Problems and Solutions on Thermodynamics and Statistical Mechanics

be considered as free electrons) in a metal with N = loz2 electrons/cm3 at
T=OK.
(UC, Berkeley)

Solution:

From the density of states

D(c)de = 4~(2rn)~/~&de/h~ ,

we get

h2 3N 2f3

2m (G)

Therefore, po = -

For N = loz2 electrons/cm3 = lo2' electrons/m3, it follows that

po = 2.7 x lo-'' J = 1.7 eV.

2095

D(E) is the density of states in a metal, and EF is the Fermi energy.

At the Fermi energy D(EF) # 0.

(a) Give an expression for the total number of electrons in the system

at temperature T = 0 in terms of EF and D(EF).

(b) Give an expression of the total number of electrons in the system

at T # 0 in terms of the chemical potential p and D(E).

(c) Calculate the temperature dependence of the chemical potential at

low temperatures, i.e., p >> kT.

(Chicago)