Normal Mode Approximation Research Articles

Theoretical study of the electron-induced vibrational excitation of H2O

This study presents calculations for cross sections of the vibrational excitation of H2O(X1A1) via electron impact. The theoretical approach employed here is based on first principles only, combining electron-scattering calculations performed using the UK R-matrix codes for several geometries of the target molecule, three-dimensional (3D) vibrational states of H2O, and 3D vibrational frame transformation. The aim is to represent the scattering matrix for the electron incident of the molecule. The vibrational wave functions were obtained numerically, without the normal-mode approximation, so that the interactions and transitions between vibrational states assigned to different normal modes could be accounted for. The thermally averaged rate coefficients were derived from the calculated cross sections for temperatures in the 10–10 000 K interval and analytical fits for rate coefficients were also provided. We assessed the uncertainty estimations of the obtained data for subsequent applications of the rate coefficients in modelling the non-local thermal equilibrium (non-LTE) spectra of water in various astrophysical environments.

Linear and Nonlinear Modes and Data Signatures in Dynamic Systems Biology Models

The particulars of stimulus–response experiments performed on dynamic biosystems clearly limit what one can learn and validate about their structural interconnectivity (topology), even when collected kinetic output data are perfect (noise-free). As always, available access ports and other data limitations rule. For linear systems, exponential modes, visible and hidden, play an important role in understanding data limitations, embodied in what we call dynamical signatures in the data. We show here how to circumscribe and analyze modal response data in compartmentalizing model structures—so that modal analysis can be used constructively in systems biology mechanistic model building—for some nonlinear (NL) as well as linear biosystems. We do this by developing and exploiting the modal basis for dynamical signatures in hypothetical (perfect) input–output (I-O) data associated with a (mechanistic) structural model—one that includes inputs and outputs explicitly. The methodology establishes model dimensionality (size and complexity) from particular I-O datasets; helps select among multiple candidate models (model distinguishability); helps in designing new I-O experiments to extract “hidden” structure; and helps to simplify (reduce) models to their essentials. These modal analysis tools are introduced to NL enzyme-regulated and protein–protein interaction biosystems via nonlinear normal mode (NNM) and quasi-steady state approximation (QSSA) analyses and unified with linear models on invariant 2-dimensional manifolds in phase space, with properties similarly informative about their dominant dynamical properties. Some automation of these highly technical aspects of biomodeling is also introduced.

Dissociative recombination of N_2hbox {H}^+: a revisited study

Dissociative recombination of N_2H^+ is explored in a two-step theoretical study. In a first step, a diatomic (1D) rough model with a frozen NN bond and frozen angles is adopted, in the framework of the multichannel quantum defect theory (MQDT). The importance of the indirect mechanism and of the bending mode is revealed, in spite of the disagreement between our cross section and the experimental one. In the second step, we use our recently elaborated 3D approach based on the normal mode approximation combined with R-matrix theory and MQDT. This approach results in satisfactory agreement with storage-ring measurements, significantly better at very low energy than the former calculations.

Electron scattering cross sections from NH3: a comprehensive study based on R-matrix method

In this study, a comprehensive cross-sectional computation on the electron scattering from NH3 was performed using the frame of the R-matrix method. Cross section sets for total elastic, differential, momentum transfer, total ionization, and electronic excitation are presented. The electron-impact dissociation of NH3 to NHH and NH+H2 was considered by summing up the cross sections for the molecular dissociative excitations to the correlated dissociation channels. The reported cross sections for vibrational excitation of NH3 remain controversial and are unable to distinguish the N–H symmetric and asymmetric stretching modes of close frequencies. Here, the excitation cross section for each vibrational mode of NH3 was computed by applying the theoretical approach combining the fixed-nuclei R-matrix method, normal mode approximation and the vibrational frame transformation. The uncertainties of the cross section sets computed in this study were estimated for convenience of use in the plasma modeling.

