Supershell Structure Research Articles

PSEUDOPOTENTIAL AND SURFACE EFFECTS ON THE ELECTRONIC SHELL STRUCTURE OF TRIVALENT-METAL CLUSTERS

Electronic shell and supershell structures in trivalent-metal clusters exhibit large discrepancies with the results of the conventional jellium model. For gallium the beat location in the supershell pattern is found around Ne=2500 valence electrons, in disagreement with the jellium model prediction Ne=1150. The discrepancies can be explained by a slight softness increase of the effective electronic potential at the surface. Theoretical calculations, taking into account ion pseudopotentials, within both local and nonlocal approaches, show that this softness arises to a large extent from the non-Coulombic short-range electron-ion interaction.

Effects of surface roughness on the electronic shell structure of metal clusters.

Electronic shell and supershell structures in spherical metal clusters with a rough surface are quantum mechanically investigated. Our nonperturbative approach involves a finite-depth step-walled spherical potential well with a sine-wave corrugation along the azimuthal and polar angular coordinates. The electronic structure is studied as a function of both the radial amplitude and the angular wavelength of the corrugation. When the spatial periods of the roughness are comparable to, or shorter than the Fermi wavelength, the shell structure related to the overall spherical symmetry survives, even in the case of a large radial corrugation amplitude. The weakening of the shell effects due to the loss of the spherical symmetry is rather minor and the electronic structure is essentially ruled by the spherical average of the rough potential, which has a soft surface profile. In consequence, the magic sizes (supershell beat locations) are slightly (noticeably) shifted towards larger sizes. These shifts depend strongly on the radial roughness amplitude. As far as the electronic structure is concerned, the results establish the equivalence between a fine-grained hard-walled rough potential and a spherical potential having a soft surface profile. Hence surface roughness could underlie partly the shift of the supershell nodes observed in trivalent metal experiments. When the wavelengths of the corrugation along the angular coordinates are large relative to the Fermi one the electronic density of states becomes quite smooth owing to the large level splitting and mixing effects. The supershell pattern and---to a lesser extent---the shell structure are considerably damped, and even may disappear. In consequence the size dependence of the shell-correction energy and of the ionization potential do not exhibit noticeable oscillations, except for small radial roughness amplitudes.

Electronic properties of large metal clusters in jellium and pseudo-jellium models

The energy-density functional approach and jellium-like models are used to examine two important electronic properties of metal (Li, Na, K) clusters: their shell and supershell structures, and the behaviour of plasmon energies with increasing cluster sizes. A comparative study is made between predictions of the usual jellium model and those of the pseudo-jellium model where pseudohamiltonians are used.

Effects of nonlocal ion pseudopotential on the electronic shell structure of metal clusters

Electronic shell and supershell structures in lithium and gallium clusters taking into account ion pseudopotentials, within both local and nonlocal approaches, are investigated. The results differ noticeably from those of the jellium background model (JBM). Some low magic numbers, underestimated in the JBM, are corrected. However, in particular, with regard to the supershell pattern, non-negligible discrepancies with experiment remain. For lithium the location of the first node in the supershell structure is found at higher sizes relative to the experimental results. For gallium the supershell node is located around ${\mathit{N}}_{\mathit{e}}$=2500 valence electrons in experiment, in striking disagreement with the JBM prediction ${\mathit{N}}_{\mathit{e}}$=1150. The pseudopotential effects shift the supershell node in the correct direction but the magnitude of the shift is insufficient to recover complete agreement with experiment. These discrepancies prove that, in addition to the pseudopotential effects, the supershell pattern observed in mass spectra depends probably on the ionic structure at the surface. Inhomogeneous ionic density at the cluster surface is then introduced in order to achieve better agreement with the experimental results. In contrast with gallium clusters for which the surface density has to be softened, the surface ionic density in lithium clusters has to be increased, suggesting a partial ion layering. Sensitivity of the supershell pattern on the pseudopotential parametrization and surface ionic density is also emphasized. In particular, the importance of the nonlocality of the pseudopotential for lithium is pointed out, in accordance with previous studies related to the optical response of small lithium clusters.

Quantum effects in many-body transport theory

Quantum extension of the many-body transport theory in the Wigner representation formalism is considered. Satisfactory description of the ground state of nuclei recovering the super-shell structure is obtained. The method is applied to analyse the role of quantum effects in low-energy nucleus-nucleus collisions. The dynamics of super-heavy systems formation and recently reported anomalies in proton transfer probability in heavy-ion reactions are discussed.

Influence of ion pseudopotential on the electronic shell structure of metal clusters.

