LAUSR

We use first-principles Density Functional Theory (DFT) to conduct an extensive structure search using the \verb|AIRSS| package for elemental sulfur in the range $50$-$550$ GPa. We then obtain the low-temperature phase diagram of sulfur in the same pressure range, including vibrational effects through the harmonic approximation. We do not find any structures lower in energy than those already reported in experiment, although the phase diagram below $100$ GPa is found to be crowded with structures separated by only a few meV. We report the transition sequence $I4_1/acd \rightarrow P\bar{1} \rightarrow$ ICM (Incommensurately Modulated) $\rightarrow C2/m \rightarrow R\bar{3}m \rightarrow Im\bar{3}m$ and obtain accurate pressures for each transition, although we find the second-order $C2/m \rightarrow R\bar{3}m$ transition particularly difficult to define. Contrary to previous first-principles works~\cite{USPEX,Rudin}, we do not reproduce a trigonal $\rightarrow$ simple cubic (SC) transition at either the static lattice or harmonic level. We also undertake a detailed analysis of the incommensurately modulated phase of sulfur (ICM) phase using a commensurate approximant found in the structure search. We find that the modulation is zero above $96$ GPa, some $50$ GPa below the experimentally reported transition to the unmodulated phase. We find that the body-centred atoms in the relaxed ICM approximant are, in addition to the dominant transverse modulation, slightly displaced longitudinally in the $b$-direction. We subsequently discover that this (small) longitudinal modulation is coupled to the transverse mode, and hence report previously unnoticed weak-mode coupling between transverse and longitudinal optical phonons in the ICM phase.