(Solved): [60 pts] In a gas of free and independent electrons in three-dimensions, the one-electron levels ar ...

[60 pts] In a gas of free and independent electrons in three-dimensions, the one-electron levels are specified by the wave vector

and spin kuantum number

with the energy given by

. At zero temperature, the Fermi energy

can be defined in a way that the levels with

are occupied (unoccupied). (e) [10 pts] To compute the heat capacity

, the specific heat should be multiplied by the volume of the system. Suppose that a mole of free electron metal contains

conduction electrons (

is the valence and

is Avogadro's number). Calculate

using

joules/(mole

calories/(mole K) and

and

. Compute

in units of

and round up to two significant figures. (f) [5 pts] In the presence of external magnetic field

, suppose that the energy of an electron with the momentum

whose spin is parallel (antiparallel) to

is given by

. What is the relevant

-factor? (g) [5 pt] Since the magnetic field induces just a constant shift of single particle energies, one can define the density of states for electrons with the spin parallel (antiparallel) to

as

and the particle density of each species is given by

satisfying

. When the Zeeman energy is much smaller than the Fermi energy, one can assume that

. Using this approximation, find the chemical potential of the system. (Note the the chemical potential is

when

.) (h) [10 pts] The magnetization density is given by

.