Screening in Dilute Alloys and Dilute Metals in the Light of Solutions of the Nonlinear Thomas-Fermi Equation

A practical procedure has been developed for solving the nonlinear Thomas-Fermi equation by quadrature, and has been applied to screening problems in dilute alloys and dilute metals. For the dilute-alloy problem, results of calculations are presented in curves and equations for factors for the effective charge of impurity ions which can be applied to any alloy system of interest. In the case of dilute metals, the effect of the gradual localization of charge as the atomic radius $r_0$ per metal atom is increased was studied. The model allows for the presence of a matrix of other atoms or molecules through a dielectric constant $K$ and effective mass ratio $M$. Also, it is necessary to take into account the exclusion of the valence electrons from the ion core region. This is done by means of an ion core radius $r_c$, within which the band electrons are excluded. Calculations were made for two effective mass ratios ${\mu }_0$ and ${\mu }_1$, which measure the effect of charge localization, respectively, in reducing the Fermi energy and in reducing the density of states at the Fermi energy. In addition to the dependence on $r_0$ and $r_c$, the ratios ${\mu }_0$ and ${\mu }_1$ depend on the "reduced" charge $Z_1$, which is the charge of the metal ion multiplied by ${\left(\frac{M}{K}\right)}^3$. The implication of the results for these models is discussed for screening as it actually occurs in dilute alloys and in dilute metals.