Electronic shell and supershell structures in spherical multiple-shell and liquidlike clusters, taking into account the electron-ion interaction through usual empty-core pseudopotentials, have been investigated for trivalent and alkali species and compared with the conventional jellium model. In the case of multiple-shell structures we find that the jelliumlike electronic shell structure is clearly developed, in spite of the strong inhomogeneities in the ion density. The supershell pattern is, however, strongly dependent on the geometrical structure and pseudopotential parametrization. In the case of a homogeneous ionic distribution, the overall effect of the non-Coulombic behavior of the pseudopotential at short range is reflected through a softness increase of the effective electronic potential, acting as a diffuse jellium. Results about the shell and supershell structures are in better agreement with experiments, as compared to the standard jellium-model predictions. This effective surface softness, arising only from the finite size of the ionic cores, could also partially correct the present disagreement between experiment and jellium calculations with regard to the cluster polarizabilities and surface plasmon frequencies.

Electronic shells and supershells in trivalent metal clusters

Electronic shell structures are observed for the first time in large gallium and aluminum clusters. In gallium, the electronic shells extend up to 7000 electrons and a nice supershell structure is obtained around 2500 electrons. This number is considerably larger than expected from standard jellium calculations and is consistent with the introduction of a small surface softness. In aluminum, the clusters are more rigid and in usual experimental conditions they have specific geometric arrangement. However by heating the source nozzle, we obtain liquid-like aluminum clusters and observe the electronic shell structure up to 1800 electrons.

Electronic shell and supershell structures in gallium clusters containing up to 7000 electrons.

Gallium clusters have been studied by near-threshold photoionization up to 2400 atoms. An electronic shell structure is clearly observed and may be interpreted by considering three electrons per atom. A quantum supershell corresponding to the interference between the contributions of triangular and square classical electronic orbits is obtained at about 2500 electrons. This number is considerably larger than expected from self-consistent homogeneous jellium calculations and is interpreted by introducing an inhomogeneous jellium approach.

Shell structure in faceted metal clusters.

We study the quantized electronic energy levels in a three-dimensional icosahedral billiard modeling a faceted metal cluster. The first 2000 levels are determined numerically. The magic numbers are compared with experimental data and with the results for a spherical model. We discuss the supershell structure and propose its study as a test of cluster sphericity. We compare our results with the predictions of the semiclassical trace formula and point out the relevance of diffractive orbits.

Influence of surface softness on supershell structure of metal clusters: Application to gallium.

In order to interpret recent experimental results on gallium clusters, we investigate the influence of the surface softness on the electronic shell structure of spherical metal clusters. We discuss the physical reasons which support the introduction of a diffuse edge in the jellium model. We then perform quantum calculations using electronic potentials slightly modified as compared to the electronic potential obtained from self-consistent homogeneous jellium calculations. We show that the main influence of the diffuseness consists in the shift of the beating region in the supershell structure. A semiclassical analysis of this effect is also performed and the shift of the beating region is interpreted in a very simple way: the softness of the surface introduces a phase shift between the contributions of the classical triangular and square orbits to the density of states. This phase shift is roughly independent of the cluster size for a given diffuseness and this allows us to predict the shift of the beating region. Finally we discuss how we have to modify the uniform positive charge density of the standard jellium model to obtain results in agreement with the experiment and we show that the modification to be used for gallium is quite reasonable.

Octupole correlations in superdeformed high-spin states

Recent theoretical works done by us on the following subjects are briefly summarized: 1) Octupole vibrations with K=0, 1, 2 and 3 built on superdeformed yrast states, 2) Octupole vibration-quasiparticle couplings in superdeformed odd-A nuclei, 3) Supershell structure in superdeformed nuclei which break the reflection symmetry, 4) Analysis of the supershell effects in terms of semiclassical theory of quantum level density.An intimate relationship among octupole instability of the superdeformed shape, the rise of supershell effects, and bifurcations of classical periodic orbits is suggested.

SHELLS AND SUPERSHELLS IN METAL CLUSTERS

We analyze the shell and supershell structure of valence electrons in large metal clusters. For this purpose we use the spherical jellium model for the ions, treat the many-body problem of the interacting valence electrons selfconsistently in local-density approximation and solve numerically the Kohn-Sham equations for up to N ~ 6000 electrons. We investigate the influence of temperature effects on the shell structure and compare our results to experimental data and other calculations performed with phenomenological Woods–Saxon potentials.

Temperature effects in the electronic shells and supershells of lithium clusters.

Mass-abundance spectra of isotopically enriched lithium clusters from a gas aggregation source have been recorded for several laser flux densities of the ionizing laser. Shell and supershell structures are clearly observed. Thermal effects are put into evidence which tend to smooth out the shell structure of large masses. At the lowest temperature sharp features are revealed in the node and antinode regions of the beating supershell pattern. These structures are discussed in the light of theoretical models, and compared to other alkalis.

