Solid State Theory
Exercise 2
FS 11
Prof. M. Sigrist
Point groups and their representations
Exercise 2.1
Energy bands of almost free electrons on the fcc lattice
Let us consider almost free electrons on a facecentered cubic (fcc) lattice. The goal of
this exercise is to compute the lowest energy bands along the Δline using degenerate
perturbation theory and the machinery of
group theory
.
Remember that in reciprocal
space, the fcc lattice transforms into a bodycentered cubic (bcc) lattice. The point group
of the cubic Bravais lattices (simple cubic, fcc, bcc) is denoted by
O
h
(symmetry group
of a cube). Its character table is given in Tab. 1.
a) We first study the Γ point (
~
k
= 0). For
free
electrons (
V
= 0) the lowest energy level
is nondegenerate and the second one has an eight fold degeneracy. We focus on the
second level and denote the eightdimensional representation of
O
h
defined on this
subspace by Γ. Find the irreducible representations contained in Γ. Compute the
group character
χ
Γ
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 Spring '11
 Sigrist
 Group Theory, Crystallography, Energy, Crystal system, Space group, character table

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