PHYS 825: PROBLEM Set IIElectron Gas (Sommerfeld Theory)

1. Show that the density of electron states, g(ϵ), in a uniform electron Fermi gas(spin 1

2) is,

g(ϵ) ∝ ϵ−12 in 1D

g(ϵ) ∝ ϵ0 in 2D

g(ϵ) ∝ ϵ12 in 3D

2. For a uniform electron gas (spin 12) at the density of electrons in Na (n = 2.65 ×1022 cm−3)

calculate the Fermi wave vector kF (in A), the Fermi energy (in eV) and the Fermitemperature TF = ϵF

kBin K.

3. Show that the average energy per electron in the gas at T = 0 K is

ϵ =3

5ϵF

=2.21

r2sin Rydberg units

(Note: We are defining rs as n−1 = VN

= (4π3)r3 with rs ≡ r

a0. This rs is the

average inter-electron spacing in units of a0. A & M use an rs ≡ r )

4. Show that the applied pressure required hold the gas at density n = NV

is

p =1

V

[2

5NϵF

]Calculate this pressure for an electron gas at the density of electrons in Na. (inJoule /m3). Calculate the Gibbs free energy & chemical potential (at T = 0 K).

5. Show that the bulk modules, B, of an electron gas is

B = κ−1 = −V

(∂p

∂V

)T

=2

3nϵF

where again n = NV

is the electron density. Calculate the B of an electron gasfor electrons at the density of electrons in Na. How does this compare with theobserved bulk modulus of Na metal?

6. Calculate the specific heat arising from electrons (an electron gas) at Na densityat temperature 300 K. How does this compare with the specific heat of classicalparticles at 300 K? Why are they different?

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7. Derive the Drude expression for the conductivity σ of electrons in an electricalfield E.

j = σE ; σne2τ

m

where n = NV

is the electron density, m is the electron mass and τ is the meanfree time between collisions (also called the relaxation time) (see Eq. 1.6 inAshcroft & Mermin). Explain why this expression is unchanged (still applicable)when classical statistics are replaced by Fermi - Dirac quantum statistics (theSommerfeld theory) (e.g. see pages 50-51 of Ashcroft & Mermin).

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