coordinate axes are assumed to be chosen with the z-axis parallel to the spin axis. In other words the spin functions and are eigenfunctions of the operator s_z with eigenvalues +1 and -1 respectively.

(a) For a 2p rotation about the z axis, we obtain the following results:

.

This gives a character of -2 for this operator .

(b) For the T_d group we will demonstrate how to obtain the character for the operations :

{C_2x}: 2-fold rotation about the x-axis; and {C_2z}: 2-fold rotation about the z-axis.

Normally we expect these two operations to belong to the same class and therefore should have the same character. In this case because of the rather unusual nature of the rotation operators we shall derive the results for both operators just to be absolutely sure.

For {C_2z} the axis of rotation is along the z-direction so n.s =s_z. As the spin wavefunctions are eigenfunctions of s_z, we find that :

.

Thus the character of {C_2z} is (-i+i)=0.

For {C_2x} the axis of rotation is along the x-direction so n.s =s_x. We note that:

s_xa= band s_xb=a while

and 

The rotation operator corresponding to {C_2x} is :

.

From the above results we can deduce that the spin wavefunctions and are eigenfunctions of the operator cos(s_x /2) with eigenvalues cos(s_x /2)=0 since an expansion of the cosine function will contain only even functions of s _x . On the other hand the operator sin(s_x /2) contain only odd functions of s_x and therefore will "flip" the spin wave functions:

and 

Hence the character of the rotation operator corresponding to {C_2x} is again 0 in agreement with the character of the rotation operator corresponding to {C_2z}.

In principle the characters for the other operators can be obtained similarly. In practice the improper rotations deserves special attention. An improper rotation can usually be expressed as the product of a proper rotation plus the inversion operator I. For example a reflection into the xy-plane or m_d can be thought as a 2-fold rotation about the z-axis followed by I. Similarly, the S₄ operation can be decomposed into a 4-fold rotation followed by I. In textbooks on quantum mechanics [see, for example, Quantum Mechanics by E. Merzbacher, p.272] one can find the result that the effect of inversion on the spin wavefunctions is equal to the product: s_xs_ys_z=I(s_z)²=i. In other words the spin wavefunctions and are eigenfunctions of the inversion operator with eignenvalues of i. Hence, the double group character of the improper rotations such as S₄ can be obtained by first calculating the character for the corresponding proper rotation and then multiply the result by i.

In the text it was pointed out on p. 68 that for some symmetry operations C_i both C_i and C_i belong to the same class. It can be shown [see The Theory of Brillouin Zones and Electronic States in Crystals by H. Jones, p.251]that this is not true if the trace of the spin operator corresponding to C_i , ie Tr[O(C_i)] is non-zero. In case Tr[O(C_i)]=0 then C_i and C_i will belong to the same class if C_i contains a 2-fold rotation about an axis perpendicular to the axis .

Copyrighted by Peter Y. Yu, 1997. You are allowed to download this material for your personal use. Reproduction for commercial sale prohibited.