= Superconducting isotope effect
{title2=$\alpha=-d\log T_c/d\log M$}
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>.<Electron-Phonon Interactions and Superconductivity by John Bardeen>[p.55, isotope-effect discovery and its significance for the electron-phonon mechanism]{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 $k_BT_c\simeq1.134\hbar\omega_De^{-1/\lambda}$ 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 $T_{c,2}/T_{c,1}=1/\sqrt{1.04}\simeq0.98058$. This is an original scaling example, not a pair of measured transition temperatures.
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