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We propose a method to generate an acoustical levitation trap at the same time as suppressing the sound in a multi-wavelength region of space. The method uses a spherical basis expansion of the sound field in the quiet zone, calculated by translating individual source expansions of elements in a transducer array. We show that it is possible to control the size of the quiet zone with the truncation order of the expansion, and explain the trade-off between field suppression in the quiet zone and stiffness loss of the levitation trap. Measurements of a generated sound field show the existence of a region of lower sound pressure. Simulations demonstrate a contrast up to 50 dB and sizes up to 60 mm for a 256 element array.

Using the solution of the problem of light scattering by a homogeneous spheroid, derived by separation of variables in spheroidal coordinates, we find the ‘spheroidal’ T -matrix. We suggest a transformation of this matrix into the ‘spherical’ T -matrix useful in applications. The procedure consists of two steps: first, transition from spheroidal to spherical basis, and then from spherical non-orthogonal to orthogonal basis.

Constrained spherical deconvolution (CSD) of diffusion‐weighted MRI (DW‐MRI) is a popular analysis method that extracts the full white matter (WM) fiber orientation density function (fODF) in the living human brain, noninvasively. It assumes that the DW‐MRI signal on the sphere can be represented as the spherical convolution of a single‐fiber response function (RF) and the fODF, and recovers the fODF through the inverse operation. CSD approaches typically require that the DW‐MRI data is sampled shell‐wise, and estimate the RF in a purely spherical manner using spherical basis functions, such as spherical harmonics (SH), disregarding any radial dependencies. This precludes analysis of data acquired with nonspherical sampling schemes, for example, Cartesian sampling. Additionally, nonspherical sampling can also arise due to technical issues, for example, gradient nonlinearities, resulting in a spatially dependent bias of the apparent tissue densities and connectivity information. Here, we adopt a compact model for the RFs that also describes their radial dependency. We demonstrate that the proposed model can accurately predict the tissue response for a wide range of b‐values. On shell‐wise data, our approach provides fODFs and tissue densities indistinguishable from those estimated using SH. On Cartesian data, fODF estimates and apparent tissue densities are on par with those obtained from shell‐wise data, significantly broadening the range of data sets that can be analyzed using CSD. In addition, gradient nonlinearities can be accounted for using the proposed model, resulting in much more accurate apparent tissue densities and connectivity metrics., Nonspherically sampled diffusion MRI data can arise either by choice of the sampling scheme (such as diffusion spectrum imaging) or due to gradient nonlinearities, precluding spherical deconvolution (SD) analysis. We adopt a compact response function model that accurately describes the b‐value dependency, enabling SD of nonspherically sampled data and the correction of gradient nonlinearities. With the proposed method, a broader range of data sets can be analyzed with SD approaches, as well as allowing more accurate downstream processing such as fiber tracking and connectomics.

Highly excited states in 156Gd, populated via the neutron pickup reaction 157Gd(3He,4He)156Gd, are investigated, and their spin–parity distribution P(Jπ,E) is examined. The cross section for one-neutron transfers to states above the neutron separation energy in 156Gd is calculated as coherent sum, using standard reaction codes that employ spherical basis states. Spectroscopic factors and form factors for the relevant states are obtained by expanding the deformed neutron wave functions in a spherical Sturmian basis. For the energy regime relevant to surrogate applications involving neutron absorption, 155Gd+n →156Gd⋆, the calculations show that the reaction 3He+157Gd →4He+156Gd⋆ induces a well-behaved formation probability P(Jπ,E) of approximately Gaussian shape. It is observed that the centroid and shape of the Gaussian distributions of the positive and negative parity states of the compound system can be significantly different from each other!

When considering light scattering from a sphere, the ratios between the expansion coefficients of the scattered and the incident field in a spherical basis are known as the Mie coefficients. Generally, Mie coefficients depend on many degrees of freedom, including the dimensions and electromagnetic properties of the spherical object. However, for fundamental research, it is important to have easy expressions for all possible values of Mie coefficients within the existing physical constraints and which depend on the least number of degrees of freedom. While such expressions are known for spheres made from non-absorbing materials, we present here, for the first time to our knowledge, corresponding expressions for spheres made from absorbing materials. To illustrate the usefulness of these expressions, we investigate the upper bound for the absorption cross section of a trimer made from electric dipolar spheres. Given the results, we have designed a dipolar ITO trimer that offers a maximal absorption cross section. Our approach is not limited to dipolar terms, but indeed, as demonstrated in the manuscript, can be applied to higher order terms as well. Using our model, one can scan the entire accessible parameter space of spheres for specific functionalities in systems made from spherical scatterers.

