Physical cluster mechanics: Statistical thermodynamics and nucleation theory for monatomic systems

We extend previous computations of mechanical stability of atomic microclusters to the realm of statistical thermodynamics, obtaining thermodynamic functions for small, solid-like Van der Waals clusters of less than some 100 atoms possessing non-crystalline structures of ‘polytetrahedral’ type. These are shown to be almost invariably at a thermodynamic advantage over alternative lattice structures of the same number of atoms, at least for the Lennard-Jones potential in the harmonic-oscillator/rigid-rotor approximation. The dependence of thermodynamic functions upon cluster size appears to be essentially monotonic in the number of internal degrees of freedom; although there are certain exceptional structures, particularly with icosahedral symmetry, there proves to be little evidence for the occurrence of ‘magic numbers’ for stability at any temperature and within the size-range considered. Particular attention is given to the heat capacity of model systems in relation to their vibrational spectra. The Debye T ³ law appears reasonably well obeyed at low temperatures with no evidence for the existence of either ‘soft modes’ or distinct surface contributions.

The results for the free energy of formation of minimal clusters ΔG _f are then applied to the computation of nucleation rates in terms of the Becker-Döring-Volmer-Zeldovitch quasi-equilibrium theory. Gibbsian behaviour in the form of a maximum in the curve of ΔG _f versus size is observed with a critical nuclear size at realistic temperatures and pressures of the order of that predicted by macroscopic liquid-drop theories. These figures and those derived for nucleation rate and critical supersaturation appear remarkably insensitive to the details of the model used, in particular to the distinction between ‘microcrystalline’ and ‘amorphous’ atomistic models.

The general status of atomistic nucleation theory is critically examined in the light of these and similar results.     

Physical cluster mechanics: Statics and energy surfaces for monatomic systems

The potential energy surfaces for clusters of some three to sixty atoms under Lennard-Jones forces have been systematically explored using numerical optimization techniques. In searching for minimum-energy configurations three particularly compact non-lattice growth schemes emerge showing tetrahedral, pentagonal (D_{5 ^h} ) and icosahedral symmetry respectively. All these systems were found to be appreciably more stable than microcrystallites based on the face-centred cubic structure while certain lattice elements were shown to be metastable for small numbers of atoms.

Some qualitative conclusions are then drawn concerning the occurrence of saddle-points for delocalized motion and the contribution of these to the generation of configurational entropy in small clusters. A crucial feature of the energy surfaces examined is the breakdown of strict local symmetry in compact clusters of more than two ‘shells’ of atoms and the possibility of delocalized motion of surface atoms around a solid-like core. These and other qualitative thresholds are pointed out as possible manifestations of entropy which could contribute to the existence of a ‘critical nucleus’ for discrete clusters with a role similar to that played in liquid-drop theories.     

Thermodynamic and structural properties of liquids modelled by ‘2-Lennard-Jones centres’ pair potentials

Molecular dynamics calculations have been carried out for model liquid systems of N (=108 or 256) molecules interacting through two Lennard-Jones (12–6) centres coinciding with the positions of the atomic masses (the ‘atom-atom’ pair potential). The objectives were (a) to study the dependence of the properties on the molecular anisotropy defined by the reduced distance l*=l/σ between the centres in the range 0·5–0·8; and (b) to compare the computed quantities with those of real liquids (F₂, Cl₂, Br₂, CO₂). This paper deals with thermodynamic and structural features. Time-dependent correlations will be treated in a future communication.

In the liquid region not too far from the triple point the energy and pressure isochores are well represented by straight lines, the slopes of which increase with density and anisotropy. Thermodynamically consistent expressions for the energy and pressure as functions of density and temperature have been obtained for each system.

With Lennard-Jones parameters adjusted so as to secure the best overall fit, the agreement between experimental and computed thermodynamic properties is very satisfactory for F₂ (l*=0·505), quite good for Cl₂ and Br₂ (l*=0·608–0·63), but rather poor for CO₂ (l*=0·793). The ‘interatomic distances’ are close to the experimental values.

