# Arrott plot

Arrott plot for a simple mean field ferromagnetic phase transition.

In condensed matter physics, an Arrott plot is a plot of the square of the magnetization ${\displaystyle M^{2}}$ of a substance, against the ratio of the applied magnetic field to magnetization ${\displaystyle H/M}$ at one (or several) fixed temperature(s). Arrott plots are an easy way of determining the presence of ferromagnetic order in a material.[1][2] They are named after American physicist Anthony Arrott who introduced them as a technique for studying magnetism in 1957.[3]

## Details

According to the Ginzburg-Landau mean field picture for magnetism, the free energy of a ferromagnetic material close to a phase transition can be written as:

${\displaystyle F(M)=-HM+a{\frac {T-T_{c}}{T_{c}}}M^{2}+bM^{4}+\ldots }$

where ${\displaystyle M}$, the magnetization, is the order parameter, ${\displaystyle H}$ is the applied magnetic field, ${\displaystyle T_{c}}$ is the critical temperature, and ${\displaystyle a,b}$ are material constants.

Close to the phase transition, this gives a relation for the magnetization order parameter:

${\displaystyle M^{2}={\frac {1}{4b}}{\frac {H}{M}}-{\frac {a}{2b}}\epsilon }$

where ${\displaystyle \epsilon ={\frac {T-T_{c}}{T_{c}}}}$ is a dimensionless measure of the temperature.

Thus in a graph plotting ${\displaystyle M^{2}}$ vs. ${\displaystyle H/M}$ for various temperatures, the line without an intercept corresponds to the dependence at the critical temperature. Thus along with providing evidence for the existence of a ferromagnetic phase, the Arrott plot can also be used to determine the critical temperature for the phase transition.[4]