Zero phonon: during emission and absorbtion there is no lattice excitation (Mössbauer effect)-here information on the hyperfine interactions (or n=0 so E=hv/2).
Phonon assisted:together with zero phonon, there will be side bands. These side bands are assisted by Mössbauer effect. Here you get information on phonon density of states..
these informations can be obtained from nuclear inelastic scattering-NIS (a.k.a. nuclear inelastic absorption, nuclear resonance vibrational spectroscopy-NRVS, phonon assisted Mössbauer effect, Nuclear resonant inelastic X-ray scattering-NRIXS).
Just try for find some quantum book, or a Mössbauer book..you will get all info there.
PS: Mössbauer effect discovered in 1957, Theorists predicted that there will be phonon assisted Mössbauer effect (side bands) in 1960, after first experiments in 1985, first NIS side band spectrum published in 1995 (help of synchrotron sources).