General three-particle scattering theory

A unified treatment of three-particle scattering theory with a three-body force in addition to the usual pair interactions is developed. The relationship of the generalized AGS and Faddeev formalisms to each other as well as distinct versions of each corresponding to the two most natural techniques for handling the three-body potential are established. It is found, just as in the case without the three-particle force, that the AGS formalism appears to be more practical for considering elastic and rearrangement scattering in two-body channels. On the other hand, for scattering amplitudes with at least one three-body channel (breakup and the 3-to-3) the Faddeev version of the theory is preferable. Other advantages of each formalism depending upon the treatment of the three-body interaction are noted.     

Perturbation theory in inverse scattering

Perturbation theory of the inverse problem is discussed for a certain class of S-matrices. The convergence of the perturbation series is proved for this class.     

Wave packet theory of Coulomb scattering

Potential scattering in a Coulomb field is treated in a wave packet formalism. The Rutherford law is derived from the off energy shell CoulombT-matrix avoiding the usual divergencies due to the long range Coulomb potential.     

Perturbation theory: Reduced scattering intensity for microemulsions

The reduced scattering intensity for a binary mixture of microemulsions has been calculated in terms of direct correlation functions introduced by Ornstein and Zernike in the Percus-Yevick (PY) approximation. Here square well and triangular well attractive tails are used as perturbations over the Percus-Yevick hard sphere model within the frame work of mean spherical model approximation. Good agreement is obtained between square well and triangular well calculations.     

Perturbation theory of scattering from irregular bodies

A perturbation solution is derived for the following problem: A time harmonic wave of amplitude ψ, propagating in a medium with wave number k, is incident on an irregular volume V, inside of which the propagation constant k′(r) can be an arbitrary function of | r |, where r is a position vector with origin inside V. The boundary conditions are that both ψ and its normal derivative ∂ψ/∂n may be discontinuous across the surface of V. Special cases of these conditions correspond to acoustic scattering, to B-wave scattering from a dielectric cylinder, or to the classical Dirichlet (ψ = 0) or Neumann (∂ψ/∂n = 0) surface conditions. An integral equation is derived that satisfies the appropriate differential equations both outside and inside the body, and satisfies the boundary conditions as well. This equation is reduced to a set of linear algebraic equations by expansion in a certain basis set and these linear equations are then solved in a perturbation approximation for the case that the surface of the body differs from a sphere or cylinder by a small parameter λ. Comparison is made with formulae in the literature, and except for some minor discrepancies, which are here corrected, there is general agreement.     

Quantal theory of Coulomb absorption in heavy-ion scattering

The component in the heavy-ion optical potential due to the Coulomb coupling to inelastic channels has been calculated using the on-energy-shell approximation for the intermediate-channel Green's functions. Closed expressions were derived for the Coulomb polarization potential representing coupling to all orders in the K = 0 rotational band. As a test of this general aproach for coupling to higher states, elastic-scattering calculations were performed with a truncated expression which included reorientation in the 2⁺ state and coupling to the 4⁺ state to all orders. Comparison with coupled-channels calculations indicate the increasing importance of off-shell effects with increasing coupling strength. An analytical estimate of offshell effects is presented. Limits on the range of validity of the optical-potential approach are determined.     

Cluster properties for Coulomb scattering

Space-like, time-like and momentum space cluster properties are examined for N-particle scattering via two-body Coulomb-like potentials.     

Perturbation theory for kinds

In this paper we prove the validity of formal asymptotic results on perturbation theory for kind solutions of the sine-Gordon equation, originally obtained by McLaughlin and Scott. We prove that for appropriate perturbations, of size in an appropriate norm, slowly varying in time in the rest frame of the kink, the shape of the kink is unaltered in theL norm toO() for a time ofO(1/). The kink parameters, which represent its velocity and centre, evolve slowly in time in the way predicted by the asymptotics. The method of proof uses an orthogonal decomposition of the solution into an oscillatory part and a one-dimensional zero-mode term. The slow evolution of the kink parameters is chosen so as to suppress secular evolution of the zero-mode.Partially supported as a graduate student at Princeton University of NSF grant 215 6211     

Bound-state solution in momentum space of three-particle equations with Coulomb interaction

