Set of supplementary notes.

30 CHAPTER 2. SOMMERFELD THEORY
This means that a small shift in the Fermi energy, δE F gives rise to a change in the number
density δn = g V (E F )δE F . The Fermi energy shift, in turn, is linked (via Eqn. 2.28) to V tot as
δE F = e(δV + V ext ), from which we obtain
δn = eg V (E F )(δV + V ext ) . (2.30)
Linearised Thomas-Fermi. When the added potential V ext is small, the induced number
density δn is small, and therefore the number density n cannot differ very much from the
density n 0 of the system without the potential (n = n 0 + δn). We may then express Eqn.2.24
in a linearised form with respect to the perturbing potential:
∇ 2 δV (r) = e2 g V (E F )
ɛ 0
(δV (r) + V ext (r)) (2.31)
Density response. This is solved by Fourier transforms, for instance by assuming an
oscillatory perturbing potential V ext = V ext (q)e iqr and a resulting oscillatory induced potential
δV = δV (q)e iqr :
δV (q) = − e2 g V (E F )/ɛ 0
V
q 2 + e 2 ext (q) = − V
g V (E F )/ɛ 0 q 2 + qT 2 ext (q) , (2.32)
F
where we have collected e 2 g V (E F )/ɛ 0 into the Thomas Fermi wave vector q T F = (e 2 g V (E F )/ɛ 0 ) 1 2
,
which for the free electron gas is calculated as
q2 T F
qT 2 F = 1 me 2
π 2 ɛ 0 h¯ k 2 F = 4 k F
= ( 2.95 √ Å −1 ) 2 . (2.33)
π a B rs
Here, a B = 4πh¯2ɛ 0
≃ 0.53 Å is the Bohr radius and r
me 2
S is the Wigner-Seitz radius, defined by
(4π/3)rS 3 = n−1 .
For the induced number density we obtain:
n ind (q) = ɛ 0q 2
e
V ext (q)
[1 + q 2 /q 2 T F ] , (2.34)
Dielectric permittivity. In general, this phenomenon is incorporated into electromagnetic
theory through the generalised wavevector dependent dielectric function ɛ(q). The dielectric
function relates the electric displacement D to the electric field E, in the form ɛ 0 ɛ(q)E(q) =
D(q). While the gradient of the total potential V tot = V 0 + δV + V ext = δV + V ext (V 0 = 0
for Jellium) gives the E− field, the gradient of the externally applied potential V ext gives the
displacement field D. As E and D are related via the relative permittivity, ɛ, the potentials
from which they derive are also connected by ɛ:
V ext (q) = ɛ(q) (δV (q) + V ext (q)) (2.35)
Using Eq. 2.32 we find
q 2
V tot (q) = V ext (q)
q 2 + qT 2 F
, (2.36)