Normal Mode Treatment Research Articles

Pulsational mode dynamics in the presence of negions

The stability dynamics, of the gravito-electrostatically driven pulsational mode of gravitational collapse (PMGC) excitable in spherically symmetric dust molecular clouds (DMCs), is systematically theorized. It actively incorporates the interplay of dust viscosity, dust charge fluctuation dynamics, and negionic (negative ionic) effects. We take the DMC constitutive electrons, positive ions, and negions as the Boltzmann species (lighter). The negatively charged dust grains behave as viscous fluids (heavier). A generalized linear cubic dispersion equation, involving a new and unique set of multi-factorial dispersion coefficients, results from the applied spherical normal mode treatment in the DMC model. The diverse growth characteristics of the PMGC dynamics are investigated illustratively. It is found mainly that the cloud size, dust charge, equilibrium electronic number density, and equilibrium dust number density play stabilizing roles in the PMGC. However, the dust mass, equilibrium positive ionic number density, equilibrium negionic number density, and dust viscosity play destabilizing roles against self-gravity. The reliability of our study is ensured by the fact that the diversified stabilizing features explored here are consistent with the previous results. The applicability of our study in astrophysical gravity-driven structure formation processes moderated actively with the negionic effects is finally outlined.

Anelasticity across seismic to tidal timescales: a self-consistent approach

In a pioneering study, Wahr & Bergen developed the widely adopted, pseudo-normal mode framework for predicting the impact of anelastic effects on the Earth's body tides. Lau et al. have recently derived an extended normal mode treatment of the problem (as well as a minor variant of the theory known as the direct solution method) that makes full use of theoretical developments in free oscillation seismology spanning the last quarter century and that avoids a series of assumptions and approximations adopted in the traditional theory for predicting anelastic effects. There are two noteworthy differences between these two theories: (1) the traditional theory only considers perturbations to the eigenmodes of an elastic Earth, whereas the new theory augments this set of modes to include the relaxation modes that arise in anelastic behaviour; and (2) the traditional theory approximates the complex perturbation to the tidal Love number as a scaled version of the complex perturbation to the elastic moduli, whereas the new theory computes the full complex perturbation to each eigenmode. In this study, we highlight the above differences using a series of synthetic calculations, and demonstrate that the traditional theory can introduce significant error in predictions of the complex perturbation to the Love numbers due to anelasticity and the related predictions of tidal lag angles. For the simplified Earth models we adopt, the computed lag angles differ by ∼20 per cent. The assumptions in the traditional theory have important implications for previous studies that use model predictions to correct observables for body tide signals or that analyse observations of body tide deformation to infer mantle anelastic structure. Finally, we also highlight the fundamental difference between apparent attenuation (i.e. attenuation inferred from observations or predicted using the above theories) and intrinsic attenuation (i.e. the material property investigated through experiments), where both are often expressed in terms of lag angles or Q−1. In particular, we demonstrate the potentially significant (factor of two or more) bias introduced in estimates of Q−1 and its frequency dependence in studies that have treated Q−1 determined from tidal phase lags or measured experimentally as being equal. The observed or theoretically predicted lag angle (or apparent Q−1) differs from the intrinsic, material property due to inertia, self-gravity and effects associated with the energy budget. By accounting for these differences we derive, for a special case, an expression that accurately maps apparent attenuation predicted using the extended normal mode formalism of Lau et al. into intrinsic attenuation. The theory allows for more generalized mappings which may be used to robustly connect observations and predictions of tidal lag angles to results from laboratory experiments of mantle materials.

A normal mode treatment of semi-diurnal body tides on an aspherical, rotating and anelastic Earth

