  Context. Barium (Ba) stars are dwarf and giant stars enriched in elements
heavier than iron produced by the slow neutron-capture process (s process).
They belong to binary systems where the primary star evolved through the
asymptotic giant branch (AGB) phase,during which it produced the s-process
elements and transferred them onto the secondary, now observed as a Ba star.
Aims. We compare the largest homogeneous set of Ba giant star observations of
the s-process elements Y, Zr, La, Ce, and Nd with AGB nucleosynthesis models to
reach a better understanding of the s process in AGB stars. Methods. By
considering the light-s (ls: Y and Zr) heavy-s (hs: La, Ce, and Nd) and
elements individually, we computed for the first time quantitative error bars
for the different hs-element/ls-element abundance ratios, and for each of the
sample stars. We compared these ratios to low-mass AGB nucleosynthesis models.
We excluded La from our analysis because the strong La lines in some of the
sample stars cause an overestimation and unreliable abundance determination, as
compared to the other observed hs-type elements. Results. All the computed
hs-type to ls-type element ratios show a clear trend of increasing with
decreasing metallicity with a small spread (less than a factor of 3). This
trend is predicted by low-mass AGB models where 13C is the main neutron source.
The comparison with rotating AGB models indicates the need for the presence of
an angular momentum transport mechanism that should not transport chemical
species, but significantly reduce the rotational speed of the core in the
advanced stellar evolutionary stages. This is an independent confirmation of
asteroseismology observations of the slow down of core rotation in giant stars,
and of rotational velocities of white dwarfs lower than predicted by models
without an extra angular momentum transport mechanism.