Energy-recurrence breakdown and chaos in disordered Fermi–Pasta–Ulam–Tsingou lattices

In this paper, we consider the classic Fermi–Pasta–Ulam–Tsingou system as a model of interacting particles connected by harmonic springs with a quadratic nonlinear term (first system) and a set of second-order ordinary differential equations with variability (second system) that resembles Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system. In the absence of variability, the second system becomes Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system (first system). Variability is introduced to Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system to take into account inherent variations (for example, due to manufacturing processes), giving rise to heterogeneity in its parameters. We demonstrate that a percentage of variability smaller than a threshold can break the well-known energy recurrence phenomenon and induce localization in the energy normal-mode space. However, percentage of variability larger than the threshold may make the trajectories of the second system blow up in finite time. Using a multiple-scale expansion, we derive analytically a two normal-mode approximation that explains the mechanism for energy localization and blow up in the second system. We also investigate the chaotic behavior of the two systems as the percentage of variability is increased, utilizing the maximum Lyapunov exponent and Smaller Alignment Index. Our analysis shows that when there is almost energy localization in the second system, it is more probable to observe chaos, as the number of particles increases.

Cross sections for vibrational excitation and dissociative recombination of the ion in collisions with low-energy electrons

Cross sections for the vibrational excitation and dissociative recombination (DR) of the ion in collisions with electrons at low scattering energies are computed using a previously-developed approach combining the normal mode approximation for the vibrational states of the target ion and the UK R-matrix code for the evaluation of the scattering matrices at fixed geometries. The obtained cross section for the DR shows excellent agreement with the experimental data from the ASTRID storage ring. Thermally-averaged rate coefficients are obtained from the cross sections for temperatures 10–3000 K.

Formation of negative-ion resonance and dissociative attachment in collisions of NO2 with electrons

The process of electron attachment to the NO2 molecule is investigated theoretically using an approach based on a study by O’Malley (1966 Phys. Rev. 150 14). The approach combines the normal mode approximation for representation of vibrational dynamics of NO2 and one-dimensional treatment, along each normal mode, of the attachment process as in O’Malley’s theory, such that only a modest computational effort is required to compute the attachment cross section. Taking into account the survival probability of the formed resonant state of , the cross section for dissociative electron attachment to NO2 is also estimated. To compare with available experimental data, the theoretical cross section is convoluted with energy distribution of NO2–e− collisions with uncertainties reported in experimental studies. Peak values of the convoluted theoretical cross section are found to be about a factor of 2–10 larger than the experimental results.

Cross Sections and Rate Coefficients for Vibrational Excitation of H2O by Electron Impact

Cross-sections and thermally averaged rate coefficients for vibration (de-)excitation of a water molecule by electron impact are computed; one and two quanta excitations are considered for all three normal modes. The calculations use a theoretical approach that combines the normal mode approximation for vibrational states of water, a vibrational frame transformation employed to evaluate the scattering matrix for vibrational transitions and the UK molecular R-matrix code. The interval of applicability of the rate coefficients is from 10 to 10,000 K. A comprehensive set of calculations is performed to assess uncertainty of the obtained data. The results should help in modelling non-LTE spectra of water in various astrophysical environments.

Tailored Subcycle Nonlinearities of Ultrastrong Light-Matter Coupling.

We explore the nonlinear response of tailor-cut light-matter hybrid states in a novel regime, where both the Rabi frequency induced by a coherent driving field and the vacuum Rabi frequency set by a cavity field are comparable to the carrier frequency of light. In this previously unexplored strong-field limit of ultrastrong coupling, subcycle pump-probe and multiwave mixing nonlinearities between different polariton states violate the normal-mode approximation while ultrastrong coupling remains intact, as confirmed by our mean-field model. We expect such custom-cut nonlinearities of hybridized elementary excitations to facilitate nonclassical light sources, quantum phase transitions, or cavity chemistry with virtual photons.

Optimizing a coarse-grained space for approximate normal-mode vibrations of molecular heterodimers.

We applied the method of coarse-graining the intermolecular vibrations to molecular heterodimers assembled by double hydrogen bonding. This method is based on principal component analysis, by which the original atomic displacement vectors are projected onto a lower-dimensional space spanned by a basis set of translations, librations, and intramolecular vibrations of the constituent molecules. Compared with homodimers, the following points are particularly noted: (1) alignment of the constituent molecules in a non-symmetric atomic arrangement of the whole system and (2) the scheme of reordering the bases to construct an optimal coarse-grained space. We tested three schemes for reordering the intramolecular vibration vectors to determine that the best one is equivalent to size reduction based on the singular value decomposition. The coarse-graining analysis affords three parameters, Φintra, Φinter, and Φapp, which are relevant to the mechanical nature of the molecular assembly. The Φintra values account for the internal stiffness of molecules, while the Φinter values are true stiffness constants of the intermolecular force and show a good correlation with the association energies of the dimers. The Φapp values are the apparent intermolecular stiffness smaller than Φinter, as a result of compensation for neglecting intramolecular vibrations. All these values are consistent with each other under the coupled oscillator model, showing that the present coarse-graining analysis is valid for heterodimers as well as homodimers.

