Number Of Multipoles Research Articles

Scattering of a Spherical Dyadic Field by a Small Rigid Sphere

A complete dyadic field, which is generated at a point and propagates within a homogeneous and isotropic elastic medium, is disturbed by a small rigid sphere. Analytic solutions for this complicated dyadic scattering problem are provided with the help of an extended theory of the Papkovich representation for elastostatic dyadic fields. Relative results obtained numerically show an amazing coincidence as long as we stay in the low-frequency regime. In contrast to the plane wave excitation case, where only a few multipole terms are needed to express the leading low-frequency approximations, the case of point source excitation provides low-frequency solutions where an infinite number of multipoles are present. An exception is offered by the first-order approximation, which enjoys a closed-form expression.

Light absorption by the clusters of colloidal gold and silver particles formed during slow and fast aggregation

Spectra of absorption (400–800 nm) by the aggregates of colloidal gold (5, 15, and 30 nm in diameter) and silver (20 nm in diameter) particles were studied experimentally and theoretically. It was revealed that, during fast aggregation corresponding to the diffusion-limited cluster aggregation (DLCA), the pattern of spectra is dependent on the size of primary particles. Spectra with the additional absorption maximum in the red region are observed for 15 and 30 nm gold and 20 nm silver particles, while the absorption spectrum for 5 nm particles is characterized by only one maximum shifted to the red region. The slow aggregation resulted in a decrease in plasmon absorption peak with no marked shift to the red region and to the broadening of long-wave absorption wing. From data on electron microscopy, typical branched DLCA-clusters were formed during fast aggregation, whereas small compact aggregates and a noticeable number of single particles were observed in a system during slow aggregation. The computer model of the diffusion-limited cluster-cluster aggregation was used to explain these results. Optical properties of aggregates were calculated using coupled dipole method (CDM or DDA) and the exact method of a multipole expansion. Corrections for the size effect were introduced into the bulk optical constants of metals for nanosized particles. It was shown that a modified version of DDA (Markel et al.,Phys. Rev. B, 1996, vol. 53, no. 5, p. 2425) allows us to explain the pattern of experimental spectra of DLCA-aggregates and their dependence on a monomer size. The exact method was applied to calculate the extinction cross sections of metallic aggregates demonstrating strong electrodynamic interaction between particles. The number of higher multipoles that are required to adequately describe this interaction is much larger than the number of terms of an ordinary Mie series and is the main obstacle to the exact calculation of the spectra of metallic aggregates with a large number of particles.

Effects of high-order multipoles on the extinction spectra of dispersive bispheres

Extinction spectra of randomly oriented pairs of dispersive spheres are calculated by the T-matrix method. The number of multipoles, L, that have to be included in the calculation in order to achieve convergence was studied for a wide range of sphere radii and inter-sphere separations. For a given sphere size, L increases with decreasing separation between the spheres. For the most severe case, i.e., touching spheres, convergence becomes more difficult to achieve as the sphere size is reduced.

Electrostatic Force on a Moist Particle Near a Ground Plane

Abstract The effect of surface moisture on the electrostatic force of attraction between an uncharged particle and a ground plane is investigated. Describing a moist sphere with an effective complex permittivity allows the image force to be calculated using complex linear multipoles. The number of multipoles needed for an accurate image force calculation is determined as a function of particle-plane separation. Adsorbed moisture introduces a relaxation in the electrostatic image force. For particles exposed to a range of relative humidities, the model predicts relaxation times that vary by several orders of magnitude. In addition, at small particle-plane separations adsorbed moisture can increase the image force by several orders of magnitude. Finally, multipolar moments are used to calculate the electric field in the gap separating the particle and ground plane to determine the onset of breakdown or field emission.

Force on two touching dielectric spheres in a parallel field

An interactive linear multipole model is used to derive equations for the electric force between two unchanged dielectric spheres aligned in a parallel field. Up to 2800 multipoles are used to compute the force between identical touching spheres. Where necessary, extrapolation to an infinite number of multipoles is performed by an empirical equation. The model and equations apply equally well to a magnetizable sphere in a parallel magnetic field. The data presented allow the calculation of the force for values of electric (or magnetic) susceptibilities up to 21 500.

Retarded treatment of substrate-related effects on granular films: I. Polarizability of a single sphere

A multipolar treatment of the retarded fields radiated by a metal sphere on a dielectric substrate under the influence of an incident plane wave described. Unlike previous treatments, the present one allows boundary conditions, both on the spherical surface and on the substrate interface, to be matched by simply solving an adequate number of linear equations. The problem is thus reduced to inverting three complex matrices, whose sizes depend on the number of multipoles included in the field forms. The solutions, which represent the object and image multipoles (both in the incidence and in the transmission media) associated with the polarized sphere, are given the form of linear combinations of the multipoles associated with the unperturbed field (incident and reflected waves).

