Derive the expressions for the density of states for the electron gas
in one, two, and three dimensions. The derivation is given in Kittel, Ch. 6
and in the lecture notes for 3 dimensions, and you can apply the same approach to
find the results in 1 and 2 dimensions. (The result is that D(E) is proportional
to (1/E)1/2 in 1 dimension and D(E) is constant in 2 dimensions.
For this exercise, you should find the expressions including all the factors.)
In this exercise you are asked to compare the heat capacities
for electrons and phonons in the metal aluminum. The Debye temperature is
given in Kittle (see also problem in homework 5), and you may consider the
electrons in Al as a free gas with parameters in Table I of Ch. 6.
Skatch the electronic and phonon heat capacities as a function of temperature
and show that there are two "crossing points", i.e., two temperatures where
the heat capacities are equal. Find approximate values for the temperatures
for the two crossing points for Al. You may use the simple forms for the
heat capacities that are valid for low and high temperature regimes.
Kittel chapter 6, problem 2. Note that pressure P = -dU/dV where
U is the energy of the system with volume V. This pressure is
sometimes called the Pauli or Fermi pressure since it is due to the
exclusion principle. In a real material this positive pressure is balanced
by a negative pressure from the Coulomb terms. (See also problem
below.)
(This question is a short form of question Kittel chapter 6, problem 8.
The definition of the density parameter rs is given there and
the values are given in Table I, where it is called rn.)
Binding in a metal requires that there be an attractive pressure balancing
the repulsive kinetic pressure considered in Kittel chapter 6, problem 2.
You may use the result that Coulomb attraction between the nuclei and the
electrons leads to an attractive energy given
approximately by -1.8/rs (You do NOT need to derive this.)
A. With this term and the repulsive kinetic energy (given in exercise 1 above),
derive the
value of rs for which the energy is minimum.
B. How does this value compare with the actual rs in Li, Na, K, Rb, Cs?
Can you think of any reason for the difference?
C. What is the binding energy relative to free nuclei and electrons?