Phonons originated in solid state theory some time ago. In a crystalline solid there are a discrete number of vibrational excitations, and each of these can be excited in discrete quanta, representing phonons.
An interesting consequence of this well-verified theory is the zero phonon line - a radioactive decay that does not cause an excitation of a phonon due to the recoil of the nucleus, since in this case the entire crystal aborbs the momentum due to “spooky action at a distance.” It’s a unique phenomenon known as the Mössbauer effect, and it has widespread technological applications.
So, how does that affect carbon dating?