Extinction of Radiation in Sterically Interacting Systems of Monodisperse Spheres. Part 2: Experimental results

The Bouguer-Lambert-Beer law (BLBL), commonly used to describe the extinction of radiation in disperse systems, is known to be valid in the limit of low concentrations only. In Part [1] it was shown that steric particle-particle interactions lead to important effects in the range of high particle concentration and two versions of a more general extinction equation including steric interactions were derived for the regimes of fraunhofer diffraction and ray propagation in this part experimental evidence for the effects of steric interactions on the extinction of radiation in suspensions of monodisperse spherical particles is given. Various experiments on light extinction and on ultrasonic extinction are combined to cover a broad range of particle size parameters, mechanisms of particle-wave interaction and receiver apertures.     

A multi-channel model of weak scattering of leptons

A multi-channel model of weak scattering is developed, based on the four-fermion theory of weak interactions using the Bethe-Salpeter equation. It leads to several relations among the crosssections of different weak scattering processes. In the low energy limit the cross-section of all included reactions are expressed with the help of a single constant.     

Generalized optical theorem to a multipole source excitation in the scattering theory

In the present paper, the Optical Theorem is generalized to the case of a penetrable obstacle excited by a multipole of arbitrary order in the presence of a transparent substrate. This generalization allows one to test computer modules when wave scattering by lossless penetrable obstacle is considered. Besides, it enables one to evaluate the absorption cross-section by subtracting the scattering cross-section from the extinction cross-section. This seems to be important because, in this particular case, the far field does not involve a Sommerfeld integral.     

The Fundamentals of Particle Size Analysis by Transmission Fluctuation Spectrometry. Part 1: A Theory of Temporal Transmission Fluctuations in Dilute Suspensions

The extinction of radiation in suspensions is traditionally described by the Bouger-Lambert-Beer law (BLBL). Based on a quasi-continuum approach, the BLBL does not account for the discrete nature of particles or their spatial extension and arrangement. If an extinction measurement is made with a high spatial and temporal resolution, the transmitted intensity signal shows significant fluctuations. The strength of fluctuation is related to the physical properties of the suspension and the process of spatial and temporal averaging. Exploiting this connection, it is possible to calculate the particle size distribution and the particle concentration from transmission measurements. This part of the series of papers provides a method for the temporal decomposition of the transmission's power spectrum, which permits the information on the particle size and concentration to be extracted from the seemingly irregular fluctuation of the transmission signal.     

On Suhl's approach to the Kondo problem

A model for the Kondo problem is studied in which the impurities are envisaged as a gas of infinitely heavy particles embedded in the gas of conduction electrons. Fors-wave interactions and low impurity concentrations the electron self-energy is expressed by the impurity-electron scattering matrix which is also shown to determine the thermodynamic potantial. Using standard Goldstone diagrams Suhl's equation is derived by the summation of leading singular graphs. For a special density of states, vanishing magnetic field, and no potential scattering the dispersion equation is solved exactly.     

Generalization of the optical theorem for an arbitrary multipole in the presence of a transparent half-space

The optical theorem is generalized to the case of excitation of a local inhomogeneity introduced in a transparent substrate by a multipole of arbitrary order. It is shown that, to calculate the generalized extinction cross section, it is sufficient to calculate the derivatives of the scattered field at a single point by adding a constant and a definite integral. Apart from general scientific interest, the proposed generalization makes it possible to calculate the absorption cross section by subtracting the scattering cross section from the extinction cross section. The latter fact is important, because the scattered field in the far zone contains no Sommerfeld integrals. In addition, the proposed generalization allows one to test computer modules for the case where a lossless inhomogeneity is considered.     

Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

Abstract

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.     

Unitary fermions and Lscher's formula on a crystal

We consider the low-energy particle-particle scattering properties in a periodic simple cubic crystal. In particular, we investigate the relation between the two-body scattering length and the energy shift experienced by the lowest-lying unbound state when this is placed in a periodic finite box. We introduce a continuum model for s-wave contact interactions that respects the symmetry of the Brillouin zone in its regularisation and renormalisation procedures, and corresponds to the nae continuum limit of the Hubbard model. The energy shifts are found to be identical to those obtained in the usual spherically symmetric renormalisation scheme upon resolving an important subtlety regarding the cutoff procedure. We then particularize to the Hubbard model, and find that for large finite lattices the results are identical to those obtained in the continuum limit. The results reported here are valid in the weak,intermediate and unitary limits. These may be used to significantly ease the extraction of scattering information, and therefore effective interactions in condensed matter systems in realistic periodic potentials. This can achieved via exact diagonalisation or Monte Carlo methods, without the need to solve challenging, genuine multichannel collisional problems with very restricted symmetry simplifications.     

The neutrino response of low-density neutron matter from the virial expansion

We generalize our virial approach to study spin-polarized neutron matter and the consistent neutrino response at low densities. In the long-wavelength limit, the virial expansion makes model-independent predictions for the density and spin response, based only on nucleon–nucleon scattering data. Our results for the neutrino response provide constraints for random-phase approximation or other model calculations, and we compare the virial vector and axial response to response functions used in supernova simulations. The virial expansion is suitable to describe matter near the supernova neutrinosphere, and this work extends the virial equation of state to predict neutrino interactions in neutron matter.     

