  Distortions of the $\sqrt3\times\sqrt3$ Sn/Ge(111) and Sn/Si(111) surfaces
are shown to reflect a disproportionation of an integer pseudocharge, $Q$,
related to the surface band occupancy. A novel understanding of the
$(3\times3)$-1U (``1 up, 2 down'') and 2U (``2 up, 1 down'') distortions of
Sn/Ge(111) is obtained by a theoretical study of the phase diagram under
strain. Positive strain keeps the unstrained value Q=3 but removes distorsions.
Negative strain attracts pseudocharge from the valence band causing first a
$(3\times3)$-2U distortion (Q=4) on both Sn/Ge and Sn/Si, and eventually a
$(\sqrt3\times\sqrt3)$-3U (``all up'') state with Q=6. The possibility of a
fluctuating phase in unstrained Sn/Si(111) is discussed.
