  Screening effects are important to understand various aspects of ion-solid
interactions and, in particular, play a crucial role in the stopping of ions in
solids. In this paper the phase shifts and scattering amplitudes for the
quantum-mechanical elastic scattering within up to the second-order Born (B2)
approximation are revisited for an arbitrary spherically-symmetric electron-ion
interaction potential. The B2 phase shifts and scattering amplitudes are then
used to derive the Friedel sum rule (FSR) involving the second-order Born
corrections. This results in a simple equation for the B2 perturbative
screening parameter of an impurity ion immersed in a fully degenerate electron
gas which, as expected, turns out to depend on the ion atomic number $Z_{1}$
unlike the first-order Born (B1) screening parameter reported earlier by some
authors. Furthermore, our analytical results for the Yukawa, hydrogenic,
Hulth\'{e}n, and Mensing potentials are compared, for both positive and
negative ions and a wide range of one-electron radii, to the exact screening
parameters calculated self-consistently by imposing the FSR requirement. It is
shown that the B2 screening parameters agree excellently with the exact values
at large and moderate densities of the degenerate electron gas, while at lower
densities they progressively deviate from the exact numerical solutions but are
nevertheless more accurate than the prediction of the B1 approximation. In
addition, a simple Pad\'{e} approximant to the Born series has been developed
that improves the performance of the perturbative FSR for any negative ion as
well as for $Z_{1}=+1$.
