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. They are named after American physicist Anthony Arrott who introduced them as a technique for studying magnetism in 1957.

Details

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

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

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

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

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

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.

Generalization

Giving the critical exponents explicitly in the equation, Arrott proposed:

$({\frac {H}{M_{z}}})^{1/\gamma }={\frac {T-T_{c}}{T_{1}}}+({\frac {M_{z}}{M_{1}}})^{1/\beta }$

Where $T_{1},M_{1},\gamma ,\beta$ are free parameters. In classical Landau theory, $\beta =1/2$ and $\gamma =1$ so this reduces to the linear $H/M$ versus $M^{2}$ plot, but allows more flexibility, since real ferromagnets do not always have critical exponents exactly consistent with a simple mean field theory ferromagnetism.

See also

• Curie–Weiss law