Theoretical Predictions of Superconductivity in Alkali Metals under High   Pressure
  We calculated the superconductivity properties of alkali metals under high pressure using the results of band theory and the rigid-muffin-tin theory of Gaspari and Gyorffy. Our results suggest that at high pressures Lithium, Potassium, Rubidium and Cesium would be superconductors with transition temperatures approaching $5-20 K$. Our calculations also suggest that Sodium would not be a superconductor under high pressure even if compressed to less than half of its equilibrium volume. We found that the compression of the lattice strengthens the electron-phonon coupling through a delicately balanced increase of both the electronic and phononic components of this coupling. This increase of the electron-phonon coupling in Li is due to an enhancement of the $s$-$p$ channel of the interaction, while in the heavier elements the $p$-$d$ channel is the dominant component. 
Lei Shi, Dimitrios A. Papaconstantopoulos