The differential scattering characteristics of an object having a uniaxial anisotropic inclusion, for incidence of an arbitrarily shaped beam, are theoretically investigated. The scattering problem is solved in a spherical basis by using the boundary conditions, method of moments technique and Schelkunoff’s equivalence theorem, which leads to a system of linear equations for the expansion coefficients of the scattered and internal fields when the description of the incident shaped beam is known. The theoretical formulation is reviewed, and the numerical problems are discussed. As examples, for a Gaussian beam, zero-order Bessel beam and Hertzian electric dipole radiation striking a spheroid with a uniaxial anisotropic spheroid inclusion and a circular cylinder with a uniaxial anisotropic circular cylinder inclusion, the normalized differential scattering cross sections are calculated, and the scattering properties are analyzed concisely.

The solution to the electromagnetic scattering of a sphere was published by Gustav Mie more than hundred years ago and is well-known as the Mie theory. In Mie theory, the ratio between the scattered and the incident field coefficients in the spherical basis is expressed in terms of the Mie coefficient. These Mie coefficients depend in an intricate manner on the spheres’ radius and the involved material properties of both the sphere and the ambient. To enable a systematic analysis of the accessible optical properties from spheres, these explicit expressions are not useful because they are too complicated. These Mie coefficients simply contain too many degrees of freedom. For fundamental research, it is of utmost importance to have easy expressions at hand that express all values of Mie coefficients that are accessible in general. The precise geometrical and material properties of a sphere that offer these coefficients can be identified in a secondary step. But if these accessible coefficients depend on the least number of degrees of freedom, they would allow for a systematic analysis of all observable effects using spherical scatterers. These desired simple expressions have been previously identified only for non-absorbing materials. However, a model for absorbing particles has never been reported. Here, while using the optical theorem we derive the generic equations to express any possible Mie coefficient of an absorbing sphere. Our model for absorbing particles can facilitate the study of absorbing systems such as perfect absorbers, optical torque calculations, cooling, thermal emitters etc. Using the proposed model, one can systematically analyse through all the possible space and search for specific functionality as it will be demonstrated at selected applications at the talk.

Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived., Same content as published version in JQSRT. Also contains content from a previous arxiv document "Relationships between spherical and toroidal harmonics"

We have examined an analog to the extended boundary conditions method (EBCM) with the standard spherical basis, which is popular in light scattering theory, with respect to its applicability to the solution of an electrostatic problem that arises for multilayer scatterers the sizes of which are smaller compared to the wavelength of the incident radiation. It has been found that, in the case of two or more layers, to determine the polarizability and other optical characteristics of particles in the far-field zone, the parameters of the surfaces of layers should obey the condition max{σ1 (j)} < min{σ2 (j)}. In this case, appearing infinite systems of linear equations for expansion coefficients of unknown fields have a unique solution, which can be found by the reduction method. For nonspheroidal particles, this condition is related to the convergence radii of expansions of regular and irregular fields outside and inside of the particle, including its shells—R 1 (j) = σ1 (j) and R 2 (j) = σ2 (j). In other words, a spherical shell should exist in which expansions of all regular and irregular fields converge simultaneously. This condition is a natural generalization of the result for homogeneous particles, for which such a condition is imposed only on expansions of the “scattered” and internal fields—R 1 < R 2. For spheroidal multilayer particles, which should be singled out into a separate class, the EBCM applicability condition is written as max{σ1 (1), σ1 (2), …, σ1 (J−1), σ1 (J)} < min{σ2 (1), σ2 (2), …, σ2 (J−1)} and parameters σ2 (j) of the surfaces of shells are not related to corresponding convergence radii R 2 j of irregular fields. Numerical calculations for two-layer spheroids and pseudospheroids have confirmed completely theoretical inferences. Apart from the EBCM algorithm, an approximate formula has been proposed for the calculation of the polarizability of two-layer particles, in which the polarizability of a two-layer particle is interpreted as a linear combination of the polarizabilities of homogeneous particles that consist of the materials of the shell and core proportionally to their volumes. The range of applicability of this formula is wider than that for the EBCM, and the calculation error is smaller than 1%.

