  Slow neutron captures at A$\gtrsim$ 85 are mainly guaranteed by the reaction
13C($\alpha$,n)16O in AGB stars, requiring proton injections from the envelope.
These were so far assumed to involve a small mass ($\lesssim 10^{-3}$
M$_{\odot}$), but models with rotation suggest that in such tiny layers
excessive 14N hampers s-processing. Furthermore, s-element abundances in
Galaxies require 13C-rich layers substantially extended in mass ($\gtrsim 4
\times 10^{-3}$ M$_{\odot}$). We therefore present new calculations aiming at
clarifying those issues and at understanding if the solar composition helps to
constrain the 13C "pocket" extension. We show: i) that mixing "from bottom to
top" (like in magnetic buoyancy or other forced mechanisms) can form a 13C
reservoir substantially larger than assumed so far, covering most of the
He-rich layers; ii) that stellar models at a fixed metallicity, based on this
idea reproduce the main s-component as accurately as before; iii) that they
make nuclear contributions from unknown nucleosynthesis processes (LEPP)
unnecessary, against common assumptions. These models also avoid problems of
mixing at the envelope border and fulfil requirements from C-star luminosities.
They yield a large production of nuclei below A = 100, so that 86,87Sr may be
fully synthesized by AGB stars, while 88Sr, 89Y and 94Zr are contributed more
efficiently than before. We finally suggest tests suitable to say a final word
on the extension of the 13C pocket.
