Order-disorder character of the $(3\mathrm{}\times 3)$ to $(\sqrt{3}\times \sqrt{3})R30\mathrm{}°$ phase transition of Sn on Ge(111)

The $\alpha$ phase of Sn/Ge(111) has been investigated from 120 K up to 500 K, using synchrotron radiation core-level photoemission. By means of photoelectron diffraction experiments, we verified that the rippled structure of the low-temperature $(3\mathrm{}\times 3)$ phase is preserved in the $(\sqrt{3}\times \sqrt{3})R30\mathrm{}°$ phase at room temperature, thus confirming the order-disorder character of the phase transition. We also found that at least two components are present in the Sn $4d$ core-level spectra up to 500 K, i.e., about 300 K above the onset of the transition from the low-temperature $(3\mathrm{}\times 3)$ phase to the $(\sqrt{3}\times \sqrt{3})R30\mathrm{}°$ phase, thus excluding the occurrence of any displacive transition.