  Si$_x$Ge$_{1-x-y}$Sn$_y$ ternary alloys are a candidate material system for
use in solar cells and other optoelectronic devices. We report on the direct
transition energies and structural properties of Ge-rich
Si$_x$Ge$_{1-x-y}$Sn$_y$ alloys with six different compositions up to 10 % Si
and 3 % Sn, lattice-matched to Ge or GaAs substrates. The direct transitions
occurring between 0.9 and 5.0 eV were investigated using spectroscopic
ellipsometry (SE), and the resulting data was used to obtain the dielectric
functions of the Si$_x$Ge$_{1-x-y}$Sn$_y$n layer by fitting a multi-layer
model. Values for the $E_0$, $E_1$, $\Delta_1$, $E_0'$ and $E_2$ transition
energies were then found by differentiating these dielectric functions to
extract the locations of critical points. Structurally, the composition of the
samples was measured using energy-dispersive X-ray measurements (EDX). The
lattice constants predicted from these compositions are in good agreement with
reciprocal space maps obtained through X-ray diffraction (XRD). The results
confirm that a 1 eV direct absorption edge can be achieved using relatively low
Si and Sn fractions ($<$ 10 % and $<$ 3 % respectively), while the
higher-energy critical points show smaller shifts relative to Ge and match
results previously observed or predicted in the literature.
