  The magnetization process and adiabatic demagnetization of the
antiferromagnetic spin-1/2 XXZ Heisenberg clusters with the shape of regular
polyhedra (tetrahedron, octahedron, cube, icosahedron and dodecahedron) are
examined using the exact diagonalization method. It is demonstrated that a
quantum (xy) part of the XXZ exchange interaction is a primary cause for
presence of additional intermediate magnetization plateaux and steps, which are
totally absent in the limiting Ising case. The only exception to this rule is
the spin-1/2 XXZ Heisenberg tetrahedron, which shows just a quantitative shift
of the level-crossing fields related to two magnetization steps. It is
evidenced that the spin-1/2 XXZ Heisenberg regular polyhedra exhibit an
enhanced magnetocaloric effect in a proximity of the magnetization steps and
jumps, which are accompanied with a rapid drop (rise) of temperature just above
(below) of the level-crossing field when the magnetic field is removed
adiabatically.
