Unified description of magic numbers of metal clusters in terms of the three-dimensionalq-deformed harmonic oscillator

Magic numbers predicted by a three-dimensional q-deformed harmonic oscillator with ${\mathrm{u}}_q\mathrm{}\left(3\right)\mathrm{}\supset {\mathrm{so}}_q\mathrm{}\left(3\right)\mathrm{}$ symmetry are compared to experimental data for atomic clusters of alkali metals (Li, Na, K, Rb, and Cs), noble metals (Cu, Ag, and Au), divalent metals (Zn and Cd), and trivalent metals (Al and 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 three-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 ${\mathrm{u}}_q\mathrm{}\left(3\right),$ which is a nonlinear extension of the u(3) symmetry of the spherical (three-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 three-dimensional q-deformed harmonic oscillator are found to be similar to the Taylor expansions of the symmetrized Woods-Saxon potential and wine bottle symmetrized Woods-Saxon potential, which are known to provide successful fits of the Ekardt potentials.