Two-body Central Force Research Articles

Molecular Dynamics Simulations of Perovskites: The Effect of Potential Function Representation on Equilibrium Structural Properties

The perovskites with general formula ABX3 have been widely used as for materials with their unique properties (ferroelectric, piezoelectric, dielectric, catalytic and so on). Hybrid organolead halide perovskites are a class of semiconductors with ABX3 (X = Cl, Br, and I) structures consisting of lead cations in 6-fold coordination (B site), surrounded by an octahedron of halide anions (X site, face centered) together with the organic components in 12-fold cub octahedral coordination. These hybrid perovskites have a direct band gap, a large absorption coefficient as well as high charge carrier mobility that represent a very attractive characteristic of cost-effective solar cells. Basically, these crystals are inorganic solids of CaTiO3 type held together by bonds that are either ionic or partially ionic and partially covalent. In spite of the partially covalent character of the Ti-O bond, the system is modeled by a two-body central force interatomic potential (the form of the Vashishta and Rahman interatomic potential), which has been used successfully for many materials with a perovskite structure. In the present work using molecular dynamics (MD) simulation method we investigate the dynamical and structural behavior of CaTiO3 perovskite at normal pressure and temperature conditions. The MD calculations were performed on a system of 16,000 particles (3200Ca + 3200Ti + 96,00O), initially in an orthorhombic-Pbnm structure. The orthorhombic MD box had edges Lx = 53.4 A, Ly = 53.4 A and Lz = 61.12 A, which provided a density matching the experimental value of ρ = 4 g/cm3. Starting with this structure and using proposed interatomic potentials the MD system stabilizes at room temperature in its initial configuration. The aim of the present study to explore the effect of potential function representations on structural equilibrium properties for the perovskite models including hybrid halide ones outlined above. Concerning the perovskite equilibrium state we elucidate the role of potential function modification on the atomic pair correlation and structural re-organization. The details of the interatomic potential representation have to be crucially important for obtaining of correct analysis data in crystallic, liquid and amorphous phases including perovskite systems.

Explicit Energy Functional for Infinite Nuclear Matter with the Tensor Force

We have applied the variational method using explicit energy functionals (EEFs) to energy calculations of infinite nuclear matter. In EEFs, the energy per nucleon is explicitly expressed with spin-isospin-dependent two-body distribution functions, which are regarded as variational functions, and fully minimized energies are conveniently calculated with the EEF. A remarkable feature of this approach is that EEFs guarantee non-negativeness of structure functions. In this study, we extend the EEF variational method so as to consider state- independent three-body forces for neutron matter at finite temperatures following the procedure proposed by Schmidt and Pandharipande. For neutron matter, the free energies obtained with the Argonne v4' two-body potential and the repulsive part of the Urbana IX (UIX) three- body potential are quite reasonable. Furthermore, we improve the EEF of nuclear matter using the two-body central and tensor forces by considering the main three-body cluster terms and guaranteeing non-negativeness of tensor structure functions. In addition, healing distances are introduced for two-body distribution functions so that Mayer's condition is satisfied. The obtained energies per neutron of neutron matter with the Argonne v6' two-body potential and the repulsive part of the UIX potential are in good agreement with those obtained by auxiliary field diffusion Monte Carlo calculations.

The importance of the three-particle correlation function in the theory of monatomic classical liquids like argon

The present study was motivated by the work of Fortov et al. (2008) on the use of experimental methods of analysis of spatial correlation of dust particles in plasma. Here, we therefore focus on the importance of the three-particle correlation function g 3 in the theory of monatomic classical liquids like argon, in the framework of two-body central forces. We cover (i) the asymptotic relation of g 3 to the Ornstein–Zernike direct correlation function c(r); (ii) the use of g 3 to determine the specific heat difference c P − c V ; and (iii) g 3 related to the pressure dependence of the pair correlation function.

Scaling of the failure stress of homophase and heterophase three-dimensional spring networks