The Rayleigh approximation and solution of the electrostatic problem are considered for pseudospheroids that are obtained as a result of inversion of usual spheroidal particles: $$r\left( \theta \right) = ab/{r_{{\text{spheroid}}}}\left( \theta \right) = a\sqrt {1 - { \in ^2}{{\cos }^2}\theta } $$ , where e2 = 1 - (b/a)2. The approach is based on using the extended boundary condition method (EBCM) for the calculation of polarizability. The approach is studied analytically. The study has been shown that, in the near zone with respect to the particle, expansions of potentials of the “scattered” and internal fields in terms of the spherical basis converge up to the surface only under the conditions $$a/b \prec \sqrt 5 $$ and $$a/b \prec \sqrt 2 $$ (the external and internal Rayleigh hypotheses, respectively). In the far zone, the ЕВСМ can be justifiably applied for the ratio of pseudospheroid parameters $$a/b \prec \sqrt 2 + 1$$ . In this case, there exists a nonempty intersection of analytical continuations of expansions in terms of the spherical basis of the “scattered” and internal fields. The radii of convergence R 1, R 2 of these expansions have been found; they are determined by the presence of singular points. The relation between singular points for pseudospheroids and corresponding spheroids is considered. Numerical analysis of the problem under consideration has completely verified results of the analytical study, in particular, the presence of corresponding singular points. To obtain the necessary accuracy in calculations of elements of linear algebraic systems, the corresponding integrals are transformed into series containing gamma functions. Numerical calculations involving the EBCM were successfully verified using the condition of symmetry of the T matrix. For the applicability area of the ЕВСМ, calculation results for the polarizability of pseudospheroids are discussed according to the exact algorithm and within the approximation of the homogeneous internal field.

The transition (T) matrix of Waterman has been very useful for computing fast, accurate acoustic scattering predictions for axisymmetric elastic objects, but this technique is usually limited to fairly smooth objects that are not too aspherical unless complex basis functions or stabilization schemes are used. To ease this limitation, a spherical-basis formulation adapted from approaches proposed recently by Waterman [J. Acoust. Soc. Am. 125(1), 42-51 (2009)] and Doicu, Eremin, and Wriedt [Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, London, 2000)] is suggested. This is implemented by simply transforming the high-order outgoing spherical basis functions within standard T-matrix formulations to low-order functions distributed along the object's symmetry axis. A free-field T matrix is produced in a nonstandard form, but computations with it become much more stable for elongated aspherical elastic shapes. Some advantages of this approach over the approaches of Waterman and Doicu, Eremin, and Wriedt are noted, and sample calculations for a 10:1 Al prolate spheroid and a 10:1 Al superspheroid of order 10 are given to demonstrate the enhanced stability.

The extended boundary condition method (EBCM) in solving the electrostatic problem for axisymmetric Chebyshev particles the surface of which is described by the equation r(θ) = a(1 + ɛcosnθ) is studied. The main attention is paid to the case n = 1. The problem is reduced to solving infinite systems of linear algebraic equations (ISLAEs) for expansion coefficients of internal and “scattered” fields in terms of a spherical basis where matrix elements of the fields are integrals of products of Legendre functions and power functions. Radii of convergence R2 and R1 of these expansions, respectively, have been found analytically. For the considered particles, depending on perturbation parameter ɛ, conditions of the applicability of the EBCM have been obtained, i.e., conditions of correct construction of the T matrix (R1 maxr(θ)) Rayleigh hypotheses under which expansions of the “scattered” and internal fields in terms of the spherical basis converge up to the surface of the particle beyond and inside it, respectively. In the particular case n = 1, numerical calculations have been performed; in this process, integral ISLAE elements have been represented as finite sums the summands of which depend on gamma functions. Calculations of matrix elements by explicit formulas have made it possible to considerably increase the dimension of the solved reduced ISLAE with preservation of the necessary accuracy. Analysis of results of numerical calculations verified their agreement with theoretical conclusions.

Author(s): Macchiavelli, AO; Crawford, HL; Campbell, CM; Clark, RM; Cromaz, M; Fallon, P; Jones, MD; Lee, IY; Salathe, M | Abstract: Spectroscopic factors in Be10, Be11, and Be12, extracted from (d,p), one-neutron knockout, and (p,d) reactions, are interpreted within the rotational model. Assuming that the ground state and first excited state of Be11 can be associated with the 12[220] and 12[101] Nilsson levels, the strong-coupling limit gives simple expressions that relate the amplitudes of these wave functions (in the spherical basis) with the measured cross sections and derived spectroscopic factors. We obtain good agreement with both the measured magnetic moment of the ground state in Be11 and the reaction data.

