  Discovery of the incommensurate structure in the element Ba under pressure 15
years ago was followed by findings of a series of similar structures in other
compressed elements. Incommensurately modulated structures of the host-guest
type consist of a tetragonal host structure and a guest structure. The guest
structure forms chains of atoms embedded in the channels of host atoms so that
the axial ratio of these subcells along the c axis is not rational. Two types
of the host-guest structures have been found so far: with the host cells
containing 8 atoms and 16 atoms; in these both types the guest cells contain 2
atoms. These crystal structures contain a non-integer number of atom in their
unit cell: tI11* in Bi, Sb, As, Ba, Sr, Sc and tI19* in Na, K, Rb. We consider
here a close structural relationship of these host-guest structures with the
binary alloy phase Au3Cd5-tI32. This phase is related to the family of the
Hume-Rothery phases that is stabilized by the Fermi sphere - Brillouin zone
interaction. From similar considerations for alkali and alkaline-earth elements
a necessary condition for structural stability emerges in which the valence
electrons band overlaps with the upper core electrons and the valence electron
count increases under compression.
