Harmonic Eigenvalues Research Articles

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T We find that the spectrum of eigenvalues [Formula: see text] has a sharp maximum near (but not at) [Formula: see text] and decreases monotonically with [Formula: see text] on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for [Formula: see text], which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T More importantly, we find an empirical correlation of the positions of the maxima of [Formula: see text] with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to [Formula: see text] from the temperature T We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

Confluent conformal blocks and the Teukolsky master equation

Quasinormal modes of usual, four dimensional, Kerr black holes are described by certain solutions of a confluent Heun differential equation. In this work, we express these solutions in terms of the connection matrices for a Riemann-Hilbert problem, which was recently solved in terms of the Painlev\'e V transcendent. We use this formulation to generate the small-frequency expansion for the angular spheroidal harmonic eigenvalue, and derive conditions on the monodromy properties for the radial modes. Using exponentiation, we relate the accessory parameter to a semi-classical conformal description and discuss the properties of the operators involved. For the radial equation, while the operators at the horizons have Liouville momenta proportional to the entropy intake, we find that spatial infinity is described by a Whittaker operator.

Fluid–structure interactions in a cylindrical layered wave guide with application in the spinal column to syringomyelia

Syringomyelia is a disease of the spinal cord in which fluid-filled cavities, called syrinxes, form and expand, compressing the surrounding neural tissue and producing neurological damage. This condition can occur following spinal injury and has limited treatment options in part because of a lack of understanding of its origins. Current pathogenic theories make predictions from localised disturbances in the cerebrospinal fluid dynamics. The poroelastic spinal cord tissues have an effective compressibility as a result of the localised displacement of the intercellular fluid. Also, despite the disease progressing over months and years, the mechanical perturbations from the heart and breathing cycles and from coughs and sneezes operate over much shorter timescales of seconds down to fractions of a second, respectively. To model these pathological features, we solve the harmonic eigenvalue problem for a two-dimensional elastic fluid–solid cylindrically layered waveguide. We analyse the dispersion behaviour and mode shapes over a wide wavelength spectrum and investigate the influence of Poisson's ratio on the wave mode characteristics. The healthy model consists of an elastic solid cylinder (spinal cord) and a surrounding annulus of inviscid fluid (cerebrospinal fluid in the subarachnoid space). To model syringomyelia we then add a cylindrical fluid space (syrinx) within the elastic cylinder. For the geometry with the inner fluid conduit present (diseased spinal cord in situ), the dispersion diagrams show three modes (0, 1, 2) that are propagatory across the wavelength spectrum. At long wavelengths, axial displacement dominates over radial, normal stress dominates over shear, but these become of the same order of magnitude as the wavelength approaches the radius of the elastic cylinder (spinal cord). Wave modes 0 and 1 induce relative motion between the elastic cylinder and the fluid contained within, which supports Williams's so-called slosh mechanism for syrinx lengthening, and all three modes involve stress concentrations (indicated by localised gradient peaks) in the inner wall of the elastic cylinder adjacent to the contained fluid, which would tend to support radial syrinx expansion. Mode 2 is the most sensitive to changes in Poisson's ratio (spinal cord tissue compressibility). We also investigate the progression of post-traumatic syringomyelia by simulating a sequence of discrete states in the gradual occlusion of the subarachnoid space and the subsequent formation and radial expansion of a syrinx. We predict that syrinx-fluid sloshing, hence the stress so induced, diminishes with the continued radial expansion of the syrinx. The largest normal stress concentrations at the syrinx boundary, which appear for modes 0 and 1, initially diminish with radial syrinx expansion; however, they reach a minimum value for intermediate syrinx diameters. On this basis we hypothesize that syrinx development may be part of a homeostatic mechanism to alleviate stress in the spinal cord. Understanding this process may aid in treatment development. The present work also has applications in industrial systems and serves as a platform for more advanced modelling of cylindrical waveguides in biological systems.

