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p848ps1 - as the highest temperature for which this...

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Physics 848: Problem Set I Due Thursday, October 6 at 11:59 PM Note: each problem is worth 10 points unless otherwise stated. 1. Use the method presented in class to obtain the exact partition of the one-dimensional antiferromagnetic Ising model in a magnetic field. As- sume that the number of spins N is very large, but is an even number. Also compute the magnetic susceptibility and the specific heat at con- stant volume. 2. An alternative way of obtaining the mean-field critical temperature of the Ising model is the following. Consider the ferromagnetic Ising model with z nearest neighbors. In an applied magnetic field B . the total effective field felt by a given spin is the sum of two parts: the applied field B, and the effective field due to the nearest neighbors. This second field to be zJ h S i , where h S i is the expectation value of the spin on one of the z nearest neighbors. (a). Use this approximation to obtain a self-consistent equation for the spin h S i . (b). Hence, obtain T c
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Unformatted text preview: as the highest temperature for which this equa-tion has a nonzero solution for h S i in the limit of zero applied magnetic field. Show that this solution is the same as that found in class using the Bragg-Williams approach. 3. Consider the “infinite range Ising model,” where the coupling J ij = J for all distinct pairs of spins, with no restriction to nearest neighbors. Thus, the Hamiltonian for a system of volume V is H =-B X i S i-J 2 X ij S i S j . (1) (a). Explain why this model only makes sense if J = J/N , where N is the number of spins in the system. 1 (b), Prove that exp ˆ ax 2 2 N ! = Z ∞-∞ dy q 2 π/ ( Na ) exp ±-Na 2 y 2 + axy ¶ , (2) provided that Re a is greater than 0. (c). Hence, show that the partition function is given by Q = Z ∞-∞ dy q 2 πk B T/ ( NJ ) e-NβL , (3) where L = Jy 2 2-1 β ln [2 cosh ( β ( B + Jy ))] . (4) Here β = 1 / ( k B T ), where T is the absolute temperature. 2...
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