Physics 301 21-Nov-2003 24-1A Simple Model of FerromagnetismRecall way back in l ecture 4 we discussed a magnetic spin system. In our discussion,we assumed spin 1/2 mag nets with an energy ±E when anti-align ed or aligned with themagnetic field. We had a total of N spins and we let 2s be the “sp in excess,” the numberof aligned minus the number of anti-alig ned magnets. We assumed that the magnets wereweakly interacting with a thermal reservoir and with each other. We found that2sN= tanhEτ,which gives small net alignment if E  τ and essentially perfect alignment if E  τ .It’s customary to speak of the magnetization which is the magnetic moment per unitvolume, and we denote magnetic moment, not the chemical potential, by µ in this section.Then the magn etization isM =NVµ tanhµBτ= nµ tanhµBτ,where n is the concentration o f elementary magnets and B is the m agnetic 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 a n effective field acting on each magnetic dipole proportional to themagnetizationBeff= λM ,where λ is a proportion ality constant. This is essentially an application of the mean fieldapproximation to get Beff. In the crystal structure of a ferromagnet (or any materia l forthat matter), the electric and magnetic fields must be quite complicated, changing bysubstantial amounts o n the scales of atoms. We are encapsula ting all our ignorance a boutwhat’s really going on in the constant λ. In any case, we now assume there is no externalmagnetic field, and we haveM = nµ tanhµλMτ,We can rewrite this in dimen sionless form with 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 kin d of remarkab le. It says that at any given temperature, a magnetiza-tion occurs spontaneously.Copyrightc 2003, Princeton University Physics Department, Edward J. GrothPhysics 301 21-Nov-2003 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 l ine) 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 i ntersect at m = 0. So there is no s ponta-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 less than about a third ofthe Curie temperature, the magne-tization is essentially compl ete—allthe magnetic moments are lined up.This is the case for iron at room tem-perature. K &K show a similar plotincluding data po ints for nickel. Thedata points follow the curve reason-ably well. Before we get too excitedabout this theory, we should plug insome numbers. For iron, the Cur iepoint is Tc= 1043 K, the satura-tion magn etic field is abou t Bs=21, 500 G, the density is ρ = 7.88 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 atom 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 theballp ark. Next, let’s calculate Tc. To do this, we have to know λ, which hasn’t enteredinto the calculations so far. If we assume that the smoothed field which we just calculatedis the effective field acting on an electron spin, then λ = 4π and Tc= 0.66 K, jus t a littleon the small side! We’re off by a factor of 2 in the over all magnetic field and a factor of2000 in the Curie tempera ture. Perhaps more than one electron per atom p articipates ingenerating the mean field. After all, i ron has 26 electrons per atom. If the electrons pairwith opposite spin s, an even number per atom have to wind up with the same spin. (Ofcourse thi s ign ores 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 off by a factor of a thousand or so, there must be some complicatedinteractions going on in order to get an effective fiel d this strong. These interactions arepresumably due to the other electrons in the atom, in nearby atoms, and in the Fermi seaCopyrightc 2003, Princeton University Physics Department, Edward J. GrothPhysics 301 21-Nov-2003 24-3of electrons in the metal. Without a detailed understanding of what’s going at the atomiclevel, we can’t say much mo re about this mod el .Superconductors, the Meissner Effect, and Magnetic E nergyAs yo u know, when some materials are cooled, they become superconductors. All re-sistance to the flow of electricity disappears. K&K state that s upercon ductivity di sappearsfor 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 r ecord hightemperatur e is aroun d 190 K. (Of course, I might be out of date, too! ) The new high Tcsupercondu ctor s are cerami cs with anisotro pic superconductivity. The old-style or nor malsupercondu ctor s are metals with isotropic superconductivity.We will be talking a bout old-style superconductors. There are two kinds of supercon-ductors , naturally called type I and type II! Type I superconductors completely excludemagnetic fields from their i nterior s when in a superconducting state. Th is is called theMeiss ner effect. Type II superconductors partiall y exclude magnetic field. Actually whathappens is the type II superconductor organizes itself into vo rtex tubes with normal con-ductor and magnetic field in the centers of the vortices and superconductor and no magneticfield between the vortices.We will consider type I supercon ductors. The Meissner effect is actually qui te amazing.You will recall from E&M (or you w ill learn when you take p hysics 304) that you can’tget a mag netic field inside a perfect conductor . If you try, then by Lenz’ law the inducedcurrents cr eate an