  Photoneutron cross sections were measured for $^{137}$Ba and $^{138}$Ba at
energies below two-neutron threshold using quasi-monochromatic $\gamma$-ray
beams produced in laser Compton-scattering at the NewSUBARU synchrotron
radiation facility. The photoneutron data are used to constrain the
$\gamma$-ray strength function on the basis of the Hartree-Fock-Bogolyubov plus
quasi-particle random phase approximation using the Gogny D1M interaction.
Supplementing the experimentally constrained $\gamma$-ray strength function
with the zero-limit E1 and M1 contributions which are unique to the
deexcitation mode, we discuss radiative neutron capture cross sections relevant
to the s-process nucleosynthesis of barium isotopes in the vicinity of the
neutron magic number 82.
