Relation between Lattice Vibration and London Theories of Superconductivity

A gas of noninteracting electrons of small effective mass, $m_{\mathrm{eff}}$ has a large diamagnetic susceptibility. It is shown, that the London phenomenological equations of superconductivity follow as a limiting case when $m_{\mathrm{eff}}$ is so small that the Landau-Peierls theory yields a $\mathrm{susceptibility}<-\frac{1}{4\pi }$. Justification is given for the use of an effective mass, $m_s\sim {10}^{-4\mathrm{}}m$, for superconducting electrons in the lattice-vibration theory of superconductivity. This value is sufficiently small to show that the theory gives the London equations and, as a consequence, the typical superconducting properties. The concentration of superconducting electrons, $n_s$, is smaller than the total electron concentration, $n$, by about the same ratio as the effective masses, so that $\frac{m_s}{n_s}\sim \frac{m}{n}$, and thus the penetration depth is of the same order as that given by the usual London expression.