Curie-Weiss Law

Magnetic susceptibility χ describes how strongly a material magnetizes in response to an applied field: M = χ H. For paramagnets and ferromagnets, χ depends strongly on temperature.

Curie Law (Paramagnets)

For non-interacting magnetic moments (paramagnet), susceptibility follows the Curie law:

$\chi = \frac{C}{T}$

The Curie constant for spin-J ions with g-factor g and number density n is:

$C = \frac{n \mu_0 (g \mu_B)^2 J(J+1)}{3 k_B}$

For spin-1/2 particles (J = 1/2, g = 2): J(J+1) = 3/4.

Curie-Weiss Law (Ferromagnets above T_c)

When magnetic moments interact (mean-field theory), susceptibility diverges at the Curie temperature T_C:

$\chi = \frac{C}{T - T_C}$

Above T_C, the material is paramagnetic but with enhanced susceptibility. At T_C, χ → ∞ (spontaneous magnetization appears for T < T_C).

Saturation Magnetization

At absolute zero in a strong field, all moments align:

$M_{sat} = n \cdot g \cdot \mu_B \cdot J$

This is the maximum magnetization achievable.

Constants

• μ₀ = 4π×10⁻⁷ T·m/A (permeability of free space)
• μ_B = 9.274×10⁻²⁴ J/T (Bohr magneton)
• k_B = 1.381×10⁻²³ J/K

Your Task

def curie_constant(n_m3, J=0.5, g=2):
    # C = n * mu0 * (g*mu_B)^2 * J*(J+1) / (3*k_B)
    pass

def curie_susceptibility(T_K, n_m3, J=0.5, g=2):
    # chi = C / T
    pass

def curie_weiss_susceptibility(T_K, T_c_K, n_m3, J=0.5, g=2):
    # chi = C / (T - T_c)
    pass

def saturation_magnetization_A_m(n_m3, J=0.5, g=2):
    # M_sat = n * g * mu_B * J
    pass