Curie–Weiss law

The Curie–Weiss law describes the magnetic susceptibility χ of a ferromagnet in the paramagnetic region above the Curie point:

$\chi = \frac{C}{T - T_{c}}$

where C is a material-specific Curie constant, T is absolute temperature, measured in kelvin, and Tc is the Curie temperature, measured in kelvin. The law predicts a singularity in the susceptibility at T = Tc. Below this temperature the ferromagnet has a spontaneous magnetization.

In many materials the Curie–Weiss law fails to describe the susceptibility in the immediate vicinity of the Curie point, since it is based on a mean-field approximation. Instead, there is a critical behavior of the form

$\chi \sim \frac{1}{(T - T_{c})^\gamma}$

with the critical exponent γ. However, at temperatures T ≫ Tc the expression of the Curie–Weiss law still holds, but with Tc replaced by a temperature Θ that is somewhat higher than the actual Curie temperature. Some authors call Θ the Weiss constant to distinguish it from the temperature of the actual Curie point.

Curie-Weiss Derivation

The Curie-Weiss Law is an adapted version of Curie's Law, which for a paramagnetic material is[1]

$\chi = \frac{M}{H} =\frac{M \mu_0}{B} =\frac{C}{T} .$

Here µ0 is the permeability of free space; M the magnetization (magnetic moment per unit volume), B=µ0H two measures of the magnetic field, and C the material-specific Curie constant:

$C = \frac{\mu_B^2}{3 k_B}N g^2 J(J+1),$

where kB is Boltzmann's constant, N the number of magnetic atoms (or molecules) per unit volume, g the Landé g-factor, μB the Bohr magneton, J the angular momentum quantum number.[2]

For the Curie-Weiss Law the total magnetic field is B+λM where λ is the Weiss molecular field constant and then

$\chi =\frac{M \mu_0}{B}$$\frac{M \mu_0}{B+\lambda M} =\frac{C}{T}$

which can be rearranged to get

$\chi = \frac{C}{T - \frac{C \lambda }{\mu_0}}$

which is the Curie-Weiss Law

$\chi = \frac{C}{T - T_{c}}$

where the Curie Temperature TC is

$T_C = \frac{C \lambda }{\mu_0}$