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Thermodynamics and Statistical Mechanics II - Home Exercise 12

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Thermodynamics and Statistical Mechanics II - Home Exercise 12
Thermodynamics and Statistical Mechanics II - Home Exercise 12
1. Density of orbitals in 1, 2 and 3 dimensions
(a) Show that the density of orbitals of a free electron in 1D is D1 (ε) =
L
π
2m
~2 ε
1/2
,
where L is the length of the line.
(b) Show that in 2D, for a square of area A, D2 (ε) =
Am
π~2
(c) Show that in 3D, for a box of volume V, D3 (ε) =
V
2π 2
independent of ε.
2m 3/2 1/2
ε
~2
2. Free electron gas in 2D
Consider a two-dimensional (2D) metal in a simple square lattice with a nearest neighbor distance of 2.5Å. The atoms have 2 valence electrons per atom. Assume that the
valence electrons are non-interacting fermions.
(a) Calculate kF and EF , at a temperature of T = 0. Express your results respectively
in units of Å−1 and eV . Also, determine the number of electrons per unit area n.
(b) Derive the density of states g(E) in states per atom per eV for the electron gas in
two dimensions.
(c) Consider the electrons to have momentum p = ~k = mv, where v is the velocity.
Determine the average hv 2 i at a temperature of T = 0.
(d) Consider now temperatures greater than zero. For T > 0, show that the chemical
h
i
πn~2
potential is given by- µ = kB T ln exp mk
−
1
.
BT
(e) Calculate the chemical potential using the equation in (d) for T = 1300◦ C. Express your result in eV . In this temperature, is the electron gas degenerate or
not?
3. Liquid 3 He as a Fermi gas
The atom 3 He has spin I = 1/2 and is a fermion.
(a) Calculate the Fermi sphere parameters vF , εF and TF for 3 He at absolute zero,
viewed as a gas of noninteracting fermions. The density of the liquid is 0.081gcm−3 .
1
(b) Calculate the heat capacity at low temperatures T TF and compare with the
experimental value CV = 2.89N kB T , as observed for T < 0.1K by A. C. Anderson,
W. Reese and J. C. Wheatly, Phys. Rev. 130, 495 (1963).
2
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