  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.