Dielectric constant of a suspension of uniform spheres.

We present computer simulation data for the dielectric constant of a system of uniform polarizable spheres randomly distributed in a homogeneous background. The deviations from the Clausius-Mossotti formula are found for seven values of the volume fraction. In addition, the spectral density which appears in the Bergman representation of the dielectric constant is determined. The spectrum obtained with a sufficient number of higher-order multipoles differs significantly from that found with just dipolar or dipole-quadrupolar interactions.

Reaction-diffusion on a periodic array of penetrable spherical sinks

In this paper an analysis is given of the reaction-diffusion process on a periodic array of penetrable spherical sinks. The effective rate coefficient is calculated for these lattices as a function of the several parameters in the system, such as the volume fraction, the reactivity of the sinks and the ratio of the diffusion constants inside and outside the sinks. In the analysis the distribution of the absorption inside and on the surface of the sinks is expanded in multipole moments. For the calculation only a finite number of these multipoles is taken into account. A careful analysis of the convergence is made by increasing the number of multipoles until the result stabilizes. This is in particular important at volume fractions close to dense packing. Earlier results by Felderhof [1] who uses a similar method for the special case of perfectly absorbing sinks are found to deviate up to 15% compared to the fully converged results. We also compare our results with the so-called spherical approximation which is found to be a good approximation for all volume fractions when the process is reaction-controlled while in the limit of perfectly absorbing sinks this approximation is only correct at low volume fractions.

The polarizability of truncated spheres and oblate spheroids on a substrate: Comparison with experimental results

A general method for the calculation of the polarizability of a truncated spherical particle on a substrate is discussed. The particle is assumed to be much smaller than the wavelength of the incident light. The method consists of a multipole expansion of the electric potential, with multipoles located at the centre of the particle and “image multipoles” at the image of this centre with respect to the surface of the substrate. In this way all effects of the particle-substrate interaction are incorporated. The polarizabilities normal and parallel to the substrate can then be calculated to any order in the number of multipoles taken into account. An analogous method, in which spheroidal multipoles are used, is discussed for an oblate spheroidal particle with axis of symmetry normal to the substrate. Using both models, explicit results are presented for a gold particle on a sapphire substrate. For a square lattice of identical gold particles of both types on a sapphire substrate the transmittance is calculated and compared with experimental values.

The exact multicenter multipolar part of a molecular charge distribution and its simplified representations

We study the problem of representing the molecular charge distribution in a convenient way for practical applications and we propose, instead of a single representation, a flexible procedure for building approximations with an arbitrary level of accuracy as concerns the long-range part of the electrostatic potential. We first discuss the splitting of the total electrostatic potential into a multipolar part (long-range) and a penetration part (shortrange) in connection with the usual one-center multipole expansion: at large enough distances, this expansion precisely converges towards the multipolar part. However, this representation is not practically efficient as soon as the molecule departs from a spherical shape, and we therefore consider a so-called multicenter multipole representation. In the MO-LCAO framework, the use of a basis of Gaussian atomic orbitals {χα} generates such a representation in a natural way: indeed, each elementary distribution χ*αχβ then reduces to a one-center distribution with a finite number of multipoles, hence an exact multipolar representation (corresponding to the approximate MO-LCGO wave function) involving a finite (albeit large, in general) number of centers, each one bearing a finite (small) number of multipoles. From this representation used as a reference, we then generate simplified multicenter multipole representations through a systematic procedure of reduction of the number of centers (having chosen a reduced set of new centers, we represent the multipoles of each suppressed old center through suitable new multipoles located on a few new centers closest to the old one under consideration). The number and locations of the new centers are arbitrary, hence the great flexibility of the method. We checked the accuracy of this procedure for molecules of different polarity (water, benzene, adenine), and we noticeably found that a representation involved as centers, the atoms plus one point per chemical bond exhibited quite a satisfactory accuracy. In the conclusion, we briefly discuss the possibility of extending the method to other kinds of basis functions (e.g., Slater orbitals), in connection with the problem of representing the charge distribution associated with the exact molecular wave function, we also discuss the choice of the number centers according to the desired accuracy and we briefly describe a way of applying the method to the problem of representing a large molecule in terms of molecular fragments.

