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Unformatted text preview: as the highest temperature for which this equation 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 BraggWilliams 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 iJ 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 ) eNβL , (3) where L = Jy 2 21 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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 Fall '05
 STROUD
 Physics, Magnetism, Magnetic Field, Fundamental physics concepts, Ferromagnetism, Ising

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