Extended boundary condition method in combination with field expansions in terms of spheroidal functions

The problem of light scattering by nonspherical particles, which arises in many applications, is nowadays most frequently solved by the method of extended boundary conditions in combination with the expansion of the fields in terms of spherical wave functions. However, such an approach encounters difficulties if the shape of particles is far from spherically symmetric, even in the simplest case of spheroids with the semiaxis ratio a/b > 5?10. A new approach to solving this problem is proposed, which also applies the extended boundary condition method but involves the expansion of the fields in terms of spheroidal functions. In this case, to obtain effective solutions for strongly prolate and oblate particles, the fields are divided in two parts with known properties and specific scalar potentials are used for each part. The basic relations of the approach are presented and some results of calculations of the optical properties of spheroids and spheroidal Chebyshev particles that are performed using computer codes realizing this approach are given. The convergence of the results for different cases and the domain of applicability of the method are discussed.     

Analysis of the modified point-matching method in the electrostatic problem for axisymmetric particles

An integral modification of the generalized point-matching method (GPMMi) in the electrostatic problem for axisymmetric particles is developed. Scalar potentials that determine electric fields are represented as expansions in terms of eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAEs), which are obtained from the requirement of a minimum of the integrated residual in the boundary conditions on the particle surface. Matrix elements of ISLAEs and expansion coefficients of the “scattered” field at large index values are analyzed analytically and numerically. It is shown analytically that the applicability condition of the GPMMi coincides with that for the extended boundary conditions method (ЕВСМ). As model particles, oblate pseudospheroids \(r\left( \theta \right) = a\sqrt {1 - {^2}{{\cos }^2}\theta } ,\;{^2} = 1 - {\raise0.7ex\hbox{${{b^2}}$} \!\mathord{\left/ {\vphantom {{{b^2}} {{a_2}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{a_2}}$}} \geqslant 0\) with semiaxes a = 1 and b ≤ 1 are considered, which are obtained as a result of the inversion of prolate spheroids with the same semiaxes with respect to the coordinate origin. For pseudospheroids, the range of applicability of the considered methods is determined by the condition \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} < \sqrt 2 + 1\). Numerical calculations show that, as a rule, the ЕВСМ yields considerably more accurate results in this range, with the time consumption being substantially shorter. Beyond the ЕВСМ range of applicability, the GPMMi approach can yield reasonable results for the calculation of the polarizability, which should be considered as approximate and which should be verified with other approaches. For oblate nonconvex pseudospheroids (i.e., at \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} \geqslant \sqrt 2 \)), it is shown that the spheroidal model works well if pseudospheroids are replaced with ordinary “effective” oblate spheroids. Semiaxes a_ef and b_ef of the effective spheroids are determined from the requirement of the particle volumes, as well as from the equality of the maximal longitudinal and transverse dimensions of particles or their lengths. As a result, the polarizability of pseudospheroids can be calculated by simple explicit formulas with an error of about 0.5–2%.     

The Condition of Applicability for the Extended Boundary Conditions Method for Small Multilayer Particles

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—R₁^(j) = σ₁^(j) and R₂^(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—R₁ < R₂. 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 R₂^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%.     

Surface-integral formulation for electromagnetic scattering in spheroidal coordinates

We derive surface-integral expressions for the Q matrices in spheroidal coordinates that allow us to compute the T matrix in spheroidal coordinates. This approach combines the advantages of the null-field method (also referred to as the extended boundary condition method) with those of the separation of variables method. For spheroidal particles we obtain explicit Q matrix expressions that display the expected symmetry properties and yield correct results in the spherical limit. Compared to surface-integral expressions for spheroids in spherical coordinates, our results are considerably simpler because the integrands do not contain radial functions.     

Light Scattering by Small Multilayer Particles: A Generalized Separation of Variables Method

