# 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 $M^2$ of a substance, against the ratio of the applied magnetic field to magnetization $H/M$ 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:

$F(M)=-H M+a (T-T_c) M^2+bM^4+\ldots$

where $M$, the magnetization, is the order parameter, $H$ is the applied magnetic field, $T_c$ is the critical temperature, and $a,b$ are arbitrary constants.

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

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

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

Thus in a graph plotting $M^2$ vs. $H/M$ 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]