Science and Technology at the Ames Laboratory
Spring 1996

According to modern science, the properties of all substances ultimately arise from the interactions among the electrons and nuclei of which they are composed. This holds not only for atoms and molecules but also for bedroom slippers and cheese. People, as well, examined closely, are nothing but the capering of electron clouds around nuclei, or a fastidious arrangement of positive and negative charges in space.

Until recently, however, this notion that the properties of the whole can be derived from the interactions among its parts was more a philosophical principle than a scientific one. Except in the case of a few simple systems, such as the hydrogen atom, no one could actually derive all of a system's properties from the capering of its electron clouds. Calculations for solids at absolute zero, or better yet, with the atomic nuclei held fixed on a periodic crystalline lattice, were first assayed in the 1950s. By the 1970s the techniques had advanced to the point that the states of simple materials at absolute zero could be reliably predicted. By the 1980s, more complicated crystal structures, such as surfaces, could be accurately treated. And in 1985 a formulation was devised that allowed the nuclei of large systems, previously frozen in space, to move simultaneously with the electrons, so that thermal vibrations and even melting could now be approached from first principles.

Few subjects in science are more difficult to understand than magnetism. -- Encyclopedia Britannica, 1987

But one quarry eluded pursuit. The new formulations, called ab initio molecular dynamics, captured the chemical and physical properties of matter but not their magnetic properties. Although the bonding energy of a copper-zinc alloy could be calculated from first principles, no one could calculate the Curie temperature of iron. (The Curie temperature, named after Pierre Curie, is the temperature above which iron ceases to be ferromagnetic.)

Last spring a team of four theoretical physicists working in Iowa, Russia and California and communicating by e-mail finally nailed magnetism. They developed a new set of coupled equations that allows the magnetic properties of materials at finite temperature to be calculated from fundamental principles. The physicists are Vladimir Antropov and Bruce Harmon of the Ames Laboratory, Mikhail Katsnelson of the Institute of Metal Physics in Ekaterinburg, Russia, and Mark van Schilfgaarde of SRI International in California.

The new formulation ends a lull in the development of the theory of magnetism. In the past 10 years, although new magnetic materials and applications continued to appear, progress in theory was slow. Theories that explained aspects of the prototypical ferromagnet, iron, were not forthcoming. Nor was much progress made with the question of what happens to atomic magnetic moments at the Curie temperature and higher temperatures. As one physicist flippantly remarked, in magnetism, at any rate, there was little pressure to rush prematurely to publication; one could write something and sit on it for 10 years, and it would still be current.

Today, however, theoretical groups are racing to convert the Ames equations into computer codes so they can tackle problems in magnetism before others do. One of the first problems to be addressed will be the magnetic structure of iron at the Curie temperature. After listening to Antropov present the Ames formulation at DOE's Oak Ridge National Laboratory (ORNL), John Cooke, head of the theory group in the Solid State Physics Division, expressed his delight that he may yet learn what happens at the Curie temperature before he retires.

The Ames method is also an ab initio method, that is, a method of calculating structure from first principles. Unlike earlier magnetics formulations, which were really models, it doesn't include built-in preconceptions or empirically determined parameters. If using a model is like using a Betty Crocker mix to make a cake, using the Ames method is like baking a cake from scratch.

First principles in this case means the quantum mechanical theory of electron interactions, but with a scruple. Although quantum mechanics is a universal method of describing the physical world, mathematical difficulties arise in attempting to apply it to systems of many atoms. Indeed it is impossible to solve the time-dependent form of a complete quantum equation for both the electrons and the nucleus of an atom more complicated than hydrogen. "That's the ab initio magnetism of the next century," says Antropov. "In this century we are limited to a quasi-first-principles magnetism achieved by squeezing slower motions out of the quantum equation to leave a solvable residue."

In quantum mechanics, magnetism has to do with the electron's spin. An electron in an atom spins around an internal axis as well as circling the nucleus, just as the earth spins on its axis as well as circling the sun. Because the spinning electron has electrical charge, it acts like a tiny magnet. Ultimately the magnetic properties of all materials, down to the tackiest refrigerator magnet, arise from the magnetism of their electrons.

At low temperatures, the magnetic moments in a ferromagnetic crystal all point in the same direction (above). At high temperatures, the atoms jiggle about and the moment orientations become more randomized (below).

