Theory Of Crystal Lattices Research Articles

A Cluster-Based Deep Learning Model Perceiving Series Correlation for Accurate Prediction of Phonon Spectrum.

The spectral properties are the most prevalent continuous representation for characterizing transport phenomena and excitation responses, yet their accurate predictions remain a challenge due to the inability to perceive series correlations by existing machine learning (ML) models. Herein, a ML model named cluster-based series graph networks (CSGN) is developed based on the dynamical theory of crystal lattices to predict phonon density of states (PDOS) spectrum for crystal materials. The multiple atomic cluster representation is constructed to capture the diverse vibration modes, while the mixture Gaussian process and dynamic time warping mechanism are compiled to project from clusters to PDOS spectrum. Accurate predictions of complicated spectra with multiple or overlapping peaks are achieved. The high performance of CSGN model can be attributed to the pertinent feature extraction and the appropriate similarity evaluation, which enable the natural perception of structure-property relation and intrinsic series correlations as confirmed in the predictive results. The transferable and interpretable CSGN model advances ML predictions of spectral properties and reveals the potential of designing ML methods based on physical mechanisms.

Research on mechanical properties of metal rubber based on planar lattice design

Metal rubber is a common vibration is olation material. Because the arrangement of the internal spiral coil will affect the mechanical properties of the metal rubber, this paper draws on the crystal lattice theory in metallurgy, changes the mesoscopic arrangement of the spiral coil through the design of the two-dimensional blank plane lattice structure, and prepares three kinds of lattice structures, and optimizes the mass ratio of the three metal rubber lattice structures, so as to improve the average stiffness and energy dissipation coefficient of the metal rubber. The quasi-static compression experiments passed. The results show that the triangular lattice structure design can reflect the good average stiffness, and the square lattice structure design can obtain the maximum energy dissipation coefficient. The study confirms that the fabrication of metal rubber materials based on planar lattice design is an effective way to improve stiffness and energy dissipation.

Retraction: “Modeling of plasma wave propagation and crystal lattice theory based on computational simulations” [AIP Adv. 13, 045223 (2023)

Retraction: “Modeling of plasma wave propagation and crystal lattice theory based on computational simulations” [AIP Adv. 13, 045223 (2023)

Solving the Fornberg–Whitham Model Derived from Gilson–Pickering Equations by Analytical Methods

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method.

Cutting-Edge Analytical and Numerical Approaches to the Gilson–Pickering Equation with Plenty of Soliton Solutions

In this paper, the Gilson–Pickering (GP) equation with applications for wave propagation in plasma physics and crystal lattice theory is studied. The model with wave propagation in plasma physics and crystal lattice theory is explained. A collection of evolution equations from this model, containing the Fornberg–Whitham, Rosenau–Hyman, and Fuchssteiner–Fokas–Camassa–Holm equations is developed. The descriptions of new waves, crystal lattice theory, and plasma physics by applying the standard tan(ϕ/2)-expansion technique are investigated. Many alternative responses employing various formulae are achieved; each of these solutions is represented by a distinct plot. Some novel solitary wave solutions of the nonlinear GP equation are constructed utilizing the Paul–Painlevé approach. In addition, several solutions including soliton, bright soliton, and periodic wave solutions are reached using He’s variational direct technique (VDT). The superiority of the new mathematical theory over the old one is demonstrated through theorems, and an example of how to design and numerically calibrate a nonlinear model using closed-form solutions is given. In addition, the influence of changes in some important design parameters is analyzed. Our computational solutions exhibit exceptional accuracy and stability, displaying negligible errors. Furthermore, our findings unveil several unprecedented solitary wave solutions of the GP model, underscoring the significance and novelty of our study. Our research establishes a promising foundation for future investigations on incompressible fluids, facilitating the development of more efficient and accurate models for predicting fluid behavior.

