Electrostatic potential: A new approach for some mixed boundary value problems

A new approach to the solution of problems of electrostatics, some of them with mixed boundary conditions, is presented. The proposed scheme can be used in cases were we have a formal solution in the form of a series in Legendre polynomials and the boundary or matching conditions are given not on the whole interval (0, π) of the polar variable, θ, but only over the interval (0, π/2) or (π/2, π). Truncation of the series after the Nth term and the projection on the subspace generated by the set of the first N even (or odd) Legendre polynomials allows us to determine the unknown coefficients of the approximate solution. The results show rapid convergence toward the exact values as we increase the number of terms, N, included in the approximate solutions. The procedure allows to solve approximately some problems whose exact solutions, we believe, are not yet known.     

Gaussian estimation of electron scattering cross sections on the basis of a Feynman path integral

A version of Gaussian estimation of a Feynman path integral is considered and its validity for a scattering problem is investigated. Test calculations of the differential and total cross sections are performed for the scattering of a plane wave by a three-dimensional spherically symmetric Gaussian potential and for the electron impact excitation of the 1s → 2s transition in the hydrogen atom. The data on the scattering by a potential are compared with the analogous results obtained using the first Born approximation and the method of phase functions (which gives almost exact results). The excitation cross sections for the transition in the hydrogen atom are compared with those obtained by the convergent close-coupling method. The validity of this approach is demonstrated. The accuracy of the method proposed is acceptable for many cases. For total cross sections, the result obtained in terms of the density matrix formalism turned out to be more exact than that derived from differential cross sections. Among all the above approaches using path integration to solve problems of scattering of electrons by atoms, the method proposed here.     

On the theory of multiple scattering

In this paper a kinetic equation is derived for the distribution function in the variable q=2 sin(ϑ/2) for the case of a scattering cross section of general form under the assumption that the region of multiple scattering (the diffusion region) is small. The limits of the kinetic equation are discussed, with no restrictions imposed on the scattering angles. It is found that the equation has a solution in the form of an integral. Finally, it is established that the solution is applicable over the entire range of angles, from 0 to 180°. Zh. éksp. Teor. Fiz. 116, 418–435 (August 1999)     

Two-photon bremsstrahlung processes in atoms: Polarization effects and analytic results for the Coulomb potential

The partial wave analysis of two-photon free-free (bremsstrahlung) electron transition cross sections during scattering by a static potential U(r), as well as by an atom with a nonzero angular momentum, is carried out. The dipole interaction with radiation is taken into account in the second order of perturbation theory for the general case of elliptic polarization of photons. The polarization and angular dependences of the two-photon potential scattering amplitude is presented as a combination of the scalar product of electron momenta and photon polarization vectors and five atomic parameters containing Legendre polynomials of the scattering angle as well as radial matrix elements depending on the initial (E) and final (E′) electron energies. The results are applicable both for spontaneous double bremsstrahlung at nonrelativistic energies and for induced absorption and emission in the field of a light wave. Specific polarization effects (circular and elliptic dichroism) are analyzed for two-photon bremsstrahlung processes associated with the interference of the Hermite and anti-Hermite parts of the amplitude and depending on the sign of photon helicity. The limiting cases of high and low photon frequencies are investigated analytically, and the asymptotic forms of radial matrix elements and amplitudes for the general form of the U(r) potential are determined. Closed analytic expressions are derived for the radial matrix elements of the Coulomb potential in the form of integrals of hypergeometric function, and singularities are singled out in explicit form for E′ → E. The methods of approximate calculation of the radial matrix elements are discussed, and the results of their exact numerical calculation, as well as angular distributions and the cross sections of induced one-and two-photon emission and absorption, are given for the case of the Coulomb potential. The numerical results show that dichroism effects are quite accessible for experimental observations.     