Normal mode treatments of the Earth's body tide response were developed in the 1980s to account for the effects of Earth rotation, ellipticity, anelasticity and resonant excitation within the diurnal band. Recent space-geodetic measurements of the Earth's crustal displacement in response to luni-solar tidal forcings have revealed geographical variations that are indicative of aspherical deep mantle structure, thus providing a novel data set for constraining deep mantle elastic and density structure. In light of this, we make use of advances in seismic free oscillation literature to develop a new, generalized normal mode theory for the tidal response within the semi-diurnal and long-period tidal band. Our theory involves a perturbation method that permits an efficient calculation of the impact of aspherical structure on the tidal response. In addition, we introduce a normal mode treatment of anelasticity that is distinct from both earlier work in body tides and the approach adopted in free oscillation seismology. We present several simple numerical applications of the new theory. First, we compute the tidal response of a spherically symmetric, non-rotating, elastic and isotropic Earth model and demonstrate that our predictions match those based on standard Love number theory. Second, we compute perturbations to this response associated with mantle anelasticity and demonstrate that the usual set of seismic modes adopted for this purpose must be augmented by a family of relaxation modes to accurately capture the full effect of anelasticity on the body tide response. Finally, we explore aspherical effects including rotation and we benchmark results from several illustrative case studies of aspherical Earth structure against independent finite-volume numerical calculations of the semi-diurnal body tide response. These tests confirm the accuracy of the normal mode methodology to at least the level of numerical error in the finite-volume predictions. They also demonstrate that full coupling of normal modes, rather than group coupling, is necessary for accurate predictions of the body tide response.

Absolute/Convective Instability Dichotomy in a Soret-Driven Thermosolutal Convection Induced in a Porous Layer by Inclined Thermal and Vertical Solutal Gradients

In this article, we extend the analysis of Diaz and Brevdo (J. Fluid Mech. 681:567–596, 2011) of the absolute/convective instability dichotomy at the onset of convection in a saturated porous layer with either horizontal or vertical salinity and inclined temperature gradients to studying the influence of the Soret effect on the dichotomy in a similar model. Only longitudinal modes are considered. We treat first normal modes and analyze the influence of the Soret effect on the critical values of the vertical thermal Rayleigh number, R v, wavenumber, l, and frequency, ω, for a variety of values of the horizontal thermal Rayleigh number R h, and the vertical salinity Rayleigh number, S v. Our results for normal modes agree well with relevant results of Narayana et al. (J. Fluid Mech. 612:1–19, 2008) obtained for a similar model in a different context. In the computations, we use a high-precision pseudo-spectral Chebyshev-collocation method. Further, we apply the formalism of absolute and convective instabilities and compute the group velocity of the unstable wavepacket emerging in a marginally unstable state to determine the nature of the instability at the onset of convection. The influence of the Soret effect on the absolute/convective instability dichotomy present in the model is treated by considering the destabilization for seven values of the Soret number: S r = −1, −0.5, −0.1, 0, 0.1, 0.5, 1, for all the parameter cases in the treatment of normal modes.

Relaxation of thermal concentration fluctuations in ternary liquids

Abstract Many small-angle neutron experiments have proven that concentration fluctuations develop in liquid alloys. The present work is devoted to the study of the relaxation of these concentration fluctuations. The mathematics associated with the analysis is close to the normal mode treatment of lattice vibrations. When applied to a ternary liquid exhibiting strong heteroatomic bonding between two components having high atomic mobilities, a long-lived clustering mode of relaxation and a short-lived substitutional mode are defined. It is found that the long-lived clustering relaxation mode is generated from a concentration fluctuation which is also the most probable. This gives physical meaning to the socalled ‘clustering’ in liquids where atomic cohesion is governed by metallic bonding. The possible influence of these long-lived clusters on crystalline nucleation is also discussed.

Relaxation of thermal concentration fluctuations in ternary liquids

Abstract Many small-angle neutron experiments have proven that concentration fluctuations develop in liquid alloys. The present work is devoted to the study of the relaxation of these concentration fluctuations. The mathematics associated with the analysis is close to the normal mode treatment of lattice vibrations. When applied to a ternary liquid exhibiting strong heteroatomic bonding between two components having high atomic mobilities, a long-lived clustering mode of relaxation and a short-lived substitutional mode are defined. It is found that the long-lived clustering relaxation mode is generated from a concentration fluctuation which is also the most probable. This gives physical meaning to the socalled ‘clustering’ in liquids where atomic cohesion is governed by metallic bonding. The possible influence of these long-lived clusters on crystalline nucleation is also discussed.

