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[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)`

is Avogadro's number). Calculate

`\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_(-))`

.