  The magnetization curves and magnetocaloric effect during the adiabatic
demagnetization of antiferromagnetic spin-1/2 $XXZ$ Heisenberg clusters with
the shape of Johnson's solids (triangular cupola, square cupola and pentagonal
cupola) are investigated using the exact numerical diagonalization for
different values of the exchange anisotropy ranging in between the Ising and
fully isotropic limit. It is demonstrated that spin-1/2 $XXZ$ Heisenberg
cupolae display, in comparison with their Ising counterparts, at least one more
magnetization plateau. The novel magnetization plateaux extends over a wider
range of the magnetic fields with increasing of a quantum part of the $XXZ$
exchange coupling at the expense of the original plateaux present in the
limiting Ising case. It is shown that the spin-1/2 $XXZ$ Heisenberg triangular
and pentagonal cupolae exhibit an enhanced magnetocaloric effect in the
vicinity of zero magnetic field, which makes these magnetic systems promising
refrigerants for cooling down to ultra-low zero temperatures.