This paper concentrates on the scaling of the failure stress of a three-dimensional spring network as a function of its volume. In particular, the influences of the geometry and the local structure are examined. Both homophase disordered three-dimensional structures and composite systems are studied, containing a more or less ordered slab. The structures are generated by starting with a node distribution. A molecular-dynamics-based algorithm uses void volume spheres, which all have Lennard-Jones interacting outer surfaces. The generated distributions of nodes form the basis of a procedure to interconnect the nodes with springs. In the calculation of the failure stress the total elastic energy is described by two-body central force, three-body bond bending, and four-body torsion contributions. The areas under uniaxial compression are varied in the range of 0.64--5.76 \ensuremath{\mu}${\mathrm{m}}^{2}$, and the height h ranges between 0.80 and 6.4 \ensuremath{\mu}m. It is found that the failure stress at constant base area could be described by ${\ensuremath{\sigma}}_{\mathrm{fail}}\ensuremath{\propto}[\mathrm{log}(h/\ensuremath{\xi}){]}^{\ensuremath{-}1/\ensuremath{\mu}},$ where \ensuremath{\xi} represents the correlation length within the sample the (logarithm is to the base e). The values of \ensuremath{\mu} are effective values. Only within the same kind of failure mechanism and microstructure does the exponent \ensuremath{\mu} become more or less universal. Actually, the modulus \ensuremath{\mu} appears to depend on the system size, but in all cases thin samples are stronger than thick samples under uniaxial compression, and the failure stress increases with increasing coordination number. The failure stress of heterophase materials differs considerably in our calculations from that of homophase materials. The composite materials exhibit an increase in strength by a factor of 4, in comparison to the disordered structures of the same size. The actual failure stress of the composite material depends critically on the layering effect of the disordered region near the ordered phase.

EFFECT OF SUBSTITUTION AND PRESSURE ON LITHIUM ION CONDUCTIVITY IN PEROVSKITES Ln[formula omitted]Li[formula omitted]TiO3 (Ln = La, Pr, Nd AND Sm)

The effect of substitution and hydrostatic pressure on lithium ion conductivity in polycrystalline Ln12Li12TiO3 (Ln = La, Pr, Nd and Sm) were investigated and the ion conduction mechanism in perovskite-type oxides was elucidated. The lithium ion conductivity was found to decrease, accompanied by an increase of the activation energy and volume for the ion conduction when the ionic radius of Ln ions was decreased or high pressure was applied for these compounds. A simple energetic calculation using a two-body central force has confirmed the importance of the bottle-neck size for ion conduction, which effectively determines the value of the repulsion potential between oxygen and lithium ions. © 1997 Elsevier Science Ltd

Refinement of the Energy Expression for Infinite Fermion Systems with Central Forces

A refinement is made on the approximate energy expression proposed previously by Takano and Yamada for infinite zero-temperature systems of spin-1/2 fermions interacting through two-body central forces which may depend on the two-body spin states. This refinement is accomplished through incorporation of a certain part of the three-body-cluster exchange terms into the energy expression. The Euler-Lagrange equation is derived from the refined energy expression and solved numerically for liquid 3He and for neutron matter. The energy per particle for liquid 3He is in better agreement with experimental data, while the effect of this refinement is relatively small for neutron matter. The three-body cluster terms not included in the refined energy expression are evaluated perturbatively and shown to be negligibly small for liquid 3He as well as for neutron matter with cenral forces.

Global optimization for molecular conformation problems

A primal-relaxed dual global optimization algorithm is presented along with an extensive review for finding the global minimum energy configurations of microclusters composed by particles interacting with any type of two-body central forces. First, the original nonconvex expression for the total potential energy is transformed to the difference of two convex functions (DC transformation) via an eigenvalue analysis performed for each pair potential that constitutes the total potential energy function. Then, a decomposition strategy based on the GOP algorithm [1–4] is designed to provide tight upper and lower bounds on the global minimum through the solutions of a sequence of relaxed dual subproblems. A number of theoretical results are included which expedite the computational effort by exploiting the special mathematical structure of the problem. The proposed approach attainse-convergence to the global minimum in a finite number of iterations. Based on this procedure global optimum solutions are generated for small Lennard-Jones and Morse (a=3) microclustersn≤7. For larger clusters (8≤N≤24 for Lennard-Jones and 8≤N≤30 for Morse), tight lower and upper bounds on the global solution are provided which serve as excellent initial points for local optimization approaches.

Violations of the strong energy condition for interacting systems of particles.

This paper generalizes the analysis of Parker and Wang regarding violations of the strong energy condition, demonstrating that they remain applicable even without the assumption that the system of interest is in thermal equilibrium. It does indeed seem possible to violate the strong energy condition even for "normal" matter described by a "reasonable" distribution function or density matrix. As a tool in deriving this result, a special relativistic analogue of Clausius's virial theorem is derived for particles interacting via instantaneous two-body central forces, either in flat space or in a Friedmann universe.

Computational complexity of the ground-state determination of atomic clusters

The authors prove that determining the ground state of a cluster of identical atoms, interacting under two-body central forces, belongs to the class of NP-hard problems. This means that as yet no polynomial time algorithm solving this problem is known and, moreover, that it is very unlikely that such an algorithm exists. It also suggests the need for good heuristics.

