Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 20.2-6

POSTLUDE: SYMMETRY AND THE CONCEPTUAL FOUNDATIONS OF THERMOSTATISTICS

Physics — Statistical Mechanics · mean-field-classical-heisenberg

Problem

In the classical Heisenberg model each spin can take any orientation in space (recall that the classical partition function of a single spin in an external field BB is zclassical=∫e−βBScos⁡θsin⁡θ dθ dϕz_{\text{classical}} = \int e^{-\beta BS\cos\theta}\sin\theta\,d\theta\,d\phi). Show that, in mean field theory, ⟨Sz⟩=Scoth⁡[β(B^+B)S]−S[β(B^+B)S]\langle S_z \rangle = S\coth\left[\beta(\hat{B} + B)S\right] - \frac{S}{[\beta(\hat{B}+B)S]}

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