  The $\gamma$-ray strength functions (GSF) and nuclear level densities (NLD)
below the neutron threshold have been extracted for $^{111-113,116-122,124}$Sn
from particle-$\gamma$ coincidence data with the Oslo method. The evolution of
bulk properties of the low-lying electric dipole response has been investigated
on the basis of the Oslo GSF data and results of a recent systematic study of
electric and magnetic dipole strengths in even-even Sn isotopes with
relativistic Coulomb excitation. The obtained GSFs reveal a resonance-like peak
on top of the tail of the isovector giant dipole resonance, centered at
$\approx$8 MeV and exhausting $\approx$2\% of the classical Thomas-Reiche-Kuhn
(TRK) sum. In contrast to predictions of the relativistic quasiparticle
random-phase and time-blocking approximation calculations (RQRPA and RQTBA), no
monotonous increase in the total low-lying $E1$ strength was observed in the
experimental data from $^{111}$Sn to $^{124}$Sn, demonstrating rather similar
strength distributions in these nuclei. The Oslo GSFs and NLDs were further
used as inputs to constrain the cross sections and Maxwellian-averaged cross
sections of $(n,\gamma)$ reactions in the Sn isotopic chain using TALYS. The
obtained results agree well with other available experimental data and the
recommended values from the JINA REACLIB, BRUSLIB, and KADoNiS libraries.
Despite relatively small exhausted fractions of the TRK sum rule, the low-lying
electric dipole strength makes a noticeable impact on the radiative
neutron-capture cross sections in stable Sn isotopes. Moreover, the
experimental Oslo inputs for the $^{121,123}$Sn$(n,\gamma)$$^{122,124}$Sn
reactions were found to affect the production of Sb in the astrophysical
$i$-process, providing new constraints on the uncertainties of the resulting
chemical abundances from multi-zone low-metallicity Asymptotic Giant Branch
stellar models.
