10. Appendix

Solution to Problem 4.1 619
the above equation can be simplified to:
‚ 〈Á〉CV
v
V Î0v0
〈Á〉U
1 ∼
v
〈Á〉Cv
1
v0 Î0
〈Á〉U
1
v0
〈Á〉CvÎ0
(3.17.18)
v0
This result shows that: the volume coefficient of thermal expansion is directly
proportional to the average mode Grüneisen parameter and hence is a measure
of the anharmonicity of the lattice, provided this anharmonicity is weak.
References
G.P. Srivastava: The Physics of Phonons. (Adamm Hilger, Bristol, 1990) p. 115.
Ryogo Kubo, H. Ichimura, T. Usui, and N. Hashizume: Statistical Mechanics. (North-
Holland Publishing Co., Amsterdam, 1965) p. 163.
Solution to Problem 4.1
(a) We have to first establish the characters of the six wave functions:
X, Y, Z, X, Y and Z with respect to the symmetry operations of the point
group Td. To do this we have to apply to them the symmetry operations belonging
to the 5 classes of Td. As an example, we will apply the two-fold rotation
C2(z) about the z-axis. This operation will leave only the functions Z
and Z unchanged. Thus the character ¯ for C2 is 2. On the other hand, a S4(z)
operation will interchange Z and Z so that the character ¯(S4) 0. The reflection
operation onto a [110] plane will not change Z and Z so its character
¯(Ûd) 2. Finally, a C3 rotation will permute all 6 wave functions so its character
¯(C3) 0.
In summary, the characters for the 6 conduction band minima wave functions
in Si with respect to the symmetry operations of Td are given by:
{E} {C2} {S4} {Ûd} {C3}
6 2 0 2 0
(b) By inspection one sees that the characters in the above table can be obtained
by taking the following direct sum of the irreducible representations of
Td listed in Table 2.3:
A1 ⊕ E ⊕ T2 .
(c) To obtain the proper linear combinations of the 6 Si conduction band wave
functions that transform according to the above irreducible representations,
we note first that a sum of all 6 functions must be invariant under all the
symmetry operations of Td and therefore belongs to the A1 irreducible representation.
The irreducible representation T2 is three dimensional and the three
basis functions should transform into each other like the coordinate axes x, y
and z. A possible choice for these three functions (with proper normalization)
is clearly: [X X]/ √ 2; [Y Y]/ √ 2 and [Z Z]/ √ 2. Notice that under C2(x)
rotation both y and z will change sign. This is also the case for [Y Y] √ 2 and
[Z Z] √ 2. Finally, according to Table 2.3 the E irreducible representation is