Physics 301 23-Nov-2005 24-1A Simple Model of FerromagnetismRecall way back in lecture 4 we discussed a magnetic spin system. In our discussion,we assumed spin 1/2 magnets with an energy ±E when anti-aligned or a ligned with themagnetic field. We had a total of N spins and we let 2s be the “spin excess,” the numb erof aligned minus the number of anti-aligned magnets. We assumed that the magnets wereweakly interacting wi th a thermal reservoir and with each other. We found that2sN= tanhEτ,which gives small net alignment if E ≪ τ and essentially perfect alig nment if E ≫ τ.It’s customary to speak of the magnetizat ion which is the magnetic moment per unitvolume, and we denote magnetic moment, not the chemical potential, by µ in this section.Then the magnetization isM =NVµ tanhµBτ= nµ tanhµBτ,where n is t he concentration of elementary magnets and B is the magnetic field. Pre-viously, we assumed that B was externally supplied. But of course, if the system has anet magnetization, it generates a magnetic field. We assume that when the magnetiza-tion is M , there is an effective field acting on each magnetic dipole proportional to themagnetizationBeff= λM ,where λ is a proportionality constant. This is essentially an application of the mean fieldapproximation to get Beff. In the crystal structure of a ferromagnet (or any material forthat matter), the electric and magnetic fields must be quite complicated, changing bysubstantial amounts on the scales of atoms. We are encapsulating all our ignorance aboutwhat’s really going on in the constant λ. In any case, we now assume t here is no externalmagnetic field, and we haveM = nµ tanhµλMτ,We can rewrite this in dimensionless form w ith the following definitions:m =Mnµ, τc= nµ2λ , t =ττc,where τcis called the Curie temperature. With these definitions, our equation becomesm = tanhmt,which is actually kind of remarkable. It says that at any given temperature, a magnetiza-tion occurs spontaneously.Copyrightc 2005, Princeton University Physics Department, Edward J. GrothPhysics 301 23-Nov-2005 24-2In order to determine the spontaneous magneti-zation, we must solve this transcendental equation form. The figure shows a plot of the left hand side (thestraight line) and several plots of the right hand sidefor various values of t. At t = 1, the right hand sideis tangent to the left hand side at m = 0. For t > 1,the curves intersect at m = 0. So there is no sponta-neous magnetization when the temperature is greaterthan the Curie temperature. For t < 1, there is anintersection at a non-zero m which moves to larger mas t gets smaller, approaching m = 1 at τ = 0.The figure shows the magneti-zation versus temperature. For tem-peratures l ess than about a third ofthe Curie temperature, t he ma gne-tization is essentially complete—allthe mag netic moment s are l ined up.This i s the case for iro n at room tem-perature. K&K show a simil ar plotincluding data points for nickel. Thedata points follow the curve reason-ably well. Before we get too excitedabout this theory, we should plug insome numbers. For iron, t he Curiepoint is Tc= 1043 K, the satura-tion magnetic field is about Bs=21, 500 G, the density is ρ = 7.8 8 g cm−3and the molecular weight is about 56 g mole−1.We might also want to know the Bohr magneton, µB= 9.27×10−21G cm3. The Bohr mag-neton is almost exactly the magnetic moment of the electron. If we assume that one elec-tron per ato m participates in generating the magnetic field, we have n = 8.47 × 1022cm−3,M = nµB= 785 G. Note also that we expect B = 4πM = 9900 G. So we are in theballpark. Next, let’s calculate Tc. To do this, we have to know λ, w hich hasn’t enteredinto the calculations so far. If we assume that the smo o thed field which we just calculatedis the effective field acting on an electron spin, then λ = 4π and Tc= 0.66 K, just a littleon the small side! We’re off by a factor of 2 in the overall magnetic field and a factor of2000 in the Curie temperature. Perhaps more than one electron per atom participates ingenerating the mean field. After all, iron has 26 electrons per atom. If the electrons pairwith opposite spins, an even number per atom have to wind up with the same spin. (Ofcourse this ignores the fact that electrons are in the conduction band of the solid.) Also, λis supposed to characterize the field acting on the aligned electrons. Since it appears that asimple estimate of λ is o ff by a factor of a thousand or so, there must be some complicatedinteractions going on in order to get an effective field this strong. These interactions arepresumably due to the other electrons in the atom, in nearby atoms, and in the Fermi seaCopyrightc 2005, Princeton University P hysics Department, Edward J. GrothPhysics 301 23-Nov-2005 24-3of electrons in the metal. Without a detail ed understanding of what’s going at the atomiclevel, we can’t say much more about this model.Superconductors, the Meissner Effect, and Magnetic EnergyAs you know, when some materials are cooled, they become superconductors. All re-sistance to the flow of electricity disappears. K&K state that superconductivity disappearsfor temperatures above about 20 K. This is a little out of date. In the last decade or so,high temperature superconductors were discovered (called high Tc) and the record hightemperature is around 140 K. (Of course, I might be out of date, too!) The new high Tcsuperconductors are ceramics with anisotropic superconductivity. The old-style or normalsuperconductors are metals with i sotropic sup erconductivity.We will be talking about old-style superconductors. There are two kinds of supercon-ductors, naturally called type I and typ e II! Type I superconductors completely excludemagnetic fields from t heir interio rs when in a superconducting stat e. This is called theMeissner effect. Type II superconductors partially exclude magnetic field. Actually whathapp ens is the type II superconductor organizes itself into vortex tubes with normal con-ductor and magnetic field in the centers of the vortices and superconductor a nd no magneticfield b etween the vortices.We will consider type I superconductors. The Meissner effect is actually quite amazing.You will recall from E&M (or you wil l learn when you take physics 304) that you can’tget a magnetic field inside a perfect conductor. If you try, then by Lenz’ law the inducedcurrents create an induced ma gnetic field opposite