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We derive a dielectric function tensor model approach to render the opticalresponse of monoclinic and triclinic symmetry materials with multiple uncoupledinfrared and farinfrared active modes. We apply our model approach tomonoclinic $\beta$-Ga$_2$O$_3$ single crystal samples. Surfaces cut underdifferent angles from a bulk crystal, (010) and ($\bar{2}$01), are investigatedby generalized spectroscopic ellipsometry within infrared and farinfraredspectral regions. We determine the frequency dependence of 4 independent$\beta$-Ga$_2$O$_3$ Cartesian dielectric function tensor elements by matchinglarge sets of experimental data using a point by point data inversion approach.From matching our monoclinic model to the obtained 4 dielectric function tensorcomponents, we determine all infared and farinfrared active transverse opticphonon modes with $A_u$ and $B_u$ symmetry, and their eigenvectors within themonoclinic lattice. We find excellent agreement between our model results andresults of density functional theory calculations. We derive and discuss thefrequencies of longitudinal optical phonons in $\beta$-Ga$_2$O$_3$. We deriveand report density and anisotropic mobility parameters of the free chargecarriers within the tin doped crystals. We discuss the occurrence oflongitudinal phonon plasmon coupled modes in $\beta$-Ga$_2$O$_3$ and providetheir frequencies and eigenvectors. We also discuss and present monoclinicdielectric constants for static electric fields and frequencies above thereststrahlen range, and we provide a generalization of the Lyddane-Sachs-Tellerrelation for monoclinic lattices with infrared and farinfrared active modes. Wefind that the generalized Lyddane-Sachs-Teller relation is fulfilledexcellently for $\beta$-Ga$_2$O$_3$.

Anisotropy, phonon modes, and free charge carrier parameters in monoclinic $\beta$-gallium oxide single crystals