Spontaneous magnetization

Temperature dependence

To a first order approximation, the temperature dependence of spontaneous magnetization at low temperatures is given by the Bloch T^3/2 law (by Felix Bloch):

${\displaystyle M(T)=M(0)\left[1-{\left(\frac{T}{T_{\mathrm{C}}}\right)}^{3/2}\right],}$

where M(0) is the spontaneous magnetization at absolute zero. The decrease in spontaneous magnetization at higher temperatures is caused by the increasing excitation of spin waves. In a particle description, the spin waves correspond to magnons, which are the massless Goldstone bosons corresponding to the broken symmetry. This is exactly true for an isotropic magnet.

Magnetic anisotropy, that is the existence of an easy direction along which the moments align spontaneously in the crystal, corresponds however to "massive" magnons. This is a way of saying that they cost a minimum amount of energy to excite, hence they are very unlikely to be excited as ${\displaystyle T\to 0}$. Hence the magnetization of an anisotropic magnet is harder to destroy at low temperature and the temperature dependence of the magnetization deviates accordingly from the Bloch T^3/2 law. All real magnets are anisotropic to some extent.

Near the Curie temperature,

${\displaystyle M(T)\propto {\left(T_{\mathrm{C}}-T\right)}^{\beta },}$

where β is a critical exponent that depends on the universality class of the magnetic interaction. Experimentally the exponent is 0.34 for iron and 0.51 for nickel.

An empirical interpolation of the two regimes is given by

${\displaystyle \frac{M(T)}{M(0)}={\left(1-(T/T_{\mathrm{C}}\right)}^{\alpha }{)}^{\beta },}$

it is easy to check two limits of this interpolation that follow laws similar to the Bloch law, for ${\displaystyle T\to 0}$, and the critical behavior, for ${\displaystyle T\to T_{\mathrm{C}}}$, respectively.