Ashcroft Mermin eng · Problem 9

Superconductivity

Physics — Statistical Mechanics · curie-law-high-temperature-expansion

Problem

If one writes the free energy (31.5) in the form

e^{-\beta F} = \sum_n e^{-\beta E_n} = \sum_n \langle n|e^{-\beta \mathcal{H}}|n\rangle = \text{Tr}\, e^{-\beta \mathcal{H}}, \tag{31.89}

then it is easy to deduce Curie's law directly at high temperature without going through the algebra of Brillouin functions, for when

\mathcal{H} \ll k_B T

, we can expand

e^{-\beta\mathcal{H}} = 1 - \beta\mathcal{H} + (\beta\mathcal{H})^2/2 - \cdots

. Evaluate the free energy to second order in the field, using the fact that

\text{Tr}(\mathbf{J}_\mu \mathbf{J}_\nu) = \tfrac{1}{3}\delta_{\mu\nu}\,\text{Tr}\, \mathbf{J}^2,

and extract the high-temperature susceptibility (31.47).