= BCS coherence length
{c}
{title2=$\xi_0=\hbar v_F/(\pi\Delta(0))$}
The clean isotropic weak-coupling <BCS coherence length> is conventionally $\xi_0=\hbar v_F/(\pi\Delta(0))$, with <Fermi velocity> $v_F$. It characterizes the spatial extent of pairing correlations.<Theory of Superconductivity by John Bardeen and others>[pp.1195–1196, equations (5.43) and (5.50)]{ref}
The scale follows from the same spectral calculation. Linearizing the normal <energy> near the <Fermi surface> gives $\xi_k\simeq\hbar v_F(k-k_F)$. The region $|\xi_k|\lesssim\Delta$ therefore has radial width $\delta k\sim\Delta/(\hbar v_F)$. <Fourier analysis> associates this width with a distance $\ell\sim1/\delta k\sim\hbar v_F/\Delta$. This argument establishes the scale, not the conventional factor $1/\pi$. A small <superconducting energy gap> consequently permits pairing correlations over many interparticle spacings.
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