Scattering of a slow quantum particle on an axially symmetric short-range potential

A linear version of the variable-phase approach in potential-scattering theory is supplemented with a new asymptotic method. This method is intended for analyzing quantum mechanically and for constructing explicit low-energy approximations for partial-wave phase shifts, amplitudes, cross sections, and radial components of the wave function for the scattering of a quantum particle on an axially symmetric short-range potential. The procedure used to construct all low-energy approximations reduces to solving a recursion chain of energy-independent sets of equations, each such set consisting of two linear first-order differential equations.     

Computational improvements in phase shift calculations of elastic electron scattering

A method of calculating the scattering phase shifts from the Dirac radial equations containing a rather general potential is described. The method takes advantage of the fact that a system of linear differential equations can be solved with the help of power series. Application of the method is demonstrated for the case of low-energy scattering (uniformly distributed nuclear charge screened by the atomic electrons) and for the case of high-energy scattering (uniform or more complicated nuclear charge distribution without atomic screening).     

Variational approximations in a path integral description of potential scattering

Using a recent path integral representation for the T -matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables --over which one has to integrate functionally-- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading). We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.     

Wilsonian Renormalization Group and the Lippmann-Schwinger Equation with a Multitude of Cutoff Parameters

The Wilsonian renormalization group approach to the Lippmann-Schwinger equation with a multitude of cutoff parameters is introduced.A system of integro-differential equations for the cutoff-dependent potential is obtained.As an illustration,a perturbative solution of these equations with two cutoff parameters for a simple case of an S-wave low-energy potential in the form of a Taylor series in momenta is obtained.The relevance of the obtained results for the effective field theory approach to nucleon-nucleon scattering is discussed.     

Scattering and active acoustic control from a submerged spherical shell

This paper is concerned with the scattering from a submerged (heavy fluid) bilaminate spherical shell composed of an outer layer of steel, and an inner layer of radially polarized piezoelectric material. The methodology used includes separation formulas for the stresses and displacements, which in turn are used (coupled with spherical harmonics) to reduce the governing equations to linear systems of ordinary differential equations. This technique uses the full equations of elasticity rather than any of the various thin-shell approximations in determining the axisymmetric scattering from a shell, normal modes of vibration for the shell, as well as voltages necessary for annihilation of a scattered pressure due to insonification of the shell by an incident plane wave.     

Elastic scattering of electrons by europium and ytterbium atoms

The method of relativistic optical potential is applied to studying elastic scattering of electrons by europium and ytterbium atoms in a wide range of collision energies up to 2 keV. The angular dependences of the scattering differential cross sections and the energy dependences of the scattering integral (total, elastic, momentum transfer, and viscosity) cross sections are calculated in both spin-polarized and spin-unpolarized approximations. It is shown that the spin-polarized approximation should be used to calculate the scattering cross sections at energies below 10 eV for a europium atom. The low-energy scattering of an electron by a europium atom is characterized by P-, D-, and F-wave shape resonances. For an ytterbium atom, the calculated cross sections are in good agreement with available experimental data and with those obtained by calculation in terms of the relativistic convergent close-coupling method.     

Microscopic calculation of low-energy deuteron-deuteron scattering on the basis of the cluster-reduction method

The cluster-reduction method is used to solve numerically the differential equations for the s-wave Yakubovsky components characterizing the nnpp system in the S=2 spin state. Elastic deuteron-deuteron scattering is analyzed for the case where nucleon-nucleon interaction is simulated by the MT I–III potentials. Effective equations describing the relative motion of clusters is derived. The scattering length and phase shifts for low-energy deuteron-deuteron scattering are calculated.     

Scattering phase shift in stochastic fields

A very simple model of a spherically symmetric potential likeμ/r, withμ varying randomly alongr, the distance from the scattering centre is analysed and it is found that ifμ changes its sign in a stochastic fashion, the average potential felt at each pointr consists of a coulomb part as well as a short range Yukawa type component. Using the non linear equations for the phase function for potential scattering of a Schrödinger particle, the average and other moments of the phase shifts can be computed in certain approximations.     

Rigorous and approximate methods for modeling wave scattering from a locally perturbed perfectly conducting surface

Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.     

