Mesoscopic and macroscopic dipole clusters: Structure and phase transitions

A two dimensional (2D) classical system of dipole particles confined by a quadratic potential is studied. This system can be used as a model for rare electrons in semiconductor structures near a metal electrode, indirect excitons in coupled quantum dots etc. For clusters of N ≤ 80 particles ground state configurations and appropriate eigenfrequencies and eigenvectors for the normal modes are found. Monte-Carlo and molecular dynamic methods are used to study the order-disorder transition (the “melting” of clusters). In mesoscopic clusters (N < 37) there is a hierarchy of transitions: at lower temperatures an intershell orientational disordering of pairs of shells takes place; at higher temperatures the intershell diffusion sets in and the shell structure disappears. In “macroscopic” clusters (N > 37) an orientational “melting” of only the outer shell is possible. The most stable clusters (having both maximal lowest nonzero eigenfrequencies and maximal temperatures of total melting) are those of completed crystal shells which are concentric groups of nodes of 2D hexagonal lattice with a number of nodes placed in the center of them. The picture of disordering in clusters is compared with that in an infinite 2D dipole system. The study of the radial diffusion constant, the structure factor, the local minima distribution and other quantities shows that the melting temperature is a nonmonotonic function of the number of particles in the system. The dynamical equilibrium between “solid-like” and “orientationally disordered” forms of clusters is considered.     

Structure, melting, and potential barriers in mesoscopic clusters of repulsive particles

This paper discusses two-dimensional mesoscopic clusters of particles that repel according to dipole, Coulomb, and logarithmic laws and are confined by an external parabolic potential. These models describe a number of physical systems, in particular, electrons in semiconductor structures or on a liquid-helium surface allowing for image forces, indirect excitons in coupled semi-conductor dots, and a small number of vortices in an island of a second-order superconductor or in superfluid helium. Two competing forms of ordering are detected in the particles in the mesoscopic clusters-the formation of a triangular lattice or of a shell structure. The temperature dependences of the potential energy, the mean-square radial and angular deviations, the radial and angular distributions of the particles, and the distribution of the particles over the local minima are studied. Melting in mesoscopic clusters occurs in two stages: at lower temperatures, there is orientation melting, from the frozen phase into a phase with rotational reorientation of “crystalline” shells with respect to each other; subsequently, a transition occurs in which the radial order disappears. Melting in dipole macroclusters occurs in a single stage. However, in Coulomb and logarithmic macroclusters, orientation melting occurs only for the outer pairs of shells. Orientation melting is also detected in three-dimensional Coulomb clusters. A connection is established between the character of the melting and the ratio of the energy barriers that describe the breakdown of the orientational and radial structure of a cluster. Zh. éksp. Teor. Fiz. 116, 2012–2037 (December 1999)     

Two-dimensional Abrikosov-vortex microclusters: Shell structure and melting

Melting of two-dimensional Abrikosov-vortex microclusters in a type-II superconductor island with thickness less than the coherence length has been studied. Equilibrium configurations corresponding to local and global minima of potential energy for clusters with N=1–50 particles are calculated. The temperature dependences of the structure and of mean-square radial and angular vortex displacements are investigated. It is shown that vortex microclusters melt in two stages: first the frozen-out phase transfers to a state corresponding to rotational reorientation of crystalline shells with respect to one another, followed by a transition to a state with no radial order at a substantially higher temperature. The reason for this is that the barrier to shell rotation is significantly lower than that to radial breakdown of shells. Fiz. Tverd. Tela (St. Petersburg) 39, 1005–1010 (June 1997)     

