Supershell Structure Research Articles

Hyperheavy spherical and toroidal nuclei: The role of shell structure

The properties of toroidal hyperheavy even-even nuclei and the role of toroidal shell structure are extensively studied within covariant density functional theory. The general trends in the evolution of toroidal shapes in the $Z\approx 130-180$ region of nuclear chart are established for the first time. These nuclei are stable with respect of breathing deformations. The most compact fat toroidal nuclei are located in the $Z\approx 136, N\approx 206$ region of nuclear chart, but thin toroidal nuclei become dominant with increasing proton number and on moving towards proton and neutron drip lines. The role of toroidal shell structure, its regularity, supershell structure, shell gaps as well as the role of different groups of the pairs of the orbitals in its formation are investigated in detail. The lowest in energy solutions at axial symmetry are characterized either by large shell gaps or low density of the single-particle states in the vicinity of the Fermi level in at least one of the subsystems (proton or neutron). Related quantum shell effects are expected to act against the instabilities in breathing and sausage deformations for these subsystems. The investigation with large set of covariant energy density functionals reveals that substantial proton $Z=154$ and 186 and neutron $N=228$, 308 and 406 spherical shell gaps exist in all functionals. The nuclei in the vicinity of the combination of these particle numbers form the islands of stability of spherical hyperheavy nuclei. The study suggests that the $N=210$ toroidal shell gap plays a substantial role in the stabilization of fat toroidal nuclei.

Electronic shell and supershell structure in graphene flakes

We use a simple tight-binding (TB) model to study electronic properties of free graphene flakes. Valence electrons of triangular graphene flakes show a shell and supershell structure which follows an analytical expression derived from the solution of the wave equation for triangular cavity. However, the solution has different selection rules for triangles with armchair and zigzag edges, and roughly 40 000 atoms are needed to see clearly the first supershell oscillation. In the case of spherical flakes, the edge states of the zigzag regions dominate the shell structure which is thus sensitive to the flake diameter and center. A potential well that is made with external gates cannot have true bound states in graphene due to the zero energy band gap. However, it can cause strong resonances in the conduction band.

Edge-dependent selection rules in magic triangular graphene flakes

The electronic shell and supershell structure of triangular graphene quantum dots has been studied using density functional and tight-binding methods. The density functional calculations demonstrate that the electronic structure close to the Fermi energy is correctly described with a simple tight-binding model, where only the ${p}_{z}$ orbitals perpendicular to the graphene layer are included. The results show that (i) both at the bottom and at the top of the ${p}_{z}$ band, a supershell structure similar to that of free electrons confined in a triangular cavity is seen, (ii) close to the Fermi level, the shell structure is that of free massless particles, (iii) triangles with armchair edges show an additional sequence of levels (``ghost states'') absent for triangles with zigzag edges while the latter exhibit edge states, and (iv) the observed shell structure is rather insensitive to the edge roughness.

Supershell structures and pairing in ultracold trapped Fermi gases

We calculate level densities and pairing gaps for an ultracold dilute gas of fermionic atoms in harmonic traps under the influence of mean field and anharmonic quartic trap potentials. Supershell nodes, which were found in Hartree-Fock calculations, are calculated analytically within periodic orbit theory as well as from WKB calculations. For attractive interactions, the underlying level densities are crucial for pairing and supershell structures in gaps are predicted.

Super-shell structure in harmonically trapped fermionic gases and its semi-classical interpretation

It was recently shown in self-consistent Hartree–Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced super-shell structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ‘magic numbers’ occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semi-classical trace formula that uniformly describes the symmetry breaking U(3) → SO(3) in a three-dimensional harmonic oscillator potential perturbed by an anharmonic term ∝r4 with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the twofold degenerate circular and (diametrically) pendulating orbits.

Supershell structure in trapped dilute Fermi gases

We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced super-shell structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking U(3) --> SO(3).

Jahn-Teller Distortions and the Supershell Effect in Metal Nanowires

A stability analysis of metal nanowires shows that a Jahn-Teller deformation breaking cylindrical symmetry can be energetically favorable, leading to stable nanowires with elliptic cross sections. The sequence of stable cylindrical and elliptical nanowires allows for a consistent interpretation of experimental conductance histograms for alkali metals, including both the electronic shell and supershell structures. It is predicted that for gold, elliptical nanowires are even more likely to form since their eccentricity is smaller than for alkali metals. The existence of certain metastable superdeformed nanowires is also predicted.