P-Matrix approach and parameters of low-energy scattering in the presence of a Coulomb potential

The scattering of two charged strongly interacting particles is described on the basis of the P-matrix approach. In the P matrix, it is proposed to isolate explicitly the background term corresponding to purely Coulomb interaction, whereby it becomes possible to improve convergence of the expansions used and to obtain a correct asymptotic behavior of observables at high energies. The expressions for the purely Coulomb background P matrix, its poles and residues, and purely Coulomb eigenfunctions in the P-matrix approach are obtained. The nuclear-Coulomb parameters of the low-energy scattering of two charged hadrons are investigated on the basis of this approach combined with the method for isolating the background P matrix. Simple explicit expressions for the nuclear-Coulomb scattering length and effective range in terms of the residual P matrix are derived. For models of short-range strong interaction, these expressions give a general form of nuclear-Coulomb parameters for low-energy scattering. Specific applications of the general expressions derived in this study are exemplified by considering, on the basis of these expressions, some exactly solvable models of strong interaction, including the hard-core model, and, for these models, the nuclear-Coulomb parameters for low-energy scattering at arbitrary values of the orbital angular momentum are found explicitly for the first time. In particular, the nuclear-Coulomb scattering length and effective range are obtained explicitly for the boundary-condition model, the model of a hard-core delta-shell potential, the Margenau model, and the model of square-well hard-core potential.     

Antisymmetrisation effects in composite particle scattering

Several approximations for the determination of the real part of the heavy-ion optical potential have been studied. The results of these approximations for α-α scattering have been compared with the equivalent local potential determined by an inverse scattering method from a fully antisymmetrised “exact” calculation and some general conclusions are given.     

Approximate solution of bound state problems through continued fractions

A method to solve ordinary linear differential equations through continued fractions is applied to several physical systems. In particular, results for the Schrödinger equation give a good accuracy for the eigenvalues of bound states in theS-wave Yukawa potential, and the lowest order approximations provide exact-values for the harmonic oscillator and Coulomb potential eigenvalues and eigenfunctions.     

Linear photoelectron diffraction: application of a rapid approximation for surface structural studies

The linear superposition approximation proposed by Wander, Pendry and Van Hove for efficient low-energy electron diffraction calculations (“linear LEED”) is applied to photoelectron diffraction (“linear PD”). As with linear LEED, linear PD works very well for calculating the effect of displaced atoms. However, due to strong forward scattering at higher energies, linear PD requires that atoms do not move into or out of alignment. This limitation can be removed by suitable simple adjustments to the basic approximation, promising to make the method effective for structural searches of complex surfaces.     

The manifold of solutions of Roy's S- and P-wave equations for pion-pion scattering: The neighbourhood of the physical amplitudes

We study the solutions of the coupled S- and P-wave Roy equations for low-energy pion-pion scattering which are in the neighbourhood of a given solution. We give a general method for constructing these solutions for fixed inelasticities and driving terms and variable S-wave scattering lengths. In the case of the neighbourhood of the physical amplitudes, our procedure leads to a seven-dimensional manifold of solutions. If we omit variations affecting only the I = 0 S-wave around the $\text{K}\text{K}$ threshold, the number of dimensions is reduced to five, in accordance with the results of the phenomenological analysis of low-energy pion-pion scattering. Our solutions allow a study of the correlations between S- and P-waves implied by the Roy equations. This work completes a previous one dealing with the single-channel problem of the I = 1 P-wave.     

A novel method for solving the three-body scattering problem

A numerical scheme for solving a three-body scattering problem within the framework of the configuration space Faddeev equations in three-dimension, i.e., without resort to explicit partial wave expansion, is presented. The method is applied to calculate the low-energy n-d observables.     

Multiple scattering, optical potential, and N-body approaches to elastic two-fragment collisions

Two-fragment elastic scattering problems are often studied using multiple scattering theories such as those due to Watson along with Feshbach-type optical potentials. These conventional methods are re-examined, rephased, and generalized using the language and techniques of contemporary N-particle scattering theory. A special realization of the latter theory is developed which is especially useful for relating the older and newer methods. This is facilitated by maintaining the same off-shell continuations of the scattering operators in both approaches. In particular, a set of connected-kernel scattering integral equations is introduced which provides a consistent N-particle framework for the calculation of that definition of the optical potential possessing the Feshbach off-shell continuation. These equations exhibit a multiple-scattering substructure and therefore allow the systematic generalization of some of the usual low-order approximations.     

A new path-integral representation of the T-matrix in potential scattering

We employ the method used by Barbashov and collaborators in Quantum Field Theory to derive a path-integral representation of the T-matrix in nonrelativistic potential scattering which is free of functional integration over fictitious variables as was necessary before. The resulting expression serves as a starting point for a variational approximation applied to high-energy scattering from a Gaussian potential. Good agreement with exact partial-wave calculations is found even at large scattering angles. A novel path-integral representation of the scattering length is obtained in the low-energy limit.