Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 8 / e / Solution

Average the exact slab solution across its thickness:For $u\ll 1$, $M\simeq -B_a{a}^2/(3{\mu }_0{\lambda }^2)$. The zero-field condensation free energy per volume is $-B_C^2/(2{\mu }_0)$. Neglecting normal-state magnetic susceptibility, the Gibbs free-energy difference at fixed applied field follows from $d(G_s-G_n)/d{B}_a=-M$:Equating it to zero gives the constant-$\lambda$ estimate for the parallel critical field of a thin London slabThe enhancement is equivalently $\sqrt{12}\lambda /d$ for total thickness $d=2a$. It is an estimate assuming an intact, spatially uniform condensate density up to the transition. Field-induced pair suppression, the paramagnetic limiting field, and failure of bulk local electrodynamics in an extremely thin film limit an unlimited extrapolation as $a\to 0$.