Inelastic Neutron Scattering: A Tool in Molecular Vibrational Spectroscopy and a Test of ab Initio Methods

Inelastic neutron scattering (INS) as discussed in this article is a technique for studies of the atomic vibrations of solids at low temperature. Unique features of this method are the absence of selection rules and the predominance of scattering due to the motion of hydrogen (but not deuterium) atoms. The scattering from hydrogen is given by a sum of intensities from individual atoms and is easily calculated from a normal mode treatment. When hydrogen is absent, the scattering is usually dominated by coherent contributions involving the motions of pairs of atoms. In either case, ab initio methods can be tested by comparison of the calculated INS spectral intensities as well as the frequencies of molecular vibrations with the observed spectra. For molecular solids, these spectra may be interpreted in molecular terms. In cases where intermolecular interactions are important, these must be included explicitly. Hydrogen-bonded systems are of greatest interest in this respect. Selective deuteration of hydrogen-bonded systems can be used to provide detailed tests of the potential surfaces. At some level of resolution, the effects of intermolecular interactions are revealed even for hydrocarbons.

Intra- and Interchain Relaxation Processes in Meshlike Polymer Networks

The dynamic properties of meshlike polymer networks are considered using the cubic network of multisegmental “bead and spring” Rouse chains. The normal mode treatment is used. Rigorous expressions for the mean-square displacements of elements of the Gaussian network (junctions and nonjunction beads) are obtained. The autocorrelation functions of chain vectors between neighboring nonjunction beads (subchain vectors) and of end-to-end chain vectors between neighboring junctions are calculated. The time dependence of the relaxation modulus of a network is also considered. Contributions of intra- and interchain relaxation processes to various dynamic characteristics of a Gaussian network are compared. The time behavior of dynamic quantities obtained is estimated for different scale motions. It is shown that the relaxation of regular polymer networks consisting of long multisegmental Gaussian chains can be considered as a superposition of four different types of relaxation processes. These are the relaxation of network chains with fixed junctions, the relaxation of chains with free ends, the relaxation processes in the intermediate region of the spectrum, and interchain cooperative network motions. The possibility of describing low-frequency dynamic properties of networks by a simplified coarse-grained network model used previously is proved.

Electronic absorption spectra of molecules and aggregates with interatomic charge transfer using a normal mode treatment of the atomic monopole–dipole interaction model

The theory of the atom monopole–dipole interaction model for electronic polarizability is extended to the case of complex frequency-dependent atom polarizabilities, allowing the prediction of absorption spectra in terms of a set of electronic normal modes. The charge constraints on the system are taken into account using Lagrangian and penalty function methods. Expressions are obtained for the complex polarizability of an aggregate of molecules in which intermolecular charge transfer is precluded. The results are illustrated by calculations for naphthalene using atom polarizability parameters which give an approximate fit to the experimentally observed absorption spectrum. We also treat helical stacked aggregates of naphthalene molecules and predict intensity shifts analogous to those observed in the stacked aromatic bases of helical nucleic acids.

Theoretical π-π* circular dichroic spectra of helical poly(glycine) and poly(L-alanine) as functions of backbone torsion angles.

Circular dichroic spectra and oscillator strengths of the π-π transition near 190 nm are calculated for helical (Gly)6 and (Ala)6 at 30° intervals of the backbone torsion angles (ϕ,ψ) over the range -180° ≤ ϕ ≤ -60°, -60° ≤ ψ ≤ 180°, using the partially dispersive normal mode treatment of the dipole interaction model. Polarizabilities of atoms and the NC'O group are those determined semiempirically in previous studies. Calculations for (Ala)6 at (ϕ,ψ) angles corresponding to the α-helix, the poly(Pro) II helix, a collagen single helix, a poly-(MeAla) helix, and single β-helices are found to agree well with most of the available experimental data.

Erratum: A normal mode treatment of optical properties of a classical coupled dipole oscillator system with Lorentzian band shapes

DeVoe’s classical coupled dipole oscillator model for molecular optical properties is specialized for the case where the individual oscillators have complex polarizabilities with Lorentzian band shapes. The optical properties of this system are derived in terms of a set of eigenvectors and eigenvalues which describe the normal modes of the system. When the real and imaginary parts of the interaction matrix can be diagonalized by the same transformation, the properties are expressed as explicit functions of frequency and do not require a matrix inversion at each frequency. This condition is met by a system in which all oscillators have the same bandwidth. For such a case the computation time is much less than that required by the more general method involving point‐by‐point matrix inversion. A further simplification is achieved for a system in which a subset of the oscillators have natural frequencies so high that their polarizabilities may be regarded as nondispersive. For this case the order of the eigenvalue problem is reduced to the number of dispersive oscillators. Sum rules are derived for the mixed dispersive/nondispersive system. The normal mode method is illustrated by calculations for a 12‐residue right‐handed α‐helix in which oscillators at 52 000 and 67 600 cm−1 are assigned to each residue. The predicted absorption and circular dichroic spectra are similar to those predicted using Gaussian band shapes for the individual oscillators by the matrix inversion method, while a 40‐fold saving in computer time was achieved by the normal mode method.

