The Curie Temperature

In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic materials, including ferromagnets (permanent magnets in which electron spins are aligned even in the absence of an external magnetic field). We write

If the temperature of the sample is raised above a critical temperature called the Curie temperature, the magnetism is destroyed.

When seeing such a sentence, my first inclination is to write a homework problem that uses a toy model to illustrate the physics behind the concept. Unfortunately, analyzing the Curie temperature is difficult, so no new homework problem appears in this post (readers are encouraged to try their hand at writing one).

Solid State Physics,
by Ashcroft and Mermin.

When faced with a difficult concept in material physics, I reach for a copy of Solid State Physics by Neil Ashcroft and N. David Mermin. Regular readers of this blog may recall that I am a big fan of Mermin (for instance, see here and here). Everything I know about solid state physics (which isn’t much) I learned from Solid State Physics. When Ashcroft and Mermin describe magnetic ordering of spins, they explain that

Quantitative theories of magnetic ordering have proved most difficult to construct near the critical temperature T_c at which ordering disappears. The difficulty is not peculiar to the problem of magnetism. The critical points of liquid-vapor transitions, superconducting transitions (Chapter 34), the superfluid transition in liquid He₄, and order-disorder transitions in alloys, to name just a few, present quite strong analogies and give rise to quite similar theoretical difficulties.

They settle for a phenomenological description of the Curie temperature.

The critical temperature T_c above which magnetic ordering vanishes is known as the Curie temperature in ferromagnets… As the critical temperature is approached from below, the spontaneous magnetization…drops continuously to zero. The observed magnetization just below T_c is well described by a power law.

M(T) ∼ (T_c – T)^β,

where β is typically between 0.33 and 0.37.

Below I plot of the spontaneous magnetization M versus the absolute temperature T for β=1/3.

The Curie temperature is interesting for two reasons. First, it is not named after Marie Curie, who plays such a big role in medical physics for isolating some of the first radioactive elements including radium and polonium. Instead, it is named after her husband Pierre Curie, who did important research on magnetism. Second, the ferromagnetic material that Russ and I discuss most in IPMB is magnetite (Fe₃O₄), which is found in magnetosomes, small magnetic particles that cause magnetotactic bacteria to align with the earth's magnetic field. The Curie temperature for magnetite is 585 °C, or 858 K, which is too hot to support life. Perhaps other substances exist for which the Curie temperature plays a role in biology and medicine, but I don’t know what they are.

I conclude with a quote from Mermin’s delightful essay “Writing Physics” in which he talks about writing Solid State Physics with Ashcroft. Enjoy!

The striking exception to my inability to write collaboratively is my eight-year collaboration with Neil Ashcroft on our 800 page book on solid state physics. We have very different prose styles. Yet the book has a clear and distinctive uniform tone, which can't be identified as belonging to either of us. I think the reason this worked was that Neil knows solid state physics much better than I do. So he would produce the first drafts. Characteristically, I would not understand them. So I would try to make sense of what he was saying, and then produce my typical kind of irritating second draft. Neil, however, would now have to correct all my mistakes in a massively rewritten third draft. I would then have to root out any new obscurities he had introduced in a fourth draft. By this kind of tennis-playing, we would go through five or six drafts, and emerge with something that was clear, correct, and sounded like a human voice. That voice, however, was neither of ours.