# Magnetization as a function of temperature in ferromagnets

Suppose a ferromagnetic material with initial magnetization $M_o$.Is there some specific formula which calculates the total magnetization $M$ as a function of $M_{o}$ and the Curie temperature $T_{c}$?

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The most famous formula of this kind is Bloch's $T^{3/2}$ law: $$M(T)=M_0 \left( 1-\left(\frac{T}{T_c}\right)^{3/2}\right)$$ It is a low-order approximation for the spontaneous magnetization in isotropic ferromagnets at low temperatures. It works well for systems like gadolinium, but is not accurate for systems with strong magnetic anisotropy. It also fails near $T_c$, where criticality instead produces $$M(T) \propto \left( T - T_c\right)^\beta,$$ with $\beta$ a critical exponent whose value depends on the universality class. Of course, for $T>T_c$ the spontaneous magnetization vanishes.

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