Higher harmonic analyses of the 252Cf source driven noise analysis method

A theoretical formula has been derived to reconstruct the ratio of spectral densities in the 252Cf source driven noise analysis (CSDNA) method from the higher harmonic eigenvalues and eigenfunctions of the α-mode neutron transport equations. The formula closely reproduces each power spectral density and the ratio of spectral densities calculated by Monte Carlo simulations, thereby verifying the theoretical formula. The reactivity or keff of a subcritical system is related to the ratio of spectral densities by the fundamental mode approximation in which the higher harmonic modes are neglected. However, the ratio of spectral densities measured in the CSDNA method yields an ambiguous reactivity or keff that depends on the locations of detectors due to the effect of the higher harmonics. A more elaborate method developed by Mihalczo and Valentine et al. infers an “experimental” keff using a measurement and accurate Monte Carlo simulation of CSDNA. This paper discusses the uniqueness of the inferred “experimental” keff.

Infrared Spectra at a Conical Intersection: Vibrations of Methoxy

The J = 0 infrared spectrum of methoxy is theoretically calculated for the ground X̃(2)E state using a quartic potential energy force field, and the quadratic dipole moment expansion is calculated ab initio at the CCSD(T) level of theory and cc-pVTZ basis. Writing these expansions with vibronic operators whose symmetry properties are defined with respect to C(3v) rotation greatly simplifies these calculations. With minor adjustments to the force field, excellent agreement with experiment is found for both the transition energies of CH(3)O and those of CD(3)O. The role of Jahn-Teller and Fermi coupling is illustrated by scaling these terms by a parameter δ that varies from 0 to 1. Plotting the eigenvalues as a function of δ yields a correlation diagram connecting the harmonic eigenvalues to those of the fully coupled problem. The spectrum for CH(3)O is determined using a combination of Davidson and Lanczos iteration schemes. The spectral features are found to be dominated by Jahn-Teller effects, but direct Fermi coupling and indirect potential couplings have important roles. The origin of the complexities in the CH stretch region are discussed.

Spatial neutronic coupling aspects in nuclear reactors

In this paper, an effort is made to gain insights about neutronic coupling and decoupling phenomena of nuclear reactors and its consequences on their safety and stability. The neutronic coupling and decoupling aspects are investigated using eigenvalue separation (EVS) methodology. Higher harmonic eigenvalues are calculated by the method of mode subtraction. The eigenvalue separation for a typical 1000 MWe PWR is calculated and its relations with reactor core shape and size and consequent effects on spatial stability are investigated. It is demonstrated quantitatively that it is necessary to optimize height to diameter ( H/ D) ratio to suppress the susceptibility to multimode oscillations and to enable ease in designing spatial control algorithm. Consequences of extreme H/ D ratio are also addressed. Optimum shape of the reactor core is investigated and the evaluation of upper limit of about 1.3 for H/ D ratio has been carried out for large PWR cores. Safety implications of neutronic loose coupling on departure from nucleate boiling ratio (DNBR) are also addressed.

Anharmonicity of water stretching vibrations in series of isomorphous crystalline hydrates: Copper and manganese saccharinates hexahydrates

On the basis of the experimentally obtained frequencies of isotopically isolated OH and OD species, the anharmonicity constants, as well as the harmonic eigenvalues of the OH and OD stretching vibrations for two members of the isomorphous series of metal(II) saccharinates hexahydrates (those of Mn and Cu) were calculated using several theoretical models [B. Berglund, J. Lindgren, J. Tegenfeldt, J. Mol. Struct. 43 (1978) 169, M.G. Sceats, S.A. Rice, J. Chem. Phys. 71 (1979) 973, H. Engstrom, J.B. Bates, L.A. Boatner, J. Chem. Phys. 73 (1980) 1073]. The anharmonicity constants and the ν(OH)/ ν(OD) isotopic ratios correlate well with the stretching frequencies of the isotopically isolated OH and OD oscillators. Both anharmonicity constants and isotopic ratios as criteria for the anharmonicity of the OH (OD) vibrations show that, with a very few exceptions, it increases with the increase in the hydrogen bond strength. The exceptions from the trend are explained in terms of local electrostatic field differences and force constant changes due to the coordination to the metal ion. The obtained regression equations were used to predict the anharmonicity constants of the ν(OD) modes in other members of the series. Within the three-particle model of the hydrogen bonded complex, the parameters characterising the coupling of the ν(OH)/ ν(OD)/modes with the low-frequency ν(O⋯O) ones were estimated. The positions of the overtones of OH and OD stretching vibrations in the compounds of copper and manganese were estimated using the calculated anharmonicity parameters. The predicted values for the OD oscillators were compared with the experimentally obtained data. Model calculations of the described type are shown to be valuable in the assignment of the second-order transitions, especially in complex systems.