A Rayleigh approximation is constructed for light scattering by small multilayer axisymmetric particles, in which their polarizability is determined by the generalized separation of variables method (SVM). In this method, scalar potentials, the gradients of which yield the electric-field strengths, are represented as expansions in spherical harmonics of the Laplace equation. Unknown coefficients of expansions are determined from the boundary conditions, which are reduced to infinite systems of linear algebraic equations (ISLAEs), since the separation of variables is incomplete. The T matrix of the electrostatic problem, principal element T₁₁ of which is proportional to the particle polarizability, is determined. The necessary condition for the ISLAEs solvability for the SVM coincides with the condition of the correct application of the extended boundary conditions method (EBCM). However, numerical calculations in which finite-dimensional (i.e., reduced) systems are solved, yield different results in ranges of variation of parameters that are close to the boundary of the range of applicability. An analysis of the numerical calculations of the scattering and absorption cross sections for two-layer confocal spheroids, an exact solution for which can be obtained using spheroidal harmonics, shows that the SVM is preferable to the EBCM. It turned out that the proposed method yields workable results in a wider range of variation of parameters. Even outside the range of applicability, in which it should be regarded as a certain approximate solution, its use in a number of cases is quite acceptable. Additional calculations for three-layer nonconfocal spheroids, as well as for three-layer similar pseudospheroids and Pascal’s snails, which can be obtained from spheroids as a result of the inversion with respect to the coordinate origin and one of the foci, respectively, confirm these inferences. We note that, for certain values of the parameters, the shapes of the latter particles are nonconvex.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Equilibrating Effects of Boundary and Collision in Rarefied Gases

We investigate the time-asymptotic behavior for rarefied gases in the spherical domain with variable boundary temperature in ${\mathbb{R}^d,}$ d=1,2,3, under the diffuse reflection boundary condition. First, we obtain an optimal convergence rate of (1 + t)^?d to the steady state for free molecular flow. Next, we use this to construct the steady state solution of the Boltzmann equation for sufficiently large Knudsen number and small boundary temperature variation. We also obtain an exponential convergence to the steady state for the Boltzmann equation for small perturbation.     

FRW in Cosmological Self-creation Theory

We use the Brans-Dicke theory from the framework of General Relativity (Einstein frame), but now the total energy momentum tensor fulfills the following condition $[\frac{1}{\phi}T^{\mu \nu M}+T^{\mu \nu}(\phi)]_{;\nu}=0$ . We take as a first model the flat FRW metric and with the law of variation for Hubble’s parameter proposal by Berman and Gomide (Nuovo Cimento B 74: 182, 1983), we find solutions to the Einstein field equations by the cases: inflation (γ=?1), radiation ( $\gamma=\frac{1}{3}$ ), stiff matter (γ=1). For the Inflation case the scalar field grows fast and depends strongly of the constant M _γ=?1 that appears in the solution, for the Radiation case, the scalar stop its expansion and then decrease perhaps due to the presence of the first particles. In the Stiff Matter case, the scalar field is decreasing so for a large time, ?→0. In the same line of classical solutions, we find an exact solution to the Einstein field equations for the stiff matter (γ=1) and flat universe, using the Hamilton-Jacobi scheme.     

Polynomial behaviour of scattering amplitudes at fixed momentum transfer in theories with local observables

It is shown that, in theories of exactly localized observables, of the type proposed byAraki andHaag, the reaction amplitude for two particles giving two particles is polynomially bounded ins for fixed momentum transfert<0. The proof does not need observables localized in space-time regions of arbitrarily small volume, but uses relativistic invariance in an essential way. It is given for the case of spinless neutral particles, but is easily extendable to all cases of charge and spin. The proof can also be generalized to the case of particles described by regularized products $$\int {\varphi (x_1 ,..., x_n ) \phi _1 } (x - x_1 ) ... \phi _n (x - x_n )dx_1 ...dx_n $$ ofWightman orJaffe fields.     

Harmonic Oscillator Trap and the Phase-Shift Approximation

The energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator (H.O.) trap is related to the free scattering phase-shifts \(\delta \) of the particles by a formula first published by Busch et al. It is here used to find an expression for the shift of the energy levels, caused by the interaction, rather than the perturbed spectrum itself. In the limit of high energy (large quantum number \(n\) of the H.O.) this shift (in H.O. units) is shown to be given by \(\Delta =-2\frac{\delta }{\pi }\) , also exact in the limit of infinite scattering length ( \(\delta =\pm \frac{\pi }{2}\) ) in which case \(\Delta =\mp 1\) . Numerical investigation shows that this expression otherwise differs from the exact result of Busch et al., by less than \(\frac{1}{2}\,\%\) except for \(n=0\) when it can be as large as \(\approx \) 2.5 %. This result for the energy-shift is well known from another exactly solvable model, namely that of two particles interacting in a spherical infinite square-well trap (or box) of radius \(R\) in the limit \(R\rightarrow \infty \) , and/or in the limit of large energy. It is in solid state physics referred to as Fumi’s theorem. It can be (and has been) used in (infinite) nuclear matter calculations to calculate the two-body effective interaction in situations where in-medium effects can be neglected. It is in this context referred to as the phase-shift approximation a term also used throughout this report.     