Coulomb clusters: melting and potential barriers

The melting of two-dimensional and three-dimensional Coulomb micro- and macroclusters is studied. Temperature dependences of radial and angular square deviations of particles are investigated. The melting of microclusters has two stages: at lower temperature there is a transition from a frozen phase to a state with a rotatory reorientation of “crystalline” shells relative to each other, different pairs of shells melting at different temperatures. In the case of large N and high triangular symmetry inside the cluster, orientational melting takes place only for external pairs of shells. In this case external shells lose their order. At higher temperature a transition with a loss of radial shell order occurs. The origin of two-stage melting is in the smallness of the barrier energy relative to the rotation of shells in comparison with the barrier corresponding to the radial disordering of shells. It is shown also that the temperatures of orientational and total melting are at 5–15 times lower than the temperatures of disappearance of corresponding potential barriers. The influence of confinement anisotropy on the character of cluster melting is considered. It is found that at some degree of anisotropy the melting becomes one stage. The last is connected with an increase of the ratios of barriers of intershell rotation to barriers of jumps of a particle between the shells.     

Structure and melting of dipole clusters

Two-dimensional clusters of particles, repelling due to dipole-dipole interactions and confined by an external parabolic potential, are considered. The model describes different physical systems, particularly electrons in semiconductor structures, or electrons above a drop of He near a metal electrode, a drop of colloid liquid etc. Two kinds of ordering are in competition in the clusters: a triangular lattice and a shell structure. The ground-state configurations corresponding to the local and global minima of the potential energy for clusters with N = 1 – 40 “particles” are calculated. The structure, the potential energy and the radial and angular r.m.s. displacements as functions of temperature are also calculated. Analysing these quantities the melting of clusters is studied. One- or two-stage melting occurs depending on the number of particles in the cluster. In the case of clusters consisting of two shells melting has two stages: at lower temperature reorientation of neighbouring shells (“orientational melting”) arises; at much higher temperatures the radial shell order disappears. In clusters consisting of more than two shells total melting occurs as a first-order one-stage transition (analogously to a dipole crystal). This is connected with the barrier of rotation being less than the barrier of interchange of particles between shells for small microclusters while the barriers are of equal order for clusters with a greater number of particles.     

Two-dimensional microclusters of vortices: Shell structure and melting

The melting of two-dimensional microclusters of “particles” which repel one another according to a logarithmic law and are confined by an external quadratic potential is investigated. The model describes Abrikosov vortices in a superconducting island of vortices in a rotating superfluid liquid and electrons in a semiconductor nanostructure surrounded by a low-permittivity medium. The structure of clusters and its dependence on temperature and melting are investigated. The melting of microclusters of vortices proceeds in two stages: 1. A transition from a frozen phase into a state corresponding to rotational reorientation of crystal shells relative to one another. 2. At a higher temperature, the radial order vanishes. This is connected with the fact that the barrier for rotation of the shells is much lower than the barrier for radial breakup of the shells. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 3, 268–273 (10 February 1997)     

Characteristic features of the melting of two-dimensional mesoscopic Wigner clusters

Two-dimensional Wigner microclusters in a semiconductor dot are studied. Their melting is investigated in detail and it is shown that, for typical mesoscopic clusters possessing a shell structure, melting occurs in two stages: orientational melting (rotation of the shells relative to one another) and total melting, where the shells start to overlap with one another and exchange particles. An example of a “magic” microstructure which has a triangular structure and melts in a single stage is presented. For this, the temperature dependences of various quantities characterizing cluster structure are investigated. The change in the distribution of cluster configurations over local minima of the potential energy with increasing temperature is investigated. At temperatures below the temperature of total melting, a cluster is always located near the configuration of a global minimum and, at temperatures above the temperature of complete melting, a cluster can be located with finite probability near configurations corresponding to various local minima of the potential energy. Fiz. Tverd. Tela (St. Petersburg) 41, 1499–1504 (August 1999)     

Two-dimensional mesoscopic vortex clusters in a superconducting ring: Shell structure and melting