A normal mode treatment of optical properties of a classical coupled dipole oscillator system with Lorentzian band shapes

DeVoe’s classical coupled dipole oscillator model for molecular optical properties is specialized for the case where the individual oscillators have complex polarizabilities with Lorentzian band shapes. The optical properties of this system are derived in terms of a set of eigenvectors and eigenvalues which describe the normal modes of the system. When the real and imaginary parts of the interaction matrix can be diagonalized by the same transformation, the properties are expressed as explicit functions of frequency and do not require a matrix inversion at each frequency. This condition is met by a system in which all oscillators have the same bandwidth. For such a case the computation time is much less than that required by the more general method involving point-by-point matrix inversion. A further simplification is achieved for a system in which a subset of the oscillators have natural frequencies so high that their polarizabilities may be regarded as nondispersive. For this case the order of the eigenvalue problem is reduced to the number of dispersive oscillators. Sum rules are derived for the mixed dispersive/nondispersive system. The normal mode method is illustrated by calculations for a 12-residue right-handed α-helix in which oscillators at 52 000 and 67 600 cm−1 are assigned to each residue. The predicted absorption and circular dichroic spectra are similar to those predicted using Gaussian band shapes for the individual oscillators by the matrix inversion method, while a 40-fold saving in computer time was achieved by the normal mode method.

Lattice vibrations of gallium metal I. Group-theoretical analysis

A basic group-theoretical treatment of normal modes of vibration is developed, and applied in detail to the structure of gallium (which has four atoms per primitive unit cell). The introduction of a family of linear spaces leads to the definition of a dynamical space of high dimensionality. The central feature of this treatment consists in the establishment of representations in the dynamical space of the point groups of the wave vector k. Various properties of these representations considerably simplify the dynamical eigenvalue problem, in particular: (i) by decomposition of the representations it is possible to classify the phonon dispersion branches; (ii) use of a certain invariance condition simplifies the dynamical matrix; (iii) projection operator techniques may be applied to derive the form of the eigenvectors describing the atomic motions. Application of these considerations to the gallium structure provides basic information which has been used for the experimental observation of phonon curves in gallium as detailed in a second paper.

Simplified normal mode treatment of long-period acoustic-gravity waves in the atmosphere

This paper deals primarily with the effects of geometric dispersion on low-frequnncy mechanical waves generated by nuclear explosions. This dispersion is the result of the stratification of the atmosphere (to be distinguished from dispersion due to changes in physical characteristics due to changing frequency). It is found that the pressure signal can be divided naturally into an early-arriving acoustic-gravity portion (treated in this report) and a later--by about five percent of the travel time--acoustic portion. In general, both portions of the signal are inversely proportional to the range (geometric spreading included), although, at very great ranges, portions of the gravity wave fall off faster by r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/6</sup> ; the effect of dispersion is to reduce the signal by r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> . Most of the signal is composed of many propagating modes which, at a given time and range, will each demonstrate a characteristic frequency. A simplified treatment of this complicated model picture is presented here. It is argued that a ray treatment for the higher-frequency portion is appropriate. It is shown that the frequency of the long-period signal increases with time. So long as the frequency of the received signal is less than the characteristic frequency of the initiating explosive impulse, it is found that the signal has a universal form for the fundamental mode. Such characteristics as yield, range, and phase velocity merely change the scaling of the signal. It is concluded that attenuation is probably not important for the low-frequency signals (below one c/s) usually observed.

Boundary value problems in plasma oscillations: The plasma capacitor

A normal mode treatment is proposed for the solution of boundary value problems in plasma oscillations. The plasma distribution function and electric field are expanded in terms of the solutions of the coupled Vlasov and Maxwell equations. These normal mode solutions are somewhat unusual in that they are singular functions of the velocity variable, and, in fact, the expansion theorem is proved by means of a direct solution of an associated singular integral equation. As an application of the method we obtain the impedance of a plasma-filled parallel plate condenser.