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N Question: Consider a magnet consisting of N dipoles with magnetic moment je in an external field H. In the Ising Model, the Hamiltonian is defined by Η = - μΗΣ si - JΣ’ss; , where si = £1 is the spin (up or down) of dipole i on a lattice, J is the magnitude of the interaction energy between spins, and the prime denotes that the sum is limited to only nearest-neighbor spins on the lattice. (a) Write down an expression for the canonical partition function Z(N, H, T) of the system as a sum over spin variables si. (b) In a mean-field approximation that ignores correlations between dipoles, show that the partition function may be written in the form Z(N, H,T) = {9(X)exp[N(uHX +qJX2/2)/kg)], where X = (1/N) Ei si is the average spin of the system, gn(X) is the degeneracy number of microstates for a given value of X), and is the coordination number (number of nearest neighbors) of the lattice. (c) Give an explicit expression for the degeneracy function g(x) as a function of X and N. (d) Derive an equation for the spontaneous magnetization M.(T; H = 0) (in zero external field) and show that M +0 for T <te. Determine the critical temperature T.