Latent symmetry versus accidental degeneracy effects in the vibrational spectra of dopant chromate anions in M 2Cr xS 1− xO 4 solid solutions (M∈{K, Rb, Cs})

Fourier transform IR and Raman spectra of chromate anions isomorphously isolated in potassium, rubidium and cesium sulfate matrices were recorded at room and low temperature (∼100 K). On the basis of the detected second-order vibrational transitions involving the dopant species ν 3 mode components, anharmonicity constants and harmonic eigenvalues for these modes were calculated. Despite the fact that, rigorously speaking, the appearance of the spectra of dopant chromate anions may be explained in terms of a C s site symmetry, the ν 3 stretching region resembles an approximate ( A 1⊕ E) splitting (characteristic for a local C 3 v symmetry). Although such findings may be rationalized in terms of the latent symmetry concept, we propose an alternative explanation, based on the concept of vibrational mode mixing (a ‘Fermi-like’ resonance) of the CrO 4 2− ν 1 mode with the ν 3 a site-group component. We also derive a quantitative model based on the degenerate case stationary perturbation theory that allows an estimation of the relative importance of the latent symmetry versus vibrational mode mixing effects. In the present case, we show that the Fermi-like resonance is predominantly responsible for the observed spectral features.

Infrared spectra of permanganate anions in potassium perchlorate matrices: vibrational anharmonicity, effective symmetry and vibrational mode mixing effects

Fourier transform infrared spectra of MnO 4 − anions isomorphously isolated in potassium perchlorate matrices were recorded at room and low temperature (LT, ∼100 K). On the basis of the detected second-order vibrational transitions involving the dopant species ν 3 mode components, anharmonicity constants and harmonic eigenvalues for these modes were calculated. Despite the fact that, rigorously speaking, the appearance of the spectra of dopant permanganate anions may be explained in terms of a C s site symmetry, the ν 3 stretching region resembles an approximate A 1⊕ E splitting (characteristic for a local C 3 v or even higher symmetry), which is not expected, even within the latent symmetry approach. We explain such spectral patterns on the basis of vibrational mode mixing (a ‘Fermi-like’ resonance) of the MnO 4 − ν 1 mode with the ν 3a site-group component. With the results of degenerate case stationary perturbation theory, we show that in the present case the Fermi-like resonance is predominantly responsible for the observed spectral features. The appearance of the region of second-order vibrational transitions in the spectra of dopant permanganate anions may be better explained in terms of the (rigorous) crystallographic C s site group (corresponding to the crystallographic Pnma space group), instead of the ‘latent’ (effective) symmetry site group C 2 v (corresponding to the latent symmetry space group Imma).

