Nanothermodynamics

Breathing life into an old model

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A classic theory of magnetism has been modernized by a novel use of thermodynamics. The theory can now describe the behaviour of ferromagnetic materials at higher temperatures.

The mean-field theory of ferromagnetism, put forward in 1907 by Pierre Weiss1, was an important milestone in the development of modern physics. For many years, the Weiss theory was the 'Standard Model' of magnetism. Apart from describing the magnetic properties of materials such as iron, it stimulated great interest in the study of interacting systems, which had an impact far beyond magnetic research.

The theory is still widely used today, even though its predictions are not accurate at certain temperatures. Because of its popularity, attempts to improve the theory's predictions are still worthwhile. One such refinement is reported on page 337 of this issue by Ralph Chamberlin2. By including the effects of small-system thermodynamics (nanothermodynamics) to describe local magnetic fluctuations, Chamberlin has extended the mean-field approach to cover the behaviour of ferromagnets at higher temperatures.

The problem tackled by Chamberlin has a long history. Work towards quantitative models of magnetism began in the late 1800s, shortly after classical electromagnetic theory was formulated by J. C. Maxwell. The first successes were in studies of paramagnets and diamagnets — materials that are weakly magnetic under an applied magnetic field (and the magnetization is aligned with the applied field or opposing it, respectively). In 1895, Pierre Curie derived a magnetization law for paramagnets from experiments. Paul Langévin, Curie's pupil, later showed that the law could be derived theoretically by treating a paramagnet as a system of classical non-interacting magnetic dipoles. He also managed to explain diamagnetism using classical electromagnetic theory. But this theory completely failed in its attack on ferromagnets — the materials in which magnetism is strongest. In ferromagnets a sizable magnetization can remain in the material even when there is no external field.

Ferromagnetism abruptly disappears from a material at a certain critical temperature called the Curie point, TC. Above this, the system behaves as a paramagnet until it is cooled back below TC and ferromagnetism is restored. Classical physics can explain such effects only if TC values are at least 1,000 times lower than those observed. Pierre Weiss solved this puzzle by postulating the existence of anomalously strong interactions between the atomic magnetic dipoles. He introduced the interactions into his calculations as a homogenized 'molecular field' proportional to the magnetization. The new model was a spectacular success, qualitatively describing observations at all temperatures.

In particular, for temperatures above TC, Weiss's model provides an equation for the magnetic susceptibility χ, so that χ(T)C/(TΘ), where C is the Curie constant and Θ is the Weiss temperature (the model predicts that Θ=TC). This is the famous Curie–Weiss law and is closely obeyed by all ferromagnets, except at temperatures close to TC. The theory was primarily the product of Weiss's incredible intuition. In 1907 he could not know the origin of the 'molecular field' he postulated, which only became clear 25 years later, when Heisenberg discovered 'exchange interactions' between atomic spins — forces of a purely quantum nature.

The simplicity of Weiss's concept was an advantage, but also a weakness. In his model each magnetic atom in the sample 'sees' an identical, homogenized environment. It ignores local fluctuations and wave-like excitations that play a crucial role in magnetic behaviour. Consequently, the model's performance is not always satisfactory, especially for T→0 (because of wave-like excitations) and around TC (because of local fluctuations). For instance, the Curie–Weiss law tells us that for temperatures above TC, where paramagnetic behaviour emerges, a plot of inverse susceptibility, 1/χ, against T should be a straight line. This is not the case for temperatures starting 20–50% above TC, and in experiments TC is lower than Θ . Also, the model predicts that ferromagnetic ordering completely disappears above TC. In reality, significant short-range-order effects persist well above TC, through the formation of nanoscopic magnetized regions (clusters) with fluctuating size.

With the development of modern experimental tools, such as NMR spectroscopy and neutron scattering, these deficiencies became clearer. In the 1960s and 1970s, powerful new theories emerged that were better at describing the critical behaviour of ferromagnets near TC (most notably, renormalization group theory3). In those days many experts believed that the usefulness of the mean-field model had been exhausted and that, like the Bohr model of the hydrogen atom, it should be consigned to the history books. But this did not happen. None of the newer models offer the same versatility as the mean-field theory: each model focuses on effects occurring at different temperatures. Moreover, their underlying formalisms are incompatible, so they cannot be united into one comprehensive picture. The mean-field model is still the only one that describes the qualitative behaviour of ferromagnets over the entire temperature range from T→0 to T→∞. For that reason, mean-field theory offers a common platform for researchers working on different types of interacting systems — one can say it has become the lingua franca of the community.

In his paper, Chamberlin2 uses the mean-field formalism to address the entire paramagnetic regime, including temperatures near TC where significant clustering effects occur. Several ways of incorporating such effects into the mean-field formalism have been proposed in the past. But previous work considered only fixed-size clusters. Imposing fixed sizes on internal fluctuations was a drastic step, used to make the models tractable at the expense of physical reality. Chamberlin shows that the mathematical difficulties can be overcome by using a thermodynamical formalism known as the 'generalized ensemble'. This was first introduced by Guggenheim4 and later used by Hill5 in his development of nanothermodymamics. The ensemble allows unrestricted fluctuations in all extensive thermodynamic parameters, including size.

Chamberlin's model gives results for χ(T) that agree very well with data from several different ferromagnets. It also produces realistic results on the temperature dependence of the local magnetic order above TC, thereby creating a robust formalism that appears to solve a very old problem. The success of the model shows that nanothermodynamics — of clear importance for dealing with small systems in many areas of nanoscience — is also crucial for describing internal fluctuations in bulk materials. At last we have a unified picture of the paramagnetic behaviour of ferromagnetic materials.

References

  1. 1

    Weiss, P. J. Phys. 6, 661–690 ( 1907).

  2. 2

    Chamberlin, R. V. Nature 408, 337–339 ( 2000).

  3. 3

    Stanley, H. E. Rev. Mod. Phys. 71, S358–S366 (1999 ).

  4. 4

    Guggenheim, E. A. J. Chem. Phys. 7, 103 (1939).

  5. 5

    Hill, T. L. Thermodynamics of Small Systems (Parts I and II) (Dover, New York, 1994).

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Correspondence to Tom Giebultowicz.

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