# A Little More on LO-TO Splitting

In my previous post, I addressed the concept of LO-TO splitting and how it results from the long-ranged nature of the Coulomb interaction. I made it a point to emphasize that while the longitudinal and transverse optical phonons are non-degenerate near $\textbf{q}=0$, they are degenerate right at $\textbf{q}=0.$ This scenario occurs because of the retarded nature of the Coulomb interaction (i.e. the finite speed of light).

What exactly goes on? Well, it so happens that in a very narrow momentum window near $\textbf{q}=0$, the transverse optical phonon is strongly coupled to light and forms a polariton. This is a manifestation of the avoided crossing or level repulsion principle that I have blogged about previously. Since light is a transverse wave, it interacts with the transverse optical phonon (but not the longitudinal one).

In a tour-de-force experiment at Bell Labs by Henry and Hopfield (the same Hopfield of Hopfield neural networks), Raman scattering was conducted at grazing incident angles to measure the dispersion of the lower polariton branch as shown below:

The dispersing solid lines represent the transverse optical (TO) phonon interacting with light. The straight solid line is the unaffected longitudinal optical (LO) phonon branch. The dotted line labelled with the angles are the incident beam angles in the Raman experiment. The remaining dotted lines represent the non-interacting TO phonon and the non-interacting light dispersion.

Usual Raman measurements are taken in a backscattering as opposed to a grazing incidence geometry, hence the momentum transfers are ordinarily too high to observe the low-$\textbf{q}$ dispersion. Because of this, the authors mentioned that some Raman exposures in this experiment took up to seven hours!

The takeaway from the plot above is that the transverse optical phonon at $\textbf{q}=0$ is degenerate with the longitudinal one and “turns into a photon” at higher momenta, while the photon branch at $\textbf{q}=0$ “turns into the transverse optical phonon” at higher momenta.

Unfortunately, the paper does not contain their raw data, only the dispersion. Publishing standards seem to have been different back then. Nonetheless, this is a very clever and illuminating experiment.