  Interactions between electrons and lattice vibrations are responsible for a
wide range of material properties and applications. Recently, there has been
considerable interest in the development of resonant inelastic x-ray scattering
(RIXS) as a tool for measuring electron-phonon (e-ph) interactions. Here, we
demonstrate the ability of RIXS to probe the interaction between phonons and
specific electronic states both near to, and away from, the Fermi level. We
performed carbon $K$-edge RIXS measurements on graphite, tuning the incident
x-ray energy to separately probe the interactions of the $\pi^*$ and $\sigma^*$
electronic states. Our high-resolution data reveals detailed structure in the
multi-phonon RIXS features that directly encodes the momentum dependence of the
e-ph interaction strength. We develop a Green's-function method to model this
structure, which naturally accounts for the phonon and interaction-strength
dispersions, as well as the mixing of phonon momenta in the intermediate state.
This model shows that the differences between the spectra can be fully
explained by contrasting trends of the e-ph interaction through the Brillouin
zone, being concentrated at the $\Gamma$ and $K$ points for the $\pi^*$ states,
while being significant at all momenta for the $\sigma^*$ states. Our results
advance the interpretation of phonon excitations in RIXS, and extend its
applicability as a probe of e-ph interactions to a new range of
out-of-equilibrium situations.
