What is the difference between Curie's law and Curie-Weiss law?

Question

Curie's Law: $$\chi_m = \frac{C}{T}$$ Curie-Weiss law: $$\chi_m = \frac{C}{T-T_c}$$ (C is Curie constant and $T_c$ is Curie temperature.)

Answer 1

Curie's law is:

$$ \chi_m = \frac{C}{T} = \frac{M\mu_0}{B} \tag{1} $$

where $B$ is the applied field. Weiss's modification was to say that the applied field needs to be replaced by $B + \lambda M$ to get:

$$ \chi_m = \frac{M\mu_0}{B + \lambda M} \tag{2} $$

We can rewrite this as:

$$ \chi_m = \frac{M\mu_0/B}{1 + \lambda M/B} \tag{3} $$

and then use equation (1) to substitute for $M\mu_0/B$ to get:

$$ \chi_m = \frac{C/T}{1 + \lambda M/B} = \frac{C}{T + \lambda MT/B} \tag{4} $$

Then rearrange equation (1) again to get $MT/B = C/\mu_0$ and substitute to get:

$$ \chi_m = \frac{C}{T + \lambda C/\mu_0} = \frac{C}{T + T_c} \tag{5} $$

where we define the constant $T_c$ as $T_c = \lambda C/\mu_0$.