The structure and phase transitions in the mesoscopic system of vortices in a quasi-two-dimensional superconducting ring are investigated. The shell structure of the mesoscopic system of vortices is studied, and its variation with the number of vortices and the parameters of the superconducting ring is analyzed. Two mechanisms of formation of new shells in vortex clusters with an increasing number of vortices in an increasing magnetic field are discovered: the generation of a new shell in a cluster and the splitting of the internal shell into two shells. The melting of vortex clusters and their thermodynamic parameters are analyzed using the Monte Carlo method. It is found that the melting of shell-type clusters occurs in two stages, orientation melting taking place at the lower temperature (during which nearly crystalline adjacent shells start rotating relative to each other) and blurring of the vortex structure occurring at the higher temperature. The shells obtained by splitting upon an increase in the number of vortices do not participate in orientational melting. The two-stage form of melting is associated with the smaller height of potential barriers being surmounted during the rotation of shells relative to one another as compared to the barrier for vortices jumping from one shell to another.     

Molecular dynamics study of orientational melting and thermodynamic properties of C<Subscript>60</Subscript>@C<Subscript>240</Subscript> nanoparticles

The barriers to relative shell rotation and other energy characteristics of C₆₀@C₂₄₀ two-shell carbon nanoparticles (“onions”) with outer shells of different shapes are calculated. The disturbance of the orientational order in the mutual arrangement of shells with an increase in temperature (orientational melting) is studied by the molecular dynamics method. The intershell orientational diffusion is represented by the Arrhenius relationship, and the Arrhenius parameters are calculated numerically. A definition is proposed for the temperature of short-range order disturbance in systems that undergo melting without structural change. The calculated temperature of orientational melting of the C₆₀@C₂₄₀ nanoparticle is approximately equal to 60 K.     

Melting behavior of large disordered sodium clusters

The melting-like transition in disordered sodium clusters Na₉₂ and Na₁₄₂ is studied by performing density functional constant-energy molecular dynamics simulations. The orbital-free version of the density functional formalism is used. In Na₁₄₂ the atoms are distributed in two distinct shells (surface and inner shells) and this cluster melts in two steps: the first one, at ≈130 K, is characterized by the development of a high intrashell atomic mobility, and the second, homogeneous melting at ≈270 K, involves diffusive motion of all the atoms across the whole cluster volume. On the contrary, the melting of Na₉₂ proceeds smoothly over a very broad temperature interval, without any abrupt change in the thermal or structural indicators. The occurrence of two steps in the melting transition is suggested to be related to the existence of a grouping of the atoms in radial shells, even if those shells present a large degree of internal disorder. It then appears that a cluster can be considered fully amorphous (totally disordered) only when there are no radial regions of low atomic density separating shells. The isomer of Na₉₂ studied here fulfills this criterion and its thermal behavior can be considered as representative of that expected for fully amorphous clusters. Disordered Na₁₄₂, on the other hand, that has a discernible structure of an inner and a surface shell, should be considered as not fully disordered. The thermal behavior of these two clusters is also compared to that of icosahedral (totally ordered) sodium clusters of the same sizes. Received 5 February 2001 and Received in final form 21 May 2001     

Wigner crystallization in mesoscopic 2d electron systems

Wigner crystallization of electrons in 2D quantum dots is reported. It proceeds in two stages: (i) via radial ordering of electrons on shells and (ii) freezing of the intershell rotation. The phase boundary of the crystal is computed in the whole temperature-density plane, and the influences of quantum effects and the particle number are analyzed.     

Shell evolution in neutron-rich nuclei: the single particle perspective

The isospin dependence of spin-orbit(SO)splitting becomes increasingly important as N/Z increases in neutron-rich nuclei.Following the initial independent-particle strategy toward explaining the occurrence of magic numbers,we systematically investigated the isospin effect on the shell evolution in neutron-rich nuclei within the Woods-Saxon mean-field potential and the SO term.It is found that new magic numbers N=14 and N=16 may emerge in neutron-rich nuclei if one changes the sign of the isospin-dependent term in the SO coupling,whereas the traditional magic number,N=20,may disappear.The magic number N=28 is expected to be destroyed despite the sign choice of the isospin part in the SO splitting,corresponding to the strength of the SO coupling term.Meanwhile,the N=50 and 82 shells may persist within the single particle scheme,although there is a decreasing trend of their gaps toward extreme proton-deficient nuclei.Besides,an appreciable energy gap appears at N=32 and 34 in neutron-rich Ca isotopes.All these results are more consistent with those of the interacting shell model when enhancing the strength of the SO potential in the independent particle model.The present study may provide a more reasonable starting point than the existing one for not only the interacting shell model but also other nuclear many-body calculations toward the neutron-dripline of the Segrèchart.     