Plenty of accurate, explicit solitary unidirectional wave solutions of the nonlinear Gilson–Pickering model

This paper proposes abundant accurate wave solutions that represent the plasma wave propagation based on crystal lattice theory and physicochemical characterization. The Gilson–Pickering ([Formula: see text]) model is analytically and numerically solved by two recent techniques. This model is a basic unidirectional wave propagation model that describes the prorogation of waves in crystal lattice theory and plasma physics. The investigated model has a deep connection with some nonlinear evolution equations under specific values of its parameters, such as the Fuchssteiner–Fokas–Camassa–Holm ([Formula: see text]) equation, the Rosenau–Hyman ([Formula: see text]) equation, and the Fornberg–Whitham ([Formula: see text]) equation. The Sardar sub-equation and He’s variational iteration techniques are employed to construct novel and accurate solitary wave solutions of the handled model. The obtained solutions are explained through some distinct graphs in contour, three-, two-dimensional. The research value is explained by comparing our results with some recently published studies. The employed methods show their simple, direct, effectiveness and their ability for handling many nonlinear evolution equations.

Computational modeling of wave propagation in plasma physics over the Gilson–Pickering equation

In this paper, the Gilson–Pickering equation is studied which enables a wave propagation in plasma physics and crystal lattice theory. Based on the auxiliary transformations, some sets of the nonlinear form of ODE along with some analytic solutions are derived. To study the characterizations of the new waves, the crystal lattice theory and plasma physics is studied by using the modified exponential Jacobi technique and generalized rational tan(ϕ/2)-expansion technique. Through the new approaches, some solitary solutions are proposed. Meanwhile, influence of the coefficients in that equation for the periodic and soliton is illustrated. The effect of the free variables on the solutions is investigated. The behavior of these solutions on qualitatively of different structural natures relying on physical parameters coefficients is analyzed. The reported bright explosive envelopes, explosive solitons, periodic blow up, bright periodic envelope and huge solitary waves are highly applied in plasma and nuclear physics, optical communications, electro-magnetic propagations, superfluid and plenty other applied sciences. For more details about the physical dynamical representation of the presented solutions, are illustrated with profile pictures using Mathematica and Matlab 18, to obtain complete configurations. The proposed approaches can be applied to several equations arising from all nonlinear applied sciences.

Modeling of plasma wave propagation and crystal lattice theory based on computational simulations

This study uses crystal lattice theory and physicochemical characterization to show a number of correct wave solutions that are like the way plasma waves move. The nonlinear time–fractional Gilson–Pickering (GP) model has been addressed using two distinct analytical and numerical techniques. This model is used in crystal lattice theory and plasma physics to show how waves move, so it is a basic model for how waves move in one direction. Utilizing the modified rational and He’s variational iteration approximations in conjunction with the β–fractional derivative principles, we give the handled model fresh and precise solitary wave solutions. Various contour, three-dimensional, and two-dimensional graphs depict the determined outcomes. Compared to other recent studies, ours indicates the importance of this research area. The presented methods show how simple, direct, and effective they are and how they can be used with a wide range of nonlinear evolution equations.

Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson–Pickering equation

In this study, we use cutting-edge analytical and numerical approaches to the Gilson–Pickering (GP) problem in order to get precise soliton solutions. This model explains wave propagation in plasma physics and crystal lattice theory. A variety of evolution equations have been developed from the GP model, including the Fornberg–Whitham (FW) equation, the Rosenau–Hyman (RH) equation, and the Fuchssteiner-Fokas-Camassa–Holm (FFCH) equation, to name a few. The GP model has been studied using these evolution equations. To investigate the characterizations of new waves, crystal lattice theory and plasma physics use the Khater II, and He’s variational iteration approaches. Many alternative responses may be achieved by utilizing various formulae; each of these solutions is shown by a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The results of this study suggest that the technique is preferred for successfully resolving nonlinear equations that emerge in mathematical physics.

Traveling wave solutions of the nonlinear Gilson–Pickering equation in crystal lattice theory

This paper focuses on obtaining the traveling wave solutions of the nonlinear Gilson–Pickering equation (GPE), which describes the prorogation of waves in crystal lattice theory and plasma physics. The solution of the GPE is approximated via the finite difference technique and the localized meshless radial basis function generated finite difference. The association of the technique results in a meshless approach that does not require linearizing the nonlinear terms. At the first step, the PDE is converted to a system of nonlinear ODEs with the help of the radial kernels. In the second step, a high-order ODE solver is adopted to discretize the nonlinear ODE system. The global collocation techniques pose a considerable computationl burden due to the calculation of the dense algebraic system. The proposed method approximates differential operators over the local support domain, leading to sparse differentiation matrices and decreasing the computational burden. Numerical results and comparisons are provided to confirm the efficiency and accuracy of the method.

