This is the first of a 2-part review for the final exam.

HREF="http://128.210.157.22:1013/Boilercast/2006/Spring/PHYS545/0101/PHYS545_2006_04_27_0900.mp3">Lecture Audio

A metal in a magnetic field has its Fermi sea sectioned into onion-like layers, shaped like cylinders. These are Landau levels, due to the harmonic oscillator motion of electrons moving in circular orbits in a magnetic field. (They're only "circular" for free electrons, and can have funny shapes for electrons in a real metal.) We show a very easy way to spot the quantum harmonic oscillator in this type of problem. The quantum nature of the harmonic oscillator leads to the quantized "Landau levels" of the electrons.

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There are many more phases of matter than solid, liquid, and gas. Superconductivity is a different phase of matter, and superconductors in the vortex state are yet again another phase of matter. We study vortices today, what they are, and how they happen.

HREF="http://128.210.157.22:1013/Boilercast/2006/Spring/PHYS545/0101/PHYS545_2006_04_20_0900.mp3">Lecture Audio

When superconductors go superconducting, the energy gain is called the condensation energy.

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The quantum stability of a superconductor ensures that electrons can carry current perfectly, without losing energy. There are 2 ingredients to this physics: 1. Electrons pair into "composite bosons"; 2. The bosonic pairs all fall into the same lowest energy wavefunction (called Bose condensation.) Since bosons don't obey the Pauli exclusion principle, they can all occupy the same wavefunction macroscopically -- that's right, you might get 10^23 bosons in the same wavefunction. Once they're there, they're very hard to disturb (that's quantum stability), and in this phase of matter, electrons can carry current without energy loss.

We show a video of a magnet levitating over a superconductor (called the Meissner effect), available at
www.fys.uio.no/super/levitation


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We finish off the low temperature corrections to the magnetization in a ferromagnet due to spin wave excitations, and also calculate the energy and heat capacity of spin waves. Now, on to antiferromagnets, where neighboring spins are antialigned. We derive the susceptibility, and the spin wave dispersion.

Due to technical difficulties, I post last year's audio:
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We started off today with a demonstration of Barkhausen Noise in ferromagnets.
(Your refrigerator magnets are ferromagnets.) If you've ever used a permanent magnet to magnetize a paperclip, you know that not all magnetic materials have a discernible north and south pole. Rather, as with paperclips, many ferromagnets have instead a "domain structure" -- there are many regions in the paperclip which are magnetized, but the many domains point in different directions, and the paperclip doesn't act like a permanent magnet. But you can magnetize it, by rubbing it with a permanent magnet. As you do so, you align domains. We used the Barkhausen experiment to hear the domains flip! We wound a pickup coil (lots of wire loops) around the object to be magnetized, and hooked the wire up to a speaker. You can find out more about this setup at www.simscience.org.

We also passed around magnets of various strengths. The weakest magnets were transition metal based (like iron), because the individual magnetic moments are weak. The strongest moment was neodymium-based. Neodymium (Nd) has a large magnetic moment, because it has unfilled f-shells. These and other "rare earth" magnets are surprisingly strong, and pinch your finders if you're not careful! You can buy your own rare earth magnets to play with at Edmund Scientifics.

Then we discussed the mean field theory of ferromagnetism. Mean fields aren't cruel.
What we mean is "average", in the sense that each spin in the system feels an average, effective field due to its neighboring spins. This modifies our equations for magnetization, and we're able to show using this "mean field theory" that when ferromagnets form as temperature is lowered from the disordered paramagnetic phase, the magnetization rises continuously.


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Ferromagnets spontaneously break a continuous symmetry -- that is, when the net magnetization develops, it must choose a particular direction to point. But raise the temperature to disorder this, then lower it again, and -- surprise! -- the magnetization will now form in a different direction. You already know that when a continuous symmetry (here, the rotational symmetry) is broken, the system has Goldstone modes. (See Lectures 1 and 3.) The Goldstone modes of a ferromagnet are called spin waves. These are waves of precession of the magnetic moments.

How many electrons get polarized when you apply a magnetic field to a metal? Is it all the electrons inside the Fermi surface? It turns out that only a small fraction of the electrons are able to respond -- most are stuck deep inside the Fermi surface, and the Pauli exclusion principle does not allow the spins to flip in response to the magnetic field. This is Pauli paramagnetism, and we derive the corresponding magnetic susceptibility (how easy it is to magnetize something).

We also begin to study ferromagnets -- these are your refrigerator magnets.

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Magnetic moments in a solid come from the electronic spin, and also its orbital angular momentum. We review how the orbital angular momentum contributes to the magnetic moment. We also use Atom in a Box by Dauger Research www.daugerresearch.com to show how this net angular momentum can arise from adding, say, p orbitals together in the right way.

We also show how diamagnetism arises from atomic cores. Every material is weakly diamagnetic (meaning it resists having a magnetic field penetrate) due to screening currents which come from the atomic cores.

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Paramagnets have magnetic moments whose directions fluctuate wildly with temperature. But, if you apply an external magnetic field, you can align the moments, and the paramagnet develops a net magnetization. Turn the external field off, and the paramagnet loses its magnetization. We calculate the Curie susceptibility -- how easy it is to magnetize a paramagnet by applying a net magnetic field.

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