Optical investigation of the thermoelectric topological crystalline insulatorPb0.77Sn0.23Se

${\mathrm{Pb}}_{0.77}{\mathrm{Sn}}_{0.23}\mathrm{Se}$ is a promising thermoelectric alloy that exhibits a temperature dependent band inversion below 300 K. Recent work has shown that this band inversion also coincides with a trivial to nontrivial topological phase transition. To understand how the properties critical to thermoelectric efficiency are affected by the band inversion, we measured the broadband optical response of ${\mathrm{Pb}}_{0.77}{\mathrm{Sn}}_{0.23}\mathrm{Se}$ as a function of temperature. We find clear optical signatures suggesting the band inversion occurs at $160\ifmmode\pm\else\textpm\fi{}15$ K, and use the extended Drude model to accurately determine a ${T}^{3/2}$ dependence of the bulk carrier lifetime, associated with electron-acoustic phonon scattering. Due to the high bulk carrier doping level, no discriminating signatures of the topological surface states are found, although their presence cannot be excluded from our data.