Analytically approximate screened calculations of atomic-field pair production cross sections

A new method is described to obtain analytically approximate screened cross sections of atomic-field pair production. The Thomas-Fermi-Csavinszky potential model is expanded at the first order and put in the place of the point Coulomb potential in the Dirac equation. That method can be very useful to calculate approximate screened cross sections for the intermediate photon energy range (5m_oc² to about 50m_oc²) where numerically exact screened cross sections are needing a prohibitive computer time and when the form factor approach based on Born approximation is not always valid.     

Numerical and asymptotic scattering from two inhomogeneous lenses

The scattering of a time-harmonic, plane electromagnetic wave by either a Maxwell fish-eye or a metal-like lens is considered. Numerical values of the monostatic cross sections are obtained by evaluating the exact, infinite series solutions of hypergeometric functions of the₂ F ₁ type. These results are compared with the high-frequency asymptotic estimates based on either geometrical optics or a modified Watson transformation.     

Study of the time correlation function of scattered light in bounded strongly inhomogeneous media

The time correlation function of the multiply scattered light by a medium comprised of finite-size scatterers is studied theoretically and experimentally. A solution of the Bethe-Salpeter equation for the time correlation function is sought in the P ₂-approximation in the form of a series of terms of the Legendre polynomials. With allowance for the boundedness of the medium, the problem is reduced to a generalized Milne equation, which is solved by the Wiener-Hopf method. The time dependence of the correlation function is studied experimentally in a concentrated latex suspension for particles of different sizes. The results of numerical calculations are in qualitative agreement with the measured dependence of the time correlation function on the scatterer size.     

Diffuse reflectance of TiO2 pigmented paints: Spectral dependence of the average pathlength parameter and the forward scattering ratio

Diffuse reflectance spectra of paint coatings with different pigment concentrations, normally illuminated with unpolarized radiation, have been measured. A four-flux radiative transfer approach is used to model the diffuse reflectance of TiO₂ (rutile) pigmented coatings through the solar spectral range. The spectral dependence of the average pathlength parameter and of the forward scattering ratio for diffuse radiation, are explicitly incorporated into this four-flux model from two novel approximations. The size distribution of the pigments has been taken into account to obtain the averages of the four-flux parameters: scattering and absorption cross sections, forward scattering ratios for collimated and isotropic diffuse radiation, and coefficients involved in the expansion of the single particle phase function in terms of Legendre polynomials.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

7Li(α,n)10B differential cross-section measurements from threshold to Eα = 5.1 MeV

Differential cross sections for the ⁷Li(α, n)¹⁰B reaction have been measured at lab angles of 0°, 20°, 31°, 50°, 60°, 70°, 80°, 90°, 100° and 114° for α-particle energies between 4.385 and 5.1 MeV. A thick natural lithium target was bombarded with a 5.2 MeV, nanosecond-pulsed ⁴He⁺ beam and neutron velocity spectra at each angle were measured by time-of-flight techniques. These data have been converted to cross sections at 10 keV intervals in α-particle energy. Angular distributions have been fitted with a series of Legendre polynomials. Angle-integrated cross sections, the 0° excitation function, and angular distributions are compared to past measurements and R-matrix calculations.     

Application of jacobi polynomial methods to the one-speed transport equation

The transport equations associated with radiation damage studies are often solved using expansions in Legendre polynomials. The radiation damage distribution functions which satisfy these equations may be sharply peaked in the forward direction, while the Legendre polynomials, as a set, are isotropic. This situation requires the use of many terms in the Legendre expansion in order to adequately represent the distribution functions. The Jacobi polynomials, on the other hand, can have strong peaking built into their associated weight function. To test the usefulness of the Jacobi polynomials we use them to solve the simple, one-speed, neutron transport equation. The results are then compared to the exact theory and to the results of applying Legendre methods to the same problem. This sample calculation demonstrates the advantage of the Jacobi polynomials in strongly non-isotropic situations.     

