Parallel critical field of a thin London slab ($B_{\parallel C}/B_C\simeq \sqrt{3}\lambda /a$)

For a slab of thickness $2a$ with parallel applied field, the local London equation gives $B(x)=B_a\cosh (x/\lambda )/\cosh (a/\lambda )$. Averaging gives $M=-B_a[1-\tanh (a/\lambda )/(a/\lambda )]/{\mu }_0$. Integrating $-Md{B}_a$ and comparing with the bulk condensation energy yields the displayed thin-slab enhancement. It assumes field-independent condensate density and can be limited by pair breaking or nonlocal electrodynamics.