The static structural correlations are discussed in terms of the pair-correlation functions (pcf) g _A(r*) for the separation between ‘atoms’, the first few functions g_ll'm (R*) which arise from the expansion of the g(R*, θ₁, θ₂, φ₁₂) in spherical harmonics, and the pcf's for certain special near-neighbour configurations. The computed atom-atom structure factor is compared with the experimental data for liquid Br₂.

Mean square forces and torques have been evaluated and are related to some experimental results.     

Relationship between the parting limit for de-alloying and a particular geometric high-density site percolation threshold

The parting limit or de-alloying threshold for electrolytic dissolution of the more reactive component from a homogeneous fcc binary alloy is usually between 50 and 60 at%. The system that has been most studied, dissolution of Ag from Ag–Au, shows a parting limit close to 55 at% Ag. Here, Kinetic Monte Carlo (KMC) simulations of ‘Ag–Au’ alloys and geometric percolation modeling are used to study the relationship between this parting limit and the high-density site percolation thresholds p _c(m) for an fcc lattice, subject to the rule that atoms with coordination greater than nine are prevented from dissolution. The value of p _c(9) is calculated from geometric considerations to be 59.97 ± 0.03%. In comparison, using KMC simulations with no surface diffusion and no dissolution allowed for ‘Ag’ atoms with more than nine total neighbors, the parting limit is found to be slightly lower (58.4 ± 0.1%). This slight discrepancy is explained by consideration of the local atomic configurations of ‘Ag’ atoms – a few of these configurations satisfy the percolation requirement but do not sustain de-alloying, while a larger number show the converse behavior. There is still, however, an underlying relationship between the parting limit and the percolation threshold, because being at p _c(9) guarantees a percolation path in which successive ‘Ag’ atoms share at least one other ‘Ag’ neighbor. With realistic kinetics of surface diffusion for ‘Au’, the parting limit drops to 54.7 ± 0.3% because a few otherwise inaccessible dissolution paths are opened up by surface diffusion of ‘Au’.     

Structures of inert atomic clusters and their magic numbers of geometry

The optimal configurations of all atoms in atomic microclusters of an inert element have been obtained from their arbitrary positions and shapes by means of a Lennard-Jones interaction potential between atoms in the clusters, calculating the binding energies of the clusters with the numbers of atoms N ⩽ 14, which have shown the magic numbers of geometry in accordance with the experimental results. The structural pictures of such clusters are also presented.     

Structure and thermodynamic characteristics of small inert gas clusters

The relaxation dynamics of small groups of identical atoms interacting according to the Lennard-Jones law was studied experimentally. It is shown that for a fixed number of atoms, the probabilities of the formation of clusters with different structures depend on the random initial distribution of atoms in space, i.e., on the initial total energy and geometry of the particle distribution. Probabilities of the emergence of different structures of clusters vary greatly and do not contradict classical statistics. Except in extraordinary cases (e.g., N = 13), distances between the nearest atoms in clusters are different and change with the addition of each subsequent atom. The thermodynamics is constructed from the canonical ensembles of clusters with different numbers of particles. The resulting dependence of the cluster energy on the number of particles proves to be a smooth function, since only pair interactions were taken into account.     

The structure of para-hydrogen clusters

The path integral Monte Carlo calculated radial distributions of para-hydrogen clusters $({\rm p}\text{-}{\rm H}_2)_N$ consisting of N = 4-40 molecules interacting via a Lennard-Jones potential at $T=1.5~{\rm K}$ show evidence for additional peaks compared to radial distributions calculated by diffusion Monte Carlo ( $T=0~{\rm K}$ ) and path integral Monte Carlo at $T \leq 0.5~{\rm K}$ . The difference in structures is attributed to quantum delocalization at the lowest temperature. The new structures at finite temperatures appear to be consistent with classical structures calculated for an effective Morse potential, which in order to account for the large zero point energy, is substantially softer than the Lennard-Jones potential.     