The Ames team began by finding separable degrees of freedom in the motion of the electron. The electron has two degrees of freedom, they explain. One is the location of the electron in space, and the other is its spin. Electrons move fast, so the coordinate degree of freedom is a fast degree of freedom. But the orientation of a spinning electron changes only slowly, and so the spin degree of freedom is slow.

Because of this difference in time scale, the team thought the complete quantum equation for the electron could be divided into two components, one describing the motion of the electron in space and the other describing the motion of the electron's spin axis.

So they rolled up their sleeves and sat down to work. Months later -- Antropov says they nearly abandoned the problem at the penultimate moment -- they wrote out a classical equation for the slow motion of the electron's spin axis, and they wrote out a one-particle wave function for the electron's fast motion. The equation for the spin axis determines the orientation of the electron's magnetic moment, and the wave function determines the magnitude of the electron's magnetic moment. Together, these coupled classical and quantum mechanical equations of motion capture the spin dynamics of the electron in a solvable form.

Although the main difference between the Ames method and earlier formulations is that the Ames method is an ab initio dynamical method, there is another important difference as well. The Ames team found a way to include temperature in the equations, making it possible to calculate the magnetic structure of a material as it is heated or cooled as well as at the surreal temperature of absolute zero. This allows the results of calculations to be compared directly with those of experiments. And as Antropov waggishly puts it, "Many more experiments are done at finite temperature than are done at absolute zero."

Malcolm Stocks, a senior scientist at ORNL, says the Ames approach "is likely to affect a large number of different areas." Magnetism, he points out, is a huge business. Every motor and transformer has a magnet in it, and most of us own several magnetic devices in the form of CD players or personal computers. But, he says, "even the magnets in common use are made by recipes that have been developed over time, because nobody really understands at a microscopic level" why one recipe works and another doesn't. Spin dynamics promises to help put the magnetics business on a more rational footing.

In addition, Stocks says there is "a community of physicists interested in magnetism largely because it includes some of the most wonderfully unsolved problems in physics." Stocks's group is one of those in hot pursuit of the Curie temperature of iron. He explains that in the mid-1980s his group simulated the magnetic phase transition in iron with a static ab initio method. Stocks says they "got a good shot at the Curie temperature, which was a sign that they were getting there," but the simulation broke down when they were trying to calculate the magnetic susceptibility of iron above the Curie temperature. The Curie-Weiss law they extracted from the simulations had the wrong slope. "We think the reason for that is dynamics, that there are time-dependent effects in there that are outside the scope of our theory," Stocks says. But the new spin dynamics "looks like a way to get at them."

Warren Pickett, a fellow at the Naval Research Laboratory, says that the equations are "definitely needed." He plans to use them to study a new magnetic phenomenon "very modestly called colossal magnetoresistance" that might one day be exploited to make magnetic sensors, such as the read heads for magnetic storage disks. "If I could do a dynamical simulation tomorrow, that's what I'd spend tomorrow doing," he says. He predicts that many simulations of magnetic materials employing the new formulation will appear in the next year.

But perhaps the best description of the new formulation's significance is the one given by Katsnelson. "I like to compare the problem of the magnetism of iron to the famous phenomenon of superconductivity," he says. For about 40 years after the discovery of superconductivity, there was no reasonable explanation for it. But once Cooper came up with a mechanism, a very satisfactory theory quickly took shape. The reason, from a technical point of view, is that superconductivity is based on long-range interactions, and you could use a technique called the mean-field approximation to study it. In effect, this approximation reduces a many-body problem to an equivalent single-body problem. So in the case of superconductivity, it was extremely difficult to find the basic idea, but once that idea was found, further steps were not very difficult.

"The magnetism of iron is just the opposite," Katsnelson continues. "The principal idea is well known. It is the exchange interaction between atoms, proposed by Werner Heisenberg and Jacob Frenkel in the 1920s. But because the exchange interactions that underlie magnetism are short-range interactions, we have no satisfactory approximation like the mean-field approximation. And so magnetism is really a many-body problem, and all many-body problems are very difficult.

"Probably not all physicists would agree," he says laughing, "but I believe that one test of our understanding of the many-body problem is the level of our understanding of the magnetism of iron. And that is why I believe all the progress in this problem will be very exciting and very important from a conceptual point of view."

For more information: Bruce Harmon, 515-294-7712, harmon@ameslab.gov
Current research funded by: Basic Energy Sciences Office, DOE

Last Revision: 4/17/98 mab

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