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The Hamiltonian of N noninteracting spin-1/2 particles in magnetic field H is given by
(1)(a) Calculate the average magnetization <M>, the average square of the magnetization <M2>, and the magnetic susceptibility $\chi$ = (d/dH)<M> at temperature T.
(b) Verify that your results obey the thermodynamic identity
(4)(c) Prove that the above identity holds even in the presence of interactions, $\mathcal{H}_0 \rightarrow \mathcal{H}_0 + \mathcal{H}_{int}$, for arbitrary $\mathcal{H}_{int}(\{\sigma_i \} )$.
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Answer
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A shorter derivation can be given if we start with part (c).
(c) The partition function is
(5)whence
(6)Now
(8)which proves the identity.
(a) We can now apply the above formulas to the problem in hand. We have
(9)Taking the requisite derivatives, we find
(10)(b) We have
(13)
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