Physics — Solid State · cohesive-energy-of-free-electron-fermi-gas
Problem
Cohesive energy of free electron Fermi gas. We define the dimensionless length rs, as r0/aH, where r0 is the radius of a sphere that contains one electron, and aH is the Bohr radius ℏ2/e2m. (a) Show that the average kinetic energy per electron in a free electron Fermi gas at 0 K is 2.21/rs2, where the energy is expressed in rydbergs, with 1 Ry =me4/2ℏ2. (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 rs is 2.45. Will such a metal be stable with respect to separated H atoms?