Circular polarization of the cosmic microwave background (CMB) arises in the standard cosmological model from Faraday conversion of the linear polarization generated at the surface of last scatter by various sources of birefringence along the line of sight. If the sources of birefringence are generated at linear order in primordial density perturbations the principal axes of the index-of-refraction tensor are determined by gradients of the primordial density field. Since linear polarization at the surface of last scatter is generated at linear order in density perturbations, the circular polarization thus arises at second order in primordial perturbations. Here, we re-visit the calculation of the circular polarization using the total-angular-momentum formalism, which allows for some simplifications in the calculation of the angular power spectrum of the circular polarization---especially for the dominant photon-photon scattering contribution---and also provides some new intuition., Comment: 6 pages

Any acoustic sensor disturbs the spatial acoustic field to certain extent, and a recorded field is different from a field that would have existed if a sensor were absent. Recovery of the original (incident) field is a fundamental task in spatial audio. For some sensor geometries, the disturbance of the field by the sensor can be characterized analytically and its influence can be undone; however, for arbitrary-shaped sensor numerical methods have to be employed. In the current work, the sensor influence on the field is characterized using numerical (specifically, boundary-element) methods, and a framework to recover the incident field, either in the plane-wave or in the spherical wave function basis, is developed. Field recovery in terms of the spherical basis allows the generation of a higher-order Ambisonics representation of the spatial audio scene. Experimental results using a complex-shaped scatterer are presented.

Deformed single particle energies obtained by averaging a particle–core Hamiltonian with a projected spherical basis depend on a deformation parameter and an arbitrary constant defining the canonical transformation relating the collective quadrupole coordinates and momenta with the boson operators. When the mentioned basis describes the single particle motion of either protons or neutrons the parameters involved are isospin dependent. An algorithm for fixing these parameters is formulated and then applied for 194 isotopes covering a good part of the nuclide chart. Relation with the Nilsson deformed basis is pointed out in terms of deformation dependence of the corresponding single particle energies as well as of the nucleon densities and their symmetries. The proposed projected spherical basis provides an efficient tool for the description of spherical and deformed nuclei in a unified fashion.

The applicability of an analog of the extended boundary condition method, which is popular in light-scattering theory, is studied in combination with the standard spherical basis for the solution of an electrostatic problem appearing for spheroidal layered scatterers the sizes of which are small as compared to the incident radiation wavelength. In the case of two or more layers, polarizability and other optical characteristics of particles in the far zone are shown to be undeterminable if the condition under which the appearing systems of linear equations for expansion coefficients of unknown fields are Fredholm systems solvable by the reduction method is broken. For two-layer spheroids with confocal boundaries, this condition is transformed into a simple restriction on the ratio of particle semiaxes a/b< \(\sqrt 2 \) + 1. In the case of homogeneous particles, the solvability condition is that the radius of convergence of the internal-field expansion must exceed that of the expansion of an analog of the scattering field. Since homogeneous spheroids (ellipsoids) are unique particles inside which the electrostatic field is homogeneous, it is shown that the solution can be always found in this case. The obtained results make it possible to match in principle the results of theoretical and numerical determinations of the domain of applicability for the extended boundary condition method with a spherical basis for spheroidal scatterers.

Using the spherical basis of the spin-ν operator, together with an appropriate normalized complex (2ν +1)-spinor on S 3 we obtain spin-ν representation of the U(1) Hopf fibration S 3 → S 2 as well as its associated fuzzy version. Also, to realize the first Hopf map via the spherical basis of the spin-1 operator with even winding numbers, we present an appropriate normalized complex three-spinor. We put the winding numbers in one-to-one correspondence with the monopole charges corresponding to different associated complex vector bundles.

The low-lying high-spin yrast band structure of neutron-rich 113 , 118 – 121 Ag has been established for the first time using prompt γ-ray spectroscopy of isotopically identified fission fragments produced in the 9Be(238U, fγ) fusion- and transfer-induced fission processes. The newly obtained level energies follow the systematics of the neighboring isotopes. The sequences of levels exhibit an energy inheritance from states in the corresponding Cd core. A striking constancy of a large signature splitting in odd-A Ag throughout the long chain of isotopes with 50 N 82 and a signature inversion in even-A Ag isotopes, which are indications of triaxiality, were evidenced. These observed features were reproduced by large-scale shell-model calculations with a spherical basis for the first time in the Ag isotopic chain, revealing microscopically their complex nature with severely broken seniority ordering. The essential features of the observed signature splitting were further examined in the light of simplified, two-orbital shell-model calculations including only two intruder orbitals π g 9 / 2 and ν h 11 / 2 from two consecutive shells above Z = 50 and N = 82 for protons and neutrons respectively, resulting in the π g 9 / 2 − 3 × ν h 11 / 2 m configurations. The newly established bands were understood as fairly pure, built mainly on unique-parity intruder configurations and coupled to the basic states of the Cd core.

