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BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

The superconducting isotope effect is the dependence of the superconducting transition temperature on isotopic mass $M$ at fixed chemical composition. Its discovery in 1950 was a crucial clue that phonons participate in conventional superconductivity.^[ref]

The model prediction can be derived from two assumptions. For harmonic motion with a fixed restoring-force constant $K$, a phonon frequency scales as $\omega \propto \sqrt{K/M}$. If the BCS dimensionless coupling is independent of $M$, the weak coupling result $k_B{T}_c\simeq 1.134\hslash {\omega }_D{e}^{-1/\lambda }$ then gives $T_c\propto M^{-1/2}$ and $\alpha =1/2$. Changes in the effective interaction can change this exponent; it is not a universal law for every material exhibiting superconductivity. For a stipulated mass ratio $M_2/M_1=1.04$, the ideal model predicts $T_{c,2}/T_{c,1}=1/\sqrt{1.04}\simeq 0.98058$. This is an original scaling example, not a pair of measured transition temperatures.