Vibrational excitation of N2O by an electron impact and the role of the Renner-Teller effect in the process

Cross sections and rate coefficients for vibrational excitation and de-excitation of the ${\mathrm{N}}_{2}\mathrm{O}$ molecule by a low-energy electron for transitions between the lowest vibrational levels are computed using a first-principles approach. The present theoretical approach employs the normal-mode approximation for the description of target vibrational states, the vibrational frame transformation to compute amplitudes of vibrational transitions, and the $R$-matrix method to compute ab initio electronic bound and continuum states. It was found that the nonadiabatic Renner-Teller effect, which couples partial waves of the incident electron with degenerate bending vibrations of ${\mathrm{N}}_{2}\mathrm{O}$, is responsible for the excitation of the bending mode. Theoretical results obtained agree reasonably well with available experimental data at low energies. Thermally averaged rate coefficients are computed for temperatures in the 10--10 000 K range.

Acoustic Noise Generated on a Shallow-Water Shelf by Vessels with Electric Motors

The results of spatial measurements of acoustic noise generated by ice-navigation diesel–electric vessels, which are used in Sakhalin Energy Investment Company Ltd. to work with the PA-B and Molikpaq oil and gas platforms installed on the Northeastern shelf of the island of Sakhalin, are presented. Using a 3D modal parabolic equation and full-scale reference measurements, calculations of anthropogenic acoustic fields that are generated by these vessels in the coastal Piltun area of summer–autumn gray-whale feeding were performed in the approximation of normal vertical modes and a narrow-angle parabolic equation in the horizontal plane. Spectral functions of the point sources of anthropogenic acoustic noise that are equivalent to vessels are constructed for the simulation. The results of the numerical simulation that was performed using these functions are consistent with full-scale measurements.

Theoretical study of electron-induced vibrational excitation of NO2

In this study, we compute cross sections for vibrational excitation of NO2 by electron impact. Calculations are performed using a theoretical approach based on a combination of the normal mode approximation for vibrational states of the target molecule, fixed-nuclei electron-NO2 scattering matrices and the vibrational frame transformation employed to evaluate the scattering matrix for vibrational transitions. Thermally-averaged rate coefficients are derived from the obtained cross sections for temperatures in the 10–10000 K interval for excitation of each normal mode of the target molecule. Analytical fits for the rate coefficients for singlets and triplets are provided. In addition, a comprehensive set of calculations are performed for assessing the uncertainty of the present calculations. The uncertainty assessments indicate that the computed observables for vibrational (de-)excitation is reasonable for later use in NO2-containing plasma kinetics modeling.

Normal Modes of Transverse Coronal Loop Oscillations from Numerical Simulations. I. Method and Test Case

Abstract The purpose of this work is to develop a procedure to obtain the normal modes of a coronal loop from time-dependent numerical simulations with the aim of better understanding observed transverse loop oscillations. To achieve this goal, in this paper we present a new method and test its performance with a problem for which the normal modes can be computed analytically. In a follow-up paper, the application to the simulations of Rial et al. is tackled. The method proceeds iteratively and at each step consists of (i) a time-dependent numerical simulation followed by (ii) the Complex Empirical Orthogonal Function (CEOF) analysis of the simulation results. The CEOF analysis provides an approximation to the normal mode eigenfunctions that can be used to set up the initial conditions for the numerical simulation of the following iteration, in which an improved normal mode approximation is obtained. The iterative process is stopped once the global difference between successive approximate eigenfunctions is below a prescribed threshold. The equilibrium used in this paper contains material discontinuities that result in one eigenfunction with a jump across these discontinuities and two eigenfunctions whose normal derivatives are discontinuous there. After six iterations, the approximations to the frequency and eigenfunctions are accurate to ≲0.7% except for the eigenfunction with discontinuities, which displays a much larger error at these positions.