Peculiar thermodynamic properties of LJ<Subscript> N</Subscript> (N = 39–55) clusters

The solid-liquid phase transitions of Lennard-Jones clusters LJ_N (N=39–55) were simulated by a microcanonical molecular dynamics method using Lennard-Jones potential, and their thermodynamic quantities were calculated. The caloric curves of clusters (except N=42) have S-bend. To understand this behaviour, configurational and total entropies were evaluated, and dents on the entropy curves were taken as a sign of negative heat capacity. The heat capacities were evaluated for N=39–55 clusters using configurational entropy data. The potential energy distributions have bimodal behaviour for all clusters in the given range at the melting temperature. The distinct melting behaviour of LJ₄₂ was explained by the topology of the potential energy surface by examining the isomer distributions at phase transitions for LJ₃₉-LJ₅₅. The isomer distributions were found to be a useful way to interpret this behaviour and melting dynamics in general. Melting temperature, latent heat and entropy change upon melting values were reported and are consistent with literature values and values calculated from bulk thermodynamic properties. The dependence of these quantities on the size of the clusters was examined and it is found that latent heat is the key quantity to determine the magic numbers.     

Melting properties of two-dimensional multi-species colloidal systems in a parabolic trap

The angular and radial melting properties of two-dimensional classical systems consisting of different types of particles confined in a parabolic trap are studied through modified Monte Carlo simulations. A universal behavior of the angular melting process is found, which occurs in multiple steps due to shell depended melting temperatures. The melting sequence of the different shells is determined by two major factors: (1) the confinement strength which each shell is subjected to, and (2) the specific structure of each shell. Further, a continuous radial disordering of the particle types forming a single circular shell is found and analyzed. This phenomenon has never been observed before in two-dimensional mono-dispersive systems. This continuous radial disordering results from the high energy barrier between different particle types in multi-species systems.     

Emergence of new magic numbers,N = 16 and 32 by tensor interaction in Skyrme–Hartree–Fock theory

Melting of N = 20 shell and development of N = 16 and 32 shells for neutron-rich nuclei have been studied extensively by including tensor interaction in Skyrme–Hartree–Fock theory optimized to reproduce the splitting Δ1f shells of ^40,48Ca and ⁵⁶Ni nuclei. Evolution of gap generated by the energy difference of single-particle levels ν2s _1/2 and ν1d _3/2 has been found to be responsible for shell closure at N = 16. The splitting pattern of spin–orbit partners 2p shell model state in Ca, Ti, Cr, Fe and Ni isotopes indicates the formation of a new shell at N = 32 region.     

Neutron and proton shell closure in the superheavy region via cluster radioactivity in <Superscript>280–314</Superscript>116 isotopes

Based on the concept of cold valley in fission and fusion, the radioactive decay of superheavy^280–314116 nuclei was studied taking Coulomb and proximity potentials as the interacting barrier. It is found that the inclusion of proximity potential does not change the position of minima but minima become deeper which agrees with the earlier findings of Gupta and co-workers. In addition to alpha particle minima, the other deepest minima occur for ⁸Be, ^12,14C clusters. In the fission region two deep regions are found each consisting of several comparable minima, the first region centred on ²⁰⁸Pb and the second is around ¹³²Sn. The cluster decay half-lives and other characteristics are computed for various clusters ranging from alpha particle to ⁷⁰Ni. The computed half-lives for alpha decay match with the experimental values and with the values calculated using Viola-Seaborg-Sobiczewski (VSS) systematic. The plots connecting computed Q values and half-lives against neutron number of daughter nuclei were studied for different clusters and it is found that the next neutron shell closures occur at N = 162, 172 and 184. Isotopic and isobaric mass parabolas are studied for various cluster emissions and minima of parabola indicate neutron shell closure at N = 162, 184 and proton shell closure at Z = 114. Our study shows that ₁₆₂²⁷⁶114 is the deformed doubly magic and ₁₈₄²⁹⁸114 is the spherical doubly magic nuclei.      

