The compression of SH$_2$ and its subsequent decomposition to SH$_3$, presumably in a cubic Im$overline{3}$m structure, has lead to the discovery of conventional superconductivity with the highest measured and confirmed $T_c$ to date, 203 K at 160 GPa. Recent theoretical studies suggest that a mixture of S with other elements of the chalcogen group could improve the superconducting temperature. Here, we present a detailed analysis of the thermodynamic properties of S and Se mixtures in the bcc lattice with Im$overline{3}$m symmetry using a cluster expansion technique to explore the phase diagram of S$_x$Se$_{1-x}$H$_{3}$. In contrast to earlier reports, we find that S$_{0.5}$Se$_{0.5}$H$_3$ is not stable in the pressure range between 150-200 GPa. However, phases at compositions S$_{0.2}$Se$_{0.8}$H$_3$, S$_{0.overline{3}}$Se$_{0.overline{6}}$H$_3$, and S$_{0.6}$Se$_{0.4}$H$_3$ are stable at 200 GPa, while additional phases at S$_{0.25}$Se$_{0.75}$H$_3$ and S$_{0.75}$Se$_{0.25}$H$_3$ are accessible at lower pressures. Electron-phonon calculations show that the values of $T_c$ are consistently lower for all ternary phases, indicating that mixtures of S and Se with H might not be a viable route towards compounds with improved superconducting properties.
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