Angular spectrum representation of the electromagnetic multipole fields, and their reflection at a perfect conductor

Angular spectrum representations are derived for electric and magnetic multipole fields of arbitrary order. The result involves generalized spherical harmonics and generalized vector spherical harmonics, and the representations are in the form of integrals over the k_∥-plane. The representations are especially useful for the study of reflection and transmission of multipole radiation by a plane interface. As an example, we have considered the reflection at a perfect conductor. The reflected field of a multipole field could be expressed in the form of an angular spectrum with a very simple relation to the angular spectrum of the source field. The radiation pattern of a multipole near the perfect conductor is obtained with the method of stationary phase. We also introduce a method for determining the mirror image of the source of an arbitrary multipole.     

Sound cancellation by the use of secondary multipoles: experiments

Theory related to global, free-field cancellation of a primary monopole field by the use of a displaced, secondary multipole was presented previously: a corresponding experimental investigation is presented here. The construction of multipole source components to octopole order is described, as are procedures for determining their source strengths. Dipoles, longitudinal quadrupoles, and longitudinal octopoles that conformed closely to their theoretical models were constructed using arrays of unbaffled loudspeakers. Two methods of calculating the multipole strengths required to cancel a primary monopole field were implemented in an open-loop manner: a "direct" approach based on a multipole expansion of the primary field, and a least-squares procedure based on fitting the secondary field to the primary field either along a circle enclosing a secondary source, or along a segment of that circle. Cancellation measurements were made on a 1-m-radius circle centered on the secondary source: the primary-to-secondary source separation was approximately 0.2 wavelengths. It was found both that a secondary multipole could provide far greater cancellation than a monopole placed at the same distance from the primary source and that the least-squares approach resulted in greater far-field cancellation than did the direct approach.     

Liénard-Wiechertsche Potentiale für Multipolquellen

Liénard-Wiechert Potentials for Multipole Sources The Liénard-Wiechert solution is generalized to the case of a point-like multipole source possessing time-dependent multipole moments of arbitrary orders and moving along any time-like world line. The resulting potentials are exact solutions to the field equations of tensor fields of arbitrary rank s. If s ? 2, the motion of the source is constrained by the field equations.     

Cardiac magnetic source imaging based on current multipole model

It is widely accepted that the heart current source can be reduced into a current multipole. By adopting three linear inverse methods, the cardiac magnetic imaging is achieved in this article based on the current multipole model expanded to the first order terms. This magnetic imaging is realized in a reconstruction plane in the centre of human heart, where the current dipole array is employed to represent realistic cardiac current distribution. The current multipole as testing source generates magnetic fields in the measuring plane, serving as inputs of cardiac magnetic inverse problem. In the heart-torso model constructed by boundary element method, the current multipole magnetic field distribution is compared with that in the homogeneous infinite space, and also with the single current dipole magnetic field distribution. Then the minimum-norm least-squares (MNLS) method, the optimal weighted pseudoinverse method (OWPIM), and the optimal constrained linear inverse method (OCLIM) are selected as the algorithms for inverse computation based on current multipole model innovatively, and the imaging effects of these three inverse methods are compared. Besides, two reconstructing parameters, residual and mean residual, are also discussed, and their trends under MNLS, OWPIM and OCLIM each as a function of SNR are obtained and compared.     

Concept of multipole magnetic field rotation in ECRIS

The conventional type of magnetic well is formed by superposition of two types of magnetic field, axial bumpy field and radial multipole field. It is used to contain plasma that consists of neutrals, ions and electrons. These particles are in constant motion in the well and energetic electrons create plasma by violent collisions with neutrals and ions. The confined electrons are constantly heated by ECR technique in the presence of magnetic field. In this paper it has been shown theoretically that how the electron motion is influenced in terms of heating, containment and azimuthal uniformity of plasma, by the axial rotation of the multipole magnetic field [1,2]. Afterwards, the feasibility of achieving a rotating magnetic multipole field is discussed to some extent. And it is seen that it is not beyond the capability of the scientific community in the present scenario of the advanced technology. Presently, it can be achieved for lesser field and slightly larger size of the multipole electromagnet and can be used for improvement of the ECR ion source (ECRIS).     