Quantum size effects in Pb islands on Cu(111): Electronic structure calculations

The appearance of ``magic'' heights of Pb islands grown on Cu(111) is studied by self-consistent electronic structure calculations. The Cu(111) substrate is modeled with a one-dimensional pseudopotential reproducing the essential features, i.e., the band gap and the work function, of the Cu band structure in the [111] direction. Pb islands are presented as stabilized jellium overlayers. The experimental eigenenergies of the quantum-well states confined in the Pb overlayer are well reproduced. The total energy oscillates as a continuous function of the overlayer thickness reflecting the electronic shell structure. The energies for completed Pb monolayers show a modulated oscillatory pattern reminiscent of the supershell structure of clusters and nanowires. The energy minima correlate remarkably well with the measured most probable heights of Pb islands. The proper modeling of the substrate is crucial to set the quantitative agreement.

Analysisof the shell- and supershell structures of metallic nanowireswith jellium models

The stability of metallic nanowires is studied byanalysing the shell- and supershell structures of total energyoscillations. Fully quantum-mechanical calculations areperformed for Cs, Na and Al nanowires covering the whole rangeof valence electron densities in simple metals. Nanowires aremodelled as infinite stabilized jellium cylinders and solving fortheir electronic structure self-consistently. The results areanalysed within a semiclassical model. A comparison between theshell structures of nanowires and atomic clusters is made.

Shell effects in alkali metal nanowires

After making a cold weld by pressing two clean metal surfaces together and then gradually separating the two pieces, a metallic nanowire is formed, which progressively thins down to a single atom before the contact is lost. We show evidence that the radii of such nanowires are influenced by electronic shell filling effects, in analogy to electronic shell structure in metal clusters. A modulation of the shell effect, known as supershell structure, is clearly observed. For sodium and potassium at larger diameters there is a crossover to crystalline facets with shell-closings corresponding to the completion of an additional atomic layer. We conclude that a complete analogy exists between shell structure in metallic clusters and nanowires.

From favorable atomic configurations to supershell structures: a new interpretation of conductance histograms.

Simulated minimum cross-section histograms of breaking Al nanocontacts are produced using molecular dynamics. The results allow a new interpretation of the controverted conductance histogram peaks based on preferential geometrical arrangements of nanocontact necks. As temperature increases, lower conductance peaks decrease in favor of broader and higher conductance structures. This reveals the existence of shell and supershell structures favored by the increased mobility of Al atoms.

Shell and supershell structures of nanowires: A quantum-mechanical analysis

The stability of sodium nanowires is studied by modeling them as infinite uniform jellium cylinders and solving self-consistently for the electronic structure. The total energy per unit length oscillates as a function of the wire radius giving a shell structure. The amplitude of the energy oscillations attenuates regularly, reflecting a supershell structure. We compare our theoretical results with recent experiments [A. I. Yanson et al., Nature 400, 144 (1999); Phys. Rev. Lett. 84, 5832 (2000)] performed by the mechanically controllable break junctions (MCB) technique. The comparison clarifies the origin of the observed shell structure and especially the formation of the quantum beats of the supershell structure and supports the conclusions based on an earlier semiclassical model. The comparison is also a quantitative test for the reliability of the simple stabilized-jellium model as well as for the accuracy of the equation used to relate the conductivity and the area of the narrowest point of the constriction.

Supershell structure in alkali metal nanowires.

Nanowires are formed by indenting and subsequently retracting two pieces of sodium metal. Their cross section gradually reduces upon retraction and the diameters can be obtained from the conductance. In previous work we have demonstrated that when one constructs a histogram of diameters from large numbers of indentation-retraction cycles such histograms show a periodic pattern of stable nanowire diameters due to shell structure in the conductance modes. Here, we report the observation of a modulation of this periodic pattern, in agreement with predictions of a supershell structure.

