Vector Spherical Harmonics Research Articles

Jahn-Teller distortion in the lowest excited singlet state of C60.

Several properties of the lowest excited singlet state of the ${\mathrm{C}}_{60}$ molecule are investigated with reference to the Jahn-Teller distortion. We find that the symmetry of the molecular structure in the relaxed excited state is approximately the ${\mathit{C}}_{2\mathit{h}}$ symmetry, though the strict symmetry is that of the inversion group ${\mathit{C}}_{\mathit{i}}$. Also it is found that the most remarkable contribution to the Jahn-Teller distortion is given by the ${\mathit{h}}_{\mathit{g}}$ mode of the lowest frequency and the next by the ${\mathit{h}}_{\mathit{g}}$ mode of the fourth-lowest frequency. Furthermore, we calculate the frequencies of the normal modes in the relaxed excited state. It is shown that the frequency splittings of the ${\mathit{h}}_{\mathit{g}}$ modes of the lowest and the fourth-lowest frequency reach several tens of ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. Also it is pointed out that the two ${\mathit{h}}_{\mathit{g}}$ modes are the radial and the tangential mode described by the second-order vector spherical harmonics.

A kinematic analysis of the high‐latitude thermospheric neutral circulation pattern

Averaged measurements from the Dynamics Explorer 2 (DE 2) satellite have been combined with theoretical model calculations to provide “synthesized” thermospheric neutral wind fields for solar maximum, December solstice conditions for both quiet (Kp ≤ 3) and active (3 + ≤ Kp ≤ 6) levels of geomagnetic activity. Specifically, high‐latitude DE 2 wind data obtained from multiorbit averages were merged with modeled winds from the National Center for Atmospheric Research (NCAR) thermosphere‐ionosphere general circulation model (NCAR‐TIGCM), using an objective analysis scheme based on the vector spherical harmonic (VSH) spectral technique of Killeen et al. (1987). A “kinematic analysis” was then performed to decompose the merged semiempirical wind fields into their respective divergent (irrotational) and nondivergent (rotational) components at high latitudes and to calculate the corresponding potential and stream functions. This decomposition provides quantitative insight into the sources of momentum that contribute to each of the wind components. The principal conclusions of the study are (1) the nondivergent component of the high‐latitude thermospheric neutral wind is representative of the convection‐driven component of the neutral wind at F region altitudes. This component of the neutral wind is primarily driven by the ion‐drag and Coriolis forces. (2) The nondivergent wind component makes up a large percentage of the total wind field for both quiet and active geomagnetic conditions. In fact, the nondivergent wind component has stronger sunward directed winds in the dawn and dusk sectors than observed in the total wind field but weaker antisunward directed winds over the polar cap. Vortex formation of the nondivergent wind component is consistently more dominant in the dusk sector than in the dawn sector. (3) The irrotational component of the high‐latitude thermospheric neutral wind is representative of the solar‐driven component of the neutral wind at F region altitudes, directed primarily along the 14 ‐ 2 MLT plane. This component of the neutral wind is mostly driven by the pressure‐gradient‐, Coriolis‐, and Pedersen‐drag (nonconvective‐ion drag) forces. (4) The irrotational component complements the nondivergent antisunward flow over the polar cap but inhibits the nondivergent sunward flow in the dawn and dusk sectors. The total wind field is therefore observed to have a strong antisunward surge over the polar cap with modified sunward flow in the flanks of the polar cap. The development of sunward winds in the dawn sector of the total wind field is capricious, owing to the near vector cancellation of the irrotational and nondivergent wind components in that sector. (5) The above results are also observed in the kinematic analysis of the NCAR‐TIGCM simulations. Poor formation of dawn sector sunward winds is observed in both simulations and indicates other nongeomagnetic forces are responsible, given knowledge of the input ion convection pattern having equal potential between convection cells. (6) Overall, the NCAR‐TIGCM simulations were in reasonable agreement with the DE 2 observations for the nondivergent wind component but typically underestimated the structure in the irrotational wind component. This analysis indicates the possible underestimate of matched power input to the convection patterns chosen to represent the two geomagnetic activity conditions in the NCAR‐TIGCM simulations, thus causing the underestimate of the irrotational wind component.

