# (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

k

and spin kuantum number

s

with the energy given by

E(k)=(ℏ^(2)k^(2))/(2m)

. At zero temperature, the Fermi energy

E_(F)

can be defined in a way that the levels with

E(k)<=E_(F),(E(k)>E_(F))

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

C=\gamma T

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

ZN_(A)

conduction electrons (

Z

is the valence and

N_(A)

\gamma

using

R=k_(B)N_(A)=8.314

joules/(mole

,K

calories/(mole K) and

(E_(F))/(k_(B))=10000K

and

Z=3

. Compute

\gamma

in units of

(J)/(molK)

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

H

, suppose that the energy of an electron with the momentum

k

whose spin is parallel (antiparallel) to

H

is given by

E_(+)(k)=E(k)-\mu _(B)H

E_(-)(k)=E(k)+\mu _(B)H

. What is the relevant

g

-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

H

as

g_(+)(E)=(1)/(2)g(E-\mu _(B)H)(g_(-)(E)=(1)/(2)g(E+\mu _(B)H))

and the particle density of each species is given by

n_(+-)=\int dEg_(+-)(E)f(E)

satisfying

n=n_(+)+n_(-)

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

g_(+-)(E)=(1)/(2)g(E+-\mu _(B)H)=(1)/(2)g(E)+-(1)/(2)\mu _(B)Hg^(')(E)

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

\mu

when

H=0

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

M=-\mu _(B)(n_(+)-n_(-))

.

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