2 76 Problems €4 Solutions on Thermodynamics €4 Statistical Mechanic8
2102
Fermi gas. Consider an ideal Fermi gas whose atoms have mass m =
5 x grams, nuclear spin I = i, and nuclear magnetic moment p =
1 x erg/gauss. At T = 0 K, what is the largest density for which the
gas can be completely polarized by an external magnetic field of lo5 gauss?
(Assume no electronic magnetic moment).
Solution:
(MZT)
After the gas is completely polarized by an external magnetic field, the
A2
Fermi energy is EF = -(6~~n)~/~, where n is the particle density.
2m
With EF 5 2pH, we have
1 4mpH 3i2
n.&F-).Hence, nmax = = 2 x 10”atoms/cm~.
2103
State and give a brief justification for the leading exponent n in the
temperature dependence of the following quantities in a highly degenerate
three-dimensional electron gas:
(a) the specific heat at constant volume;
(b) the spin contribution to the magnetic moment M in a fixed mag-
netic field H.
(MITI
Solution:
Let us first consider the integral I: