Terms Of Normal Modes Research Articles

One-dimensional thermal shock problem for a semi-infinite hygrothermoelastic rod

The present research article deals with the study of a boundary value problem of a one-dimensional semi-infinite hygro-thermoelastic rod of length l. The deformation of the rod is under consideration when the left boundary of the hygro-thermoelastic rod is subjected to a sudden heat source. The solutions of the considered variables are decomposed in terms of normal modes. Analytical expressions of displacement, moisture concentration, temperature field, and stresses are obtained and presented graphically for different periods. By studying the one-dimensional thermal shock problem for a semi-infinite hygrothermoelastic rod, the authors aim to gain insights into the fundamental behavior of materials subjected to rapid temperature changes and moisture effects.

Inverse Design of Phononic Crystal with Desired Transmission via a Gradient-Descent Approach

We propose a general approach based on the gradient descent method to study the inverse problem, making it possible to reversely engineer the microscopic configurations of materials that exhibit desired macroscopic properties. Particularly, we demonstrate its application by identifying the microscopic configurations within any given frequency range to achieve transparent phonon transport through one-dimensional harmonic lattices. Furthermore, we obtain the phonon transmission in terms of normal modes and find that the key to achieving phonon transparency or phonon blocking state lies in the ratio of the mode amplitudes at ends.

Niobium and tantalum octahedral Halides: Vibrational properties and Intra–Cluster interactions

IR and Raman spectroscopies are important characterisation methods for Nb and Ta octahedral halide clusters [M6Xi12Xa6]x (M = Nb, Ta; X = Cl, Br; x = 4–, 3–, 2–; i – “inner”, bridging ligand belonging to the cluster core, a – “ausser” or “apical” terminal ligand) and the polymeric Ta6I14. The structural and spectroscopic data of these compounds (x = 4–) and their corresponding {M6X12}2+ core clusters were systematically investigated and analysed by means of quantum chemical calculations. The intra–cluster interactions were studied in details by means of Natural Bond Orbital and Bader’s Atoms–In–Molecules theory analyses, and the results were related to the spectroscopical features, in particular to the composition of the main vibrational normal modes in terms of the contribution of the internal coordinates.

Simulation of Ab Initio Optical Absorption Spectrum of β-Carotene with Fully Resolved S0 and S2 Vibrational Normal Modes.

The electronic absorption spectrum of β-carotene (β-Car) is studied using quantum chemistry and quantum dynamics simulations. Vibrational normal modes were computed in optimized geometries of the electronic ground state S0 and the optically bright excited S2 state using the time-dependent density functional theory. By expressing the S2-state normal modes in terms of the ground-state modes, we find that no one-to-one correspondence between the ground- and excited-state vibrational modes exists. Using the ab initio results, we simulated the β-Car absorption spectrum with all 282 vibrational modes in a model solvent at 300 K using the time-dependent Dirac-Frenkel variational principle and are able to qualitatively reproduce the full absorption line shape. By comparing the 282-mode model with the prominent 2-mode model, widely used to interpret carotenoid experiments, we find that the full 282-mode model better describes the high-frequency progression of carotenoid absorption spectra; hence, vibrational modes become highly mixed during the S0 → S2 optical excitation. The obtained results suggest that electronic energy dissipation is mediated by numerous vibrational modes.

Shear and Breathing Modes of Layered Materials.

Layered materials (LMs), such as graphite, hexagonal boron nitride, and transition-metal dichalcogenides, are at the center of an ever-increasing research effort, due to their scientific and technological relevance. Raman and infrared spectroscopies are accurate, non-destructive approaches to determine a wide range of properties, including the number of layers, N, and the strength of the interlayer interactions. We present a general approach to predict the complete spectroscopic fan diagrams, i.e., the relations between frequencies and N for the optically active shear and layer-breathing modes of any multilayer comprising N ≥ 2 identical layers. In order to achieve this, we combine a description of the normal modes in terms of a one-dimensional mechanical model, with symmetry arguments that describe the evolution of the point group as a function of N. Group theory is then used to identify which modes are Raman- and/or infrared-active, and to provide diagrams of the optically active modes for any stack composed of identical layers. We implement the method and algorithms in an open-source tool to assist researchers in the prediction and interpretation of such diagrams. Our work will underpin future efforts on Raman and infrared characterization of known, and yet not investigated, LMs.

