Set of supplementary notes.

28 CHAPTER 2. SOMMERFELD THEORY
Response to an external potential
The aim of this calculation is to estimate the response of a free electron gas to a perturbing
potential. The perturbing potential could be caused by charges outside the metal, but it could
also be due to extra charges placed inside the metal.
We begin with the free electron gas in a metal, without an externally applied perturbing
potential. The electrostatic potential in the metal, V 0 (r) is connected to the charge distribution
ρ 0 (r) via
∇ 2 V 0 (r) = − ρ 0(r)
ɛ 0
(2.23)
In the simplest model of a metal, we consider the positive background charge to be smeared
out homogeneously throughout the metal. The electron gas moves on top of this positive
background. This is the plasma or ’Jellium’ model for a metal. ρ 0 = 0 everywhere in this case. 1
In the presence of a perturbing potential V ext (r), the electron charge density ρ(r) will redistribute,
ρ(r) = ρ 0 (r) + δρ(r), causing a correction to the potential V (r) = V 0 (r) + δV (r):
∇ 2 δV (r) = − δρ(r)
ɛ 0
(2.24)
In order to make progress, we need to link the charge density redistribution δρ to the applied
potential V ext . For long-wavelength perturbations, it is plausible that in a region surrounding
the position r the perturbing potential effectively just shifts the free electron energy levels,
which is equivalent to assuming a spatially varying Fermi energy. This is the essence of the
Thomas Fermi approximation.
Thomas-Fermi approximation
The Thomas-Fermi theory of screening starts with the Hartree approximation to the Schrödinger
equation. The Hartree approximation is to replace the many-body pairwise interaction between
the electrons by a set of interactions between a single electron and the charge density made up
from all the other electrons, i.e. by a one-body potential for the i th electron
U coul (r) = −
e ∫
4πɛ 0
dr ′ ρ(r ′ )
|r − r ′ | = e2 ∑
∫
4πɛ 0
j≠i
dr ′ |ψ j(r ′ )| 2
|r − r ′ |
, (2.25)
where the summation is over all the occupied states ψ j . This clearly takes into account
the averaged effect of the Coulomb repulsion due to all the other electrons. This introduces
enormous simplicity, because instead of needing to solve an N-body problem, we have a (selfconsistent)
one-body problem. It contains a lot of important physics, and turns out to be an
approximation that is good provided the electron density is high enough. We will discuss better
theories later, in the special topic of the electron gas.
We shall treat the case of “jellium”, where the ionic potential is spread out uniformly to
neutralise the electron liquid. Note: the average charge density is therefore always zero! The
1 The correction to the charge density, δρ, does not include those charges (ρ ext ) which may have been placed
inside the metal to set up the perturbing potential. They obey ∇ 2 V ext = −ρ ext /ɛ 0 .