Folding model analysis of pion elastic and inelastic scattering from 6Li and 12C

π ^±-Nucleus scattering cross sections are calculated applying the Watanabe superposition model with a phenomenological Woods-Saxon potential. The phenomenological potential parameters are searched for π ^± scattering from ⁶Li and ¹²C to reproduce not only differential elastic cross sections but also inelastic and total and reaction cross sections at pion kinetic energies from 50 to 672 MeV. The optical potentials of ⁶Li and ¹²C are calculated in terms of the alpha particle and deuteron optical potentials. Inelastic scattering has been analyzed using the distorted waves from elastic-scattering data. The values of deformation lengths thus obtained compare very well with the ones reported earlier.     

Polarization effects on heavy-ion scattering in the distorted-wave method

On the basis of an exact formalism for DWBA methods we calculate the distorting potentials to be used in a standard distorted-wave Born approximation for systems with strongly coupled channels. We examine for a practically useful model some simple approximate treatments through comparisons of the polarization potentials in the case of elastic and inelastic ¹⁶O + ¹⁶O scattering. The adiabatic approximation omitting the radial kinetic energy in the propagator is found to lead to satisfactory agreement with the exact coupled-channel cross sections.     

Summation of partial waves for large-angle scattering

For large-angle elastic scattering different methods of summing partial wave amplitudes are investigated for their accuracy and simplicity of computation. It is found that among the approximations considered, the method of expanding the T-matrix in terms of the weighted orthogonal polynomials proposed by Brysk is the most accurate way of calculating the scattering amplitude in the backward direction. If the two-particle interaction is assumed to be a Yukawa potential, then the lth partial sum of the T-matrix with the weighted polynomials can be expressed as the lth partial sum with the Legendre polynomials and a correction term which depends on the phase shift for the lth partial wave.     

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Dirac-delta function approximations to the scattering phase function

Dirac-delta function approximations are used to represent the single scattering phase function of large spherical particles or voids. The phase function for a spherical particle or void can be represented by a series of Legendre polynomials; however, as the diameter is increased, forward scattering becomes dominant and the number of terms in the series becomes very large. A Dirac-delta function approximation consists of a Dirac-delta function in the forward direction plus a finite series of Legendre polynomials. The Dirac-delta function accounts for strong forward scattering. Particular attention is given to large ice spheres and spherical voids in ice. The Dirac-delta function is shown effective in reducing the number of terms needed to describe the phase function.     

The (γ, p0) reactions in the giant dipole resonance region of 51V and 52Cr nuclei

The differential cross sections at 90° for the ⁵¹V(e, p₀)⁵⁰Ti and ⁵²Cr(e, p₀ + p₁)⁵¹V reactions have been measured over the giant dipole resonance region. These cross sections were used to obtain the differential cross sections of the ⁵¹V(γ, p₀)⁵⁰Ti and ⁵²Cr(γ, p₀ + p₁)⁵¹V reactions. The results show two peaks that appear at the same energies as the main peaks of the (γ, n) and (γ, p) cross section for both nuclei. The angular distributions of protons from the (e, p) reaction have also been measured at several points of the incident electron energy. The coefficients A₂ obtained by fitting with a series of Legendre polynomials, W(θ) = 1 + A₁P₁(cos θ)+A₂P₂(cos θ), varies with excitation energy. These results are discussed in terms of the direct-semidirect process considering isospin effects in the giant dipole resonance.     

Approximate scattering solutions of the dirac equation for electron-nucleus processes in light nuclei

Scattering solutions of the second-order Dirac equation for the case of the Coulomb potential and which are correct to first order in the coupling constantZe ²/hc are investigated and found to describe pure Coulomb scattering equally well as the Sommerfeld-Maue wave functions. Errors introduced by the use of these solutions are studied in a numerical calculation of cross sections for nuclear electric-quadrupole excitation by high-energy electrons. The use of these wave functions is suggested for simplified calculations of lowest-order Coulomb corrections to Born approximation results for various electron-nucleus processes.