Scattered and internal intensity of a sphere illuminated with a Gaussian beam

A method of calculating the internal and scattered electric fields of a spherical dielectric object illuminated with a Gaussian beam is presented. The vector nature of the beam is considered. The fields satisfy Maxwell's equations, and the beam can be located arbitrarily with respect to the object. A polarized Gaussian beam is first represented as an angular spectrum of plane waves. These waves are then expanded in vector spherical harmonics. Although the details of the expansion are presented for a lowest-order Gaussian beam, the method can be applied to any wave which can be expressed as a sum of homogeneous plane waves. The interaction of an arbitrarily located Gaussian beam with a spherical object is analyzed using the T-matrix method. Calculated results for beams having waists much smaller than the radius of the sphere help in visualizing how a narrow beam reflects and refracts at the spherical dielectric interfaces. The combination of the plane-wave spectrum technique and the T-matrix method can be applied to the problem of an arbitrary beam interacting with an axisymmetric, nonspherical, homogeneous or layered object. >

Vector spherical spline interpolation—basic theory and computational aspects

AbstractVector spherical interpolation is discussed from both the theoretical and computational points of view. The theory of vector spherical harmonics is an essential tool. An estimate is given for the absolute error of the interpolation process; an efficient algorithm is developed for the computation of a vector spherical interpoiant. The displacement boundary value problem of determining the elastic field from a finite number of discretely given displacement vectors is solved by the use of vector splines.

Transition operators in electromagnetic-wave diffraction theory. II. Applications to optics.

This paper is the second of a series. The first [G. E. Hahne, Phys. Rev. A 45, 7526 (1992)] developed a general theory of the transition operator approach to diffraction of time-harmonic electromagnetic waves from fixed obstacles, such that the response of the obstacle, denoted by \ensuremath{\Omega}, to an impinging electromagnetic signal with wave number ${\mathit{k}}_{0}$ was simulated by nonlocal, homogeneous Leontovich (i.e., impedance) boundary conditions on the obstacle's surface, which surface is called \ensuremath{\partial}\ensuremath{\Omega}. Moreover, the exterior region, called ${\mathrm{\ensuremath{\Omega}}}^{\mathrm{ex}}$, was presumed to be unbounded empty space, and has an electromagnetic response that can be expressed in terms of the so-called radiation impedance operator, denoted Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$; Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ is a certain invertible, linear functional operator that maps the space of complex tangent-vector fields on \ensuremath{\partial}\ensuremath{\Omega} into itself.The matching of the limiting tangential electric and magnetic fields on \ensuremath{\partial}\ensuremath{\Omega} yielded a functional-operator expression for the transition operator and thereby a formal reduction to quadratures of the entire direct-scattering problem. This paper is intended to serve as an illustration and elaboration of the formalism presented in the first paper. After a brief recapitulation of the theory, the following topics are dealt with. First, an infinite sum in terms of vector spherical harmonics is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ for \ensuremath{\partial}\ensuremath{\Omega} a sphere. Second, a quasiplanar approximation, based directly on the exact result for planar \ensuremath{\partial}\ensuremath{\Omega}, is obtained for Z${\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}_{\mathit{k}0}^{+}$ when \ensuremath{\Vert}${\mathit{k}}_{0}$\ensuremath{\Vert} is large compared to the local surface curvatures of \ensuremath{\partial}\ensuremath{\Omega} and attention can be restricted to small neighborhoods on \ensuremath{\partial}\ensuremath{\Omega}; when augmented by the method of stationary phase, this approximation is shown to lead to the familiar physical optics method for smooth, convex surfaces. Third, the latter method, supported by further interventions of the method of stationary phase, is applied in a well-established manner to secure the results of geometrical optics for the complete Green's function for the time-harmonic Maxwell field in the presence of a smooth-surfaced, convex, perfectly electrically conducting obstacle.One feature of the latter computations is that the original source currents from which the free-space Maxwell Green's function is constructed are presumed to generate ordinary outgoing (``causal'') electromagnetic waves for electric current sources, and purely ingoing (``anticausal'') waves for magnetic current sources (i.e., sinks) of electromagnetic radiation. Fourth, a formal construction is derived for mapping the Leontovich boundary conditions on an inner surface into a set of Leontovich boundary conditions on an outer surface, in the circumstance that a layer comprised of a material medium, which has possibly nonuniform and nonsymmetric tensors representing its constitutive properties, fills the domain between the surfaces; the construction presumes that a Green's function is available for the corresponding Maxwell equations in the medium when the medium is extended appropriately to fill all space. The main body of the paper concludes with a brief discussion of directions of possible future work and applications. The first appendix shows how to apply the method of stationary phase in the present context, in particular for the mixed case that the original radiation source is anticausal and the response currents generated in the obstacle are causal sources of radiation. A second appendix develops an acoustic (Helmholtz equation) analog to the fourth topic mentioned above, and exhibits an interplay of the theory with symplectic transformations in the case that the principle of reciprocity holds; it is noted that a symplectic connection also exists in the electromagnetic case when the propagation medium's constitutive properties are such that reciprocity holds.

