= Solution
In <BCS theory>, an <electron> distorts the positively charged lattice, and the retarded response can attract another <electron>. Within a phonon-energy shell around the <Fermi surface>, the effective attraction can overcome the screened Coulomb repulsion and destabilize the normal Fermi sea toward <Cooper pairs>. In conventional elemental <superconductors> the pairs predominantly join time-reversed states of opposite momentum and <spin>, forming a <spin> singlet.
In the weak-coupling approximation,
$$
\boxed{k_BT_c\simeq1.14\hbar\omega_D\exp[-1/(N(0)V_{\rm eff})]}.
$$
Here $N(0)$ is the appropriate single-spin <density of states> and $V_{\rm eff}>0$ the effective attractive interaction in a consistent normalization. Both the <phonon> scale and the exponential coupling matter. A large Fermi-level <density of states>, often associated with partially filled narrow $d$ bands, can raise $T_c$; weak coupling or a small <density of states> can suppress it even in a good conductor. With force constants and coupling fixed, $\omega_D\propto M^{-1/2}$ gives the conventional isotope trend. This is not a universal monotonic ranking of elements by atomic mass: <phonon> stiffness, <electron>–<phonon> matrix elements, Coulomb effects and magnetism vary between elements. Magnetic pair breaking helps explain why some elements do not follow a naive density-of-states expectation.
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