Phonon dispersion and low energy anomaly in CaC$_6$.

We report measurements of phonon dispersion in CaC$_6$ using inelastic X-ray and neutron scattering. We find good overall agreement, particularly in the 50 meV energy region, between experimental data and first-principles density-functional-theory calculations. However, on the longitudinal dispersion along the $(1 1 1)$ axis of the rhombohedral representation, we find an unexpected anti-crossing with an additional longitudinal mode, at about 11 meV. At a comparable energy, we observe also unexpected intensity on the in-plane direction. These results resolve the previous incorrect assignment of a longitudinal phonon mode to a transverse mode in the same energy range. By calculating the electron susceptibility from first principles we show that this longitudinal excitation is unlikely to be due to a plasmon and consequently can probably be due to defects or vacancies present in the sample.