= Microscopic Ginzburg-Landau coefficients from BCS theory
{title2=$\beta_\Delta=7\zeta(3)N(0)/(8\pi^2(k_BT_c)^2),\quad K_\Delta=\beta_\Delta\hbar^2v_F^2/6$}
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 from
$$
\beta_\Delta=N(0)k_BT_c\sum_{n\in\mathbb Z}\int_{-\infty}^{\infty}
\frac{d\xi}{(\omega_n^2+\xi^2)^2},\qquad\omega_n=(2n+1)\pi k_BT_c.
$$
The 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 $K_\Delta=\beta_\Delta\hbar^2v_F^2/6$. The pair charge is $q=-2e$ in $D=\nabla-iq\mathbf A/\hbar$. 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.<Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Superconductivity by Lev Gor'kov>[Microscopic near-transition expansion; pair charge]{ref}
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