New recursive solution of the problem of scattering of electromagnetic radiation by multilayer spheroidal particles

A new recursive algorithm for the solution of the problem of scattering of light (of an arbitrarily polarized plane electromagnetic wave) by multilayer confocal spheroidal particles is constructed. This approach preserves the advantages of the two approaches proposed earlier by us for single-layer and two-layer spheroids (special choice of scalar potentials and utilization of the basis of wave spheroidal harmonics) and for homogeneous axially symmetric particles (formulation of the problem in terms of surface integral equations, calculation of the potentials inside the particle from the potentials of the incident radiation, and calculation of the potentials of the scattered radiation from the potentials inside the particle). In the case of multilayer particles, the potential inside each shell is a sum of two terms. The first has the properties of the incident radiation (no singularities inside the volume enclosed by the external boundary of the shell), whereas the second term has the properties of the scattered radiation (satisfies the radiation conditions at infinity). Therefore, as the calculation progresses from one layer to the next (from the core to the outer shell), the dimensionality of the reduced linear matrix equations for the unknown expansion coefficients of the scattered field potentials does not increase with respect to the case of a homogeneous spheroid. The algorithm is particularly simple and lucid (as far as possible for such a complex problem). In the case of spherical multilayer particles, the solution can be found explicitly.     

Lyapunov Exponents for Particles Advected in Compressible Random Velocity Fields at Small and Large Kubo Numbers

We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo numbers $\operatorname {Ku}$ (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to $\operatorname {Ku}\sim 1$ by resummation of a perturbation expansion in $\operatorname {Ku}$ . At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. The Lyapunov exponent is always negative. In two spatial dimensions the sign of the maximal Lyapunov exponent λ ₁ may change, depending upon the degree of compressibility of the flow and the Kubo number. For small Kubo numbers we compute the first four non-vanishing terms in the small- $\operatorname {Ku}$ expansion of the Lyapunov exponents. By resumming these expansions we obtain a precise estimate of the location of the path-coalescence transition (where λ ₁ changes sign) for Kubo numbers up to approximately $\operatorname{Ku} = 0.5$ . For large Kubo numbers we estimate the Lyapunov exponents for a partially compressible velocity field by assuming that the particles sample those stagnation points of the velocity field that have a negative real part of the maximal eigenvalue of the matrix of flow-velocity gradients.     

On the theory of galvanomagnetic properties of composites

The conductivity of composites in the presence of a magnetic field H is considered. The galvanomagnetic characteristics for a weakly inhomogeneous medium are determined in explicit form in an approximation quadratic in the deviations of conductivity tensor $\hat \sigma $ (r) from its mean value 〈 $\hat \sigma $ 〉. The contribution to the effective conductivity tensor $\hat \sigma _e $ linear in concentration c of inclusions for a composite with a small value of c is expressed in terms of the dipole polarizability of an individual inclusion, which is defined in the transformed system in which it is surrounded by an isotropic matrix with a scalar conductivity. Transition to this system is performed using a symmetry transformation that does not change the dc equations. An approximate approach proposed for describing the galvanomagnetic properties of composites in the wide range of parameters appearing in the problem generalizes the standard theory of an effective medium to the case of anisotropic systems with inclusions of arbitrary shape in field H ≠ 0.     

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ?³. The set of homogeneous isotropic connections forms a line l in ${\cal A}$ . We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding $l \hookrightarrow {\cal A}$ does not continuously extend to an embedding $\overline{l} \hookrightarrow \overline{\cal A}$ of the respective compactifications. Here, the Bohr compactification $\overline{l}$ corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and $\overline{\cal A}$ to that of loop quantum gravity. Analogous results are given for the anisotropic case.     

The spherical hierarchical model

The spherical version of Dyson's hierarchical model is analyzed. A particular case which is designed to simulate the long-range Ising problem is dealt with in detail. A phase transition is found with critical temperature $$\beta _c = \tfrac{1}{2}(2^\alpha - 2)(4 - 2^\alpha )^{ - 1} $$ wheren ^th neighbor spins interact with a strength ofn ^?α. Critical exponents are calculated for this particular case and are found to be identical with the critical exponents of the long-range spherical Ising model.     

