Supershells in deformed harmonic oscillators and atomic clusters

Abstract

ABSTRACT:From the mathematical point of view, the appearance of supershells is a general feature of potentials having relatively sharp edges. In physics, supershells have been observed in systems of metal clusters, which are also known to exhibit an underlying shell structure with magic numbers intermediate between the magic numbers of the 3-D isotropic harmonic oscillator and those of the 3-D square well. In the present study, Nilsson's modified harmonic oscillator (without any spin-orbit interaction), as well as the 3-D q-deformed harmonic oscillator with u q (3) ʛ so q (3) symmetry, are considered. The former model has been used for an early schematic description of shell structure in metal clusters, while the latter has been found to successfully reproduce the magic numbers of metal clusters up to 1500 atoms, the expected limit of validity for theories based on the filling of electronic shells. The systematics of the appearance of supershells in the two models will be considered, putting emphasis on the differences between the spectra of the two oscillators. While the validity of Nilsson's modified harmonic oscillator framework is limited to relatively low particle numbers, the 3-D q-deformed harmonic oscillator gives reliable descriptions of the first supershell in metal clusters, which lies within its region of validity.

Deformed Harmonic Oscillators for Metal Clusters and Balian-Bloch Theory

“…2 and 3 (although in these cases, for reasons of clarity, E shell is plotted vs. N and not vs. N 1/3 ). The fact that the predictions of the VFO corresponding to the 3D q-HO approximately exhibit the right periodicity features does not come as a surprise, since the 3D q-HO is known to show this feature [6,7], while, as we have seen above, the addition of the second term in Eq. (10) does not influence the position of the minima.…”

“…In terms of the 3D q-HO a good description of the magic numbers of alkali clusters (up to 1500 atoms, which is the limit of validity for theories based on the filling of electronic shells [19,20]), as well as of Al clusters, has been obtained [5]. In addition, it has been proved that supershells occur naturally in this model [6,7], which is characterized by only one parameter (the deformation parameter q = e τ , with τ being real) in addition to the overall scale.…”

“…An algebraic approach to the structure of metal clusters [3,4] has been introduced recently [5,6,7,8,9], taking advantage of the three-dimensional q-deformed harmonic oscillator (3D q-HO) [10], which is constructed using the techniques of quantum algebras (quantum groups) [see [11] and references therein]. The symmetry of this oscillator is u q (3)⊃so q (3) [12,13,14,15,16], while its derivation involves use of irreducible tensor operators under so q (3) [15,17,18].…”

“…The last expression to leading order bears great similarity to the modified harmonic oscillator suggested by Nilsson [26,27] (with the spin-orbit term omitted). Comparisons between the predictions of the 3D q-HO and Nilsson's modified oscillator for the magic numbers and supershells of metal clusters have been given in [5,6,7]. One of the main differences between the two models is that the term τ n(n + 1) in the eigenvalues of the 3D q-HO given in Eq.…”