We developed a deformed quasi-particle random phase approximation (DQRPA) to describe the Gamow–Teller (GT) transitions on even–even neutron-rich nuclei. To describe deformed nuclei, we exploited the deformed axially symmetric Woods–Saxon potential, the deformed BCS, and the deformed QRPA with realistic two-body interaction calculated by Brueckner G-matrix based on Bonn CD potential. The deformed single particle states are expanded in terms of the spherical harmonic oscillator basis in order to take the realistic G-matrix stored in the spherical basis. We calculated GT strength distributions, B(GT), of two nuclei 12,14Be for many different deformation parameter β 2 values as a function of the excitation energy E ex w.r.t. the ground state of a parent nucleus. Our results for 12Be predict to prefer a prolate shape and B(GT) results of 14Be turn out to be independent of the β 2 values.

Probe Correction (PC) in Spherical Near Field (SNF) measurements is typically performed involving the so called first order PC algorithm [1]–[3] which assumes probes with limited |μ|=1 spherical wave spectrum. Lot of effort is typically put in the developments of probes with such requirement [4]–[5] but their design is in many case very challenging especially for wide-band applications. In order to have less restrictions in the selection of the probe, different full probe correction techniques have been recently proposed [6]–[9]. In this paper the accuracy and limitation of the first and higher order PC algorithms are investigated considering the SNF measurement of a wideband horn measured with a first order probe and with a standard gain antenna with higher modal content. The full PC algorithm here considered is based on the modification of the spherical basis functions that are properly elaborated taking into account the effect of the probe.

Several relevant properties of the Na clusters were studied by using a projected spherical single particle states.The proposed model is able to describe in an unified fashion the spherical and deformed clusters. Photoabsorbtion cross section is realistically explained within an RPA approach and a Shiff dipole moment as a transition operator, 11 pages, 7 figures

The separation of variables method for coordinate system, the extended boundary condition method, and the point-matching method that are used to solve the problem of light scattering by nonspherical particles are considered from a unified viewpoint. It is shown that, if the mathematical correctness condition (the Rayleigh hypothesis) holds, these methods are interrelated and are equivalent. The applicability ranges of the methods in the near- and far-field zones are analyzed, discussed, and compared on both analytical (based on analytical investigations) and practical (based on numerical calculations) grounds.

A recently proposed computationally tractable version of the Bohr collective model is developed to the extent that we are now justified in describing it as an algebraic collective model. The model has an SU ( 1 , 1 ) × SO ( 5 ) algebraic structure and a continuous set of exactly solvable limits. Moreover, it provides bases for mixed symmetry collective model calculations. However, unlike the standard realization of SU ( 1 , 1 ) , used for computing beta wave functions and their matrix elements in a spherical basis, the algebraic collective model makes use of an SU ( 1 , 1 ) algebra that generates wave functions appropriate for deformed nuclei with intrinsic quadrupole moments ranging from zero to any large value. A previous paper focused on the SO ( 5 ) wave functions, as SO ( 5 ) (hyper-)spherical harmonics, and computation of their matrix elements. This paper gives analytical expressions for the beta matrix elements needed in applications of the model and illustrative results to show the remarkable gain in efficiency that is achieved by using such a basis in collective model calculations for deformed nuclei.

1Advanced Science Research Center, Japan Atomic Energy Research Institute (JAERI), Ibaraki 319-1195, Japan 2Senior High School of Waseda University, Tokyo 177-0044, Japan 3Advanced Research Institute for Science and Engineering, Waseda University, Tokyo 169-8555, Japan 4Ministry of Education, Culture, Sports, Science and Technology, Tokyo 100-8959, Japan 5The Institute of Physical and Chemical Research (RIKEN), Saitama 351-0198, Japan

Since the early 1990s the radial basis function (RBF) method has become a well established tool for reconstructing functions and for solving partial differential equations based on data prescribed at scattered locations throughout a domain in \({{{\mathbb R}}^{d}}\).

A projected spherical basis is defined with the scope of treating deformed metallic clusters. It depends on a parameter that simulates the system deformation. When the deformation goes to zero the spherical-shell model states are obtained. The basis is tested, with positive results, in several applications for Na clusters. The present formalism predictions are compared with those obtained within other models. Such a basis is suitable for a realistic and a unified fashion description of spherical and deformed metallic clusters.

