Polyatomic Lattices Research Articles

Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source

In the paper we deal with ballistic heat transport in a graphene lattice subjected to a point heat source. It is assumed that a graphene sheet is suspended under tension in a viscous gas. We use the model of a harmonic polyatomic (more exactly diatomic) lattice performing out-of-plane motions. The dynamics of the lattice is described by an infinite system of stochastic ordinary differential equations with white noise in the right-hand side, which models the point heat source. On the base of the previous analytical unsteady analysis an analytical formula in continuum approximation is suggested, which allows one to describe a steady-state kinetic temperature distribution in the graphene lattice in continuum approximation. The obtained solution is in a good agreement with numerical results obtained for the discrete system everywhere excepting a neighbourhood of six singular rays with the origin at the heat source location. The continuum solution becomes singular at these rays, unlike the discrete one, which appears to be localized in a certain sense along the rays. The factors, which cause such a directional localization and the mismatch between the continuum and discrete solutions are discussed. We expect that the suggested formula is applicable for various damped polyatomic lattices where all particles have equal masses in the case of universal for all particles external viscosity.

On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals

The work is devoted to the description of unsteady thermal processes in low-dimensional structures. To obtain the relationship between the microscopic and macroscopic descriptions of solids, it is necessary to understand the heat transfer mechanism at the micro-level. At the latter, in contrast to the macro-level, analytical, numerical, and experimental studies demonstrate significant deviations from the Fourier’s law. The paper bases on the ballistic heat transfer model, according to which the heat is carried by the thermal waves. This effect can be applied, for example, to signal transmission and heat removal problems. The influence of non-nearest neighbors on processes in discrete media, as well as processes in polyatomic lattices, is investigated. To describe the evolution of the initial thermal perturbation, the dispersion characteristics and group velocities in one-dimensional crystal are analyzed for (i) a diatomic chain with variable masses or stiffnesses and for (ii) a monatomic chain with regard for interaction with second neighbors. A fundamental solution to the heat distribution problem for the corresponding crystal models is obtained and investigated. The fundamental solution allows to obtain a description of waves traveling from a point source, and can serve as the basis for constructing all other solutions. In both cases, the solution consists of two thermal fronts moving one after another with different speeds and intensities. Quantitative estimates of the intensity of the thermal wave front are given, and the dynamics of changes in the velocities and intensities of the waves depending on the parameters of the problem is analyzed. Thus, a simple method for estimating the wavefronts intensity is proposed and tested on two models. These results can be used to identify and analyze those parts of the wave processes that are of interest in terms of interpretation of the effects observed in the experiments.

Dispersion and topological characteristics of permutative polyatomic phononic crystals

This work presents a comprehensive mathematical treatment of phononic crystals (PCs) which comprise a finite lattice of repeated polyatomic unit cells. Wave dispersion in polyatomic lattices is susceptible to changes in the local arrangement of the monatoms (subcells) constituting the individual unit cell. We derive and interpret conditions leading to identical and contrasting band structures as well as the possibility of distinct eigenmodes as a result of cyclic and non-cyclic cellular permutations. Different modes associated with cyclic permutations yield topological invariance, which is assessed via the winding number of the complex eigenmode. Wave topology variations in the polyatomic PCs are quantified and conditions required to support edge modes in such lattices are established. Next, a transfer function analysis of finite polyatomic PCs is used to explain the formation of multiple Bragg band gaps as well as the emergence of truncation resonances within them. Anomalies arising from the truncation of the infinite lattice are further exploited to design mirror symmetrical edge modes in an extended lattice. We conclude with a generalized explanation of the band gap evolution mechanism based on the Bode plot analysis.

On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals

Unsteady thermal processes in low-dimensional structures are considered. Understanding the heat transfer at the micro-level is necessary to obtain a link between micro- and macroscopic descriptions of solids. At the macroscopic level, heat propagation is described by the Fourier law. However, at the microscopic level, analytical, numerical and experimental studies show significant deviations from this law. The previously created model of heat transfer at the microlevel, which has a ballistic character, is used in the work. The influence of non-nearest neighbors on the thermal processes in discrete media is studied, as well as the heat distribution in polyatomic lattices is considered. To describe the evolution of the initial thermal perturbation, the analysis of dispersion characteristics and group velocities in a one-dimensional crystal for a diatomic chain with alternating masses or stiffnesses and a monoatomic chain with regard for interaction with second neighbors is carried out. The fundamental solution of the heat propagation problem for the corresponding crystal models is obtained and studied. The fundamental solution allows us to obtain a description of the waves running from a point source and to use it as a basis for the construction of all other solutions. For both chains, the solution consists of two fronts moving one after another with different velocities and intensity characteristics. Quantitative estimations of the thermal wavefront intensity coefficients are given, and the dynamics of changes in speeds and intensities of the waves depending on the parameters of the problem is analyzed. Two mechanisms of evolution of the heat wavefront in one-dimensional discrete systems are revealed. The presented results can be used for correct interpretation of experiments on unsteady ballistic heat transfer in crystals.

