  We employ a combined experimental-theoretical study of the first- and
second-order Raman modes of monoclinic $\beta$-Ga$_{2}$O$_{3}$. The
investigated materials is of particular interest due to its deep-UV bandgap
paired with a high critical field strength, offering promising applications in
power-electronics. A crucial prerequisite for the future development of
Ga$_{2}$O$_{3}$-based devices is a detailed understanding of the lattice
dynamics as they are important for the elasticity (through acoustic phonons),
thermal conductivity (through the heat transferred by phonons), the
temperature-dependence of the bandgap (impacted by electron-phonon coupling) or
the free carrier transport (via phonon scattering). Polarized micro-Raman
spectroscopy measurements on the (010) and ($\bar{2}01$) planes enable the
determination of the phonon frequencies of all 15 first-order and more than 40
second-order Raman modes. The experimental results are correlated with
calculations of the mode frequencies, phonon dispersion relation and phonon
density of states using density functional perturbation theory (DFPT). By
applying a group-theoretical analysis, we are able to distinguish between
overtones and combinational modes and identify the high symmetry points in the
Brillouin zone which contribute to the observed second order modes. Based on
these information, we demonstrate the simultaneous determination of Raman-,
IR-, and acoustic phonons in $\beta$-Ga$_{2}$O$_{3}$ by second-order Raman
spectroscopy.
