Experiments searching for neutrinoless double beta decay ($0 ubetabeta$) require precise energy calibration and extremely low backgrounds. One of the most popular isotopes for $0 ubetabeta$ 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 $0 ubetabeta$ 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.
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