  The structure of the $(\sqrt{31}\times \sqrt{31})R\pm9^\circ$ reconstructed
phase on sapphire (0001) surface is investigated by means of a simulation based
on the energy minimization. The interaction between Al adatoms is described
with the semi-empirical many-body Sutton-Chen potential, corrected for the
charge transfer between the metallic overlayer and the substrate. The
interactions between the Al adatoms and sapphire substrate are described with a
simple three-dimensional potential field which has the hexagonal periodicity of
sapphire surface. Our energy analysis gave evidence that the structure which is
observed at room temperature is in fact a frozen high-temperature structure. In
accordance with the X-ray scattering, a hexagonal domain pattern separated by
domain walls has been found. The Al adatoms, distributed in two monolayers, are
ordered and isomorphic to metallic Al(111) in the domains and disordered in the
domain walls. The main reason for the rotational reconstruction is the lattice
misfit between the metallic Al and sapphire.
