Statistical Mechanics - Physics at Oregon State University

116 CHAPTER 5. FERMIONS AND BOSONS
volume of 22,4 liters and the number of molecules in this volume is one mole.
Assume that Tr ≈ 175 K and Tv ≈ 6, 500 K.
A() Calculate Tq
(B) Calculate an approximate value for Cv at T = 50 K
(C) Calculate an approximate value for Cv at T = 4, 000 K
(D) Calculate an approximate value for Cv at T = 50, 000 K
Problem 2.
Using the expansion
4
f 3 (z) =
2 3 √
(log z)
π
3
2 + π2
1
(log z)− 2 +
8 7π4
640
5
(log z)− 2 · · ·
for large values of z, calculate the low temperature behavior of µ(T ), U(T ),
S(T ), and p(T ) up to fourth order in T.
Problem 3.
At high temperatures we have λ → 0 for both bosons and fermions. Use
the formula for N
V
small!
and expand f 3
2
and g 3
2
up to second order. Note that n
nQ is
(A) Find the correction term to µ(T ) due to the second term in that expansion
for bosons and fermions.
(B) Calculate the pressure including this second term for bosons and fermions.
Problem 4.
Pauli Paramagnetism.
The energy of non-relativistic electrons in a small magnetic field is given
by ɛp,s = p2
2m − sµ0B where s = ±1 and µ0 is the magnetic moment of the
electron. Assume µ0B ≪ ɛF . Note that in this problem we ignore the effect
of the magnetic field on the orbit of the electron, that turns out to be OK in
first approximation. Evaluate the magnetic susceptibility χ in the following four
cases:
(A) For T = 0.
(B) For kBT ≪ ɛF , one more term beyond (A).