  We report on the spin waves and crystal field excitations in single crystal
LiFePO$_4$ by inelastic neutron scattering over a wide range of temperatures,
below and above the antiferromagnetic transition of this system. In particular,
we find extra excitations below $T_N=50$ K that are nearly dispersionless and
are most intense around magnetic zone centers. We show that these excitations
correspond to transitions between thermally occupied excited states of
Fe$^{2+}$ due to splitting of the $S=2$ levels that arise from crystal field
and spin-orbit interaction. These excitations are further amplified by the
highly distorted nature of the oxygen octahedron surrounding the iron atoms.
Above $T_N$, magnetic fluctuations are observed up to at least 720~K, with
additional excitation around 4 meV, likely caused by single-ion splittings
through spin-orbit and crystal field interactions. The latter weakens slightly
at 720~K compared to 100~K, which is consistent with calculated cross-sections
using a single-ion model. Our theoretical analysis, using the MF-RPA model,
provides both detailed spectra of the Fe $d-$ shell and estimates of the
average ordered magnetic moment and $T_N$. By applying the MF-RPA model to a
number of existing spin-wave results from other Li$M$PO$_4$ ($M=$ Mn, Co, and
Ni), we are able to obtain reasonable predictions for the moment sizes and
transition temperatures.