Fourier transform infrared study of perchlorate (35ClO4− and 37ClO4−) anions isomorphously isolated in potassium permanganate matrix. Vibrational anharmonicity and pseudo-symmetry effects

Fourier transform infrared spectra of isotopomeric 35ClO4− and 37ClO4− anions isomorphously isolated in potassium permanganate matrix were recorded at room and low temperature (LT, ∼100K). On the basis of the detected second-order vibrational transitions involving the dopant species ν3 mode components, anharmonicity constants and harmonic eigenvalues for these modes were calculated. Although the overall appearance of the region of fundamental vibrational transitions in the spectra of dopant perchlorate anions may be better explained in terms of a pseudo-symmetry (the so-called ‘latent’ symmetry) site group C2v, corresponding to the pseudo-symmetry space group Imma, instead of the (rigorous) crystallographic Cs one (corresponding to the crystallographic Pnma space group), an opposite statement seems to be valid for the region of the second-order vibrational transitions. The vibrational mode mixing (a ‘Fermi-like’ resonance) of the ClO4− ν1 mode with the ν3a site group component is almost negligible.

Elasticity and melting of vortex crystals in anisotropic superconductors: Beyond the continuum regime

The elastic moduli of vortex crystals in anisotropic superconductors are frequently involved in the investigation of their phase diagram and transport properties. We provide a detailed analysis of the harmonic eigenvalues ~normal modes! of the vortex lattice for general values of the magnetic field strength, going beyond the elastic continuum regime. The detailed behavior of these wave-vector-dependent eigenvalues within the Brillouin zone ~BZ!, is compared with several frequently used approximations that we also recalculate. Throughout the BZ, transverse modes are less costly than their longitudinal counterparts, and there is an angular dependence which becomes more marked close to the zone boundary. Based on these results, we propose an analytic correction to the nonlocal continuum formulas which fits quite well the numerical behavior of the eigenvalues in the London regime. We use this approximate expression to calculate thermal fluctuations and the full melting line ~according to Lindeman’s criterion ! for various values of the anisotropy parameter.

Vibrational anharmonicity and vibrational Stark effect of sulfate ions trapped in potassium, rubidium and cesium chromate

Room and low temperature (≈100 K) FT-IR and Raman spectra of the sulfate doped K 2CrO 4, Rb 2CrO 4 and Cs 2CrO 4 were recorded. The positions of the ν 1, ν 3 and ν 4 fundamental mode components of the dopant anions were measured. Nine (out of possible ten) second-order stretch-stretch vibrational transitions of the dopant anions were detected. On the basis of these data, the anharmonicity constants and the corresponding harmonic eigenvalues were calculated for several vibrational transitions using second-order perturbation theory expressions. The anharmonicity of the studied second-order transitions of the type ν 1+ ν 3i increases in the order (SO 4/K 2CrO 4)<(SO 4/Rb 2CrO 4)<(SO 4/Cs 2CrO 4), while for those of the type ν 3j+ ν 3i it basically follows the trend: (SO 4/K 2CrO 4)>(SO 4/Rb 2CrO 4)<(SO 4/Cs 2CrO 4). The measured relative Stark splittings of the ν 3 and ν 4 mode components of the dopant SO 4 2− anions, as well as the average X 13i/ X 3i3j values decrease in the order (SO 4/K 2CrO 4)>(SO 4/Rb 2CrO 4)>(SO 4/Cs 2CrO 4). In all cases, the splitting is larger for ν 3 than for ν 4 modes, indicating a smaller angular than the bond length distortion. The theory of vibrational Stark effect suggests that the observed frequency shifts of the ν 1, ν 3 and ν 4 mode components may be attributed to the increase of the field strength at the doped anion site going from K 2CrO 4 to Cs 2CrO 4. The Stark splitting of these modes, on the other hand, implies that the internal crystalline field vector is almost parallel to the (hypothetical) C 2 axis of the slightly distorted dopant tetrahedral anions.