Normal ordering normal modes

In a soliton sector of a quantum field theory, it is often convenient to expand the quantum fields in terms of normal modes. Normal mode creation and annihilation operators can be normal ordered, and their normal ordered products have vanishing expectation values in the one-loop soliton ground state. The Hamiltonian of the theory, however, is usually normal ordered in the basis of operators which create plane waves. In this paper we find the Wick map between the two normal orderings. For concreteness, we restrict our attention to Schrodinger picture scalar fields in 1+1 dimensions, although we expect that our results readily generalize beyond this case. We find that plane wave ordered n-point functions of fields are sums of terms which factorize into j-point functions of zero modes, breather and continuum normal modes. We find a recursion formula in j and, for products of fields at the same point, we solve the recursion formula at all j.

Memory responses in a three-dimensional thermo-viscoelastic medium

ABSTRACT Due to the shortcomings of power law distributions in the heat transfer laws of fractional calculus, some other forms of derivatives with few other kernel functions have been proposed. This literature survey focuses on the mathematical model of thermo-viscoelasticity which investigates the transient phenomena in a three-dimensional thermoelastic medium in the context of two-temperature Kelvin–Voigt three-phase-lag model of generalized thermoelasticity, defined in integral form on a slipping interval incorporating the memory-dependent heat transport law. The bounding plane is subjected to a time-dependent thermal loading and is free of tractions. Incorporating normal mode as a tool, the problem has been solved analytically in terms of normal modes and the physical fields have been depicted graphically for a copper-like material. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative, effect of delay time and viscosity also.

Computational notes on the electronic properties of carboxylic acid

The present work describing the electronic properties and vibrational characteristics of carboxylic acids. Acetic acid is chosen as model molecules then optimized at B3LYP/6-31g(d,p) level of theory. The vibrational frequencies were calculated at the same level of theory. Band assignments which were calculated as 18 normal modes were assigned as one compare the normal mode coordinates with original one. Band assignments were described indicating the directions of normal modes in terms the vibrating atoms of the acetic acids. It could be concluded that DFT could be a useful tool for elucidation both the structural and vibrational features of carboxylic acids and then further utilized for assignment of the structures contains carboxyl groups which are known as most reactive structures in chemistry, biology and environment.

Elastodynamical disturbances due to laser irradiation in a microstretch thermoelastic medium with microtemperatures

Abstract This research is concerned with the study of mechanical disturbances due to presence of ultra-short laser pulse as input heat source in a microstretch thermoelastic medium with microtemperatures. The medium is subjected to normal and tangential forces. The solution of the problems is developed in terms of normal modes. Mathematical expressions have been obtained for normal stress, tangential stress, microstress and temperature change. The numerically computed results are shown graphically. A mathematical model has been developed and various stress quantities have been analyzed. Some particular cases are also derived from the present investigation.

Sound Radiation from the Perforated End of a Lined Duct

Radiation of sound wave through a lined duct with perforated end is analyzed rigorously. The problem considered is axisymmetric. By using the Fourier transform technique in conjunction with the Mode Matching method, the related boundary value problem is formulated as a Wiener-Hopf (W-H) equation. The Mode-Matching technique allows us to express the field component defined in the waveguide region in terms of normal modes. The solution involves a set of infinitely many expansion coefficients satisfying an infinite system of linear algebraic equations. The numerical solution of this system is obtained for different parameters of the problem such as the surface impedances, specific impedance of the perforated screen and their effects on the radiation phenomenon are shown graphically.

Generalized hydrodynamics of the classical Toda system

We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorized scattering of Toda particles, we obtain the generalized free energy and exact average currents and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and, in particular, the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorized scattering of Toda particles.

Explaining event-related fields by a mechanistic model encapsulating the anatomical structure of auditory cortex

Event-related fields of the magnetoencephalogram are triggered by sensory stimuli and appear as a series of waves extending hundreds of milliseconds after stimulus onset. They reflect the processing of the stimulus in cortex and have a highly subject-specific morphology. However, we still have an incomplete picture of how event-related fields are generated, what the various waves signify, and why they are so subject-specific. Here, we focus on this problem through the lens of a computational model which describes auditory cortex in terms of interconnected cortical columns as part of hierarchically placed fields of the core, belt, and parabelt areas. We develop an analytical approach arriving at solutions to the system dynamics in terms of normal modes: damped harmonic oscillators emerging out of the coupled excitation and inhibition in the system. Each normal mode is a global feature which depends on the anatomical structure of the entire auditory cortex. Further, normal modes are fundamental dynamical building blocks, in that the activity of each cortical column represents a combination of all normal modes. This approach allows us to replicate a typical auditory event-related response as a weighted sum of the single-column activities. Our work offers an alternative to the view that the event-related field arises out of spatially discrete, local generators. Rather, there is only a single generator process distributed over the entire network of the auditory cortex. We present predictions for testing to what degree subject-specificity is due to cross-subject variations in dynamical parameters rather than in the cortical surface morphology.