ASCA observations of the spiral galaxy M108 with HI supershell structure

Astronomische NachrichtenVolume 320, Issue 4-5 p. 303-303 Original Paper ASCA observations of the spiral galaxy M108 with HI supershell structure Y. Nagayama, Y. Nagayama Kobe Univiversity, 3-11 Tsurukabuto, Nada-ku, Kobe 657-8501, JapanSearch for more papers by this authorM. Itoh, M. Itoh Kobe Univiversity, 3-11 Tsurukabuto, Nada-ku, Kobe 657-8501, JapanSearch for more papers by this authorK. Mitsuda, K. Mitsuda ISAS, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, JapanSearch for more papers by this authorR. Fujimoto, R. Fujimoto ISAS, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, JapanSearch for more papers by this authorK. Okada, K. Okada SPring-8, Mihara 323-3, Mikazuki-cho, Sayo-gun, Hyogo 679-5198, JapanSearch for more papers by this author Y. Nagayama, Y. Nagayama Kobe Univiversity, 3-11 Tsurukabuto, Nada-ku, Kobe 657-8501, JapanSearch for more papers by this authorM. Itoh, M. Itoh Kobe Univiversity, 3-11 Tsurukabuto, Nada-ku, Kobe 657-8501, JapanSearch for more papers by this authorK. Mitsuda, K. Mitsuda ISAS, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, JapanSearch for more papers by this authorR. Fujimoto, R. Fujimoto ISAS, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, JapanSearch for more papers by this authorK. Okada, K. Okada SPring-8, Mihara 323-3, Mikazuki-cho, Sayo-gun, Hyogo 679-5198, JapanSearch for more papers by this author First published: 03 July 2001 https://doi.org/10.1002/1521-3994(199908)320:4/5<303::AID-ASNA303>3.0.CO;2-FAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume320, Issue4-5August 1999Pages 303-303 RelatedInformation

ASCA observations of the spiral galaxy M108 with HI supershell structure

ASCA observations of the spiral galaxy M108 with HI supershell structure

Supershell structure of magnetic susceptibility

The magnetic susceptibility of electrons confined to a spherical cavity or a circular billiard shows slow oscillations as a function of the number of electrons, which are a manifestation of the supershell structure found in the free energy of metal clusters. The relationship of the oscillations of the two different quantities is analyzed by means of semiclassical calculations, which are in quantitative agreement with quantal results. The oscillations should be observable for ensembles of circular ballistic quantum dots and metal clusters.

Periodic orbit theory for realistic cluster potentials

The formation of supershells observed in large metal clusters can be qualitatively understood from a periodic orbit expansion for a spherical cavity. To describe the changes in the supershell structure for different materials, one has, however, to go beyond that simple model. We show how periodic orbit expansions for realistic cluster potentials can be derived by expanding only the classical radial action around the limiting case of a spherical potential well. We give analytical results for the leptodermous expansion of Woods-Saxon potentials, and show that it describes the shift of the supershells as the surface of a cluster potential becomes softer.

Influence of the ion-pseudopotential on the electronic shell structure and the dipole strength distribution in metal clusters

Using the pseudo-Hamiltonian technique we investigate, through complete self-consistent ground-state calculations, nonlocal ion-pseudopotential effects on the dipole strength distribution and the electronic shell structure of alkali clusters. The results obtained for sodium clusters are compared to the predictions of the standard jellium model (JM) and to experiments. Some wrong JM-predictions — shift of some low magic sizes, insufficient redshift of the surface plasmon frequency relative to the Mie frequency — are corrected, but a noticeable discrepancy remains with regard to the supershell structure. In the second part of this paper we restore the reliability of the pseudo-Hamiltonian approximation, on which some doubts have been recently cast, because of incorrect predictions concerning the bulk properties of lithium. From a careful analysis of the nonlocal effects in a smoothed ionic background, we show that a specific pseudo-Hamiltonian parametrization has to be systematically selected for ensuring maximum accuracy. We successfully test the suitability of this parametrization in the case of lithium clusters, through a comparison with recently published results obtained using accurate norm-conserving nonlocal pseudopotentials.

Density dependence of the electronic supershells in the homogeneous jellium model.

We present the results of self-consistent calculations of the electronic shell and supershell structure for clusters having up to 6000 valence electrons. The ionic background is described in terms of a homogeneous jellium. The calculations were performed for a series of different electron densities, resembling Cs, Rb, K, Na, Li, Au, Cu, Tl, In, Ga, and Al, respectively. By analyzing the occupation of the energy levels at the Fermi energy as a function of cluster size, we show how the shell and supershell structure for a given density arises from the specific arrangement of energy levels. We investigate the electronic shells and supershells obtained for different electron densities. Using a scaling argument, we find a surprisingly simple dependence of the position of the supernodes on the electron density.

Simple metal clusters in magnetic fields.

The electronic supershell structure of simple metal clusters in a uniform magnetic field is studied with a spherical infinite-well potential as a simple yet realistic model for confining the valence electrons. It is found that there is little perceivable change in the supershells with experimentally achievable field strengths. If yet stronger fields are applied, the supershell structure gets destroyed and then new beat structures come about as the field strength increases further. These phenomena are explained by a recently derived semiclassical trace formula for broken symmetry that has been applied to the present system. \textcopyright{} 1996 The American Physical Society.