Statistical Mechanics - Physics at Oregon State University

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182 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE. expect that if we take the limit of an infinite cluster we might be able to get the exact solution, as long as we do take limits in the appropriate order. In our case we have short range (nearest neighbor) interactions only. So we would expect rapid convergence with cluster size. But that is not necessarily true, since near a phase transition the correlation length becomes infinitely large. The cluster we consider here consists of a central atom and its q neighbors. This is the approach taken by Bethe. Because of symmetry we assume that all neighboring sites are equivalent. We label the central spin 0 and the neighbors 1,2,..,q. The energy of this cluster is Ec(σ0, · · · , σq) = −Jσ0 q σi − h i=1 q i=0 σi − h ′ q σi + f(h ′ ) (8.109) The last term represents the interactions with the environment. We assume that the environment is in some average thermodynamic state and that the effects of the environment on the outer spins are noticeable through an effective magnetic field h ′ acting on the surface of the cluster. The average field due to the outside acts only on the atoms that are in contact with the outside. Our result can be generalized to larger and non-symmetric clusters, which makes the mathematics much more complicated. For example, it would introduce effective fields that are site specific. In our case we use the same effective field h ′ on all neighbors, since the cluster is symmetric. For this cluster we now have to impose translational symmetry, and we need 〈σi〉 = 〈σ0〉. This gives an equation which determines the value of h ′ . We did already make one choice here, and we assumed that the effective field on the central atom is zero. That makes sense. If we make the cluster larger, however, we have to introduce an effective field for each shell around the center, and we get an equation equation magnetic moments for each shell, and can again determine all fields. Only the central effective field can be set equal to zero! An individual atom in the nearest neighbor shell is not at a symmetric position with respect to the surface of the cluster, and hence not equivalent to the central cell. It therefore needs an effective field. Next, we need to calculate the partition function. In order to do so, we drop the term f(h ′ ) in the energy, since it will give a factor in the partition function that does not depend on the values of σ and hence does not influence the spin averages. We do need it at the end again, however, to calculate the energy. The partition function for the cluster is Zc = {σ0,···,σq} i=1 e −βEc(σ0,···,σq) (8.110) The derivation is a bit tedious. First we introduce the energy formula and separate the sum on the central spin from the others. Zc = {σ1,···,σq} e βJσ0 σ0 q i=1 σi e βh q i=0 σi e βh ′ q i=1 σi (8.111)
8.6. BETHE APPROXIMATION. 183 Next, we perform the sum over the central spin Zc = {σ1,···,σq} Zc = e βJ {σ1,···,σq} q i=1 σi e βh e β(h+h′ ) e βh q i=1 q i=1 σi + e −βJ e β(J+h+h′ )σi + e −βh q i=1 q i=1 σi e −βh e β(h+h′ ) e β(−J+h+h′ )σi q i=1 σi (8.112) (8.113) The sums and products can now be interchanged, since x1,x2,··· f(x1)f(x2) · · · = [ x f(x)]N . This leads to and Zc = e βh [2 cosh(β(J + h + h ′ ))] q + e −βh [2 cosh(β(−J + h + h ′ ))] q The spin averages are calculated from the expressions 〈σ0〉 = 1 〈σj〉 = 1 Zc {σ0,···,σq} Zc {σ0,···,σq} σ0e −βEc(σ0,···,σq) = S0 Zc σje −βEc(σ0,···,σq) = Sj Zc (8.114) (8.115) (8.116) The additional sums are not that hard to evaluate once we have done the partition function. S0 = and S0 = {σ1,···,σq} S0 = {σ1,···,σq} e βJ {σ1,···,σq} σ0e βJσ0 σ0 q i=1 σi e βh e β(h+h ′ ) e βh q i=1 σi e βh q i=1 σi − e −βJ q e β(J+h+h′ )σi −βh − e i=1 q i=0 σi q βh′ e i=1 σi (8.117) q i=1 q i=1 σi e −βh e β(h+h ′ ) e β(−J+h+h′ )σi S0 = e βh [2 cosh(β(J + h + h ′ ))] q − e −βh [2 cosh(β(−J + h + h ′ ))] q q i=1 σi (8.118) (8.119) (8.120)
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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44 CHAPTER 3. VARIABLE NUMBER OF PA
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90 CHAPTER 5. FERMIONS AND BOSONS
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94 CHAPTER 5. FERMIONS AND BOSONS w
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120 CHAPTER 6. DENSITY MATRIX FORMA
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232 APPENDIX A. SOLUTIONS TO SELECT
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