Multipole expansion calculation of slow viscous flow about spheroids of different sizes

The multipole representation technique of Gluckman, Pfeffer & Weinbaum for slow, viscous, axisymmetric motion has been applied to a system of two spheroids. The two particles may have different shapes and volumes. The limitations of this method have been studied through calculations of the drag forces and velocities of particles with aspect ratios from 0·1 to 10 and relative volumesVfrom 1 to 103. The multipole-expansion convergence is quite slow for large relative volumes and requires on the order of$4 \times V^{\frac{1}{3}}$multipoles. Wild fluctuations in the drag as the number of multipoles increases were found for large oblate spheroids. Relative velocities are given quite accurately for separations of the centres greater than 1·02 times the minimum separation. Comparisons are made with previous results and approximate theories for two spheres, for non-identical particles at large separation, and for a sphere near a large spheroid.

Simplified derivation of the amplitudes for the photodisintegration of the deuteron with arbitrary number of electromagnetic multipoles

By employing the interaction Hamiltonian in the operator form as derived by Breit and Rustgi, a simplified derivation of the amplitudes for the photodisintegration of the deuteron is reported, and the same expressions as given in an earlier paper are obtained. The derivation makes use of the Wigher-Eckart theorem and commutation relations between the spin matrices, but does not employ Racah coefficients. As in the earlier paper, the nonrelated approximation is used; the finalstate interaction of the outgoing neutron and proton is included. The effect of adding various electromagnetic multipoles on the cross-section and polarization is studied.

Cross-section and polarization in the photodisintegration of the deuteron with arbitrary number of electromagnetic multipoles

By employing the expressions for the electric- and magnetic-multipole interaction energy between the radiation field and nucleons as derived in the paper by Breit and Rustgi, and adopting the general precedure of a previous paper by Rustgi, Zernik, Breit and Andrews, simple expressions for the transition amplitudes of the reaction2H(γ, n)1H have been derived for all electromagnetic multipoles in the nonretarded approximation. The final-state interaction of the outgoing neutron and proton is taken into account.

A neutron small-angle scattering study of bovine fibrinogen

Neutron small-angle scattering of fibrinogen in various H 2O 2H 2O mixtures is strictly proportional to the square of the contrast at a resolution of less than 15 Å. Therefore, the excess scattering density with respect to the solvent is positive everywhere. The maximum scattering density is only a quarter of that encountered with globular proteins. It is not so low that fibrinogen could be regarded as a statistical coil. The shape is resolved in terms of multipoles. A stepwise increase in the resolution is achieved by taking into account an increasing number of multipoles; on addition of a dipolar term to the spherical average a cup-shaped model results. A still better description is obtained if the quadrupolar term is derived from the scattering curve. The superposition of the first three multipoles leads to a banana-shaped fibrinogen model. This result, from the analysis of the scattering pattern in terms of multipoles, is compared with other physicochemical and biochemical properties of fibrinogen.

Magnetic scattering of neutrons

The magnetic scattering of neutrons by an arbitrary system of particles has been examined by exploiting its similarity to the radiation problem in spectroscopy. It has been shown, in fact, that the magnetic scattering amplitude can be expressed in terms of the multipole moments of the scattering system. The number of multipoles, which contribute to the scattering amplitude, is limited by selection rules based on the symmetry properties of the states of the system, in particular, parity and angular momentum conservation. The formalism has been applied to the magnetic scattering of neutrons by an atom in the ${l}^{n}$ electronic configuration. If the spin---other-orbit and orbit-orbit interactions in the atomic Hamiltonian can be neglected, only even-order electric and odd-order magnetic multipoles, whose order of multipolarity is less than or equal to $2l+1$, contribute to the scattering amplitude. In this case the calculation of the magnetic scattering amplitude is reduced to evaluating matrix elements of the Racah double tensors ${W}^{(0,k)}$ and ${W}^{(1,{k}^{\ensuremath{'}})}({k}^{\ensuremath{'}} \mathrm{even})$. The former tensors are associated with the convection current and the latter with the spin magnetization contribution to the magnetic scattering amplitude. The calculation of the maxtrix elements of these tensors is simplified by selection rules based on the groups Sp($4l+2$), R($2l+1$), R(3), ${\mathrm{G}}_{2}$ used in the classification of the atomic states. The contribution to the magnetic scattering amplitude of the convection current, associated with the spin-orbit and mass correction terms of the atomic Hamiltonian, has been examined in some detail.

Gamma-ray linear-polarization and angular-distribution formulas for mixtures of three multipoles

Gamma-ray linear-polarization and angular-distribution formulas for mixtures of three multipoles are presented in terms of the phase-defined reduced matrix elements of Rose and Brink. These formulas are applicable to the simple angular distribution as well as to the particle-gamma angular correlation known as Method II of Litherland and Ferguson. In addition, a “recipe” has been worked out for transforming an angular distribution formula into a linear polarization formula in the general case of a γ radiation consisting of an arbitrary number of multipoles.