10. Appendix

Partial solution to Problem 2.10 591
(e) The next higher energy plane wave states at the X point are obtained by
adding the reciprocal vectors [0, ±2, 0] and [0, 0, ±2] to the vector [±1, 0, 0].
This gives rise to eight additional plane wave states:
|120〉, |120〉, |120〉, |120〉, |102〉, |102〉, |102〉 and |102〉
while the additional form factors are: v11, v12, v16 and v18. These four form factors
correspond to: g11 (1, 3, 1) etc; g12 (2, 2, 2) etc; g16 (0, 4, 0) etc; and
g18 (2, 4, 0) etc (using the notation on p. 58). For the face-centered cubic
lattice all form factors with reciprocal lattice vector (h, k, l) where the components
are a mixture of even and odd integers vanish. As a result we only need
to include g11, g12, g16 and g18.
Solution to Problem 2.10
Let us denote the operator exp[i£Û · n/2] by O(ı). The operators Ûx, Ûy and
Ûz are the Pauli matrices given in 2.6.2. The coordinate axes are assumed to
be chosen with the z-axis parallel to the spin axis. In other words, the spin
functions (to be denoted by · and ‚) are eigenfunctions of the operator Ûz
with eigenvalues 1 and 1, respectively.
(a) For a 2apple rotation about the z axis, we obtain the following results:
O(2apple)· exp[i2appleÛz/2]· ·.
O(2apple)‚ exp[i2appleÛz/2]‚ ‚.
This gives a character of 2 for this symmetry operation.
(b) For the Td group we will demonstrate how to obtain the character for the
operations: {C2x}: 2-fold rotation about the x-axis; and {C2z}: 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 when applied to
spin functions (which can change sign upon a 2apple rotation), we shall derive the
results for both operators just to be absolutely sure. For {C2z} the axis of rotation
is along the z-direction so n · Û Ûz. As the spin wave functions are
eigenfunctions of Ûz, we find that:
O(apple)· exp[iappleÛz/2]· i·.
O(apple)‚ exp[iappleÛz/2]‚ i‚.
Thus the character of {C2z} is (i i) 0.
For {C2x} the axis of rotation is along the x-direction so n · Û Ûx. We
note that: Ûx· ‚ and Ûx‚ · while (Ûx) 2 · Ûx‚ · and similarly
(Ûx) 2 ‚ Ûx· ‚. From these results we can conclude that all even functions
of (Ûx) will leave · and ‚ unchanged while all odd functions of (Ûx) will
interchange · and ‚.
The rotation operator corresponding to {C2x} is: exp[iappleÛx/2]
cos[appleÛx/2] i sin[appleÛx/2]. From the above results we can deduce that the spin
wave functions · and ‚ are eigenfunctions of the operator cos(appleÛx/2) with