Elastic electron scattering off Si-30 and P-31 was studied in an effective momentum-transfer range of 1.8-3.0 fm(-1). The form-factor data were analyzed together with existing data sets for these nuclei and for S-32 in a model-independent Fourier-Bessel expansion. For P-31 the M1 contribution was subtracted following an established parametrization. Results of Hartree-Fock (HF) calculations, performed for these three nuclei in a spherical basis and in an axially deformed basis, are compared to experiment. Occupancies have been determined which, when used in the spherical-basis HF calculations, lead to a good description of the elastic form-factor data. The deformed-basis calculations have been used to study the influence of the deformation on the calculated binding energies and ground-state charge densities. In all calculations the influence of using different effective nucleon-nucleon interactions was investigated. The resulting differences in 2s(1/2) in occupancy are combined with results from previous existing (e,e'p) experiments to yield ''absolute occupancies'' for the 2s(1/2) orbital. The deduced 2s(1/2) occupancies for Si-30 and S-32 are 0.24(4) and 1.35(19), respectively.

A null-field T-matrix formalism is developed for scattering a pressure wave from a stationary elastic shell immersed in a homogeneous and isotropic fluid half-space and in proximity to a rough fluid–elastic interface. Helmholtz–Kirchhoff integral representations of the various scattered pressure and displacement fields are constructed. The surface fields are required to satisfy the elastic tensor boundary conditions and the scattered fields are required to satisfy the extended boundary condition. Spherical basis functions are used to construct a free-field T-matrix for the elastic shell and rectangular vector basis functions are used to construct a representation of the free-field T- matrix for the rough fluid–elastic interface. The free-field T-matrices are introduced into the Helmholtz–Kirchhoff and the null-field equations for the shell-interface system and a general system of equations for the spectral amplitudes of the various fields is obtained. The general system of equations is specialized to scattering from periodic surface roughness and an exact solution for the scattered pressure field in the fluid is obtained. Then the general system of equations is specialized to scattering from small-amplitude arbitrary roughness profiles and a perturbative solution is obtained. It is shown that the formalism contains multiple scattering effects on the rough surface and between the rough surface and the shell.

This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The problem of interbasis expansions of the wavefunctions is completely solved. A connection between the generalized oscillator system (projected on the z-line) and the Morse system (in one dimension) is discussed., Comment: 23 pages, Latex File, to be published in International Journal of Quantum Chemistry

The scattering of focused laser beams by arbitrarily shaped dielectric bodies is investigated theoretically. The beam description is based on Davis third-order beam approximation for the field components. The scattering problem can be solved on a spherical basis by the extended boundary condition method or by the so-called modified version of the extended boundary condition method. For spheroidal particles with small eccentricities, a perturbation technique for the internal subproblem of the extended boundary condition method is described. This procedure is applied to analyse the influence of the particle shape and orientation on the response of a phase-Doppler system.

Acoustic radiation modes divide the velocity patterns into groups with effective and ineffective radiation efficiencies. It suggests a promising way in the computation of sound power and near field acoustic holography. However, obtaining the radiation modes based on eigenvalue analysis suffers from issues of CPU time and memory, especially for large-scale problems. Based on the idea of equivalent source method and boundary integral equation, it is mathematically and numerically proved that the spherical basis functions is a set of linearly independent patterns on an arbitrary surface. They are termed as mapped radiation modes. The analytical sound power of a structure vibrating in its mapped radiation modes is obtained. An efficient and accurate method to compute the sound power based on the mapped radiation modes is proposed. Based on the relationship between the mapped radiation modes of a vibrating structure and its field sound pressure distribution patterns on a sphere, an improved near field acoustic holography is also developed. It does not need the inverse process but requires microphones to locate at the integral points on a sphere. Numerical examples are presented to validate advantages of applying mapped radiation modes in the sound power evaluation and near field acoustic holography.

We investigate the coupling of an axial rotor and a single particle with spin and with orbital wave functions restricted to a single harmonic-oscillator shell. Coupling coefficients for interesting subgroup chains are derived. Eigenstates of a quadrupole-quadrupole interaction are obtained analytically in a spherical basis and we show that these eigenstates are related to su(3) basis states in the limit of large representations. Other interesting limits are also discussed.

We discuss intermediate-energy single-nucleon removal reactions from deformed projectile nuclei. The removed nucleon is assumed to originate from a given Nilsson model single-particle state and the inclusive cross sections, to all rotational states of the residual nucleus, are calculated. We investigate the sensitivity of both the stripping cross sections and their momentum distributions to the assumed size of the model space in the Nilsson model calculations and to the shape of the projectile and residue. We show that the cross sections for small deformations follow the decomposition of the Nilsson state in a spherical basis. In the case of large and prolate projectile deformations the removal cross sections from prolate-like Nilsson states, having large values for the asymptotic quantum number nz, are reduced. For oblate-like Nilsson states, with small nz, the removal cross sections are increased. Whatever the deformation, the residue momentum distributions are found to remain robustly characteristic of the orbital angular momentum decomposition of the initial state of the nucleon in the projectile.

