Zur elektronischen Struktur ungeordneter eindimensionaler Systeme. I. Streuformulierung

The general scattering properties of an one-dimensional disordered system of arbitrarily distributed non-oberlapping single potentials are investigated by means of the phase theory. The scattering matrix of the single potentials is characterized by three energy dependent parameters. The application to periodic structures is especially directed to the calculation of forbidden energy gaps and yields important information on the energy dependence of the phase transfer function. In a further paper the results will be generalized on disordered systems.     

Zur elektronischen Struktur ungeordneter eindimensionaler Systeme. II. Verlauf der Streuparameter als Funktion der Energie

Statements on the energy dependence of the three scattering parameters are made for general energy dependent and non-local potentials used in a preceding paper [1] in a one-dimensional model for the electronic structure of disordered systems. All results are also valid in the relativistic case. The possible types for the behaviour of the wave function within the gaps and the form of the scattering parameters for δ-potentials are discussed.     

On the nucleon–nucleon scattering phase shifts through supersymmetry and factorization

By exploiting the supersymmetry-inspired factorization method through a judicious use of deuteron ground state wave function, higher partial wave nucleon–nucleon potentials, both energy independent and energy dependent, are generated. We adopt the phase function method to deal with the scattering phase shifts and demonstrate the usefulness of our constructed potentials by means of model calculation.     

A class of exactly solvable potentials. I. One-dimensional Schrödinger equation

The one-dimensional Schrödinger equation is solved for a new class of potentials with varying depths and shapes. The energy eigenvalues are given in algebraic form as a function of the depth and shape of the potential. The eigenfunctions and scattering function are also given in closed form. For certain shapes these potentials resemble the mean field of an atomic nucleus.     

Application of the incoming wave boundary condition to16O+16O and12C+12C elastic scattering

The elastic scattering of¹²C+¹²C and¹⁶O+¹⁶O has been studied in the framework of an incoming wave boundary condition model. Different logarithmic derivatives for the incoming waves have been tested and their effects investigated with the help of Fourier analyses. It is shown that a logarithmic derivative obtained from a JWKB approximation leads to strong absorption of the low partial waves while a logarithmic derivative constant for all partial waves causes reflections. These reflections are necessary to describe the high energy elastic scattering of¹²C+¹²C. The fits thus obtained with shallow real potentials are comparable to those obtained with deep folding potentials. It is shown that not the lowest partial waves, but those within a window just below the grazing angular momentum are most important for the higher energy¹²C+¹²C angular distributions.     

New approaches to the study of properties of the generalized WKB approximation. IV

The problem of evaluating partial scattering phases of particles by spherically symmetric potentials is treated by the new formulation of the generalized WKB method. The error committed in calculating scattering phases in the zeroth approximation is estimated.     

Positron scattering from hydrogen atom embedded in weakly-coupled plasmas

Elastic scattering of positrons from the hydrogen atoms in weakly-coupled plasmas has been studied using an expression for partial wave scattering amplitude that has been derived within the framework second order distorted wave Born approximation. The interactions among the charged particles in the plasma have been represented by Debye-Hückel potentials. A detailed study has been made on differential and total cross sections in the energy range 20–300 eV. To the best of our knowledge such a study on the differential and total cross sections for elastic positron-hydrogen collisions in a weakly-coupled plasma environment is reported for the first time in the literature.     

Nucleus-nucleus scattering in the high-energy approximation and optical folding potential

A microscopic complex folding-model potential that reproduces the scattering amplitude of Glauber-Sitenko theory in its optical limit is obtained. The real and imaginary parts of this potential are dependent on energy and are determined by known data on the nuclear-density distributions and on the nucleon-nucleon scattering amplitude. For the real part, use is also made of a folding potential involing effective nucleon-nucleon forces and allowing for the nucleon-exchange term. Three forms of semimicroscopic optical potentials where the contributions of the template potentials—that is, the real and the imaginary folding-model potential—are controlled by adjusting two parameters are constructed on this basis. The efficiency of these microscopic and semimicroscopic potentials is tested by means of a comparison with the experimental differential cross sections for the elastic scattering of heavy ions ¹⁶O on nuclei at an energy of E ～ 100 MeV per nucleon.     

Configuration-Space Faddeev Calculation for Proton�CDeuteron Elastic Scattering Observables

A new computational method for solving the nucleon?Cdeuteron breakup scattering problem has been applied to study the elastic nucleon?Cdeuteron scattering on the basis of the configuration-space Faddeev?CNoyes?CNoble?CMerkuriev equations. The Merkuriev?CGignoux?CLaverne approach has been generalized for arbitrary nucleon?Cnucleon potentials and with an arbitrary number of partial waves. The nucleon?Cdeuteron observables at the incident nucleon energy 3?MeV have been calculated using the charge-independent AV14 nucleon?Cnucleon potential including the Coulomb force for the proton?Cdeuteron scattering. Results have been compared with those of other authors and with experimental proton?Cdeuteron scattering data.     

Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Eikonal expansion in electron scattering: I. Elastic scattering

A systematic eikonal expansion for the scattering of high-energy electrons from nuclei is derived which starts from the iterated Dirac equation. The resulting scattering amplitude is written in an impact parameter representation depending on eikonal phases which are proportional to inverse powers of the energy. The first two correction terms to the leading Glauber-Baker amplitude are calculated. For a Coulomb potential they agree with a sinθ-expansion of the relativistic Coulomb scattering amplitude. In the case of scattering from an extended charge distribution at sufficiently high energies numerical partial wave calculations are accurately reproduced.     

