Demagnetization cooling

I've been meaning for some time to write a post about demagnetization cooling, a technique that is readily explained in an undergrad stat mech class, but has to be seen to be believed.  I was finally inspired to write this post by seeing this preprint.  Here's the basic idea.  Start with an ensemble of magnetic moments in what we will call a "demag stage".  The sample of interest will be thermally connected to this demag stage.  When I worked on this stuff in grad school, we used the nuclear magnetic moments of copper nuclei, but it's also possible to use electron magnetic moments in a paramagnetic salt of some kind.  Anyway, apply a large magnetic field to these magnetic moments (spins) while attached to a refrigeration source of some kind. It's energetically favorable for the moments to align with the applied field. When they flip to align, the energy that is released is carried away by the refrigerator.  Likewise, in the case of a metal like copper, the ramping up of the magnetic field can generate heat via eddy currents; that heat is also carried away by the refrigerator.  Now, once the spins are basically aligned, unhook the thermal connection between the demag stage and the refrigerator, and gently lower the applied magnetic field.  What happens?

First, the formalistic explanation.  Basic statistical physics tells us that the entropy of an ensemble of magnetic moments like those in our demag stage is only a function of the ratio B/T, where B is the applied magnetic field and T is the temperature of the moments.  If we are gentle in how we lower B, so that the entropy remains constant, that implies that lowering B by a factor of two also lowers T by a factor of two.  When I first did this as a grad student, it seemed like magic.  We thermally isolated the demag stage (plus sample), and I used an ancient HP calculator to tell a power supply to ramp down the current in a superconducting magnet.  Voila - like magic, the temperature (as inferred via the capacitance of a special pressure transducer looking at a mixture of liquid and solid ³He) dropped like a stone, linear in B.  Amazing, and no moving parts!  

So, physically, what's really going on, and what are the limitations?  Well, the right way to think about the ensemble of magnetic moments is as an entropic "sink" of energy.  Equilibrium statistical physics is based on the idea that all microscopic states of a system that have the same total energy are equally likely.  When you create an ensemble of 10²³ magnetic moments all pointed in the same direction (that is, with an aligned population much greater than what one would expect in equilibrium based on the new value of B), the most likely place for thermal energy in your system to go is into flipping those spins, to try and bring the aligned population down and back into the new equilibrium.  That means that heat will flow out of your sample and out of, e.g., the lattice vibrations of the demag stage, and into flipping those spins.  The fortuitous thing is that for reasonable numbers of moments (based on volumes of material) and accessible initial values of B and T, you can get lots of cooling.  This is the way to cool kilogram quantities of copper down to tens of microKelvin, starting from a few milliKelvin.  It's a way to cool a magnetic salt (and attached sample) down from 4.2 K to below 100 mK, with no messy dilution refrigerator, and people sell such gadgets.  

There are practical limitations to this, of course.  For example, there is no point in reducing the external B below the value of the effective internal magnetic field due to spin-spin interactions or impurities.  Also, when demag-ed, the system is a closed box with a finite (though initially large) heat capacity.  Any measurement done on an attached sample will dump some heat into the stage, even if only through stray heat leaks from the rest of the world, limiting the amount of time the stage and sample remain cold before needing another demag cycle.  Finally, and most relevant to the preprint linked above, there are real issues with establishing thermal equilibrium.  For example, it is not hard to get the nuclei of copper to have a much lower effective temperature than the conduction electrons, with an effective equilibration time longer than the demag-ed spin system can be kept cold.  In other words, while the nuclei can get very cold for a while, the electrons are never able to reach similar temperatures.  Still, the whole concept of cooling through demagnetization is very interesting, and really brought home to me that all the abstract concepts I'd learned about entropy and spins had real consequences.