ar X iv : n uc l - th / 9 71 00 31 v 1 1 3 O ct 1 99 7 Shell Effects in Mesoscopic Systems

A major property of confined Fermi system is the quantisation of single particle motion. It leads to a bunching of levels in the single particle spectrum, known as shells, and gives rise to magic numbers in finite Fermi systems. Consequently, a spherical symmetry leads to very strong shell effects manifested in the stability of the noble gases, nuclei and metallic clusters [1, 2]. Recently the sequence of magic numbers of a two-dimensional harmonic oscillator has been observed in the addition energy for a vertical quantum dot, i.e. in the energy needed to place the extra electron into the dot [3]. In contrast to nuclei and metallic clusters the properties of quantum dots can be controlled by men. The main aim of this talk to discuss the manifestation of shell effects in the mean field approach in different mesoscopic systems, i.e. in nuclei, metallic clusters and quantum dots. Periodic motion and higher multipoles in nuclei and metallic clusters. When a spherical shell is only partially filled, a breaking of spherical symmetry, resulting in an energy gain, can give rise to a deformed equilibrium shape. Super-and hyperdeformed nuclei are among the most fascinating examples where deviations from the spherical shape are a consequence of strong shell closures giving rise to largest level bunching (largest degeneracy or lowest level density). Nowadays it is recognized that the fine structure in the mass spectra between magic numbers in metallic clusters could be explained via symmetry breaking mechanisms similar to the situation in nuclear physics. It is therefore accepted that cluster deformations can exist, and it is actually confirmed at least for clusters with A ≤ 40 either by the Clemenger–Nilsson (CN) model (introduced by Nilsson [4] in nuclear physics and applied by Clemenger [5] for clusters) or by a self-consistent Kohn–Sham density– functional method [6] (KS) with deformed jellium backgrounds [7]–[8]. The need for multipole deformations higher than the quadrupole in the mean field approach has been recognized in nuclei and in metallic clusters in numerous calculations to explain experimental data. For instance, the octupole deformed shapes still constitute an intriguing problem of nuclear structure, experimental as well as theoretical (see for review [9]). The hexadecapole deformation is essential for the understanding of equilibrium shapes and the fission process of super-and hyperdeformed nuclei [10]. In the case of metallic clusters, the axial hexadecapole deformation is important for the interpretation of experimental data in simple metals …