# Arrott plot

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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]

## References

1. ^ Yeung, I.; Roshko, R.; Williams, G. (1986). "Arrott-plot criterion for ferromagnetism in disordered systems". Physical Review B 34 (5): 3456–3457. Bibcode:1986PhRvB..34.3456Y. doi:10.1103/PhysRevB.34.3456.
2. ^ du Trémolet de Lacheisserie, E., ed. (2005). Magnetism. Springer. ISBN 978-0-387-22967-6.
3. ^ Arrott, A. (1957). "Criterion for Ferromagnetism from Observations of Magnetic Isotherms". Physical Review 108 (6): 1394–1396. Bibcode:1957PhRv..108.1394A. doi:10.1103/PhysRev.108.1394.
4. ^ Hadimani, R. L.; Melikhov, Y.; Snyder, J. E.; Jiles, D. C. (2008). "Determination of Curie temperature by Arrott plot technique in Gd5(SixGe1−x)4 for x>0.575". Journal of Magnetism and Magnetic Materials 320 (20): e696–e698. Bibcode:2008JMMM..320E.696H. doi:10.1016/j.jmmm.2008.04.035.