$\mathrm{Si}(111)\text{\ensuremath{-}}(5\ifmmode\times\else\texttimes\fi{}2)\text{\ensuremath{-}}\mathrm{Au}$ represents a prototype system for an adsorbate-induced spontaneous symmetry break by formation of quasi-one-dimensional atomic chains that are metallic. Surprisingly, the geometric structure of these chains is still under debate. In this paper, we show that examination of the unoccupied band structure by plasmon spectroscopy in combination with low energy electron diffraction contains sufficient information to discriminate between suggested models, favoring an optimal concentration of 6 atoms per $(5\ifmmode\times\else\texttimes\fi{}2)$ unit cell. Furthermore, we tested the stability of this structure and found that higher concentrations of Au tend to destabilize the single-domain structure on the slightly misoriented Si(111) surface, favoring formation of $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})R{30}^{\ensuremath{\circ}}\text{\ensuremath{-}}\mathrm{ordered}$ islands with a local concentration of 1 ML already at a surplus of 0.08 ML. The precise role of Au atoms at concentrations exceeding 0.60 ML that seem to stabilize formation of three domains, however, must remain open.
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