Vector spherical harmonics and scattering integrals

We consider scalar and vectorial non-relativistic bound-free transition form-factors. These integrals are spatial three-dimensional Fourier transforms of the product of the bound and continuum hydrogenlike wavefunctions ϕnlm(r)ϕK–(r), weighted with operators 1, r−1, r or ∇. Hamilton gradient operator ∇r may be applied to either ϕnlm(r) or ϕK–(r). The same type of matrix elements is also calculated by using Slater-type orbitals for ϕnlm(r). Concise general results are obtained for all the form-factors under study in terms of finite quadruple summations. Quantization axis for ϕnlm(r) is held arbitrary and the exact analytical calculations are carried out for any triple nlm of quantum numbers. Vectorial integrals are most elegantly treated by employing circular Cartesian coordinates and the vector spherical harmonics Yjlm(r).

On the dynamics and composition of the high-latitude thermosphere

Recent experimental measurements of the dynamics of the neutral upper thermosphere have demonstrated the important roles of ion-drag and Joule heating processes in establishing the basic neutral wind morphology and controlling neutral composition, particularly in the high-latitude region. Instruments on the Dynamics Explorer-2 spacecraft (DE 2), for example, were capable of measuring the three-dimensional vector neutral wind and ion drift in the thermosphere along the orbital track, together with constituent densities and temperatures. Ground-based optical and radar measurements of winds and temperatures from observatories in Greenland have contributed additional measurements of thermospheric neutral wind velocities and ionospheric parameters. The comprehensive nature of these various data sets has enabled more stringent experimental constraints to be placed on the numerical models of the region (thermosphere-ionosphere general circulation models, TIGCMs), leading to an improved theoretical understanding of the important physical processes that control thermospheric circulation and variability. In addition, the measurements have enabled the development of semi-empirical models of thermosphere dynamics which can be used in various theoretical studies. The Vector Spherical Harmonic (VSH) model, for example, provides a description of global thermospheric state variables (wind, temperature and density), using a combination of empirical data and NCAR-TIGCM calculations. This paper presents a brief review of some of the more recent progress made in this area by the team of researchers at the University of Michigan, with emphasis on the interpretation of experimental measurements made from DE 2 and from ground-based observatories in Thule and Søndrestrømfjord, Greenland. Comparisons between individual data sets from these sources and the VSH model are also presented.

Vector spherical harmonics and their application to classical electrodynamics

A set of vector spherical harmonics which was earlier introduced and shown to be applicable to magnetostatics problems, is employed to decompose and solve Maxwell equations. Expressions for the radiation fields for the transverse electric and magnetic modes, the angular distribution of multipole radiation, and the total power radiated in terms of the set of vector spherical harmonics are also derived. The interesting and important Rayleigh scattering problem of electromagnetic plane waves scattering by a conducting sphere is discussed in some detail within the approach. Numerous other problems particularly well adapted for the vector spherical harmonic approach advocated in the paper are suggested.

Analysis of radiative scattering for multiple sphere configurations

An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the ‘order of scattering’ solution technique for determining the scattered field coefficients for each sphere.

Electron transfer from hydrogenlike atoms to partially and completely stripped projectiles: CDW approximation

Electron transfer from hydrogenlike target systems to multiply charged projectiles is examined within the Continuum Distorted Wave (CDW) approximation. Using vector spherical harmonics, the most concise analytical expressions are derived for the transition amplitude in the case of arbitrary initial and final hydrogenic bound states. The resulting algorithm is superior to any of the computational devices previously proposed in connection with the CDW method. The present code is used to provide comprehensive tables of the state-selective and total cross sections for charge exchange in collisions between atomic hydrogen and completely stripped projectiles at incident energies ranging from 25 keV to 10 MeV. These data also yield results for partially stripped projectiles by applying the appropriate mass and charge scaling. Thorough comparisons between the CDW theory and experiments on electron capture from atomic hydrogen by completely and partially stripped projectiles reveal systematically good agreement.

