Here we break down how the different [[critical exponent]]s relate to the [heat capacity](heat%20capacity%20of%20magnetic%20matter.md) as well as [magnetization,](magnetization%20of%20magnetic%20matter.md#magnetic%20phase%20transitions) which is the [[order parameter]].
Here $t$ refers to the [reduced temperature](reduced%20temperature.md) and $H$ is the [conjugate field](magnetic%20matter.md#magnetic%20momentum%20and%20magnetization%20of%20magnetic%20matter) - that is, an external field that couples with the magnetic matter.
| critical exponent | reduced temperature and conjugate field | function |
| ----------------- | --------------------------------------- | ------------------------------- |
| $\alpha$ | $(t>0, H=0)$ | $C_H \propto t^{-\alpha}$ |
| | $(t<0,H=0)$ | $C_H \propto (-t)^{-\alpha'}$ |
| | | |
| $\beta$ | $(t<0, H=0)$ | $M\propto (-t)^{\beta}$ |
| | | |
| $\gamma$ | $(t>0, H=0^+)$ | $\chi_T \propto(t)^{-\gamma}$ |
| | $(t<0, H=0^+)$ | $\chi_T \propto(-t)^{-\gamma'}$ |
| | | |
| $\delta$ | $(t=0)$ (critical point) | $M\propto H^{1/\delta}$ |
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A few important things to notice are:
* despite the fact that $t>0$ and $t<0$ correspond with different [[phase]]s (and consequently significantly completely different physical properties), these exponents are the same regardless, thus $\gamma = \gamma'$ and $\alpha = \alpha'$
* The notation $H=0^+$ indicates that we're considering the system as $H$ _approaches_ $0.$ That is, we introduce an infinitessimal value for $H.$
* This is needed in order to model [[symmetry breaking]].
# Example Systems
## Curie-Weiss ferromagnets
[[critical exponents of a Curie-Weiss ferromagnet]]
#StatisticalPhysics