Advanced Solid State Physics

9.3.2 Paramagnetic Response

Whenever there is a non-zero angular momentum in the z-direction (spin), there are more than one
states (remember different magnetic quantum numbers), which split into different energy levels due
to the Zeeman effect. So first one has to figure out what the energies of these different states are and
then a partition function^7 has to be constructed (lower states are occupied more likely).
Again the energy can be calculated asE=−μ·B, but now usingμ=−gJμBJ, withJthe total
angular momentum (J=L+S).

Brillouin functions
The average magnetic quantum number can be calculated as following:

〈mJ〉=

∑J

−JmJe

−E(mJ)/kBT

∑J

−Je−E(mJ)/kBT

=

∑J

−JmJe

−mJgJμBB/kBT

∑J

−Je−mJgJμBB/kBT

=−

1

Z

dZ

dx

Z is the partition function andx=gJμBB/kBT.

Z=

∑J

−J

e−mJx=

sinh

(

(2J+ 1)x 2

)

sinh

(x

2

)

The total magnetization then reads

M = −

N

V

gJμB〈mJ〉=

N

V

gJμB

1

Z

dZ

dx

=

N

V

gμBJ

(

2 J+ 1

2 J

coth

(

2 J+ 1

2 J

gμBJH

kBT

)

−

1

2 J

coth

(

1

2 J

gμBJH

kBT

))

It is possible to plot the temperature dependence of the magnetization for different values of the total
angular momentum as depicted in fig. 60. When no magnetic field is applied, x equals zero and the
magnetization is zero too. For very low temperatures or very high fields, the saturation region is
reached and (nearly) all spins get aligned.
For high temperatures there is a linear region, because here the Curie law holds. (m=χH=> m∝
H
T => χ∝

1

T) So it turns out, that forμB << kBTthe susceptibility can be written asχ≈

μ 0 M

B =

C
T
with the Curie constant C.

9.4 Free Particles in a Weak Magnetic Field

When calculating the paramagnetic response (calledPauli paramagnetism) of free electrons to a
magnetic field, it turns out to be much smaller than it would be for individual atoms. This happens
because free electrons fill up all states to the fermi surface and only those electrons, which are at the
fermi surface participate in paramagnetism. As said before, the two states for spin up and spin down
split up in a magnetic field and the state with spin parallel to the field has the lower energy (see
fig. 61). So one has to calculate the number of electrons that fill up those states in order to get the
magnetization (eqn. (71)).

(^7) Zustandssumme