Infrared spectra of sulphate ions trapped in rubidium and cesium selenate: harmonic frequencies and anharmonicity constants for antisymmetric SO4 stretching vibrations

Room and low temperature (∼100K) IR spectra of the sulphate doped Rb2SeO4 and Cs2SeO4 were recorded. The positions of the ν3 and ν4 fundamental mode components of the dopant anions, as well as of the ν1 mode, were precisely measured. Nine (out of possible ten) second-order vibrational transitions of the dopant anions were detected. The anharmonicity constants and the harmonic eigenvalues were calculated for several second-order transitions on the basis of the second-order perturbation theory expressions. Comparison with our previous results show that the anharmonicity of practically all of the studied second-order transitions decreases in the order (SO4/K2SeO4)>(SO4/Rb2SeO4)>(SO4/Cs2SeO4). According to the measured relative splitting of the ν3 and ν4 mode components, both angular and bond-length distortion of the dopant SO42− anions decrease in the same order. In all cases, the angular distortion seems to be smaller than the bond length one. The correlation of intensity of the bands due to the second order transitions vs. the corresponding anharmonicity constants suggests that the significance of the electrical anharmonicity increases in the order (SO4/K2SeO4)<(SO4/Rb2SeO4)<(SO4/Cs2SeO4).

Anharmonicity of water stretching vibrations in series of isomorphous crystalline hydrates. II. Metal(II) saccharinates hexahydrates (metal(II)∈{Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd})

Using three theoretical models, the anharmonicity constants and the harmonic eigenvalues of the O–H and O–D stretching vibrations were calculated for several members of the isomorphous series of metal(II) saccharinates hexahydrates (those of Cr, Mn, Fe, Co, Ni, Cu, Zn and Cd), on the basis of the experimentally obtained (by FTIR spectroscopy) frequencies of isotopically isolated OH and OD species. With a very few exceptions, the anharmonicity of the νOH(OD) vibrations increases with the increase in the hydrogen bond strength. A good correlation was found to exist between the anharmonicity constants and the νOH(OD) frequencies. The positions of the νOH(OD) overtones were estimated using the calculated anharmonicity parameters and the predicted values for the OD oscillators were compared with the experimentally obtained data.

Harmonic perturbation theory for the analysis of boiling water reactor regional instabilities

The onset of regional instabilities in the Boiling Water Reactor has been directly related to the reactor subcritical reactivity. Because the calculation of higher harmonic modes for practical reactor problems is a significant computational expense, the research here investigated the application of perturbation methods to predict the subcritical reactivity. The method developed here is based on a second order perturbation theory for the first harmonic eigenvalue. An original contribution of this work is an innovative method to predict the rotation of the first harmonic flux after a perturbation. This provides a second order estimate of the harmonic eigenvalue, which combined with a first order estimate of the fundamental eigenvalue, yields a perturbation estimate of the subcritical reactivity that is within a few percent of exact calculations for a series of two-dimensional benchmark problems.

On Standard Eigenvalues of Variable-Coefficient Heat and Rod Equations

Forms of the variable-heat-conductivity coefficient function in the one-dimensional heat equation are determined which yield a standard harmonic eigenvalue sequence as in the case of homogeneity. The continuous case is found to correspond to a four-thirds power law dependence on coordinate. For the stepped case, the condition on the ratio of segmental heat conductivities in terms of the junction location is presented.

General theory of dynamic light scattering from cylindrically symmetric particles with translational-rotational coupling

The general solution to the scattered electric field time autocorrelation function for a dilute system of cylindrically symmetric, optically anisotropic particles undergoing coupled translational-rotational diffusion is presented. The solution is exact within the Rayleigh–Gans–Debye approximation. The scattered time autocorrelation function is an infinite series of decaying exponentials with time constants τ−1Ms =q2D+[s(s+1)+wMs(γ)−γ/3]Θ⊥, (M=0,1,2; s=0,1,2, ⋅ ⋅ ), containing both translational and rotational diffusion coefficients and spheroidal harmonic eigenvalues. The latter depend on the coupling parameter γ=q2 (ΔD)/(Θ⊥), where ΔD is the anisotropy in the translational diffusion coefficient. The dynamical terms are weighted by structure factors that depend on scattering geometry, the particle geometry, and the coupling parameter. These structure factors are given for arbitrary geometries and polarization of the radiation. In general, the weights for two to six exponentials will be significant depending on the scattering geometry and particle size. The domain of applicability of the theory is also discussed.