Ashcroft Mermin eng · Problem 12

Superconductivity

Physics — Statistical Mechanics · pauli-susceptibility-electron-electron-interactions

Problem

Because of electron-electron interactions, the shift in an electron's energy due to the interaction of its spin magnetic moment with a field HH will have an additional term expressing the change in the distribution of electrons with which a given electron interacts. In the Hartree-Fock approximation (see, for example, Eq. (17.19)) this will be of the form E±(k)=E0(k)±μBH−∫dk′(2π)3v(∣k−k′∣)f(E±(k)).(31.95)\mathcal{E}_{\pm}(\mathbf{k}) = \mathcal{E}_0(\mathbf{k}) \pm \mu_B H - \int\frac{d\mathbf{k}'}{(2\pi)^3}v(|\mathbf{k}-\mathbf{k}'|)f(\mathcal{E}_{\pm}(\mathbf{k})). \tag{31.95} Show that when kBT≪EFk_B T \ll \mathcal{E}_F this leads to an integral equation for E+−E−\mathcal{E}_+ - \mathcal{E}_- that has the solution: [E+(k)−E−(k)]k=kF=2μBH1−v0g(EF),(31.96)[\mathcal{E}_+(\mathbf{k}) - \mathcal{E}_-(\mathbf{k})]_{k=k_F} = \frac{2\mu_B H}{1 - v_0 g(\mathcal{E}_F)}, \tag{31.96} where v0v_0 is an average of vv over all solid angles, v0=12∫−11dx v(2kF2(1−x)).(31.97)v_0 = \frac{1}{2}\int_{-1}^{1}dx\, v(\sqrt{2k_F^2(1-x)}). \tag{31.97} By what factor is the Pauli susceptibility now modified?

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