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One-dimensional ferromagnets

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Ferromagnetism and spin wave theory

Freeman Dyson
Scientist

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The problem is that the ferromagnet is a collection of atoms and each of them has a strong magnetic moment, and the magnetic moments interact with each other, so that neighbours tend to point in the same direction, they like to pull each other to point parallel to each other; and against this ordering force which tends to make them all point in the same direction, of course, you have just the thermal fluctuations which tend to disorder the system of magnets. So it's a problem of the balance between order and disorder, and there's a certain transition temperature called the Curie point at which the ferromagnet stops being a ferromagnet and the spins all point at random, and below the Curie point you have long range order and the spins are all lined up. So I'm talking about the low temperature region where it is ferromagnetic, where the spins to a first approximation are all parallel. But then you have waves of spin moving through the crystal, the spins just moving around their average position in a random fashion, but in fact the spins tend to propagate in waves in a coherent fashion at low temperatures, and so you can actually describe the ferromagnet as a system of waves, which is rather similar to the way you describe the electromagnetic field by a system of electromagnetic waves. You can use the same mathematical schemes. So it's a field theory of spins in a solid, so you sort of ignore the discrete structure of the solid and represent the spins as a continuous fluid. And the question is, how good as an approximation is that? How can you deal with the finiteness of the atoms? So to the first approximation, of course, you just treat the thing as a simply linear fluid, that was the original Heisenberg model of a ferromagnet which Heisenberg invented. Then you want go beyond that and consider the effects of finiteness of the atoms. So what I did was to develop a systematic analysis that could enable you to go beyond the linear approximation, so you had actually interactions between spin waves which you could calculate. I did that for the first time, actually to calculate the interaction between one spin wave and another, which turns out to be very, very weak at low temperatures, but you can still calculate it. And the interesting question was the power-law; with what power of the temperature does the interaction decrease? It turns out it goes down with the fourth power of the temperature, whereas most of the previous theories had got much lower power-dependences which turned out to be wrong. In fact there were three previous attempts at a theory which gave I think, three halves, and two and two, and five halves... and I mean they were all wrong and they all gave different power-laws, The correct power-law is T 4. So anyhow, it was great fun to do this, and I mean, the pleasure for me is just always to have a problem which is clearly defined, where elegant mathematics actually is useful. This was a great example. So I worked through the spin wave theory in two months and actually then took it back to Princeton to write up the paper. But that was in 1955, the second time I was in Berkeley. Then the third time I was there was in '57 which was this summer when Yang and Lee discovered parity violation, and then all my interests had gone back to particle physics temporarily because parity violation was so important, so I told Kittel, "I'm sorry, but I can't think about your problems - parity violation is too exciting." So I didn't go to Berkeley any more after that. It was, to me, one of the happiest periods of my life and it did actually set a trend. I mean the way condensed matter physics has developed since that time has been very largely simply to take this spin wave method and generalise it to all kinds of oscillations in condensed matter, so the same kind of techniques work, it turns out, for all kinds excitations at low temperatures. So it's a general theory of low temperature behaviour of condensed matter systems.

Born in England in 1923, Freeman Dyson moved to Cornell University after graduating from Cambridge University with a BA in Mathematics. He subsequently became a professor and worked on nuclear reactors, solid state physics, ferromagnetism, astrophysics and biology. He has published several books and, among other honours, has been awarded the Heineman Prize and the Royal Society's Hughes Medal.

**Title: **Ferromagnetism and spin wave theory

**Listeners:**
Sam Schweber

Silvan Sam Schweber is the Koret Professor of the History of Ideas and Professor of Physics at Brandeis University, and a Faculty Associate in the Department of the History of Science at Harvard University. He is the author of a history of the development of quantum electro mechanics, "QED and the men who made it", and has recently completed a biography of Hans Bethe and the history of nuclear weapons development, "In the Shadow of the Bomb: Oppenheimer, Bethe, and the Moral Responsibility of the Scientist" (Princeton University Press, 2000).

**Tags:**
Princeton University, 1955, 1957, Berkeley, Werner Heisenberg, CN Yang, TD Lee, Charles Kittel

**Duration:**
5 minutes, 7 seconds

**Date story recorded:**
June 1998

**Date story went live:**
24 January 2008