Scattering by two and three spheres computed by the generalized multipole technique

An analytic solution to the problem of scattering by multiple spheres was given by J.H. Bruning and Y.T. Lo (ibid., vol.AP-19, p.378-90 and 391-8, May 1971). Results from this analytic solution are compared to results obtained using the generalized multipole technique (GMT), a numerical method that uses vector spherical harmonics as basis functions. The boundary conditions are imposed at a discrete set of points and the solution is obtained by numerically inverting a system of linear equations. The problem is particularly well suited to the GMT because the surfaces are very smooth.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A unified approach to the helioseismic forward and inverse problems of differential rotation

view Abstract Citations (153) References (38) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Unified Approach to the Helioseismic Forward and Inverse Problems of Differential Rotation Ritzwoller, Michael H. ; Lavely, Eugene M. Abstract A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion. Publication: The Astrophysical Journal Pub Date: March 1991 DOI: 10.1086/169785 Bibcode: 1991ApJ...369..557R Keywords: Clebsch-Gordan Coefficients; Helioseismology; Solar Oscillations; Solar Rotation; Angular Momentum; Perturbation Theory; Spherical Harmonics; Stellar Models; Solar Physics; SUN: OSCILLATIONS; SUN: ROTATION full text sources ADS |

The whiplash problem: an application of vector spherical harmonics

A brief discussion of an alternate method for finding the displacements of an elastic, homogeneous, isotropic sphere under a rotational loading is given. The method is shown to be applicable to nonconservative loadings and asymmetric geometry. Expressions for the strains and stresses are also given.

Asymmetric vibrations of an elastic sphere

Historically, the vector Navier equation governing the dynamic response of an elastic, homogeneous, isotropic sphere has been solved using the Helmholtz decomposition of the displacement vector. Further, many of the problems in the literature have been restricted to ones involving axisymmetric geometry. In this presentation, the time-dependent Navier equation is solved using a set of vector spherical harmonics which, previously, has been used primarily in quantum mechanics studies but which seems particularly useful in solving asymmetric problems with nonconservative body forces. Expressions for the displacements, strains, and stresses and a discussion of the vibrations of an elastic sphere are given.

The Mie Scattering at an Inhomogeneous and Arbitrary Shaped Inclusion

AbstractWe consider the scattering of an electromagnetic wave at an inhomogeneous and arbitrary shaped inclusion describing the inclusion as a non‐spherical perturbation. Thus, inhomogenety and arbitrarity in the shape are treated on the same footing. The fields are developed into vector spherical harmonics. The radial parts must be calculated from a coupled set of equations using the methods known from the scattering of a scalar field at a non‐spherical potential. Using an operator notation the lengthly expressions can be avoided, which contain the Clebsch‐Gordan coefficients.

Non-linear three dimensional spectral model of the venusian thermosphere with super-rotation—II. Temperature, composition and winds

We discuss the results from a three dimensional non-linear spectral model of the Venusian thermosphere with CO 2, O and He. As described in an accompanying paper (Stevens-Rayburn et al., 1989, Planet. Space Sci. 37, 701), an expansion in terms of vector spherical and Fourier harmonics is used to represent the latitude and Local Time dependencies, treated as perturbations of a globally uniform atmosphere taken from the empirical model of Hedin et al. (1983, J. geophys. Res. 88, 73). A rigid shell super-rotation rate, uniform in altitude, is adopted. Standard heating rates with an efficiency of about 20% are taken from the work of Fox (1988, Planet. Space Sci. 36, 37). Based on the results from earlier analyses of Pioneer Venus neutral composition data (Niemann et al., 1980, J. geophys. Res. 85, 7817; von Zahn et al., 1980, J. geophys. Res. 85, 7829), height dependent eddy diffusion coefficients around 3 ×10 7cm 2s −1 are adopted, assuming a Prandtl number of 1. Following Bougher et al. (1986, Icarus 68, 284), wave drag in the form of Rayleigh friction is introduced to slow the horizontal winds and to increase the temperature contrast between day and night. With minimal adjusting of the parameterizations for eddy diffusion and Rayleigh friction, the model is successful in reproducing the major observed thermospheric features: the broad daytime maxima in CO 2 and O, with significantly larger values at dusk than at dawn, and the “saw tooth” like density maximum in He around 05:00 LT. To provide a better understanding of the dynamic conditions influencing the temperature and composition, numerical experiments were carried out. (1) The diurnal variations in He are most sensitive to thermospheric super-rotation, and calculations were performed with different rotation periods from 4 to 8 days. The longer the period is, the larger is the day-night increase in the He density and the shorter is the time delay in the density buildup after midnight. We can fit the data best with a super-rotation period of 6 days. Super-rotation also accounts for the dawn-dusk asymmetries in the major species. (2) Given a globally uniform atmosphere as input, larger heating rates yield larger temperature contrast between day and night. The temperature contrast can also be increased by lowering the rate of energy advection from day to night. Thus, an alternative model was constructed, with lower heating rates (artificially reduced by a factor of 2 from the standard values) and enhanced Rayleigh friction to slow the winds, reproducing the observed variations in the temperature and in the major species CO 2 and O. However, in this case, the winds were too small and produced He variations much smaller than observed. (3) With lower heating rate and enhanced Rayleigh friction, a reduced eddy diffusion coefficient for He increased the day-night variations of this species, but produced also a large decrease in its density from the pole to the equator, which is not observed. From these parametric studies we conclude that lower u.v. heating rates with an efficiency significantly less than 20% cannot reproduce simultaneously the observed diurnal variations in the temperature, in the heavier species and in helium. With this heating efficiency, the Venusian thermosphere is strongly non-linear; our model would not converge if it were much larger.

