Ashcroft Mermin eng · Problem 11

Superconductivity

Physics — Statistical Mechanics · pauli-susceptibility-temperature-correction

Problem

Show that if

T

is small compared with the Fermi temperature, the temperature-dependent correction to the Pauli susceptibility (31.69) is given by

\chi(T) = \chi(0)\left(1 - \frac{\pi^2}{6}(k_B T)^2\left[\left(\frac{g'}{g}\right)^2 - \frac{g''}{g}\right]\right) \tag{31.93}

where

g

g'

, and

g''

are the density of levels and its derivatives at the Fermi energy. Show that for free electrons this reduces to

\chi(T) = \chi(0)\left(1 - \frac{\pi^2}{12}\left(\frac{k_B T}{\varepsilon_F}\right)^2\right). \tag{31.94}