Nuclei near the closed shells N = 20 and N = 28

The present work reviews the properties of the neutron-rich isotopes near the closed shells N = 20 and N = 28. The changes in nuclear structure appearing as one goes away from the β-stability line are discussed. The location of the neutron drip line and questions about the stability of nuclides with Z ≥ 8 are considered in connection with the weakening or even vanishing of the shell effects at the magic numbers 20 and 28, and the discovery of the new neutron magic numbers at N = 16 and N = 32. These properties are extremely interesting from the point of view of laser experiments as well as for all other experimental methods giving access to this region.     

Thermodynamic properties of noble metal clusters:molecular dynamics simulation

The thermodynamics properties of noble metal clusters Au_N, Ag_N, Cu_N, and Pt_N (N = 80, 106, 140, 180, 216, 256, 312, 360, 408, 500, 628, 736, and 864) are simulated by micro-canonical molecular dynamics simulation technique. The potential energy and heat capacities change with temperature are obtained. The results reveal that the phase transition temperature of big noble metal clusters (N ⩾ 312 for Au, 180 for Ag and Cu, and 360 for Pt) increases linearly with the atom number slowly and approaches gently to bulk crystals. This phenomenon indicates that clusters are intermediate between single atoms and molecules and bulk crystals. But for the small noble clusters, the phase transition temperature changes irregularly with the atom number due to surface effect. All noble metal clusters have negative heat capacity around the solid-liquid phase transition temperature, and hysteresis in the melting/freezing circle is derived in noble metal clusters.     

The problem of the structure (state of helium) in small He<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>-CO clusters

A second-order perturbation theory, developed for calculating the energy levels of the He-CO binary complex, is applied to small He _N -CO clusters with N = 2−4, the helium atoms being considered as a single bound object. The interaction potential between the CO molecule and HeN is represented as a linear expansion in Legendre polynomials, in which the free rotation limit is chosen as the zero approximation and the angular dependence of the interaction is considered as a small perturbation. By fitting calculated rotational transitions to experimental values it was possible to determine the optimal parameters of the potential and to achieve good agreement (to within less than 1%) between calculated and experimental energy levels. As a result, the shape of the angular anisotropy of the interaction potential is obtained for various clusters. It turns out that the minimum of the potential energy is smoothly shifted from an angle between the axes of the CO molecule and the cluster of θ = 100° in He-CO to θ = 180° (the oxygen end) in He₃-CO and He₄-CO clusters. Under the assumption that the distribution of helium atoms with respect to the cluster axis is cylindrically symmetric, the structure of the cluster can be represented as a pyramid with the CO molecule at the vertex.     

Quantum orientational melting and the phase diagram of a mesoscopic system

The “phase diagram” of a two-dimensional mesoscopic system of bosons is investigated. An example of such a system is a system of indirect magnetoexcitons in semiconductor double quantum dots. Quantum Monte Carlo calculations show the existence of quantum orientational melting. At zero (quite low) temperature, as quantum fluctuations of the particles intensify, two quantum disordering phenomena occur with increasing de Boer parameter q. First, at q≈10⁻³ the system passes to a radially ordered but orientationally disordered state, where different shells of a cluster rotate relative to one another. Then at q≈0.16 a transition to a superfluid state occurs. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 11, 817–822 (10 December 1998)