  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.