This paper deals with a dynamical system that generalizes the Kepler-Coulomb system and the Hartmann system. It is shown that the Schr\"odinger equation for this generalized Kepler-Coulomb system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for the expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations., Comment: 24 pages, Latex, LYCEN 9415

The electronic structure of C60 molecule is calculated in a spherical representation, starting from the self-consistent solution of the Kohn-Sham equation in the field of a positively charged shell. When an icosahedral perturbing field is added to the Hamiltonian, it is possible to reproduce quite accurately the spectrum obtained with self-consistent calculations in a large basis. The advantage of our wave function is its relative simplicity due to the separation of radial and angular coordinates. We apply the model to the dipole response, obtain more structure and a more spread out strength function than in previous treatments.

We have expressed the elastic tensors in an irreducible spherical basis from a Cartesian basis in order to deal more easily with rotation operations. This allows us to calculate, in a very effective way, the elastic mean values of polycrystals and to determine the number of independent tensor components.

We suggest a separation of variables method based on expansions of the electromagnetic fields in terms of spheroidal wave functions. Our approach includes splitting the fields in two specific parts and choice of proper scalar potentials for each of them. An earlier analysis of similar methods in the case of a spherical basis has shown that the SVM has a wider practical applicability range than other methods. It was also found that when using a spheroidal basis our general approach gave reliable results for particles of high eccentricity for which other approaches were inapplicable. Thus, the suggested SVM should be highly efficient in calculations of optical properties of various nonspherical scatterers in a wide range of parameter values.

Relativistic Hartree calculations are performed for axially symmetric solutions of deformed nuclei in the sd shell. Both even-even and selected odd- A nuclei are included. The systematic behavior of quadrupole moments and binding energies are reproduced; the deformations being solely determined by self-consistency. The states are expanded in a spherical basis of states obtained by solving the same or neighboring nucleus with a constraint to retain spherical symmetry. The continuum states are artificially discretized by applying a boundary condition; both the positive and negative energy states are included in the expansion. The effects of tensor coupling of the rho meson and of the pion are included. The contribution of the former are small in the nuclei studied here; however, the pion has a large effect on the magnetic moment of odd- A nuclei. The binding energies, quadrupole moments, magnetic moments and single particle spectra are presented and compared with similar calculations, both relativistic and non-relativistic.

The deformation in rhodium isotopes is investigated using adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) approaches. The triaxial deformations are found in the ground states of (102,104,106,108,110)Rh, which is consistent with triaxial Skyrme Hartree-Fock calculations. Several minima with triaxial deformation in (104,106,108,110)Rh are obtained by the configuration-fixed constrained calculations. The corresponding configurations are characterized by the quantum numbers vertical bar nljm > obtained by transforming wave functions from a Cartesian basis to a spherical basis. The possible existence of multiple chiral doublets (M chi D) is demonstrated in (104,106,108,110)Rh isotopes, based on different particle-hole configurations and triaxial deformations.

A data-adaptive strategy for the design of spherical basis function (SBF) networks is presented. The strategy comprises a data-driven selection of the SBFs prior to parameter estimation and an automatic selection of the bandwidth of the SBFs. The selection procedure takes the data distribution and the data noise into account. The strategy can be applied to all interpolation and approximation problems that use SBFs. It is particularly suited for the local inversion of magnetic and gravity data acquired by terrestrial, airborne or space-borne sensors. It is shown that for inhomogeneous and scattered data distribution, the method requires significantly less basis functions than commonly used strategies, which range from template networks of SBFs (e.g. from hierarchical subdivision schemes or equal angular grids) to SBFs located below each data point or below a subset of the data points. Moreover. the system of normal equations is much better conditioned, which often make regularization superfluous. Finally, it is shown that the approximation errors are smaller than when using Standard SBF networks.

The idea of an analogy between a ring-shaped potential and a Coulomb potential is advanced. It is shown that the expansion of the parabolic basis with respect to the spherical basis in the problem of a ring-shaped potential is determined by the Clebsch-Gordan coefficients of the group SU(2) continued to the region of arbitrary real indices. The connection between these coefficients and the functions 3F2 is found, and it is shown that they have a symmetry property under substitution of the parabolic quantum numbers.

