Critically shielded potential in a three-dimensional electron gas: The induced charge density at the origin

The induced electron density at the position of a critically shielded single point charge embedded in a three-dimensional degenerate electron gas is studied at low densities, in a comparative manner. The partial-wave expansion method for scattering states and averaging over the occupied Fermi sphere are used for an attractive potential energy of Hulthen form. The third-order differential equation of March and Murray [N.H. March, A.M. Murray, Phys. Rev. 120 (1960) 830] for the diagonal of the one-particle density matrix is also investigated, for the same continuous set of energy eigenvalues. The comparative study, performed for the s-type component of the total electron density, explicitly demonstrates the equivalence of the two methods.