Triggering piezoelectricity directly by heat to produce alternating electric voltage

Piezoelectricity has proved itself a promising mechanism for energy conversion and signal sensing by taking advantage of its ability to convert mechanical energy into electricity. Here, we demonstrate that the piezoelectricity in free-standing non-centrosymmetric nanowires can also be triggered directly by heat to produce electricity. The feasibility of the idea is first analyzed by the dynamic theory of crystal lattices and then confirmed by molecular dynamics simulations. The most salient point is that the heat-induced voltage drop across the cross section of the free-standing nanowires alternates periodically with the vibration of the nanowire. Moreover, the electric potential induced by heat here (as large as 0.34 V) is proved to be comparable with the previously reported potentials induced by mechanical energy, and the maximum value can be tuned by controlling the size of the nanowire and the applied heat.

Self-similarity and self-inversion of quasicrystals

The discovery of quasicrystals played a revolutionary role in the condensed matter science and forced to renounce the dogma of the classical crystallography that the regular filling of the space by identical blocks is reduced solely to the Fedorov space groups. It is shown that aperiodic crystals, apart from the similarity, exhibit the self-inversion property. In a broadened sense, the self-inversion implies the possible composition of the inversion with translations, rotations, and homothety, whereas pure reflection by itself in a circle can be absent as an independent symmetry element. It is demonstrated that the symmetry of aperiodic tilings is described by Schottky groups (which belong to a particular type of Kleinian groups generated by the linear fractional Mobius transformations); in the theory of aperiodic crystals, the Schottky groups play the same role that the Fedorov groups play in the theory of crystal lattices. The local matching rules for the Penrose fractal tiling are derived, the problem of choice of the fundamental region of the group of motions of a quasicrystal is discussed, and the relation between the symmetry of aperiodic tilings and the symmetry of constructive fractals is analyzed.

Common misconceptions about the dynamical theory of crystal lattices: Cauchy relations, lattice potentials and infinite crystals

The requirement of rotational invariance for lattice potential energies is investigated. Starting from this condition, it is shown that the Cauchy relations for the elastic constants are fulfilled if the lattice potential is built from pair interactions or when the first-neighbour approximation is adopted. This is seldom recognized in widely used solid-state textbooks. Frequently, pair interaction is even considered to be the most general situation. In addition, it is shown that the demand of rotational invariance in an infinite crystal leads to inconsistencies in the symmetry of the elastic tensor. However, for finite crystals, no problems arise, and the Huang conditions are deduced using exclusively a microscopic approach for the elasticity theory, without making any reference to macroscopic parameters. This work may be useful in both undergraduate and graduate level courses to point out the crudeness of the pair–potential interaction and to explore the limits of the infinite-crystal approximation.

Volume–temperature relationship for iron at 330 GPa and the Earth's core density deficit

Estimates of core density deficit (cdd) of the Earth's outer core recently reported by Anderson and Isaak [Another look at the core density deficit of Earth's outer core, Phys. Earth Planet Int. 131 (2002) 19–27] are questionable in view of the serious errors in the pressure–volume and bulk modulus data due to an inadequacy in the calibration process used by Mao et al. [Static compression of iron to 300GPa and Fe0.8Ni0.2 alloy to 200GPa: implications for the core, J. Geophys. Res. 94 (1990) 21737–21742]. The data used by Anderson and Isaak deviate significantly from the corresponding values derived from seismology. In the present study we have used the input data on density, isothermal bulk modulus and its pressure derivative from Stacey and Davis [High pressure equations of state with application to lower mantle and core, Phys. Earth Planet Int. 142 (2004) 137–184] which are consistent with the seismological data. Volumes of hexagonal close-packed iron have been calculated at different temperatures under isobaric conditions at P = 330GPa, the inner core boundary (ICB) pressure using the relationship between thermal pressure and volume expansion based on the lattice potential theory originally due to Born and Huang [Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954, p. 50]. The formulation for thermal pressure used by Anderson and Isaak has been modified by taking into account the variations of thermal expansivity α and isothermal bulk modulus KT with temperature. Values of cdd are then estimated corresponding to different temperatures ranging from 4000 to 8000K. The results for cdd at different temperatures obtained in the present study are significantly higher than those estimated by Anderson and Isaak suggesting that the cdd for the Earth's outer core is nearly 10%. The effects of nickel when an Fe–Ni alloy replaces Fe are estimated and found to be insignificant.

