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1. Consider an ideal, non-relativistic three-dimensional Bose gas with spin zero,
so that its number and pressure are given by
N =
V
`
3
Q
G3/2(e
βµ)
P =
T
`
3
Q
G5/2(e
βµ)
(a) Show that the isothermal compressibility κT and the adiabatic compressibility κS above the condensation temperature are given by
κT =
V
NT
G1/2(λ)
G3/2(λ)
, κS =
3V
5NT
G3/2(λ)
G5/2(λ)
,
where
Gν(λ) = 1
Γ(ν)
Z ∞
0
dx x
ν−1
λ−1e
x − 1
are the Bose-Einstein functions and λ = e
βµ. Note the relationship
λ
dGν(λ)
dλ
= Gν−1(λ),
which holds for ν > 1 and can be found by directly taking the derivative
and integrating by parts.
(b) In the grand canonical ensemble, study the fluctuation in the number of
particles N and discuss what happens to the number fluctuations as the
system approaches the critical temperature.
2. Carry through the analysis of a Bose gas of ultra-relativistic particles (i.e. e =
pc) with µ 6= 0 and find the lower critical dimension for Bose-Einstein condensation for this gas. Also, find the critical temperature in dimensionalities
in which condensation occurs. [The lower critical dimension is the highest
dimension for which condensation does not occur. For example, in the nonrelativistic gas the lower critical dimension is two.]
3. (Relies on material from class on Friday) Solid aluminum has a transverse
speed of sound cs,t = 3.0 × 105
cm/s, a longitudinal speed of sound cs,l =
6.4 × 105
cm/s and a density of 2.7 g/cc. Each aluminum atom contributes
three conduction electrons to the metal, while the rest of the electrons are
bound to the ions.
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(a) Calculate the transverse and longitudinal Debye temperatures ΘD,t and
ΘD,l of the ion lattice.
(b) Determine the temperature at which the contribution to the heat capacity
CV from the phonons is equal to the contribution to CV from the conduction electron (which you can assume to form a free gas inside the aluminum) . Assume the low-temperature limit for both heat capacities.
4. (Relies on material from class on Friday) Consider a solid which has a weird
dispersion relation for sound, so that the frequency is related to the wave number by ω = ak2 and only longitudinal waves can be excited.
(a) Find an expression for the phonon heat capacity.
(b) Show that in the low-temperature limit the heat capacity goes as CV ∝ T
α
and find the exponent α.
(c) Show that in the high-temperature limit CV goes to the result expected
from the equipartition theorem.
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