We consider the relativistic quantum mechanics of a two interacting fermions system. We first present a covariant formulation of the kinematics of the problem and give a short outline of the classical results. We then quantize the system with a general interaction potential and deduce the explicit equations in a spherical basis. The case of the Coulomb interaction is studied in detail by numerical methods, solving the eigenvalue problem for J=0, J=1, J=2 and determining the spectral curves for a varying ratio of the mass of the interacting particles. Details of the computations, together with a perturbative approach in the mass ratio and an extended description of the ground states of the Para- and Orthopositronium are given in Appendix., Comment: 26 pages, 6 figures

This paper describes the results of our theoretical and numerical studies of hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls, under creeping-flow conditions. We propose a novel algorithm for accurate evaluation of the many-particle friction matrix in this system—no such algorithm has been available so far. Our approach involves expanding the fluid velocity field into spherical and Cartesian fundamental sets of Stokes flows. The interaction of the fluid with the particles is described using the spherical basis fields; the flow scattered by the walls is expressed in terms of the Cartesian fundamental solutions. At the core of our method are transformation relations between the spherical and Cartesian basis sets. These transformations allow us to describe the flow field in a system that involves both the walls and particles. We used our accurate numerical results to test the single-wall superposition approximation for the hydrodynamic friction matrix. The approximation yields fair results for quantities dominated by single particle contributions, but it fails to describe collective phenomena, such as a large transverse resistance coefficient for linear arrays of spheres. r 2005 Elsevier B.V. All rights reserved.

A null field T‐matrix formalism is developed and used to calculate plane‐wave scattering from a fluid loaded elastic spherical shell in proximity to sound hard, sound soft, fluid–fluid, and fluid–elastic interfaces with periodic surface roughness. For each type of interface, the Helmholtz–Kirchhoff integral representation of the various scattered pressure and displacement fields are constructed; the surface fields are required to satisfy the appropriate boundary conditions and the scattered fields are required to satisfy the extended boundary condition. Spherical basis functions are used to construct a free field T‐matrix for the elastic shell and rectangular vector basis functions are used to construct a representation of the free field T‐matrix for the rough interface. The free field T‐matrices are introduced into the Helmholtz–Kirchhoff equations for the scattered fields and the null field equations for the shell‐interface system and an ‘‘exact’’ analytical solution is obtained. Numerical results are o...

Experimentally observed heaviest $N \approx Z$ nuclei, Ru isotopes, are investigated by the shell model on a spherical basis with the extended $P+QQ$ Hamiltonian. The energy levels of all the Ru isotopes can be explained by the shell model with a single set of force parameters. The calculations indicate enhancement of quadrupole correlations in the $N=Z$ nucleus $^{88}$Ru as compared with the other Ru isotopes, but the observed moments of inertia seem to require much more enhancement of quadrupole correlations in $^{88}$Ru. It is discussed that the particle alignment takes place at $8^+$ in $^{90}$Ru but is delayed in $^{88}$Ru till $16^+$ where the simultaneous alignments of proton and neutron pairs take place. The calculations present interesting predictions for $^{89}$Ru that the ground state is the $1/2^-$ state and there are three $\Delta J=2$ bands with different particle alignments including the T=0 $p-n$ pair alignment.

Some nuclear mass formulas are reviewed and applied for the heavy and superheavy nuclei. A new mass formula composed of the gross term, the even‐odd term, and the shell term is also presented. The new mass formula is a revised version of the spherical basis mass formula published in 2000, that is, the even‐odd term is treated more carefully, and a considerable improvement is brought about. The root‐mean‐square deviation of the new formula from experimental masses is 641 keV for Z ⩾ 8 and N ⩾ 8. Properties on systematics of the neutron‐separation energies, alpha‐deacy Q‐values in the superheavy nuclidic region, and Q‐values of the fusion reaction for the heavy nuclei are compared with some mass formulas. Furthermore, a method of estimation for fission half‐lives for the superheavy and neutron‐rich nuclei is presented. With this estimation, the chart of nuclides for the nuclear decays are drawn, and the prediction of limit of the existence in the nuclidic region is discussed.

This paper deals with the dynamical system that generalizes the MIC-Kepler system. It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for this expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expresses through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the prolate spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations., Comment: 12 pages

A many body Hamiltonian comprising pairing, quadrupole-quadrupole and spin-spin interaction is treated within a projected spherical basis with the aim of describing the detailed structure of the magnetic states of orbital and spin-flip nature in $^{144, 148-154}$Sm. The mean field for the single particle motion takes care of the volume conservation of a phenomenological core which results in having a complex dependence on deformation for single particle energies. Several experimental features like the total orbital and total spin strengths, the shape of the orbital strength distribution, the hump structure of the spin strength distribution, the quadratic dependence of the orbital strength on nuclear deformation, the saturation effect of the orbital strength with respect to the Casten's factor P are very well described within a quasiparticle random phase approximation formalism. Comparison of present results with those obtained in some previous treatments are also given., Comment: 31 pages, RevTex, 10 Postscript figures