Microscopic Ginzburg-Landau coefficients from BCS theory (${\beta }_{\Delta }=7\zeta (3)N(0)/(8{\pi }^2(k_B{T}_c{)}^2),K_{\Delta }={\beta }_{\Delta }{\hslash }^2{v}_F^2/6$)

For a clean isotropic three-dimensional weak-coupling s-wave model, use the energy-valued gap $\Delta$, a density of states $N(0)$ per spin, and free-energy density ${\alpha }_{\Delta }|\Delta {|}^2+{\beta }_{\Delta }|\Delta {|}^4/2+K_{\Delta }|D\Delta {|}^2$. Near $T_c$, ${\alpha }_{\Delta }\simeq N(0)(T-T_c)/T_c$. The quartic coefficient follows fromThe integral is $\pi /(2|{\omega }_n{|}^3)$ and the positive odd-integer sum is $7\zeta (3)/8$, yielding the displayed coefficient. The isotropic Fermi-velocity average gives $K_{\Delta }={\beta }_{\Delta }{\hslash }^2{v}_F^2/6$. The pair charge is $q=-2e$ in $D=\nabla -iq\mathbf{A}/\hslash$. These values change under a rescaling of $\Delta$, different density-of-states convention or different material model; the local expansion is a near-transition approximation.^[ref]