  Recent models for evolved Low Mass Stars (with $M \lesssim 3M_\odot$),
undergoing the AGB phase assume that magnetic flux-tube buoyancy drives the
formation of $^{13}$C reservoirs in He-rich layers. We illustrate their crucial
properties, showing how the low abundance of $^{13}$C generated below the
convective envelope hampers the formation of primary $^{14}$N and the ensuing
synthesis of intermediate-mass nuclei, like $^{19}$F and $^{22}$Ne. In the
mentioned models, their production is therefore of a purely secondary nature.
Shortage of primary $^{22}$Ne has also important effects in reducing the
neutron density. Another property concerns AGB winds, which are likely to
preserve C-rich subcomponents, isolated by magnetic tension, even when the
envelope composition is O-rich. Conditions for the formation of C-rich
compounds are therefore found in stages earlier than previously envisaged.
These issues, together with the uncertainties related to several nuclear
physics quantities, are discussed in the light of the isotopic admixtures of
s-process elements in presolar SiC grains of stellar origin, which provide
important and precise constraints to the otherwise uncertain parameters. By
comparing nucleosynthesis results with measured SiC data, it is argued that
such a detailed series of constraints indicates the need for new measurements
of weak interaction rates in ionized plasmas, as well as of neutron-capture
cross sections, especially near the N = 50 and N = 82 neutron magic numbers.
Nontheless, the peculiarity of our models allows us to achieve fits to the
presolar grain data of a quality so far never obtained in previously published
attempts.
