  Ammonium sulfate is a giant inverse barocaloric material that is cheaply and
commercially available. Exploiting its potential for cooling applications
requires an understanding of the mechanism driving the entropy change. Here we
report an investigation by inelastic neutron scattering and density functional
theory of the phonons under working conditions of temperature and pressure. We
find excellent agreement between the experimental and calculated results. The
ammonium librational modes that are crucial to the entropy change are
identifiable by their negative Gr\"{u}neisen parameter. Our results connect the
differences in structure across the phase transition to those in the atomic
dynamics, suggesting a route towards designing new caloric materials.