Dynamic problem in piezo-electric microstretch thermoelastic medium under laser heat source

Purpose The purpose of this paper is to study the thermal and mechanical disturbances in a piezo-electric microstretch thermoelastic medium due to the presence of ultra-short laser pulse as input heat source. Design/methodology/approach The medium is subjected to normal force, tangential force and thermal source. The solution of the problems is developed in terms of normal modes. Mathematical expressions have been obtained for normal stress, tangential stress, microstress, dielectric displacement vector and temperature change. Findings The numerically computed results are shown graphically. The effect of time and laser radius on temperature distribution is also shown graphically and comparison to theoretical results has been discussed. A mathematical model has been developed for the system of equations and various stress quantities have been analyzed. Some computer programs have also been written for this study. Two particular cases are also derived from the present investigation. Originality/value The effect of laser heat source is studied in piezo-electric microstretch thermoelastic medium. It is observed from the figures that the laser heat source has significant role on the values of coupled tangential stress.

A comprehensive study of the molecular vibrations in solid-state benzylic amide [2]catenane.

The interpretation of vibrational spectra is often complex but a detailed knowledge of the normal modes responsible for the experimental bands provides valuable information about the molecular structure of the sample. In this work we record and assign in detail the infrared (IR) spectrum of the benzylic amide [2]catenane, a complex molecular solid displaying crimped mechanical bonds like the links of a chain. In spite of the large size of the unit cell, we calculate all the vibrational modes of the catenane crystal using quantum first-principles calculations. The activity of each mode is also evaluated using the Born effective charges approach and a theoretical spectrum is constructed for comparison purposes. We find a remarkable agreement between the calculations and the experimental results without the need to apply any further empirical correction or fitting to the eigenfrequencies. A detailed description in terms of the usual internal coordinates is provided for over 1000 normal modes. This thorough analysis allows us to perform the complete assignment of the spectrum, revealing the nature of the most active modes responsible for the IR features. Finally, we compare the obtained results with those of Raman spectroscopy, studying the effects of the rule of mutual exclusion in vibrational spectroscopy according to the different levels of molecular symmetry embedded in this mechanically interlocked molecular compound.

Improvement of Detection and Localization Performance Using the Receiving Array Response Difference Between Ocean Noise and Signal in Shallow Water

It was observed that when the short vertical line array (SVLA) is located in the deeper part of the water column, where sound velocity is lower, a groove always exists in the receiving array response in the horizontal direction for distant sound sources in the shallower part of the water column, where the sound velocity is higher. Normal mode modeling is used to explain this result. According to the normal mode theory, the receiving array response of the SVLA to a distant sound source can be expressed in terms of modal beams weighted in accordance with the normal mode amplitude. This modal representation offers a physical interpretation of the receiving array response to a distant sound source in terms of normal modes. The environmental effects of the shape of the sound velocity profile and geo-acoustic properties of the seabed on the receiving array response are analyzed. Based on the results, three conditions for the existence of the groove in the receiving array response are obtained: 1) a gradient in the sound velocity profile, 2) an SVLA in a water column in which the sound velocity is lower and low-order normal modes are trapped, and 3) a distant sound source in a shallow water column in which the sound velocity is higher, and acoustic source couples weakly with low-order normal modes and strongly with high-order normal modes. Finally, the receiving array response of the SVLA to ocean noise and distant sound source are analyzed and discussed using the Mediterranean Sea data. It is shown that the receiving array response to ocean noise differs from that to a distant sound source. Utilizing this difference, the array can be steered carefully to improve the output signal-to-noise ratio and increase the passive detection range against a submerged target in shallow water.