Elastic properties of the lower mantle inferred from rigid ion lattice models

Simple theoretical rigid ion lattice models, utilizing two-body central force exponential type repulsive potentials, for first and second nearest neighbor interactions, have been applied to estimate the higher order elastic constants of several alkali halides and NaCl-structure oxides. The results provided by the theoretical models for the first and second pressure derivatives of the bulk modulus ( K′ 0and K″ 0) for the alkali halides are generally consistent with available experimental data. Model calculations of K″ 0 for the oxides MgO, CaO, SrO, and BaO, are about an order of magnitude less than those for the alkali halides. Depending on the details of the lattice model, the present results suggest a value of K″ 0 for MgO in the range −1.7–−3.1 Mbar −1. In addition, the theoretical lattice models indicate values for the shear-related elastic constants C″ s, C″ 44, and μ″ 0 of the NaCl-structure oxides which are markedly lower than those of the alkali halides. The range of values for K″ 0 and the second pressure derivative of the rigidity modulus μ″ 0, which appear from the present results to characterize close-packed oxides, are consistent with estimates of these parameters for the Earth's lower mantle, assuming an adiabatic temperature gradient and compositional homogeneity. Therefore, it is not necessary to invoke a vertical compositional gradient in the deep mantle to account for the observed variation in elastic properties. However, the range of the theoretical values for K′ 0and μ″ 00 which appear to characterize lower mantle oxide materials, make it necessary to use higher order elastic properties in the equation of state in order to obtain accurate and meaningful evaluations of the composition and state of the Earth's deep interior.

An exact quantum statistical formulation of the elastic constants of a solid

A new formula is derived for elastic constants using a rigorous quantum statistical treatment based on obtaining the derivatives of the Helmholtz free energy with respect to strain elements. The stress tensor and the second-order elastic constants are formulated exactly for any homogeneous stress state by using a perturbation method applied to the Schrodinger equation. Third- and higher-order elastic constants may also be obtained in the same way. The formula for the elastic constants contains four terms, all of which are thermal averages. Two of these terms are the momentum term and the quantum stress fluctuation term. Another term involves the thermal average of the potential energy of a lattice and it reduces, in the case of two-body central forces, to the well known formula for the elastic constants of Born and Huang (1954) without making any of the following three assumptions: zero stress, sublattice equilibrium, every lattice point at a centre of inversion. The remaining term is a second-order perturbation term with energy difference denominators, and is the contribution of internal strain to the elastic constants.

Morphology and statistical statics of simple microclusters

Abstract In this paper we discuss the statics of small assemblies of soft, rare-gas type atoms (N=4 to 13) interacting under simple two-body central forces such as the Lennard-Jones and Morse type. Our main concern is to characterize the problem of isomer multiplicity in small packed structures and to devise practical algorithms for the discovery of a representative majority, if not all, stable soft-packing structures for clusters of atoms in the above size-range. We illustrate one such possible ‘Aufbau algorithm’ by demonstrating the existence of no less than 988 distinct, stable minimum configurations for 13 Lennard-Jones atoms and of correspondingly smaller numbers in the range N=7 to 12. The minimal structures obtained may be classified broadly into ‘crystallographic’ and ‘non-crystallographic’ types, the latter predominating among those of greatest binding energy. A surprising result is that, when the softer (α=3) Morse potential is used, the great majority of the Lennard-Jones minima are not support...

On symmetric variations of interfacial energies

Abstract Energy surfaces for unrelaxed stacking faults, twin boundaries and other interfaces calculated using two-body central forces often exhibit symmetric minima. The prediction of the location of these minima, which may correspond to relaxed equilibrium structures, is discussed.

Semiclassical theory for low-energy molecular collisions II. Atomic exchange reactions

A form of semiclassical theory of molecular collisions, previously formulated and applied to H+, H2 vibrational excitation in I, is applied to H, H2 atomic exchange. The formulation involves separation of the potential into two parts, a ``trajectory'' potential along which only elastic scattering can occur and ``coupling'' potential which can produce inelastic or reactive scattering. The first part is chosen such that the classical two-body central force formulas apply. This determines the coupling potential in the parameters, the classical trajectory R (t), and the classical deflection function θ(t). The time-dependent Schrödinger equation is then solved to produce the reactive probabilities. The problem is formulated in a coordinate system in which initial and final scattering coordinates are constrained to remain the same throughout the reaction. This procedure produces asymptotic momentum transfer or ``recoil'' factors of the form exp[(i/ℏ)μν · R] which multiply the vibrational-rotational wavefunctions, but it preserves orthogonality in the scattering wavefunctions and permits the coupled equations in the two-state approximation to be approximately decoupled in the symmetric process. Calculations are carried out to illustrate effect of trajectory on reaction probabilities.