Forty-year review of developments in underwater acoustic modeling

This is the sixth paper in a series of reviews presented at eight-year intervals starting in 1979 [J. Acoust. Soc. Am. 65, S42 (1979); 82, S102 (1987); 97, 3312 (1995); 114, 2430 (2003); 129, 2631 (2011)]. All surveys covered basic acoustic models and sonar performance models. Basic acoustic models included 143 propagation, 29 noise, and 32 reverberation models. Propagation models were categorized according to ray theory, multipath expansion, normal mode, fast field, and parabolic approximation formulations; further distinctions were made between range-independent and range-dependent solutions. Noise models included ambient noise and beam-noise statistics models. Reverberation models included cell and point-scattering formulations. The 44 sonar performance models included active sonar models, model operating systems, and tactical decision aids. Active sonar models integrated basic acoustic models, signal-processing models, and supporting databases into cohesive operating systems organized to solve the sonar equations. Model operating systems provided a framework for the direct linkage of data-management software with computer-implemented codes of acoustic models. Tactical decision aids represented a form of engagement-level simulation that blended environmental and sonar performance information with tactical rules. The overall inventory increased by 5 models per year; historical references are maintained in the fifth edition of Underwater Acoustic Modeling and Simulation.

Pressure response of the THz spectrum of bulk liquid water revealed by intermolecular instantaneous normal mode analysis.

The radial distribution functions of liquid water are known to change significantly their shape upon hydrostatic compression from ambient conditions deep into the kbar pressure regime. It has been shown that despite their eye-catching changes, the fundamental locally tetrahedral fourfold H-bonding pattern that characterizes ambient water is preserved up to about 10 kbar (1 GPa), which is the stability limit of liquid water at 300 K. The observed increase in coordination number comes from pushing water molecules into the first coordination sphere without establishing an H-bond, resulting in roughly two such additional interstitial molecules at 10 kbar. THz spectroscopy has been firmly established as a powerful experimental technique to analyze H-bonding in aqueous solutions given that it directly probes the far-infrared lineshape and thus the prominent H-bond network mode around 180 cm-1. We, therefore, set out to assess pressure effects on the THz response of liquid water at 10 kbar in comparison to the 1 bar (0.1 MPa) reference, both at 300 K, with the aim to trace back the related lineshape changes to the structural level. To this end, we employ the instantaneous normal mode approximation to rigorously separate the H-bonding peak from the large background arising from the pronounced librational tail. By exactly decomposing the total molecular dynamics into hindered translations, hindered rotations, and intramolecular vibrations, we find that the H-bonding peak arises from translation-translation and translation-rotation correlations, which are successively decomposed down to the level of distinct local H-bond environments. Our utmost detailed analysis based on molecular pair classifications unveils that H-bonded double-donor water pairs contribute most to the THz response around 180 cm-1, whereas interstitial waters are negligible. Moreover, short double-donor H-bonds have their peak maximum significantly shifted toward higher frequencies with respect to such long H-bonds. In conjunction with an increasing relative population of these short H-bonds versus the long ones (while the population of other water pair classes is essentially pressure insensitive), this explains not only the blue-shift of the H-bonding peak by about 20-30 cm-1 in total from 1 bar to 10 kbar but also the filling of the shallow local minimum of the THz lineshape located in between the network peak and the red-wing of the librational band at 1 bar. Based on the changing populations as a function of pressure, we are also able to roughly estimate the pressure-dependence of the H-bond network mode and find that its pressure response and thus the blue-shifting are most pronounced at low kbar pressures.

Time evolution of localized solutions in 1-dimensional inhomogeneous FPU models

We study energy localization in a quartic FPU model with spatial inhomogeneity corresponding to a site-dependent number of interacting neighbors. Such lattices can have linear normal modes that are strongly localized in the regions of high connectivity and there is evidence that some of these localized modes persist in the weakly nonlinear regime. The present study shows examples where oscillations can remain localized for long times. Nonlinear normal modes are approximated by periodic orbits that belong to an invariant subspace of a Birkhoff normal form of the system that is spanned by spatially localized modes [F. Martinez-Farias et al., Eur. Phys. J. Special Topics 223, 2943 (2014), F. Martinez-Farias et al., Physica D 335, 10 (2016)]. The invariant subspace is suggested by the dispersion relation and also depends on the overlap between normal modes. Numerical integration from the approximate normal modes suggests that spatial localization persists over a long time in the weakly nonlinear regime and is especially robust in some disordered lattices, where it persists for large, (1), amplitude motions. Large amplitude localization in these examples is seen to be recurrent, i.e. energy flows back and forth between the initial localization region and its vicinity.