Application of spheroidal functions in theory of light scattering by nonspherical particles: Separation of variables method

The separation of variables method (SVM), which uses a spheroidal basis, is proposed. According to this method, fields are presented in the form of expansion in terms of spheroidal functions. The previously conducted analysis of various methods using a spherical basis showed that the SVM is applicable in a broader area for numerical calculations, while the proposed approach using a spheroidal basis yields reliable results in the case of spheroids with a high degree of asphericity where other methods and approaches cannot be used. Importantly, the method includes an SVM that uses a spherical basis as the limiting case. Thus, the proposed method has all chances of being highly efficient for calculation of optical characteristics of various nonspherical particles in a wide range of parameters of the formulated problem.     

Can particle shape information be retrieved from light-scattering observations using spheroidal model particles?

We address the question if and how observations of scattered intensity and polarisation can be employed for retrieving particle shape information beyond a simple classification into spherical and nonspherical particles. To this end, we perform several numerical experiments, in which we attempt to retrieve shape information of complex particles with a simple nonspherical particle model based on homogeneous spheroids. The discrete dipole approximation is used to compute reference phase matrices for a cube, a Gaussian random sphere, and a porous oblate and prolate spheroid as a function of size parameter. Phase matrices for the model particles, homogeneous spheroids, are computed with the T-matrix method. By assuming that the refractive index and the size distribution is known, an optimal shape distribution of model particles is sought that best matches the reference phase matrix. Both the goodness of fit and the optimal shape distribution are analysed. It is found that the phase matrices of cubes and Gaussian random spheres are well reproduced by the spheroidal particle model, while the porous spheroids prove to be challenging. The “retrieved” shape distributions, however, do not correlate well with the shape of the target particle even when the phase matrix is closely reproduced. Rather, they tend to exaggerate the aspect ratio and always include multiple spheroids. A most likely explanation why spheroids succeed in mimicking phase matrices of more irregularly shaped particles, even if their shape distributions display little similarity to those of the target particles, is that by varying the spheroids’ aspect ratio one covers a large range of different phase matrices. This often makes it possible to find a shape distribution of spheroids that matches the phase matrix of more complex particles.     

Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle

The topics of confinement, average forces, and the Ehrenfest theorem are examined for a particle in one spatial dimension. Two specific cases are considered: (i) A free particle moving on the entire real line, which is then permanently confined to a line segment or ‘a box’ (this situation is achieved by taking the limit V ₀?→?∞ in a finite well potential). This case is called ‘a particle-in-an-infinite-square-well-potential’. (ii) A free particle that has always been moving inside a box (in this case, an external potential is not necessary to confine the particle, only boundary conditions). This case is called ‘a particle-in-a-box’. After developing some basic results for the problem of a particle in a finite square well potential, the limiting procedure that allows us to obtain the average force of the infinite square well potential from the finite well potential problem is re-examined in detail. A general expression is derived for the mean value of the external classical force operator for a particle-in-an-infinite-square-well-potential, $\hat{F}$ . After calculating similar general expressions for the mean value of the position ( $\hat{X}$ ) and momentum ( $\hat{P}$ ) operators, the Ehrenfest theorem for a particle-in-an-infinite-square-well-potential (i.e., $\mathrm{d}\langle\hat{X}\rangle/\mathrm{d}t=\langle\hat{P}\rangle/M$ and $\mathrm{d}\langle\hat{P}\rangle/\mathrm{d}t=\langle\hat{F}\rangle$ ) is proven. The formal time derivatives of the mean value of the position ( $\hat{x}$ ) and momentum ( $\hat{p}$ ) operators for a particle-in-a-box are re-introduced. It is verified that these derivatives present terms that are evaluated at the ends of the box. Specifically, for the wave functions satisfying the Dirichlet boundary condition, the results, $\mathrm{d}\langle\hat{x}\rangle/\mathrm{d}t=\langle\hat{p}\rangle/M$ and $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\mathrm{b.t.}+\langle\hat{f}\rangle$ , are obtained where b.t. denotes a boundary term and $\hat{f}$ is the external classical force operator for the particle-in-a-box. Thus, it appears that the expected Ehrenfest theorem is not entirely verified. However, by considering a normalized complex general state that is a combination of energy eigenstates to the Hamiltonian describing a particle-in-a-box with v(x)?=?0 ( $\Rightarrow\hat{f}=0$ ), the result that the b.t. is equal to the mean value of the external classical force operator for the particle-in-an-infinite-square-well-potential is obtained, i.e., $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t$ is equal to $\langle\hat{F}\rangle$ . Moreover, the b.t. is written as the mean value of a quantity that is called boundary quantum force, f _B. Thus, the Ehrenfest theorem for a particle-in-a-box can be completed with the formula $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\langle{{f_\mathrm{B}}}\rangle$ .