The Li enrichment in the Universe still presents various puzzles to astrophysics. One open issue is that of obtaining estimates for the rate of e-captures on 7Be, for T and rho conditions different from solar. This is important to model the Galactic nucleosynthesis of Li. In this framework, we present a new theoretical method for calculating the e-capture rate in conditions typical of evolved stars. We show how our approach compares with state-of-the-art techniques for solar conditions, where various estimates are available. Our computations include: i) traditional calculations of the electronic density at the nucleus, to which the e-capture rate for 7Be is proportional, for different theoretical approaches including the Thomas--Fermi, Poisson--Boltzmann and Debye--Hueckel (DH) models of screening, ii) a new computation, based on a formalism that goes beyond the previous ones, adopting a mean-field adiabatic approximation to the scattering process. The results obtained with our approach as well as with the traditional ones and their differences are discussed in some detail, starting from solar conditions, where our method and the DH model converge to the same solution. We then analyze the applicability of the various models to a rather broad range of T and rho values, embracing those typical of red giant stars. We find that, over a wide region of the parameter space explored, the DH approximation does not stand, and the more general method we suggest is preferable. We then briefly reanalyze the 7Li abundances in RGB and AGB stars of the Galactic Disk using the new Be-decay rate. We also underline that the different values of the electron density at the nucleus we find should induce effects on electron screening (for p-captures on Li itself, as well as for other nuclei) so that our new approach might have wide astrophysical consequences.
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