Spherical Multipole Expansion Research Articles

Numerical evaluation of dyadic diffraction coefficients and bistatic radar cross sections for a perfectly conducting semi-infinite elliptic cone

In this paper, the scattering of electromagnetic waves by a perfectly conducting semi-infinite elliptic cone is treated. The exact solution of this boundary value problem in problem-adapted spheroconal coordinates in the form of a spherical multipole expansion is of poor convergence if both the source point and the field point are far away from the cone's tip. Therefore, an appropriate sequence transformation of these series expansions (we apply the Shanks transformation) is necessary to numerically determine the dyadic diffraction coefficients and bistatic radar cross sections (RCS) for an arbitrary elliptic cone. Our far-field data for an elliptic cone, a circular cone, and a plane angular sector are compared with some other results obtained with the aid of quite different methods.

Diffraction and scattering of electromagnetic waves by spheres with elliptic slit apertures and interior loading

An analytical method is presented for solving the problem of diffraction of electromagnetic waves by a perfectly conducting sphere with an elliptic slit aperture. The interior of the sphere is filled with two different media which are radially stratified. The treatment of this canonical shielding problem may provide a better insight into the coupling mechanism of electromagnetic waves through slit apertures in realistic shielding structures. The solution of this three-dimensional diffraction problem is based on the spherical multipole expansion of the electromagnetic field.

The radar cross section of the semi-infinite elliptic cone: numerical evaluation

The purpose of the present paper is to evaluate the radar cross section (RCS) of a semi-infinite perfectly conducting elliptic cone, including the circular cone and the plane angular sector. The RCS has been obtained from a spherical multipole expansion of the electromagnetic field. The analytical expression for the RCS is given in the form of an infinite series with poor convergence. Euler's summation technique has been employed to evaluate the series numerically.

Self-consistent full-potential total-energy Korringa-Kohn-Rostoker band-structure method: Application to silicon.

The self-consistent full-potential total-energy Korringa-Kohn-Rostoker (KKR) coupled channel equation method for full-potential electronic band-structure determinations is applied to the case of silicon. For efficiency, the coupled channel equations are partially decoupled by using the point group symmetry of each basis atom, and solved on a linear radial mesh by using the Coulomb functions close to the singularity of the potential at the origin. In the KKR energy search, Chebyshev approximations for the scattering matrices as functions of energy save solving the coupled channel equations at many different energies. The results include the self-consistent anisotropic potential, the charge density, the energy band structure, the equilibrium lattice constant, and the bulk modulus. As expected, the energy bands agree well with results from other band-structure methods. We find that the cutoff of the spherical multipole expansion of potential and charge density must be chosen independently of the cutoff for the scattering matrices that determines the dimension of the secular matrix. In the case of silicon, we find that an ${\mathit{l}}_{\mathrm{max}}$ of 8 is sufficient for the potential and charge density, while the scattering matrices need at least ${\mathit{l}}_{\mathrm{max}}$=6. The equilibrium lattice constant converges to the experimental value with increasing ${\mathit{l}}_{\mathrm{max}}$ and the bulk modulus converges to 96 GPa at ${\mathit{l}}_{\mathrm{max}}$=6.

Ellipsoidal angular distribution of electrons emitted from Rydberg atoms.

We investigate the energy and angular distribution of electrons emitted in fast collisions with neutral atoms from Rydberg states of the projectile into low-lying continuum states of the projectile. We find highly anisotropic, approximately ellipsoidal angular distributions for large principal quantum numbers (n\ensuremath{\simeq}10). The corresponding spherical multipole expansion contains contributions from high-order multipoles ${\mathrm{\ensuremath{\beta}}}_{\mathit{k}}$ at least up to k\ensuremath{\simeq}8 and large values of low-order multipoles ${\mathrm{\ensuremath{\beta}}}_{2}$ well beyond the limit for dipole-allowed transitions. Comparison with recent data for ion-solid collisions shows surprisingly good agreement.

Scattering of an arbitrary plane electromagnetic wave by a finite elliptic cone

The vector spherical multipole analysis is applied to determine the diffraction of a plane electromagnetic wave by a perfectly conducting finite elliptic cone, which is bounded by a spherical cap. With respect to the boundary- and continuity-conditions of the tangential fields the two-domain-problem is reduced to a system of linear equations. Numerical data are presented such as scattering cross sections for the finite elliptic cone, including the half-sphere and the finite plane angular sector. In addition, a new derivation of the spherical multipole expansion of an arbitrary plane electromagnetic wave is presented.

Scattering of a plane electromagnetic wave by a spherical shell with an elliptical aperture

The vector spherical multipole expansion is applied to determine the electromagnetic field of a perfectly conducting spherical shell with an elliptical aperture, which is illuminated by a plane wave. Using problem-adapted sphero-conal coordinates the authors are able to treat this diffraction problem mostly analytically. They solve the electric field integral equation (EFIE) by Galerkin's method with entire domain expansions and a bilinear spherical multipole expansion of the dyadic Green's function as the integral equation kernel. Numerical results such as the scattering cross section and the energy density at the center of the spherical shell with an elliptical aperture are computed both as functions of frequency and/or of the aperture ellipticity. From the numerical data it is seen that the resonance frequencies of the energy density at the center are identical to those appearing in the backscattering cross section.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Explicit formulae for the electrostatic energy, forces and torques between a pair of molecules of arbitrary symmetry

The spherical multipole expansion has been used to derive explicit expressions for the terms up to and including R -5 in the R -1 expansion of the electrostatic energy of two molecules of arbitrary symmetry, in a simple form which is suitable for use in model potentials. We show also how the corresponding forces and torques can be readily derived from these energy expressions, and give explicitly the forces and torques which would be required for a molecular dynamics simulation of a fluid of linear molecules.

Spherical delta functions and multipole expansions

The Cartesian–Taylor series for an analytic function in three dimensions is rewritten as a series of solid spherical harmonics. A discussion of the distribution theory definition of singular spherical harmonics is given, which leads to a definition of spherical delta functions. An expansion of source functions in spherical delta functions and their derivatives leads to multipole expansions for the fields which, in a distribution theory sense, are valid everywhere.

On recurrence relations for multipole coefficients

General recurrence relations between the coefficients in thenth and (n+1)th order spherical harmonic multipole expansions are derived. The particular application presented here is the derivation of the equations concerned with representing the geomagnetic field by magnetic multipoles. The equations up to the 3rd order multipole are given as an example of the method. The main advantage in using these recurrence relations rather than other methods is that the mathematics is reduced to merely a matter of successive substitutions and this allows a fast step by step generation of the required equations, in a form for which there is a simple numerical program for solution.