  Experiments searching for neutrinoless double beta decay ($0\nu\beta\beta$)
require precise energy calibration and extremely low backgrounds. One of the
most popular isotopes for $0\nu\beta\beta$ experiments is $^{136}$Xe. In
support of these experiments, the neutron inelastic scattering properties of
this isotope have been measured at the GErmanium Array for Neutron Induced
Excitations (GEANIE) at the Los Alamos Neutron Science Center. Time-of-flight
techniques are utilized with high-purity germanium detectors to search for
inelastic scattering $\gamma$ rays for neutron energies between 0.7 and 100
MeV. Limits are set on production of yet-unobserved $\gamma$ rays in the energy
range critical for $0\nu\beta\beta$ studies, and measurements are made of
multiple $\gamma$-ray production cross sections. In particular, we have
measured the production of the 1313 keV $\gamma$ ray which comes from the
transition of the first-excited to ground state of $^{136}$Xe. This
neutron-induced $\gamma$ line may be useful for a novel energy calibration
technique, described in this paper.