Acousto-electric well logging by eccentric source and extraction of shear wave

The nonaxisymmetric acousto-electric field excited by an eccentric acoustic source in the borehole based on Pride seismoelectric theory is considered. It is shown that the acoustic field inside the borehole, converted electric and magnetic fields and coupled fields outside the borehole are composed of an infinitude of multipole fields with different orders. The numerical results show that both the electromagnetic waves and the seismoelectric field in the borehole, and the three components of both electric field and magnetic field can be detected. Measurements on the borehole axis will be of advantage to determining shear velocity information. The components of the symmetric and nonsymmetric acoustic and electromagnetic fields can be strengthened or weakened by adding or subtracting the two full waveforms logged in some azimuths. It may be a new method of directly measuring the shear wave velocity by using the borehole seismoelectric effect.     

A note on moving multipole sources of sound

The object of this note is to draw attention to the need, when speaking of a moving multipole source of sound, to state whether it is based on the wave equation for the velocity potential or the wave equation for the perturbation pressure. The theory of moving multipole sources of sound is firmly established, but, owing to lack of precision, some seemingly contradictory results are to be found in the literature. The reasons for these seemingly contradictory results are examined, and various basic relationships between a moving multipole sound field and its method of generation are established.     

Contribution from cosmological scalar perturbations to the angular velocity spectrum of extragalactic sources

The possibility of the influence of adiabatic scalar perturbations on the angular velocity spectrum of extragalactic sources is considered. The multipole expansion coefficients of the angular velocity field in terms of vector spherical harmonics are calculated. We show that there is no contribution from adiabatic perturbations to the angular spectrum for a spatially flat Universe at the dusty stage, while there is a contribution only to the electric multiple coefficients at the stage of ??-term domination. The cases of long-wavelength and short-wavelength perturbations are considered separately. The relationship between the multipole angular velocity spectrum and the primordial scalar perturbation spectrum is discussed.     

Static gravitational multipoles — The connection between field and source in general relativity

Multipole moments have already been defined in general relativity independently for both field and source, but owing to the non-linearity of the theory these two concepts have not hitherto been related. Recently Sciama, Waylen and Gilman have found a pseudo-linear integral representation of Einstein's field equations. This representation is here used to find a relation between the multipole moments of field and source in the static case. The technique is to expand the Green's function of the representation in a series of eigenfunctions of an associated linear differential operator.     

Simulation of scattered fields: Some guidelines for the equivalent source method

Three different approaches of the equivalent source method for simulating scattered fields are compared: two of them deal with monopole sets, the other with multipole expansions. In the first monopole approach, the sources have fixed positions given by specific rules, while in the second one (ESGA), the optimal positions are determined via a genetic algorithm. The ‘pros and cons’ of each of these approaches are discussed with the aim of providing practical guidelines for the user. It is shown that while both monopole techniques furnish quite good pressure field reconstructions with simple source arrangements, ESGA requires a number of monopoles significantly smaller and, with equal number of sources, yields a better precision. As for the multipole technique, the main advantage is that in principle any precision can be reached, provided the source order is sufficiently high. On the other hand, the results point out that the lack of rules for determining the proper multipole order necessary for a desired precision may constitute a handicap for the user.     

A NEW PROJECT OF MULTIPOLE PERMANENT MAGNETS

A newly developed multipole permanent magnet structure is suggested. In this structure, the required multipole field can be formed by gradually varing the depth of the permanent magnet pieces. 2-dimensional space expression of the magnet field of this structure is also given.     