Interaction induced light scattering in orientationally disordered crystals: the translational phonon region

We have derived an expression for the light scattering spectrum of a crystal in which the mechanically regular sites are occupied by point polarizable orientationally disordered molecules when the polarizabilities are assumed to depend on the positions of the surrounding atoms (interaction induced polarizability fluctuations).

Owing to the ‘electrical’ disorder properties of the system all phonons can contribute to the anisotropic scattering measured in all polarization configuration with wavevector, branch index and polarization dependent coefficients. Assuming short range interaction induced polarizabilities we show that the temperature reduced intensity I(ω)/[n(ω) + 1] is given by a superposition of the Brillouin zone centre symmetry ‘projected’ density of states with polarization dependent coefficients. These coefficients are found to be essentially frequency independent for all the projections, exception made for those corresponding to the acoustic phonons. For the acoustic branches the coefficients vanish in a first approximation. They can however be treated on more rigorous grounds and, as already found by other authors, their contribution is proportional to the density of states multiplied by ω².

In addition zone centre (k?0) totally symmetric phonons can also be present in the ‘isotropic’ term (which appears only in the polarized VV configuration).

A procedure is suggested to obtain the total density of states from the spectra measured in different polarization configurations.     

Theory of the liquid-vapour interface of a binary mixture of Lennard-Jones fluids

We present results of calculations of the equilibrium surface tension and density profiles for the liquid-vapour interface of a binary mixture of Lennard-Jones 12-6 fluids. The calculations are based on a density-functional theory for the Helmholtz free energy of the inhomogeneous mixture. This is a ‘microscopic’ generalization of the van der Waals-Cahn-Hilliard theory for the interface of a binary mixture.

Our calculations cover the full range of liquid-vapour coexistence and the whole range of concentration. We find a correlation between the excess surface tension of the mixture and the surface segregation (adsorption) of the species with the lower surface tension. The ways in which segregation and excess surface tension depend on the Lennard-Jones parameters of the pure components are briefly discussed. Our results for the excess surface tension of mixtures of Ar and N₂ and Ar and CH₄ are compared with experiment; the agreement is reasonable.     

An algebraic description of anharmonic diatom–diatom inelastic collisions in the semiclassical approximation

An algebraic model to describe inelastic collisions between two anharmonic diatomic molecules in the semiclassical approximation is presented. The interactions for the diatomic systems are modelled in terms of Morse potentials, while an exponential repulsive potential is taken for the interactions between the nearest atoms of the diatomic systems. This problem is treated in the interaction potential framework, where an approximation in terms of the generators of three SU(2) groups is proposed, two corresponding to the Morse oscillators and the other to the interaction. The transition probabilities are given in terms of a sum of the products of three Wigner's d(β) functions corresponding to the three SU(2) groups. As an example the systems N₂?+?N₂ and H₂?+?H₂ are described and compared with exact quantum mechanical calculations.     

Quantum effects in rutherford back-scattering and emission of electrons near a crystal axis

Abstract

By increasing the strength of the interaction potential for a reasonable range of values, we show that the angular pattern of the electron intensity, about a crystallographic axis, changes from one dominated by Bragg peaks to an essentially structureless curve. This retains a significant dependence on h, and by that token cannot be identified with the ‘classical envelope’.     

