Electronic wave functions round a vacancy in a metal
A calculation of the wave functions round a vacancy in a metal has been carried out for a simple model. This corresponds to the case of a finite number of particles in a free-electron metal, with the boundary condition that the wave functions vanish on the surface of the spherical metal. Calculations are reported for particle numbers of approximately 2 × 102, 2 × 103 and 106. The density fluctuations in the unperturbed metal are found to be serious for the first two cases, but by consideration of the difference between the electronic density distributions in the perturbed and the unperturbed cases, extrapolation from the case of a relatively small number of particles to obtain results valid for very large numbers is shown to be feasible in the region surrounding the vacancy. The possibility of obtaining an accurate density difference by consideration only of states of low orbital angular momentum is exploited. Comparison is made between the finite metal results, existing Thomas–Fermi calculations, and a self-consistent von Weizsäcker calculation which is also reported here. By examination of the solutions of Poisson's equation, it is shown that the total charge density obtained by summing the squares of the normalized wave functions for the assumed Thomas–Fermi potential in the 106 particle case is such as to make this potential self-consistent to a good approximation. Our calculations also reveal long-range oscillations in the total perturbed electron density for finite metals. Although the amplitude of these oscillations decreases with increasing volume, extrapolation to the case of an infinite metal does not seem justified here. However, our calculations should provide an upper bound to the amplitude for the assumed potential, and the oscillations are small. Finally, brief reference is made to applications of the present results.
