Advanced Solid State Physics

This additional magnetic field may now be used for statistical considerations to calculate the mag-
netization as a function of temperature. The magnetization is the magnetic moment of the electrons
times the differnce in spin occupation (see figure 66).

M= (N 1 −N 2 )μ (83)

We can calculateN 1 andN 2 in thermal equilibrium

N 1

N

=

e

μB

kBT

e

μB

kBT +e

−μB

kBT

(84)

N 2

N

=

e−

μB

kBT

e

μB

kBT +e

−μB

kBT

(85)

Combining these equations leads to a magnetization of

M = (N 1 −N 2 )μ (86)

M = Nμ

e

μB

kBT−e

−μB

kBT

e

μB

kBT+e

−μB

kBT

(87)

M = Nμtanh(

μB

kBT

) (88)

In this result for the magnetization we can plug in the magnetic field from the mean field calculation

M=Mstanh(

Tc

T

M

Ms

) (89)

with the saturation magnetization

Ms=

NgμB

2 V

(90)

and the Curie temperture

Tc=

nnnJ

4 kB

(91)

. A well known approximation to this implicit equation is the Curie-Weiss law which is valid for
B << T(see figure 67). Expanding the hyperbolic tangent aroundTcyields

M~ = χ

μ 0

B, T > T~ c (92)

χ =

C

T−Tc

(93)

Summing up