  We use variable-temperature x-ray thermal diffuse scattering and inelastic
scattering to investigate the lattice dynamics in single crystals of
multiferroic CaMn7O12 which undergo a series of orbital and magnetic
transitions at low temperatures. Upon approaching the charge and orbital
ordering temperature To=250 K from above, we observe intense diffuse scattering
features and a pronounced optical phonon softening centered around the
superstructure reflections of the incommensurately modulated structure. The
phonon anomaly appears well above To and continuously increases upon cooling,
following a canonical power-law temperature dependence that confirms the
transition at To to be of second order and related to a soft-phonon lattice
instability. Microscopic mechanisms for the incommensurate charge and orbital
ordering based on competing interactions and on momentum dependent
electron-phonon coupling could both account for the observed extended momentum
width of the phonon softening. Our results highlight the importance of the
lattice interactions in the physics of this magnetically induced ferroelectric
system.
