= Constant-shell BCS gap equation
{title2=$\Delta=\Omega_D/\sinh(1/(\nu_sV))$}
An attractive interaction $V_{k,k'}=-V/\mathcal V$ within $|\xi_k|,|\xi_{k'}|<\Omega_D$ gives a constant gap inside that energy shell and zero outside. With approximately constant <single-spin density of states> $\nu_s$ per volume,
$$
1=\nu_sV\int_0^{\Omega_D}\frac{d\xi}{\sqrt{\xi^2+\Delta^2}}=\nu_sV\operatorname{arsinh}(\Omega_D/\Delta),\qquad \Delta=\frac{\Omega_D}{\sinh(1/(\nu_sV))}.
$$
The <weak coupling> limit is $\Delta\simeq2\Omega_De^{-1/(\nu_sV)}$. A density counting both spin species is twice $\nu_s$ and must be divided by two. The shell cutoff is a relative-energy condition, even when every <Cooper pair> has zero total momentum.
Back to article page
