Arrott plot for a simple mean field ferromagnetic phase transition. Here, as in the Arrott's paper,^[1] the relationship is demonstrated as ${\displaystyle M^{3}}$ versus ${\displaystyle H}$, but Arrott plots are usually rearranged to ${\displaystyle M^{2}}$ versus ${\displaystyle H/M}$. The value of ${\displaystyle \epsilon }$ denotes whether the temperature is above, below, or at the Curie temperature (see text).

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.^[2][3] They are named after American physicist Anthony Arrott who introduced them as a technique for studying magnetism in 1957.^[1]

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:

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

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][5]

Transition order

Magnetic phase transitions can be either first order or second order. The nature of the transition can be inferred from the Arrott plot based on the slope of the magnetic isotherms. If the lines are all positive slope, the phase transition is second order, whereas if there are negative slope lines, the phase transition is first order. This condition is known as the Banerjee criterion.^[8][9]

The Banerjee criterion is not always accurate for evaluating inhomogeneous ferromagnets, since the slopes can all be positive even when the transition is first-order.^[10]

See also

• Curie–Weiss law

References