Weiss molecular field ($H_E=\lambda M$)

The Weiss molecular field replaces exchange coupling to neighbouring moments by an effective local field proportional to the average magnetization. For two-state moments and $B_E={\mu }_0\lambda M$, self-consistency gives $m=\tanh (m/t)$ at zero applied field, where $m=|M|/(n\mu )$ and $t=T/T_C$, with $k_B{T}_C={\mu }_0\lambda n{\mu }^2$. This field represents a quantum exchange energy, not generally the actual magnetostatic field generated by the moments.

For $m=\tanh (m/t)$ and positive temperature, the positive solution exists uniquely for $0<t<1$: the slope at zero is $1/t>1$, while the hyperbolic tangent is concave and below one. For $t\ge 1$ only $m=0$ remains. Expanding to cubic order near $t=1$ gives $m^2\simeq 3t^2(1-t)$ and $m\simeq t\sqrt{3(1-t)}$, with mean-field critical exponent $1/2$. The prefactor $t$ follows from the truncated equation; it is not controlled beyond leading critical order.