Relativistic Jahn-Teller effect in triplet electronic states of tetrahedral molecules

We consider the relativistic multi-mode 3E×(t2+e) Jahn-Teller effect for tetrahedral atomic complexes in a triplet electronic state. Our analysis is based on the symmetry properties of the electronic Hamiltonian and includes spin-orbit coupling terms. The Taylor series expansion of the electronic Hamiltonian in terms of normal modes is restricted to second order in the electrostatic contributions and to first order in the spin-orbitspin-orbit contributions. The main result of this work is a 6×6 vibronic matrix which depends on the t2 and e vibrational normal modes and is specified by six electrostatic and three spin-orbit coupling constants.

Filament instability under constant loads

Buckling of semi-flexible filaments appears in different systems and scales. Some examples are: fibers in geophysical applications, microtubules in the cytoplasm of eukaryotic cells and deformation of polymers freely suspended in a flow. In these examples, instabilities arise when a system’s parameter exceeds a critical value, being the Euler force the most known. However, the complete time evolution and wavelength of buckling processes are not fully understood. In this work we solve analytically the time evolution of a filament under a constant compressive force in the small amplitude approximation. This gives an insight into the variable force scenario in terms of normal modes. The evolution is highly sensitive to the initial configuration and to the magnitude of the compressive load. This model can be a suitable approach to many different real situations.

Double-diffusive convection of compressible rotating Walters’ (B′) fluid with Hall currents saturating a porous medium

Keeping in view the conflicting tendencies of rotation and Hall currents (magnetic field) while acting together; combined effects of Hall currents and rotation are considered on the hydromagnetic stability of a compressible Walters’ (Model B′) elastico-viscous fluid heated and soluted from below saturating a porous medium. Boussinesq approximation is used to simplify the complex hydromagnetic equations and the perturbation equations are analyzed in terms of normal modes. A dispersion relation governing the effects of visco-elasticity, salinity gradient, rotation, Hall currents and medium permeability is derived. It has been found that for stationary convection, Walters’ (Model B′) fluid behaves like an ordinary Newtonian fluid due to the vanishing of the visco-elastic parameter. Compressibility, solute gradient, rotation and magnetic field postpone the onset of instability as such their effect is to stabilize the system. Hall currents and medium permeability are found to hasten the onset of instability for permissible range of values of various parameters. The dispersion relation is analyzed numerically and the effects of various parameters for permissible range of values are depicted graphically. The visco-elasticity, solute gradient and Hall currents (hence magnetic field) introduce oscillatory modes in the system which were non-existent in their absence. Also the case of overstability is discussed and sufficient conditions for the non-existence of overstability are derived.Keywords: Walters’ (Model B′) fluid, rotation, Hall currents, thermosolutal instability, compressibility, porous medium.

An ab initio investigation of non-adiabatic couplings and conical intersections among the lowest five electronic states of the NO3 radical

ABSTRACTIn order to explore spectroscopic properties of a floppy molecule, it is necessary to compute accurate diabatic potential energy surfaces (PESs) around the Franck–Condon region of nuclear configuration space. In such cases, it is desirable to include the non-adiabatic coupling terms (NACTs) through Jahn–Teller (JT) and pseudo Jahn–Teller (PJT) effects as well as other accidental conical intersections (CIs) prevailing among the electronic states of the molecule. In this work, we investigate NACTs among the five lowest doublet electronic states of NO3 radical [ and 22B2)], in which all those effects (JT, PJT and CIs) have prominent contributions. The study is carried out in terms of normal modes (Qis) of NO3 and the CIs (JT and accidental) are located in a plane of nuclear configuration space of two such coordinates (Qi and Qj). We have calculated ab initio adiabatic PESs and NACTs of those five states of NO3 at the MRCI level as functions of pairwise Qis. Further, to explore the nature of a given CI in terms of normal coordinates contributing to NACTs within a particular plane Qi-Qj, we have performed CP-MCSCF study computing analytic NACTs and characterised their contributions along various Qks.

Analytical solution of thermoelastic interaction in a half-space by pulsed laser heating

In this article, we consider the problem of a two-dimensional thermoelastic half-space in the context of generalized thermoelastic theory with one relaxation time. The surface of the half-space is taken to be traction free and thermally insulated. The solution of the considered physical quantity can be broken down in terms of normal modes. The nonhomogeneous basic equations have been written in the form of a vector-matrix differential equation, which is then solved by an eigenvalue approach. The exact analytical solution is adopted for the temperature, the components of displacement and stresses. The results obtained are presented graphically for the effect of laser pulse to display the phenomena physical meaning. The graphical results indicate that the thermal relaxation time has a great effect on the temperature, the components of displacement and the components of stress.