Atomic vibrations in a metal consider point ions of mass M and charge e immersed in a uniform sea of conduction electrons. The ions are imagined to be in stable equilibrium when at regular lattice points. If one ion is displaced a small distance r from its equilibrium position, the restoring force is largely due to the electric charge within the sphere of radius r centered at the equilibrium position. Take the number density of ions (or of conduction electrons) as 3/4πR³, which defines R.

(a) Show that the frequency of a single ion set into oscillation is w = (e²/MR³)^1/2.

(b) Estimate the value of this frequency for sodium, roughly.

(c) From (a), (b), and some common sense, estimate the order of magnitude of the velocity of sound in the metal.