The propagation of infinitesimal disturbances in an ultrarelativistic gas according to the method of elementary solutions

It has recently been shown that a linearized relativistic BGK model can be reduced, in the ultrarelativistic limit, to a system of three uncoupled transport equations for thermal, sound, and shear waves. The equation describing the propagation of thermal waves is the well-known one-speed neutron transport with isotropic scattering in the conservative case. In this paper the solution of the half-space problem for the equation describing the propagation of shear and sound waves is given according to Case's elementary solutions method.     

Theory of scattering of electromagnetic waves of the microwave range in a turbid medium

The coefficient of extinction of electromagnetic waves of the microwave range due to their scattering from clusters suspended in an amorphous medium and responsible for turbidity is calculated. Turbidity resembles the case when butter clusters transform water into milk. In the case under investigation, the clusters are conductors (metallic or semiconducting). The extinction coefficient is connected in a familiar way with the cross section of light scattering from an individual cluster. A new formula is derived for the light scattering cross section in the case when damping of oscillations of an electron is due only to spontaneous emission of light quanta. In this case, the resonant scattering cross section for light can be very large. It is shown that this can be observed only in a whisker nanocluster. In addition, the phonon energy on a whisker segment must be higher than the photon energy, which is close to the spacing between the electron energy levels in the cluster.     

Radiative properties of scattering and absorbing dense media: theory and experimental study

We investigate the validity of the radiative transfer equation to model transmission of light through an absorbing and scattering medium. Assuming that radiative transfer equation is valid, the inverse scattering problem for non-polarized radiative transfer in one-dimensional absorbing and scattering media is solved using a parameter identification method. We discuss how to identify the albedo, phase function and extinction coefficient of the medium. We present experimental data that confirm that this approach is robust and can be used to make reliable predictions of the behavior of scattering absorbing systems.     

Lattice gases with static disorder: Renormalization of mean field theory

Lattice gas automata are used to model transport phenomena in random media with static disorder. If the interactions are repulsive, there is a large probability of backscattering or retracing collision sequences. In that case the Boltzmann equation or mean field theory breaks down, even in the limit of a low concentration of scatterers. Here sequences of uncorrelated and retracing collisions are of equal importance. The repeated ring approximation is used to resum the retracing trajectories, and the renormalized transport coefficients are calculated in the low-density limit, not only for hard core scatterers (diamonds, hexagons, triangles), but also for mixed point scatterers (mirrors, rotators, reflectors). The results are compared with extensive computer simulations.     

Multi-patch model for transport properties of cuprate superconductors

A number of normal state transport properties of cuprate superconductors are analyzed in detail using the Boltzmann equation. The momentum dependence of the electronic structure and the strong momentum anisotropy of the electronic scattering are included in a phenomenological way via a multi-patch model. The Brillouin zone and the Fermi surface are divided in regions where scattering between the electrons is strong and the Fermi velocity is low (hot patches) and in regions where the scattering is weak and the Fermi velocity is large (cold patches). We present several motivations for this phenomenology starting from various microscopic approaches. A solution of the Boltzmann equation in the case of N patches is obtained and an expression for the distribution function away from equilibrium is given. Within this framework, and limiting our analysis to the two patches case, the temperature dependence of resistivity, thermoelectric power, Hall angle, magnetoresistance and thermal Hall conductivity are studied in a systematic way analyzing the role of the patch geometry and the temperature dependence of the scattering rates. In the case of Bi-based cuprates, using ARPES data for the electronic structure, and assuming an inter-patch scattering between hot and cold states with a linear temperature dependence, a reasonable agreement with the available experiments is obtained. Received 3 August 2001 and Received in final form 1st November 2001     

Diffusion of tagged interacting spherically symmetric polymers

We consider the diffusion of two species of spherically symmetric macromolecules in solution under the influence of short range central pair potential interactions as well as two body hydrodynamic interactions. Starting from the N-particle Smoluchowski equation and using Felderhof's approach we derive, to linear order in densities, a pair of coupled diffusion equations for the single particle number densities. There are two independent diffusional modes each with an effective diffusion constant dependent in general upon both the interparticle potentials as well as the hydrodynamic model used for each type of macromolecule. However, in the limit that one species is present at very low density compared with the other species, one of the effective diffusion constants is dominated by hydrodynamic interactions. By tagging these tracer particles to observe their diffusion by light scattering, one can test both the mixed stick-slip boundary condition model and the permeable sphere model of the macromolecules.     

Electromagnetic-wave scattering from chiral spheres in chiral media

Summary The scattering of electromagnetic waves in chiral (optically active) media from chiral spheres is studied. Mie-scattering techniques are used to find the exact solution for plane-electromagnetic-wave scattering from a chiral sphere of arbitrary size in an infinitely extended chiral medium of arbitrarily different permeability, permittivity, and chirality, and the scattering and extinction efficiencies for chiral spheres in chiral media are derived. Special cases of achiral exterior medium-chiral sphere and achiral exterior medium-achiral scatterer are considered and in the latter case well-known results of Mie scattering are recovered. Simplified results for small spheres are also found for the limit of Rayleigh scattering.     

The range of validity of the Rayleigh limit for computing mie scattering and extinction efficiencies

For the small particle limit for scattering by either absorbing or dielectric spheres, values of the size parameter are determined below which various approximate expressions for the scattering and extinction efficiencies are accurate to within 1%. Simple, analytical criteria for ensuring 1% accuracy are given in terms of the size parameter and the complex refractive index. A much broader range of refractive indices is considered than has been studied previously.