Path integral treatment for the generalized Ginocchio potentials

The solution for the generalized Ginocchio potentials is exactly constructed by the path integral approach combined to the summation of the spectral representation. The energy spectrum and the wave functions of bound and scattering states are explicitly evaluated. The scattering functionS _l for each angular momentuml is also deduced.     

Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials

For the two-dimensional Schrödinger equation $$[ - \Delta + v(x)]\psi = E\psi , x \in \mathbb{R}^2 , E = E_{fixed} > 0 (*)$$ at a fixed positive energy with a fast decaying at infinity potentialv(x) dispersion relations on the scattering data are given. Under “small norm” assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization of scattering data for the potentials from the Schwartz classS=C _∞ ^(∞) (?²). For the potentials with zero scattering amplitude at a fixed energyE _fixed (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parametrized by a function of one variable) of two-dimensional spherically-symmetric real potentials from the Schwartz classS transparent at a given energy. For the two-dimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing, at infinity, potentials transparent at a fixed energy. For any dimension greater or equal to 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdV-type equations in dimension 2+1 related with the scattering problem (*) (the Novikov-Veselov equations) do not preserve, in general, these dispersion relations starting from the second one. As a corollary these equations do not preserve, in general, the decay rate faster than |x|^?3 for initial data from the Schwartz class.     

The energy dependence of 6Li optical potentials

The energy dependence of ⁶Li optical potentials for elastic scattering from ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr and ²⁰⁸Pb targets has been investigated by simultaneously fitting several data sets over a wide energy range. Both Saxon-Woods and double-folded real potentials have been used. In general, there is no requirement for energy dependence in either the real or imaginary potential for any of these targets, whether Saxon-Woods or folded real potentials are used. The main exception is ²⁸Si when energy dependence is required to simultaneously fit low-energy data and high-energy rainbow scattering with folded potentials. The degree to which the potentials are determined is discussed.     

Improved theoretical pion-nucleus optical potentials

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC¹² scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π^?C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.     

Reaction matrix for nonlocal potentials II. Effective interaction fora=18 nuclei

The effective interaction of two valence nucleons in s-d shell is calculated for the Tabakin and the Mongan separable potentials. The resulting matrix elements are compared in detail with similar quantities obtained for local potentials. The differences observed are explained by various on energy shell behaviour of the respective free scattering amplitudes. Also the effect of various choices of the Pauli operator is studied and the resulting spectra of the¹⁸O an¹⁸F nuclei are compared.     

Real part of the optical potential for composite particles

The optical potential for a composite particle is most simply approximated by the sum of the optical potentials of the constituent nucleons. Restricting ourselves to the real parts of the potentials we use this model as a first approximation in a calculation of the potentials for d, ³He, α and ¹²C. We add corrections for (i) the energy dependence of the nucleon potentials, (ii) three-body terms, (iii) the Pauli principle. All corrections can be important and that for the Pauli principle can be very large. We obtain a good explanation of the following phenomena: (a) the deuteron potential is nearly the sum of the neutron and proton potentials, (b) the potential for ³He is about 20 % less than the sum of the potentials of the nucleons in the ³He projectile, (c) the volume integral of the potential for ³He falls at both high and low energies in the energy range 20–100 MeV, (d) shallow potentials with large radii are found for low energy (30 MeV) scattering of α-particles, (e) deeper potentials are found for higher energy α-particle scattering. We predict shallow potentials for ¹²C scattering from light targets but deeper potentials for heavier targets.     

Classical theory of atom–surface scattering: The rainbow effect

The scattering of heavy atoms and molecules from surfaces is oftentimes dominated by classical mechanics. A large body of experiments have gathered data on the angular distributions of the scattered species, their energy loss distribution, sticking probability, dependence on surface temperature and more. For many years these phenomena have been considered theoretically in the framework of the “washboard model” in which the interaction of the incident particle with the surface is described in terms of hard wall potentials. Although this class of models has helped in elucidating some of the features it left open many questions such as: true potentials are clearly not hard wall potentials, it does not provide a realistic framework for phonon scattering, and it cannot explain the incident angle and incident energy dependence of rainbow scattering, nor can it provide a consistent theory for sticking. In recent years we have been developing a classical perturbation theory approach which has provided new insight into the dynamics of atom–surface scattering. The theory includes both surface corrugation as well as interaction with surface phonons in terms of harmonic baths which are linearly coupled to the system coordinates. This model has been successful in elucidating many new features of rainbow scattering in terms of frictions and bath fluctuations or noise. It has also given new insight into the origins of asymmetry in atomic scattering from surfaces. New phenomena deduced from the theory include friction induced rainbows, energy loss rainbows, a theory of super-rainbows, and more. In this review we present the classical theory of atom–surface scattering as well as extensions and implications for semiclassical scattering and the further development of a quantum theory of surface scattering. Special emphasis is given to the inversion of scattering data into information on the particle–surface interactions.     

An efficient method for the summation of partial wave amplitudes for long-range potentials

Using the concept of Abel summation we show that it is possible to obtain the scattering amplitude for Coulomb-type potentials for any value of the scattering angle (except forward direction), although the ordinary sum of partial waves diverges. Efficient methods to accelerate the convergence of partial wave sums are developed.     

Scattering of massless Dirac particles by oscillating barriers in one dimension

We study the scattering of massless Dirac particles by oscillating barriers in one dimension. Using the Floquet theory, we find the exact scattering amplitudes for time-harmonic barriers of arbitrary shape. In all cases the scattering amplitudes are found to be independent of the energy of the incoming particle and the transmission coefficient is unity. This is a manifestation of the Klein tunneling in time-harmonic potentials. Remarkably, the transmission amplitudes for arbitrary sharply-peaked potentials also become independent of the driving frequency. Conditions for which barriers of finite width can be replaced by sharply-peaked potentials are discussed.