Transverse oscillating field current drive in spherical plasmas

The technique of driving steady Hall currents in plasmas using a transverse oscillating magnetic field is studied numerically under the approximation of negligible ion flow. The authors model a uniformly dense and resistive spherical plasma bounded by an insulating wall and immersed in a uniform magnetic field which has both an oscillating component (of frequency Omega ) and an orthogonal constant 'vertical' component. The authors formulate the (3-D time-dependent) problem in terms of a spatial expansion in vector spherical harmonics and a temporal expansion in Fourier harmonics.

The elastic sphere under nonsymmetric loading

Existing solutions to boundary value problems arising from an elastic sphere subjected to a body force have been primarily restricted to axisymmetric, conservative loading. In this paper, a method for solving the displacement equations governing the static equilibrium of an elastic sphere subjected to an arbitrary body force and surface displacement is presented. The solutions are obtained in terms of three vector spherical harmonics and expressions for the displacement and stress fields are presented. Additionally, a short discussion indicating extension of these solutions to dynamic problems is included.

A Comparison of Three Numerical Methods for Solving Differential Equations on the Sphere

Abstract We compare three numerical methods for solving vector differential equations on a sphere. A composite mesh finite-difference method using overlapping stereographic coordinate systems is compared to transform methods based on scalar and vector spherical harmonics. The methods are compared in terms of total computer time, memory requirements, and execution rates for relative accuracy requirements of two and four digits in a five-day forecast. The computational requirements of the three methods were well within an order of magnitude of one another. In most of the cases that are examined, the time step was limited by accuracy rather than stability. This problem can be overcome by the use of a higher order time integration scheme, but at the expense of an increase in the memory requirements.

On the structure and circulation of the polar thermosphere under magnetically quiet conditions

A theoretical study is presented bearing on the thermospheric circulation and composition at polar latitudes. The observed motions and density perturbations in N 2, O and He have signatures which may be understood in terms of two different source mechanisms. We consider electric field momentum coupling and Joule heating as well as interactions between both processes. A spectral model in terms of vector spherical harmonics (with magnetic coordinates) is used, delineating the diurnal and mean (time independent) components. The important non-linearities are evaluated in configuration space. The electric field model of Volland and the global average density and temperature variations of Hedin (MSIS) are adopted as input. Our analysis leads to the following conclusions. (1) The vortex type double cell polar circulation (zonal wave number m = 1) is primarily driven by collisional momentum transfer from electric field induced ion convection. (2) Because of the thermospheric low pass filter, a large time independent component (zonal wave number m = 0) is produced by Joule heating; the heavier species (N 2) being concentrated where the lighter ones (O, He) are depleted, and vice versa due to wind induced diffusion (3) The electric field driven vortex circulation redistributes the mass and energy in the time independent density and temperature variations (from Joule heating), producing primarily diurnal variations ( m = 1) in the temperature and composition near the pole and at auroral latitudes, again the heavier and lighter species varying out of phase. The above results are in substantial agreement with observations. It is worth noting that momentum rectification associated with the diurnally varying electric field and conductivity induces a weak zonally symmetric (single cell) prograde polar vortex. However, this motion is partially compensated by a retrograde vortex from geostrophic balance due to Joule heating, which dominates near the pole. These motions are small compared with the diurnally varying component in the polar circulation.