Elastic anisotropy and magnetic phase transition of yttrium garnet ferrite

Temperature dependences of elastic moduli describing elastic anisotropy in the rage from room temperature to 600 °C are shown. These temperature dependences are in agreement with available values of elastic moduli and temperature coefficients of the elastic moduli at room temperature. They are in agreement with modern quasi-harmonic theory of crystal lattices . Near 285 °C critical changes of effective elastic moduli due to magnetic phase transition are observed. A correlation was made between experimental and theoretical critical changes of longitudinal and transverse elastic wave velocities for different crystallographic directions. It is proved that the critical change of ultrasound wave velocities of this crystal is connected with the single-ion magnetostriction .

On the computation of molecular auxiliary functions A n and B n

Molecular auxiliary functions A n (p) and B n (pt), arising in the Hartree-Fock-Roothaan (HFR) approximation for molecules, Ewald’s crystal lattice theory, electromagnetic stopping theory, and other approximate methods, are evaluated and analysed in the range of 17≤n≤60 and 25≤pt≤60.

Adhesive Contact of Flat-Ended Wedges: Theory and Computer Experiments

We give the closed-form solutions for the two-dimensional adhesive contact of a flat-ended wedge with an elastic half-space, including contact pressure distribution and load-contact width relationship. The approach is derived from contact mechanics in plane-strain elasticity and fracture mechanics concepts. The contact pressure has stress singularities both at the edges of contact due to molecular attractive forces and at the wedge corners, and is compared with those without adhesion. Under zero load, we find the contact strip has a finite width which is greater than that of the wedge end, and the central region of contact is under compression, similar to that of a flat punch problem, while the regions near the contact edges are under tension. Unlike the usual experiments with smooth and low modulus materials, we conduct molecular dynamics (MD) experiments via embedded-atom method (EAM), brownian dynamics algorithm and dynamical theory of crystal lattices. The results, including the “pull-off” force for contacting surfaces to peel apart, conform reasonably well with those derived from a continuum model.

Nonlinear acoustics of solids: History and modern trends

The author had the pleasure to be involved in the pioneering experiments on the nonlinear acoustic wave propagation in crystals and metals (1960). It was found that even for small wave amplitudes acoustic harmonics generation took place for longitudinal waves. The results obtained were interpreted from the point of view of Hooke’s law nonlinearity due to nonparabolic interaction potential taking into account accumulation effects in nondispersive media. Besides that, transversal acoustic waves under some external effects like pressure, heating, etc., also demonstrated nonlinear behavior despite that it was forbidden in a framework of crystal lattice theory. It was accounted for by the development of structural disturbations in a material and seemed to be the first attempt to demonstrate the importance of acoustic nonlinear nondestructive testing of different materials. It was shown later that some construction materials and rocks have much higher nonlinear parameter values than those for crystals and metals. The paper will also present an historical review of recent developments in other areas of nonlinear acoustics in solids: measurements of nonlinear acoustic parameters, surface acoustic waves nonlinearity, ‘‘clapping’’ nonlinearity, defective solids and rocks, etc.

Harmonic wave propagation in materials with periodic beam-structure

The characteristics of harmonic waves propagating in periodic beam structures are investigated. For this purpose, a very effective method for the calculation of dispersion relations of elastic waves in these materials is developed by applying dynamic theory of crystal lattices to discrete models of periodic beam structures. This method is applicable to general three-dimensional periodic beam structures. Results presented show that the solutions converge to the exact solution as the number of atoms in the discrete models increases. The dispersion relations of plane harmonic waves in the micropolar continuum model, developed in a previous study, are also calculated. They are compared with the exact solutions to examine the applicability of the continuum models to dynamic problems.

GENERAL EXPRESSIONS OF THE CHARACTERISTIC FREQUENCY AND GRüNEISEN COEFFCIENT FOR SOLIDS

Starting from the general expresson of frequency given by the dynamical theory of crystal lattices, we have derived three different expressions of the characteristic frequency for three kinds of solids, and re-obtained rS,rD-M and rf. We pointed out that rS,rD-M and rf. belong to three different models of solids. Finally, the general expressions of the characteristic frequency ω and Grüneisen coefficient γ for solids were found.