10. Appendix

Partial solution to Problem 2.8 589
The characters of these 6 wave functions under the operation S4 are, therefore,
¯(æ1) 0 ¯(æ2); ¯(æ3) 1; ¯(æ4) 1; ¯(æ5) 1; and ¯(æ6) 1.
We note that the characters of æ1 and æ2 are consistent with those of the
irreducible representation X5 we obtained for these two wave functions based
on the operation C2 4 (y′ ). The additional characters now allow us to determine
uniquely the irreducible representation of æ3 and æ6 as X3 and those of æ4
and æ5 as X1. The reader should try to determine the characters for the remaining
symmetries of the group of X so as to complete the character table
for these 6 wave functions.
(b) To calculate the matrix elements of the above six wave functions we shall
first derive the following results. First we notice that with the conventions in
Section 2.5 the pseudopotential form factor Vg is defined for GaAs by (2.33)
as:
Vg V s g cos(g · s) iVa g sin(g · s)
where s (a/8)(1, 1, 1) and g is a reciprocal lattice vector. If g
(2apple/a)(g1, g2, g3) then g · s (apple/4)(g1 g2 g3). Limiting ourselves to: g
(2apple/a)(2, 0, 0); (2apple/a)(2, 2, 0); (2apple/a)(1, 1, 1) we obtain
V (220) V a (220) cos apple Vs (220) sin apple Vs (220) vs g V (220) vs g (2.8.9a)
V (220) V (220) V (220) ... V (022) v s g
(2.8.9b)
V (200) iV a (200) iva4 and its cyclic permutations (2.8.9c)
V (200) iv a 4 V (200)
V (111) 1 √ 2
(2.8.9d)
s
1
v iVa √2 (v (111) (111)
s 3 iva3 ) (2.8.9e)
V (111) 1 √ (v
2 s 3 iva3 ) (2.8.9f)
V (111) 1 √ (V
2 s 3 iva3 ) (2.8.9g)
v (111) 1 √ (v
2 s 3 iva3 ) (2.8.9h)
We shall now demonstrate the calculation of a few selected matrix elements
of the pseudopotential V : 〈æi|V|æj〉 where i and j 1, ..., 6. In principle there
are 36 matrix elements. The number of inependent elements is roughly halved
by noting that 〈æi|V|æj〉 〈æj|V|æi〉 ∗ .
〈æ1|V|æ1〉 1
〈011 011|V|011 011〉
1
2
2
〈011|V|011〉 〈011|V|011〉 〈011|V|011〉 〈011|V|011〉
(2.8.10)