Green’s function method in the problem of complex configurations in fermi systems with pairing

The Green’s function method is used to derive general equations for describing effects of pairing in Fermi systems where there are two types of interaction, two-particle and quasiparticle-phonon interaction. These equations generalize Bardeen-Cooper-Schrieffertheory to the case of complex configurations involving “strong” phonons. In the approximation of weak coupling to phonons, realistic equations that make it possible to describe excited states of nonmagic even-even nuclei with allowance for a single-particle continuum and complex configurations of the two quasiparticles ? phonon type are formulated for the first time. These equations are solved for an isovector E 1 resonance in the stable isotope ¹²⁰ Sn and in the unstable isotopes ^104,132Sn. It is shown that complex configurations must be taken into account in order to describe E1 excitations—in particular, in a broad energy region around the nucleon binding energy.

     

Structures and autocorrelation functions of liquid Al and Mg modelled via Lennard-Jones potential from molecular dynamics simulation

The structures and autocorrelation functions of Al and Mg in the liquid state are investigated through the pair distribution functiong(r), the diffusion coefficients as well as the shear viscosity via the Green-Kubo and Einstein relations. From the structure and the Enskog relation we determined the frequency of collisions of atoms in the first shell ofg(r) in the systems. We also discovered that the packing fraction of Lennard-Jones liquids should be approximately half the reduced density value. This approximation is accurate to within 99%. The temperature dependence of the pair distribution function and the atomic mean square displacement are investigated by performing simulations at various experimental temperatures and corresponding densities. The structures of the systems are affected by temperature via movements of atoms in the first minimum ofg(r). The Lennard-Jones model shows that density dependence of the shear viscosity is in agreement with what is expected of simple liquids in the range of investigated temperatures and densities. In the gas limit, the Stoke-Einstein relationDη =K _BT /2πσ is grossly overestimated by Lennard-Jones model. This could not be attributed to deficiencies in the model, as other investigators using first principle method could not obtain the gas limit of the Stoke-Einstein relation.     

Comparisons between statistics of low energy collision cascades generated by full molecular dynamics and by its binary collision approximation

Abstract

Collision cascades in Cu, Au and Cu₃Au are generated by full molecular dynamics (MD) and by its binary collision approximation (BCA) with the Marlowe program. Cu and Au primaries have 1 keV initial energy.

The same Molière repulsive potential is used in both models for close encounters. In the MD model, this potential is carefully splined to the pair component of the N-body potential developed by Ackland and Vitek. In the BCA, this N-body interaction is roughly modeled by a constant isotropic 4 eV binding energy of the target atoms to their rest positions.

Time distributions of the number of atoms moving with a total energy higher than a threshold value E _d are compared and discussed. Recoil range distributions during the cascade development are discussed as well. The agreement between MD and BCA is fairly good in all cases for E _d larger than about 3 eV. In the case of smaller E _d-values, the BCA may result in an overestimate of the number of moving atoms in the late development of the cascades. This discrepancy is suggested to originate in the lack of attractive forces between the moving particles and the surrounding atoms in the BCA.     

Effect of Configuration Mixing and Core Excitation on Predominantly (f 7/2)n States in the Lower fp Shell Nuclei

Shell model calculations in the lower fp shell region, (for ^44,45Ca and ⁴⁷Sc nuclei) have been performed, with different model spaces, to probe the effect of configuration mixing from the p^3/2, p^1/2 and f_5/2 orbitals on the predominantly (f_7/2)ⁿ states and the contribution arising from the excitation of the N = Z = 16 core. Our calculations indicate that excitation of nucleons across the N = Z = 16 magic shell closure do contribute significantly towards the wavefunctions of the observed level structures of ^44,45Ca nuclei. However the inclusion of these configurations did not result in a better agreement for the observed level structure of ⁴⁷Sc nulcei. A plaussible explaination for this phenomena could be attributed to the two-body matrix elements used and calls for a detailed micro-scopic calculations involving fundamental interactions substantiated by additional spectroscopic data such as lifetime measurements to have an unambigious understanding of the intrinsic configurations of nuclei in this region.     