Surface states in defect-free polyatomic lattices described by a tight-binding model

We report about a mechanism for surface localization, present in finite defect-free polyatomic lattices described by a tight binding model. Numerical diagonalization and degenerated perturbation theory show that there is a minimum number of atoms within each unit cell in the lattice for which surface states may exist, provided the local energy of the surface atom is different from the rest in the unit cell. It is shown that the appearance of surface states is a second-order effect in the hopping parameter. Other kinds of surface states are identified in the two-dimensional case.

Comparison of nuclear irradiation parameters of fusion breeder materials in high flux fission test reactors and a fusion power demonstration reactor

Nuclear irradiation parameters relevant to displacement damage and burn-up of the breeder materials Li 2O, Li 4SiO 4 and Li 2TiO 3 have been evaluated and compared for a fusion power demonstration reactor and the high flux fission test reactor (HFR), Petten, the advanced test reactor (ATR, INEL) and the Japanese material test reactor (JMTR, JAERI). Based on detailed nuclear reactor calculations with the MCNP Monte Carlo code and binary collision approximation (BCA) computer simulations of the displacement damage in the polyatomic lattices with MARLOWE, it has been investigated how well the considered HFRs can meet the requirements for a fusion power reactor relevant irradiation. It is shown that a breeder material irradiation in these fission test reactors is well suited in this regard when the neutron spectrum is well tailored and the 6Li-enrichment is properly chosen. Requirements for the relevant nuclear irradiation parameters such as the displacement damage accumulation, the lithium burn-up and the damage production function W( T) can be met when taking into account these prerequisites. Irradiation times in the order of 2–3 full power years are necessary for the HFR to achieve the peak values of the considered fusion power Demo reactor blanket with regard to the burn-up and, at the same time, the dpa accumulation.

Reply to "Comment on 'Thermodynamic and elastic properties of a many-body model for simple oxides' "

The comment discussed four issues: (1) The sensitivity of thermoelastic properties from approximate lattice dynamics (ALD) to the adjustable parameter; (2) the applicability of normalized lattice dynamics (NLD) to shear thermoelasticity; (3) the applicability of NLD to low-symmetry lattices; (4) Iab initioR versus empirical-potential models. This reply will reaffirm the following: (1) The mathematical derivation of ALD contains serious flaws that show in the results. The use of an adjustable parameter serves to minimize the errors, but cannot completely eliminate them. (2) Our NLD formulation puts the calculation on a physically justified ground, while eliminating the need in adjustable parameters and the ensuing inaccuracies. That formalism does not depend on the cubic symmetry, and so there is no room for unclarity in the reported computation of shear thermoelastic moduli. (3) NLD is valid to polyatomic lattices of any symmetry. Therefore NLD completely solves the problem of polarization shifts in potential-induced-breathing models. (4) At the present state of the art, Iab initioR potentials cannot compete with empirical potentials on accuracy and reliability.

Applications of stochastic mechanics to polyatomic lattices.

Stochastic quantization in the sense of Nelson provides an alternative interpretation of some aspects of quantum mechanics in the coordinate representation, and it was combined recently with the Ford, Kac, and Mazur (FKM) approximation [J. Math. Phys. 6, 504 (1965)] for large lattices to construct a quantum analog to the Brownian motion process. In this paper a similar approach is applied to model the effect of temperature fluctuations in a one-dimensional ordered chain of atoms with nearest-neighbor linear forces. However, we do not make use of the FKM approximation, and as a consequence the statistical properties of the involved processes are exactly determined by the lattice force field. In particular, we evaluate the covariance matrix for the fluctuations, and we examine its high- and low-temperature behavior. Because of the translation invariance of the interaction potential, the covariance matrix for the fluctuations becomes singular implying that the associated probability density has equal density along the zero eigenvector of the interaction matrix. This behavior is readily interpreted in terms of the motion of the center of mass of the system, which corresponds to a stochastically perturbed translation, while all other modes are bounded with a probability of 1. As is well known, the transformation to internal (bondlength) coordinates leads to a Hamiltonian specified by a nonsingular interaction matrix. We examine the variance of the fluctuations for the internal coordinates, and we show that in the high-temperature limit the result agrees with that of classical statistical mechanics. Both the position and bondlength of the surface atom decrease with time as is expected for a semi-infinite lattice. However, the position of the surface atom is less dependent on substrate-atom positions than is the surface bondlength on substrate bondlengths. Finally, the autocorrelation function of the surface bondlength in the case of a semi-infinite lattice limit is investigated for low- and high-temperature limits.

XII. A variation principle for electronic wave functions in crystals

Abstract A variation principle giving the energy levels of electrons in polyatomic lattices is presented. When used in conjunction with the cellular method it includes the matching of the wave functions across cell boundaries without the need of making ad hoc assumptions. Values of ∇ k E(k) can be obtained from quantities used to determine E(k) without carrying out any numerical differentiation. The results of some calculations on the valency bands of lead sulphide are reported.