BCS coherence length (${\xi }_0=\hslash v_F/(\pi \Delta (0))$)

The clean isotropic weak-coupling BCS coherence length is conventionally ${\xi }_0=\hslash v_F/(\pi \Delta (0))$, with Fermi velocity $v_F$. It characterizes the spatial extent of pairing correlations.^[ref]

The scale follows from the same spectral calculation. Linearizing the normal energy near the Fermi surface gives ${\xi }_k\simeq \hslash v_F(k-k_F)$. The region $|{\xi }_k|\lesssim \Delta$ therefore has radial width $\delta k\sim \Delta /(\hslash v_F)$. Fourier analysis associates this width with a distance $\ell \sim 1/\delta k\sim \hslash v_F/\Delta$. This argument establishes the scale, not the conventional factor $1/\pi$. A small superconducting energy gap consequently permits pairing correlations over many interparticle spacings.