Statistical Mechanics - Physics at Oregon State University

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188 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE. 8.7 Problems for chapter 8 Problem 1. A model of a binary alloy can be constructed similar to an Ising model. Assume that the possible atomic sites i are fixed and that the actual atom on site i is either type A or type B. The effects of the chemical bond extend only to nearest neighbors. The energy of an AA-bond is ɛAA, of an AB-bond ɛAB, and of a BB-bond ɛBB. The parameter ɛ = 1 2ɛAA + 1 2ɛBB − ɛAB is useful. The concentration of A atoms is cA = NA/ N and the concentration of B atoms is cB = 1 − cA. Introduce variables σi related to the number of A atoms nAi on site i by nAi = 1 2 (1 + σi). Obviously, σi = ±1 and nBi = 1 2 (1 − σi). A. Calculate the energy of the binary alloy in a state {σ1, · · · , σN}. B. Define variables J and h in such a way that this expression looks like the Ising model. C. If J > 0 one finds a critical temperature Tc. What happens below Tc in this case? D. Suppose J < 0. What is the structure of the alloy at low temperatures? Problem 2. Consider a one-dimensional Ising model. Assume that J < 0. Introduce new spin-variables τi related to the σi variables by τi = (−1) i σi. A. Calculate Tc for this system. B. What is happening below Tc? Problem 3. Consider the Heisenberg model for spin 1 2 particles. The Hamiltonian is given by H = −J Si • Sj The spin operators 0 1 0 −ı S are the standard Pauli matrices , , 1 0 ı 0 1 0 γ and . The state of each individual spin is a two-spinor . The 0 −1 µ state of the whole system is now given by the direct product of N spinors |i >, in the form |1 > |2 > · · · |N >. Assume that the density matrix ρ can be written as a direct product of 2 × 2 matrices ρ like we did for the Ising model. Use the same parametrization for ρ as for the Ising model. A. Calculate the internal energy as a function of a and m. What is the difference compared with the result for the Ising model?
8.7. PROBLEMS FOR CHAPTER 8 189 B. Find three equations relating the matrix elements log(ρ)ij and a and m by minimizing the free energy with respect to a and m. C. A fourth equation can be found using log det A = Tr log A for arbitrary matrices. Use this together with the results of B to find an equation for a and m of the form log ρ = matrix containing a and m. D. Show that a = 0 corresponds to the thermodynamic equilibrium state and that the results for the Heisenberg and Ising model in this approximation are the same. Problem 4. Probably the simplest cluster one can imagine in a square lattice is a square of four atomic sites. Treat the interactions between these four sites exactly but treat the interactions with the remainder of the lattice in a mean-field approach. A. How many inequivalent sites are there within this cluster? B. The effective field on each site is found by assuming that the spin on all neighboring sites that are not inside the cluster is equal to m, the average magnetization. What is the cluster Hamiltonian? C. What is the self-consistency condition? D. Calculate Tc. How does this compare with the mean-field value kBTc = 4J and the cluster value kBTc = 2.885J? Problem 5. A more complicated cluster in a square lattice is a square of nine sites. A. How many inequivalent sites are there in this cluster? B. We need to impose conditions of the form < σi >=< σj >. How many constraints does this give? C. What is the cluster Hamiltonian? How many effective fields are there? D. Indicate how you would try to solve this cluster problem. Problem 6. Consider a linear chain with alternating spin one-half and spin one atoms. The state of this system is given by {s1, s2, · · ·} where si = ±1 if i is odd and si = −2, 0, 2 for i even. The energy of such a state is given by E {s1, s2, · · ·} = −J sisi+1 − h The values of the average magnetization on the two types of sites are given by m 1/2 and m1. Generalize the mean field approach to construct two coupled equations for these quantities and calculate Tc for this system. Problem 7. i i si
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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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