We solve the bound-state Faddeev equations in the momentum representation for a system of three identical bosons interacting through Yamaguchi forces. Two of the particles are then given an electric charge. The choice of interaction parameters is inspired by the trinucleon systems. In this model we are able to compute the Coulomb energyE _c of³He without further numerical or analytical approximations by solving the corresponding equations of Veselova and Alt, Sandhas, and Ziegelmann. We then are able to test commonly made approximations: We find that all three possible types of diagrams involving the CoulombT matrixT _c contribute substantially. ReplacingT _c byV _c induces an error of a few percent inE _c , and simplifies the numerical computations by orders of magnitude. ReplacingV _c by itsS-wave projection induces only a small error.Presented at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.We are grateful to Drs. D. J. Struik and J. E. Holwerda for carrying out part of the numerical computations. This work is part of the project The Coulomb Potential in Quantum Mechanics and Related Topics.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

Perturbation theory for bending potentials

An efficient perturbative method is developed to facilitate the treatment of the anharmonicity of bending degrees of freedom. The Rosen-Morse and sec² potentials are transformed so that perturbation theory may be applied easily. For a model problem it is found that the difference between the energy levels obtained from perturbation theory and the exact energy levels is less than 0·1 cm^-1 for the first six energy levels.     

Perturbation theory for velocity-dependent potentials

In the presence of a velocity-dependent Kisslinger potential, the partial-wave, time-independent Schr?dinger equation with real boundary conditions is written as an equation for the probability density. The changes in the bound-state energy eigenvalues due to the addition of small perturbations in the local as well as the Kisslinger potentials are determined up to second order in the perturbation. These changes are determined purely in terms of the unperturbed probability density, the perturbing local potential, as well as the Kisslinger perturbing potential and its gradient. The dependence on the gradient of the Kisslinger potential stresses the importance of a diffuse edge in nuclei. Two explicit examples are presented to examine the validity of the perturbation formulas. The first assumes each of the local and velocity-dependent parts of the potential to be a finite square well. In the second example, the velocity-dependent potential takes the form of a harmonic oscillator. In both cases the energy eigenvalues are determined exactly and then by using perturbation theory. The agreement between the exact energy eigenvalues and those obtained by perturbation theory is very satisfactory. Received: 24 May 2002 / Accepted: 15 July 2002 / Published online: 3 December 2002 RID="a" ID="a"e-mail: mij@hu.edu.jo Communicated by V. Vento     

The complex coordinate method for three-particle scattering above the break-up threshold

The complex coordinate method is used to evaluate the elastic scattering above the break-up threshold in a model three-body system. The results are compared with previous estimates.     

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Abstract

Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering caused by the surface plasmon polariton mechanism. Theoretical results are presented for silver surfaces at optical wavelengths.     

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering caused by the surface plasmon polariton mechanism. Theoretical results are presented for silver surfaces at optical wavelengths.     

Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

The Hartree-Fock theory for Coulomb systems

For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.Research partially supported by U.S. National Science Foundation Grant MCS-75-21684Research partially supported by U.S. National Science Foundation under Grants MPS-75-11864 and MPS-75-20638. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, NJ08540, USA     

Off-shell phenomena in Coulomb scattering

In the framework of off-shell quantum electrodynamics — the quantum field theory of a covariant symplectic mechanics, in which events evolve according to a Poincaré-invariant parameter τ — we study the low energy scattering of identical scalar particles. It is shown that exchange of mass is permitted in the formalism, and we calculate scattering cross sections for this case. In these cross sections, the usual forward pole of standard scalar QED splits into two poles and a zero, slightly offset from the forward direction. As mass exchange vanishes, a pole-zero pair cancel, the remaining pole moves to θ = 0, and the standard cross section is recovered.     

Coulomb effects in quasi-free scattering

Using the exact three-body scattering theory for Coulomb-like potentials a new result for the quasi-free breakup amplitude is obtained. It consists in adding an easily calculable factor to the existing expression for the modulus of the quasi-free breakup amplitude. The importance of this factor is discussed by illustrative examples for the breakup reactions.     

Simplified universal screened Coulomb potential for ion scattering

Coulomb's interaction of ions is complicated by the presence of electronic shells. A major variety of more or less complex screening functions has been proposed over a time span of about 80 years. A simple universal screening function with only one fitting parameter is presented here.