  We compare the magnitudes of the single-particle relaxation time accurately
computed by the variable phase approach are with those computed in the first
Born approximation for doped semiconductors such as Si and GaAs, assuming that
the Coulomb impurities are randomly distributed centers. We find that for
typical dopant concentrations in Si the Born approximation can overestimate the
single-particle relaxation time by roughly 40\% and underestimate it by roughly
30\%. It is shown that in the case of GaAs the discrepancies are typically less
severe. Our analysis shows that in general these large discrepancies in Si
arise from strong violations of the Friedel sum rule. This breakdown occurs in
a range of doping densities for which the random phase approximation starts to
break down. Moreover, our results suggest that a multi-ion correction to the
electron-impurity scattering is needed for high dopant concentrations.
