Physics 301 18-Nov-1999 17-1Landau Theory of Phase TransitionsIn this section we’ll have a quick look at Landau’s theory of phase transitions. Wewill be looking at a general approach rather than any specific system.We will consider systems at constant temperature and volume, so the Helmholtz freeenergy is a minimum when in thermal equilibrium. We suppose that the phase of a systemis determined by some parameter. For example, in the ferromagnetic system we discussedlast time, the magnetization, M, was the parameter of interest. It had a value different fromzero for temperatures below the Curie temperature, went to zero at the Curie temperature,and stayed 0 for higher temperatures. We will denote the parameter by ξ; in the genericcase, it’s called the order parameter. Sometimes I call it “wiggly.”At thermal equilibrium, the order parameter will have some value, ξ0(τ), but wesuppose that we can calculate the free energy for other values of ξ.Thatis,wewriteFL(ξ,τ)=U(ξ,τ) − τσ(ξ,τ) ,where the subscript L indicates the “Landau free energy.” Of course the free energy, energy,and entropy depend on the volume and number of particles as well, but we’re assumingthey’re constant and we’re not bothering to show them.When the system is in thermal equilibrium, the order parameter will take on the valuewhich minimizes the free energy and we haveF (τ)=FL(ξ0,τ) ≤ FL(ξ,τ) ,for all ξ 6= ξ0.GivenFL(ξ,τ), we can determine its minimum. It’s the appearance of anew minimum as τ is varied that is a sign of phase transition.Suppose FLis an even function of the order parameter. Also suppose FLcan beexpanded in a power series in the order parameter. (Both assumptions are reasonable, butare certainly not guaranteed to always be true!) We haveFL(ξ,τ)=g0(τ)+12g2(τ)ξ2+14g4(τ)ξ4+16g6(τ)ξ6+ ··· ,where the indices on the g’s indicate the corresponding power of ξ, and the fractions areput in for later convenience.Now we need to look at this power series to see under what conditions there is aminimum which moves around as we change the temperature. First of all, suppose all theg functions are positive. Then all of them increase the Landau free energy for all values ofξ2. So the minimum occurs when the order parameter is zero. Not very exciting!Copyrightc 1999, Edward J. GrothPhysics 301 18-Nov-1999 17-2To get a minimum somewhere other than ξ = 0, we need at least one of the g functionsto be negative and for this function to be large enough compared to the other functionsthat it can shift the minimum away from zero. Also, to have a transition, we would likethe minimum to depend on the temperature. We suppose thatg2(τ)=α(τ − τ0) ,where α is a positive constant. The temperature τ0is the temperature at which thephase transition will occur and this expression for g2should only be taken as valid in aneighborhood of τ0. It clearly can’t work all the way down to τ = 0! We suppose thatg4(τ) > 0andthatα and g4(τ) are sufficiently large that all the interesting behavior occursfor small ξ so we don’t have to worry about g6and higher order functions. With all thesecaveats, the Landau free energy isFL(ξ,τ)=g0(τ)+12α(τ −τ0)ξ2+14g4(τ)ξ4.Tofindtheminimum,wesetthederiva-tive with respect to the order parameter tozero,0=∂FL∂ξτ= α(τ − τ0)ξ + g4(τ)ξ3,which has the rootsξ =0, and ξ = ±r(τ0− τ )αg4(τ).The root at 0 is always at least an extremumof the free energy. The other roots are imag-inary and unphysical if τ>τ0. For temper-atures lower than the transition temperature, τ0, the roots are real, physical and areactually the global minima of the free energy. The figure shows plots of the free energyversus the square of the order parameter for several values of the temperature. For τ>τ0the minimum is at ξ =0. Forτ<τ0, the minimum moves away from 0 with decreasing τ .Copyrightc 1999, Edward J. GrothPhysics 301 18-Nov-1999 17-3The figure shows the minimum value ofthe free energy as a function of the tempera-ture. Both ξ0and the minimum free energyare continuous functions of temperature. Theminimum free energy below τ0joins smoothlyto the minimum curve g0(τ)atτ = τ0Pre-sumably the entropy is also a continuous func-tion of temperature through τ0. This meansthere is no latent heat and we are dealing witha second order transition. The ferromagneticsystem we discussed last time is a phase tran-sition of this sort. To show this, we need toexpress the energy and the entropy as func-tions of the magnetization which will be our order parameter. Recall that M is the mag-netic dipole moment per unit volume, so we will express these quantities per unit volume.The energy of an aligned magnet in a magnetic field is −µB. In terms of magnetization,this becomes U(M)=−MB. With our mean field approximation where B = λM,thisbecomes U(M)=−λM2. But... It is the magnetic dipoles which are the source of thefield so we have a double counting problem. Each magnet contributes once in the B factorof −µB andonceintheµ factor. We need a factor of 1/2. Then the energy density isU(M)=−λM2/2. What about the entropy? In homework 1, problem 3, you workedout the entropy of our spin system. When the magnetization is small (that is, near thetransition temperature), the entropy isVσ(s)=Vσ0−2s2N,where N is the number of dipoles and 2s is the spin excess as usual. We’ve written thevolume explicitly, so σ refers to entropy density. We haveM = nµ2sN,soσ = σ0−M22nµ2,where n is the concentration of magnetic moments, µ. Then the free energy as a functionof M isFL(M,τ)=−λM22− τσ0+ τM22nµ2+ higher order terms .The higher order terms arise because our approximation for the entropy is only accuratewhen the magnetization is close to zero. When the magnetization is large, then the entropyis small. Ignoring the higher order terms, we can rearrange this expression asFL(M,τ)=−τσ0|{z}g0+121nµ2τ −nµ2λ| {z }g2=α(τ−τ0)M2,Copyrightc 1999, Edward J. GrothPhysics 301 18-Nov-1999 17-4and the higher order terms will produce M4and higher powers. So the Landau freeenergy for our mean field approximation for a ferromagnet has exactly the form we’vebeen considering for a second order phase transition. This method of analysis correctlyyields the existence of a Curie temperature and the functional dependence of the orderparameter (magnetization) on temperature near the Curie point, but does not tell