Piezoelectric properties of non-uniform polymeric structures under static load

ABSTRACT

Piezo-electrical properties of structures containing ‘soft’ and ‘hard’ dielectric layers with charge stored on the interfaces are described in the paper. The piezo-activation process of structures containing layers with gas voids by partial discharges is described. The influence of the mechanical properties of the ‘soft’ layer on the piezoelectric parameter d₃₃ value and its dependence on the static pressure p are also discussed. It was found experimentally, that for the fibrous type of ‘soft’ dielectric layer, the dependence of the piezoelectric parameter d₃₃ (p) can be described by the function d₃₃ ÷ p^?n, where n ≤ 1.     

Neutron Drip Line for Nuclei in the Vicinity of the Neutron Magic Number N = 184

Ground-state properties of even-even nuclei were calculated over a broad region of mass numbers, including nuclei that contain a neutron excess in the vicinity of the neutron drip line. The calculation of the properties of such nuclei relied on the method of the relativistic and the nonrelativistic mean field and took into account the axial deformation of nuclei. Particular attention was given to nuclei beyond the theoretical neutron drip line, which form a peninsula of nuclei in the (N, Z) space at N = 184 that are stable against the emission of one or two neutrons.

     

Study of near surface layer of graphite produced by nitrogen ion bombardment at high doses

To study the modified surface layers of graphites and deposited films of sputtered material, the dependences of sputtering yield Y , and ion-electron emission coefficient γ on ion incidence angle and target temperature under high dose 30 keV N⁺ ₂ ion irradiation have been measured. In the angular range θ=0-80° Y and γ increase approximately as inverse cosθ, Y of POCO-AXF-5Q are 1.5 times larger than of MPG-LT. The dependences of γ (T) manifests a step-like behaviour typical for the radiation induced phase transitions. EPR analysis shows that at near room temperatures the point electron defects are typical of carbon and the defects due to carbon atoms interacting with 14 N nuclei. At elevated temperatures (≥ 300°C) there are the defects typical of graphite-like structures. The films deposited on glass collectors shows for cold targets only the defects typical of carbon, for the heated graphites - also the defects associated with C-¹⁴N nuclei interaction.     

Group leaders optimization algorithm

We present a new global optimization algorithm in which the influence of the leaders in social groups is used as an inspiration for the evolutionary technique which is designed into a group architecture. To demonstrate the efficiency of the method, a standard suite of single and multi-dimensional optimization functions along with the energies and the geometric structures of Lennard-Jones clusters are given as well as the application of the algorithm on quantum circuit design problems. We show that as an improvement over previous methods, the algorithm scales as N ^2.5 for the Lennard-Jones clusters of N-particles. In addition, an efficient circuit design is shown for a two-qubit Grover search algorithm which is a quantum algorithm providing quadratic speedup over the classical counterpart.     

Dynamics of α-clusters in N = Z nuclei

The systematics for binding energies per α-particle in N = Z nuclei, E _Bα/N _α, are studied up to ¹⁶⁴Pb. It is shown that, although a geometrical model can be used to explain the systematics for light nuclei, the binding energy per α-particle exhibits structures which are due to the well-known shells of the mean field of nucleons in nuclei. The overall dependence of E _Bα/N _α on N _α in N = Z nuclei (for the ground-state masses) can be described in a liquid-drop model of α-particles. Conditions for a phase change with the formation of an α-particle condensate, a dilute Bose gas in excited compound nuclei are discussed for E _Bα/N _α = 0, at the thresholds. This is achieved when the binding energy per nucleon in nuclei is equal to or smaller than in the α-cluster. At somewhat smaller excitation energies the appearance of a Bose gas with a closed-shell core (N = Z, e.g. of ⁴⁰Ca) is proposed within the same concept. The experimental observation of the decay of such condensed α-particle states is proposed with the coherent emission of several correlated α-particles not described by the Hauser-Feshbach approach for compound-nucleus decay. This decay will be observed by the emission of unbound resonances in the form of ⁸Be and ¹²C ^* (0⁺ ₂) clusters.