Lattice dynamics at zero wave vector and elastic constants of spinel in the rigid ion approximation

The long wavelength limit of the optical frequencies and of the microscopic Grueneisen parameters, the elastic constants and their first and second pressure derivatives are calculated for the spinel structure from a modified rigid ion model. The model includes Coulomb, first and second neighbor two-body central force interaction, and the three-body central force interaction of Basu and Sengupta for the tetrahedral and octahedral cation-anion units. Application to Al 2MgO 4 is made on the basis of (a) a rigid ion model and (b) the above modified rigid ion model with the three-body forces included for the MgO 4 tetrahedra. In addition to two parameters which are determined by the equilibrium conditions the models contain three and four parameters, respectively, which are fitted to experimental values of one Raman and two i.r. frequencies and, for model (b), to one elastic constant. Good to fair agreement with experimental data of the remaining optical frequencies, elastic constants and their first pressure derivatives and the macroscopic Grüneisen parameter is obtained. Both models results in new assignments for the Raman and i.r. active modes.

Theory of polarized thermal neutron scattering on 19F (I)

Spin-spin amplitudes of thermal neutron scattering on 19 F are calculated in the framework of the microscopic shell model with two-body central forces. It is shown that limited information on the nature of nucleon-nucleon exchange forces can be extracted from such a theoretical analysis of the data. The ratio of the spin-spin amplitude to the spin-independent one is predicted to be negative and the experimental cross sections are accounted for, within the large present experimental errors.

Properties of Crystalline Argon, Krypton, and Xenon Based upon the Born and Huang Method of Homogeneous Deformations. I. Zero-Pressure Thermal and Elastic Data

The thermodynamic properties of solid argon, krypton, and xenon have been calculated for a two-body central force potential. Anharmonic contributions due to thermal expansion have been taken into account via a quasiharmonic calculation based upon the Born and Huang method of homogeneous deformations. The nonarbitrarily adjustable parameters of the Morse potential have been obtained from the solid-state bulk properties in a recursive refinement procedure. For those thermal properties sensitivity dependent upon the dilatation of the lattice the calculations become valid only for temperatures less then $\frac{1}{3} {T}_{M}$.

Derivation of the Vlasov Equation

The solution of the Liouville equation is expressed in the interaction representation. We show that the early-time behavior of the single-particle distribution function ${F}_{1}$ for a classical gas consisting of particles interacting through weak two-body central forces is governed by the Vlassov equation. The two-body potentials are assumed to be "good" functions. The derivation fails if initial correlations are present. The argument is carried out in configuration space. The analysis is extended to the derivation of differential equations governing the time evolution of the multiparticle distribution functions ${F}_{k}$. In this "Vlasov approximation" to the solution of the Liouville equation, we find that the Vlassov equation for ${F}_{1}$, together with the differential equations for the ${F}_{k}$ ($k<1$), amounts to the statement that the multiparticle distribution functions are factorable into products of single-particle distribution functions.

Clustering of carbon atoms in α-iron

Binding and migration energies and atomic configurations have been calculated for small carbon clusters in α-iron. A mathematical model was used which consisted of a spherical crystallite containing about 530 atoms which were treated as individual particles, surrounded by an elastic continuum with atoms imbedded in it. A two-body central force was used to simulate the interaction between nearest and next nearest neighboring iron atoms in the crystallite and another was used for the iron carbon interaction. The dicarbon, tricarbon, and tetracarbon binding energies were 0.14, 0.36, and 0.66 eV respectively, and the binding energy continued increasing for larger clusters by approximately 0.31 eV per additional carbon atom. The most stable configurations for clusters were very thin platelets on {001} planes in which the individual carbon atoms occupied octahedral sites. The axes of the carbon atoms were parallel to each other and perpendicular to the platelet. The migration energy of the platelets was higher than the single carbon migration energy, and the migration mechanism consisted of peripheral atoms moving around the edges of the platelet.

Quasiparticle Model forLa140

A model is proposed for odd-odd nuclei in which the independent excitations are quasiparticles defined by a special Bogoliubov transformation and in which a finite-range two-body central force acts as the residual interaction. The model is applied to ${\mathrm{La}}^{140}$, whose energy levels, wave functions, ($d,p$) spectroscopic factors, and $M1$ transition rates are computed and compared with experiment.