Isotope Effect in the Bardeen-Cooper-Schrieffer and Bogoliubov Theories of Superconductivity

It is shown that the Bardeen-Cooper-Schrieffer and Bogoliubov theories of superconductivity predict an isotope effect which is the same for all superconductors, so long as the Coulomb interaction is neglected. This is demonstrated by writing the system of integral equations in a mass-invariant form, and it does not involve finding actual solutions. The theories predict that H₀, Tc, and the energy gap at $T=0$ are proportional to M^-1/2. The inclusion of the Coulomb interaction destroys the invariance of the equations and introduces deviations from the -{} in the exponent. The magnitude of the deviation depends on the particular superconductor considered.