Reverberation of Wideband Signals in Shallow Water when Using Sound Focusing

Within the framework of the normal mode approximation, expressions are obtained for calculating bottom reverberation signals recorded by a horizontal array in an inhomogeneous shallow-water waveguide in a wide frequency band. These expressions can be used to simulate bottom-scattered signals both for a monostatic and bistatic geometry, as well as in the case when sound focusing is applied. The constructed model is used to numerically study the structure of bottom reverberation in a waveguide with different parameters and characteristics of the receiver and source systems. The considered bottom inhomogeneities are the slope of the bottom, change in thermocline depth, and wind waves. The study demonstrates the promise of using sound focusing as time reversal using a single receiver–transmitter element to enhance the reverberation signal arriving from a given bottom area.

Circular Geodesics, Paczyński-Witta Potential and QNMs in the Eikonal Limit for Ayón-Beato-García Black Hole

We investigate the comprehensive geodesic structure of a spherically symmetric, static charged regular Ayón-Beato and García black hole (BH). We derive the equation of innermost stable circular orbit (ISCO), marginally bound circular orbit (MBCO) and circular photon orbit (CPO) of said BH, which are most relevant to BH accretion disk theory. Using time-like geodesic properties, we derive Paczyński-Witta potential form for this BH which are very relevant to determine the general relativistic effects on the accretion disk. We show that at a certain radius (For example in case of Reissner-Nordstrøm (RN) BH, r ∗ = Q 2 M ), there exists zero angular momentum (ZAM) orbits due to the repulsive gravity. We also show that in the eikonal approximation, the real and imaginary parts of the quasi normal modes (QNM) of the regular BHs can be expressed as in terms of the frequency of the BH and the instability time scale of the unstable null circular geodesics. Moreover, we study the Bañados, Silk and West effect for this BH. We show that the center-of-mass (CM) energy of colliding neutral test particles near the infinite red-shift surface of the regular BHs have the finite energy. In the Appendix section, we have discussed the possibility of a regular ABG BH can act as particle accelerators when two charged test particles of different energies are colliding and approaching to the horizon of the BH provided that one of charged test particle has a critical value of charge.

Axisymmetric viscous interfacial oscillations – theory and simulations

We study axisymmetric, free oscillations driven by gravity and surface tension at the interface of two viscous, immiscible, radially unbounded fluids, analytically and numerically. The interface is perturbed as a zeroth-order Bessel function (in space) and its evolution is obtained as a function of time. In the linearised approximation, we solve the initial value problem (IVP) to obtain an analytic expression for the time evolution of wave amplitude. It is shown that a linearised Bessel mode temporally evolves in exactly the same manner as a Fourier mode in planar geometry. We obtain novel analytical expressions for the time varying vorticity and pressure fields in both fluids. For small initial amplitudes, our analytical results show excellent agreement with those obtained from solving the axisymmetric Navier–Stokes equations numerically. We also compare our results with the normal mode approximation and find the latter to be an accurate representation at very early and late times. The deviation between the normal mode approximation and the IVP solution is found to increase as a function of viscosity ratio. The vorticity field has a jump discontinuity at the interface and we find that this jump depends on the viscosity and the density ratio of the two fluids. Upon increasing the initial perturbation amplitude in the simulations, nonlinearity produces qualitatively new features not present in the analytical IVP solution. Notably, a jet is found to emerge at the axis of symmetry rising to a height greater than the initial perturbation amplitude. Increasing the perturbation amplitude further causes the jet to undergo end pinch off, giving birth to a daughter droplet. This can happen either for an advancing or a receding jet, depending on the viscosity ratio. A relation is found between the maximum jet height and the perturbation amplitude. Hankel transform of the interface demonstrates that at large perturbation amplitudes higher wavenumbers emerge, sharing some of the energy of the lowest mode. When these additional higher modes are present, the interface has pointed crests and rounded troughs.