  Magic numbers predicted by a 3-dimensional q-deformed harmonic oscillator
with Uq(3)>SOq(3) symmetry are compared to experimental data for atomic
clusters of alkali metals (Li, Na, K, Rb, Cs), noble metals (Cu, Ag, Au),
divalent metals (Zn, Cd), and trivalent metals (Al, In), as well as to
theoretical predictions of jellium models, Woods-Saxon and wine bottle
potentials, and to the classification scheme using the 3n+l pseudo quantum
number. In alkali metal clusters and noble metal clusters the 3-dimensional
q-deformed harmonic oscillator correctly predicts all experimentally observed
magic numbers up to 1500 (which is the expected limit of validity for theories
based on the filling of electronic shells), while in addition it gives
satisfactory results for the magic numbers of clusters of divalent metals and
trivalent metals, thus indicating that Uq(3), which is a nonlinear extension of
the U(3) symmetry of the spherical (3-dimensional isotropic) harmonic
oscillator, is a good candidate for being the symmetry of systems of several
metal clusters. The Taylor expansions of angular momentum dependent potentials
approximately producing the same spectrum as the 3-dimensional q-deformed
harmonic oscillator are found to be similar to the Taylor expansions of the
symmetrized Woods-Saxon and wine-bottle symmetrized Woods-Saxon potentials,
which are known to provide successful fits of the Ekardt potentials.
