The physical interpretation of the spectral line polarization produced by the
joint action of the Hanle and Zeeman effects offers a unique opportunity to
obtain empirical information about hidden aspects of solar and stellar
magnetism. To this end, it is important to achieve a complete understanding
of the sensitivity of the emergent spectral line polarization to the presence
of a magnetic field. Here we present a detailed
theoretical investigation on the role of resonance scattering and
magnetic fields on the polarization signals of the Ba~{$\!$\sc ii} D$_1$
and D$_2$ lines of the Fraunhofer spectrum, respectively at 4934~{\AA}
and 4554~{\AA}.
We adopt a three-level model of Ba~{$\!$\sc ii}, and we take into account the
hyperfine structure that is shown by the $^{135}$Ba
and $^{137}$Ba isotopes. Despite of their relatively small abundance (18\%),
the contribution coming from these two isotopes is indeed fundamental for the
interpretation of the polarization signals observed in these lines.
We consider an optically thin slab model, through which we can investigate in a rigorous way the essential physical mechanisms involved (resonance
polarization, Zeeman, Paschen-Back and Hanle effects), avoiding complications
due to radiative transfer effects. We assume the slab to be illuminated from
below by the photospheric solar continuum radiation field, and we investigate
the radiation scattered at 90$^\circ$, both in the absence and in
the presence of magnetic fields, deterministic and microturbulent.
We show in particular the existence of a differential magnetic
sensitivity of the three-peak $Q/I$ profile that is observed in the D$_{2}$
line in quiet regions close to the solar limb, which is of great interest for
magnetic field diagnostics.