Electronic Supershells in Metal Clusters

We present selfconsistent calculations of the supershell structure of sodium, lithium, indium and aluminium clusters, by using the spherical jellium model in LDA and solving the Kohn-Sham equations for up to N ≈ 3000 electrons. While the agreement with the experimental results is very good for alkali metal clusters (Na and Li), it is rather poor in the case of aluminium. We also present quantum-mechanical calculations for a recently suggested phenomenological Woods-Saxon potential for the valence electrons of aluminium clusters.

Spherical supershells in metal clusters and the transition to protocrystalline structure.

One-electron calculations of spherical supershell structure are presented for metal clusters with up to 8000 valence electrons over the range of metallic densities 2\ensuremath{\le}${\mathit{r}}_{\mathit{s}}$\ensuremath{\le}6, using potentials adapted from self-consistent bulk surface calculations. Good agreement is obtained with detailed trends in the experimental data and with corresponding self-consistent cluster jellium calculations throughout. In the T=0 limit, aspherical perturbations anticipated from the incipient ionic lattice are shown to eradicate the supershell structure at a cluster size corresponding to near-degeneracy at the Fermi level between valence-electron states differing by \ensuremath{\Delta}l=4, where l is the angular-momentum quantum number. This critical size is on the order of N\ensuremath{\sim}${10}^{3}$ valence electrons for all metallic densities; in the specific case ${\mathit{r}}_{\mathit{s}}$=4, the result is N=1640, in excellent agreement with the experimentally observed transition in Na clusters to ``shells of atoms'' in the range 1400N2000.

Temperature dependence of supershells in large sodium clusters.

We present self-consistent calculations of the electronic shell and supershell structure of sodium clusters with up to N\ensuremath{\sim}1600 atoms at finite temperatures and N\ensuremath{\sim}2800 at zero temperature, employing the spherical jellium model in the local-density approximation and numerically solving the Kohn-Sham equations. The finite temperature of the valence electrons is included by treating them as a canonical subsystem embedded in the heat bath of the ions. We present sensitive quantities for the effects of temperature and self-consistency on the supershell structure, and compare our results with recent experiments and former calculations based on phenomenological mean-field potentials.

Superdeformations and fermion dynamical symmetries

In this talk, I will present a link between nuclear collective motions and their underlying fermion dynamical symmetries. In particular, I will focus on the microscopic understanding of deformations. It is shown that the SU 3 of the one major shell fermion dynamical symmetry model (FDSM) is responsible for the physics of low and high spins in normal deformation. For the recently observed phenomena of superdeformation, the physics of the problem dictates a generalization to a supershell structure (SFDSM), which also has an SU 3 fermion dynamical symmetry. Many recently discovered features of superdeformation are found to be inherent in such an SU 3 symmetry. In both cases the dynamical Pauli effect plays a vital role. A particularly noteworthy discovery from this model is that the superdeformed ground band is not the usual unaligned band but the D-pair aligned (DPA) band, which sharply crosses the excited bands. The existence of such DPA band is a key point to understand many properties of superdeformation. Our studies also pose new experimental challenge. This is particularly interesting since there are now plans to build new and exciting γ-ray detecting systems, like the GAMMASPHERE, which could provide answers to some of these challenges.

Small-scale Properties of HI in Nearby Galaxies

The instrumental requirements and current observational results are outlined for the study of resolved neutral hydrogen structures in both emission and absorption in external galaxies. Neutral super-shell structures exterior to ionized shells around OB associations seem to be a common phenomenon. Further analysis of existing data should allow quantification of the degree and rate of energy deposition in the ISM by massive stars. HI absorption studies offer great promise in constraining the physical properties of the neutral ISM, including an estimate of the gas pressure. Together these data will provide hard constraints on the existence and prevalence of chimney formation, and hence disk – halo interactions in a variety of galactic environments.

Supershells in metal clusters.

Assuming a spherical mean field for electrons in metal clusters, single-particle level densities and electronic binding energies are calculated for clusters with up to 4000 valence electrons. Two phenomenological mean-field potentials, simulating microscopically calculated ones, are used. A global beating pattern, which envelopes individual shell oscillations, emerge from the calculations. The semiclassical interpretation of such a supershell structure, as proposed by Balian and Bloch in terms of interference of amplitudes associated with classical closed orbits, is found to be valid in the present case. Thermal effects, which tend to smear out shell and, therefore, supershell structures, are investigated qualitatively. Consequences of the shell structure are not obscured for cluster sizes up to several thousand atoms under the experimentally accessible temperature of 100--1000 K.