Cohesive energy of free electron Fermi gas, we define the dimensionless length r1, as r0/aH, where r0 is the radius of a sphere that contains one electron, and aH is the bohr radius h2/e2m.

(a) Show that the average kinetic energy per electron in a free electron Fermi gas at 0 K is 2.21/r2s, where the energy is expressed in rydbergs, with 1 Ry = me4/2h2.

(b) Show that the coulomb energy of a point positive charge e interacting with the uniform electron distribution of one electron in the volume of radius r0 is – 3e2/2r0, or – 3/rs in rydbergs.

(c) Show that the coulomb self-energy of the electron distribution in the sphere is 3e2/5r0, or 6/5rs in rydbergs.

(d) The sum of (b) and (c) gives – 1.80/rs for the total coulomb energy per electron. Show that the equilibrium value of r is 2.45. Will such a metal be stable with respect to separated El atoms?