Active noise cancellation of a spherical multipole source using a radially vibrating spherical baffled piston

The creation of quiet zones in a diffuse sound field due to a multipole spherical primary source by means of a radially vibrating surface set in the side of a rigid sphere (secondary source) is investigated in this article. The formulation utilizes the appropriate wave field expansions along with the translational addition theorems for spherical wave functions to develop a closed-form solution in the form of an infinite series. The numerical results reveal that using a baffled spherical piston model as a secondary source instead of a monopole control source will obviously improve the sound minimization efficiency of such noise-control systems in all cases, especially for a dipolar primary source.     

Magnetic multipole moments

Literature definitions of magnetic multipole moment operators are shown to be at variance, and new definitions are formulated which are consistent with a general multipole interaction hamiltonian and with the radiation field of a dynamic charge distribution. The applicability of traceless multipole moments is examined.

The multipole hamiltonian is used to derive expressions for some magnetic quadrupole distortion tensors. For those describing the quadrupole moment induced by a magnetic field and by a field gradient the number of independent components for various molecular symmetries is evaluated.     

光斑的形状因子及其在光纤定解问题中的应用

通过比较面光源与多极子点源两种辐射场,定义了光斑的形状因子,它决定于光斑上的源强分布.给出了形状因子的确定方法,求出了几种常见光斑的形状因子.把形状因子应用于光纤理论,补充了入射端面上的定解条件,完整地设定了定解问题.利用形状因子简单讨论了光纤端面的源强分布对光纤模式特性的影响 关键词： 光纤光学 多极子面光源 形状因子     

Crystal field effects in rare-earth ferroborates RFe3(BO3)4 with R = Nd, Tb, Dy, or Er

Crystal field effects in rare-earth ferroborates RFe₃(BO₃)₄ with various rare-earth elements are studied theoretically. The rare-earth magnetoelastic interaction Hamiltonian is written in a multipole approximation. The field and temperature dependences of the multipole moments and strain susceptibilities of rare-earth ions in the ferroborate structure are calculated. A comparative analysis of anomalies in the thermal expansion and elastic constants of rare-earth ferroborates with Tb and Dy ions having identical magnetic structure but differing in the degree of anisotropy of the rare-earth ion is performed.     

RPA instabilities in finite nuclei at low density

Early development of the instabilities in a dilute nuclear source is investigated using a finite temperature quantal RPA approach for different systems. The growth rates of the unstable collective modes are determined by solving a dispersion relation, which is obtained by parametrizing the transition density in terms of its multipole moments. Under typical conditions of a dilute finite system at moderate temperatures the dispersion relation exhibits an ultraviolet cut-off. As a result, only a finite number of multipole modes becomes unstable, and the number of the unstable collective modes increases with the size of the source. Calculations indicate that for an expanding source, unstable modes show a transition from surface to volume character.     

Multipole polarizability of a graded spherical particle

We have studied the multipole polarizability of a graded spherical particle in a nonuniform electric field, in which the conductivity can vary radially inside the particle. The main objective of this work is to access the effects of multipole interactions at small interparticle separations, which can be important in non-dilute suspensions of functionally graded materials. The nonuniform electric field arises either from that applied on the particle or from the local field of all other particles. We developed a differential effective multipole moment approximation (DEMMA) to compute the multipole moment of a graded spherical particle in a nonuniform external field. Moreover, we compare the DEMMA results with the exact results of the power-law graded profile and the agreement is excellent. The extension to anisotropic DEMMA will be studied in an Appendix.     

Contribution of the multipole polarization to the elastic constants in fluorite structure crystals

Numerical calculations are presented for the contribution of the multipole polarization to the elastic constants in fluorite structure crystals. The multipole polarizability is calculated by the self-consistent field treatment of the local density approximation and the spherical solid model. The elastic constants are significantly affected by the multipole polarization of the ions. The contributions of the multipole polarization of the ions to the elastic constant C₁₁, C₁₂, and C₄₄ are about 25%, 40% and 20% of the experimental values, respectively. The calculated values of the deviation from the Cauchy relation are in good agreement with the experimental values.