The scattering of the Manning-Rosen potential with centrifugal term

In this Letter the approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Manning-Rosen potential are carried out by a proper approximation to the centrifugal term. The normalized radial wave functions of l-wave scattering states are presented and the calculation formula of phase shifts is derived. It is well shown that the poles of the S-matrix in the complex energy plane correspond to bound states for real poles and scattering states for complex poles in the lower half of the energy plane. We consider and verify two special cases: the l=0 and the s-wave Hulthén potential.     

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.      

Extrapolation of the differential cross section for triplet <Emphasis Type="Italic">pp</Emphasis> scattering to low energies

It has been shown that the differential cross section for low-energy triplet pp scattering that is caused by the sum of the nuclear and magnetic interactions in the Coulomb field of the protons is a rapidly oscillating function and has second-order poles in the forward and backward directions. In order to extrapolate such a cross section to the energy region below 10 MeV, a simple low-energy approximation has been proposed. New phenomena—proton-proton analogues of the Mott and Schwinger effects—are discussed.     

Scaling of differential cross-section in high energy proton-helium scattering

Possible occurrence of scaling of differential cross-section for high energy hadronnucleus elastic scattering is demonstrated takingp-⁴He scattering as an example and using three well-known scaling variables proposed earlier for hadron-hadron scattering. The available data on differential cross-section ratio betweenE _lab=45 and 393 GeV are found to scale in all the three variables reasonably well and the positions of the dip and the secondary maximum are found to follow the predicted patterns of behaviour as a function of energy. Extrapolating the fits to the available slope-parameter data onto higher energies and using the scaling curves, the positions of the dip and the secondary maximum and the differential cross-section ratio as a function of |t| are predicted for higher energies.     

Scattering States of l-Wave Schrödinger Equation with Modified Rosen-Morse Potential

Within a Pekeris-type approximation to the centrifugal term, we examine the approximately analytical scattering state solutions of the l-wave Schrödinger equation with the modified Rosen-Morse potential. The calculation formula of phase shifts is derived, and the corresponding bound state energy levels are also obtained from the poles of the scattering amplitude.     

A meson-exchange N-N potential

A meson-theoretic model of the intermediate range nucleon-nucleon potential is presented with emphasis placed on the two-pion exchange contribution. The Bethe-Salpeter equation is reduced, by the Blankenbecler-Sugar technique, to a Lippmann-Schwinger equation, from which an approximate nonlocal, energy-dependent potential is obtained. The nucleon-antinucleon pair contribution, which plagues meson-theoretical two-pion calculations, is suppressed by the complex poles of the one-nucleon Green's function. The importance of the retention of the explicit energy dependence of the potential is demonstrated by calculating the off-shell scattering matrices. The potential is presented in a linearized (in energy) form with the core region adjusted to produce a fit to low energy data.     

Exact solutions of the Schrödinger equation for zero energy

Some new exact solutions of the Schrödinger equation for zero energy are presented for certain nontrivial model potentials. Exact expressions for the different scattering lengths are derived and their differences and similarities are worked out. In particular, the respective distributions of the zeros and poles of the scattering lengths are characterized in detail.     

Optical constants of Cu measured with reflection electron energy loss spectroscopy (REELS)

Two energy loss spectra of 1000 and 3000 eV electrons reflected from a Cu surface are analysed to give the normalized distribution of energy losses in a single surface and volume inelastic scattering process. These single scattering loss distributions are subsequently fitted to theoretical expressions for the differential inverse inelastic mean free path (DIIMFP) and differential surface excitation probability (DSEP) providing the real and imaginary part of the dielectric function in terms of a set of Drude-Lorentz oscillators. The optical constants obtained in this way are subjected to several sum rule checks and compared with other experimental data and with density-functional-theory (DFT) calculations. The present optical data agree excellently with the DFT-results, while the earlier optical data deviate significantly from these two data sets for energies below 30 eV. The mean free path for inelastic electron scattering for energies below 2000 eV is derived from the dielectric data and is found to agree satisfactorily with values reported earlier.     

A new technique for calculating virtual photon spectra

A first-order matrix differential equation in energy is used to propagate radial matrix elements arising in DWBA of relativistic electron scattering from nuclei. Given an initial set of matrix elements at some value of the energy, this equation permits the evaluation of the radial matrix elements over the complete energy transfer range. A computer code has been written for this new procedure and the virtual photon spectra accompanying electron scattering from a point nucleus is calculated as a function of photon energy for various multipoles, nuclear charges, and incident electron energies.     

Special analytic properties of the amplitude for scattering on the Woods-Saxon potential

It is shown that the s-wave partial amplitude f(k) for scattering on the real-valued Woods-Saxon potential V(r)=?V ₀/[1+exp((r?R)/d)] has very special analytic properties: the trajectories of the poles of the function k cotδ [of the zeros of the amplitude f(k)] coincide with the lines of the dynamical singularities [spurious poles of f(k)], so that the zeros and the poles compensate each other. In contrast to what is obtained for Yukawa-like potentials, the scattering length does not vanish here at zero energy. The results reported in this article were obtained analytically under the assumption that exp(-R/d)?1. The problem of revealing the poles of the function k cotδ in a partial-wave analysis of neutron scattering on nuclei is discussed.     

On the Connection between Particles and Poles of the S-Matrix

In axiomatic S-matrix theory it is usually assumed that stable particles give rise to simple poles of the S-matrix for real negative energies while unstable particles give rise to poles close to the real axis on an unphysical sheet of the energy Riemann surface. The stable particle — pole association has been known for a long time not to be always true. For example in potential scattering what is relevant in this case in fact is not the S-matrix but the Jost function. The zeroes of this function for real negative energies are in fact in one-to-one correspondence with the bound states, while the correspondence may break down for the poles of the S-matrix. On the other hand it has recently been pointed out that there also is in general no connection between unstable particles and poles of the S-matrix.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

The Relativistic Scattering States of the Hulthén Potential with an Improved New Approximate Scheme to the Centrifugal Term

The approximately analytical scattering state solutions of the l-wave Klein-Gordon equation with the unequal scalar and vector Hulthén potentials are carried out by an improved new approximate scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Hulthén potentials are presented and the corresponding calculation formula of phase shifts is derived. It is well shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and two special cases for s-wave (l=0) and for l=0 and equal scalar and vector Hulthén potentials are also studied briefly.     

Singularities in the Vertex Function in Two-Dimensional Tight-Binding Fermi Systems

The singularities in the vertex function are studied within the ladder approximation for a non-half-filled two-dimensional tight- binding Fermi system. The location of all the possible poles of the vertex function has been clarified for both repulsive and attractive interacting cases.. The scattering phase shift is examined and found not always zero at Fermi energy. The Kohn-lut tinger-like singularity is also briefly discussed.     

Certain general questions of fast-electron transport II. Kinetic equations and conditions governing the accelerated motion of fast electrons in dielectrics in an electric field

A general kinetic equation for the differential density of fast particles moving in a medium in an external field is derived on the basis of the continuity equation in phase space. An equation is written for the differential flux in the case of fixed target particles. This equation is used to derive equations for fast electrons; account is taken of the coupling of energy-loss and scattering events in an electric field for various particular problems analogous to those studied in the theory of electron transport in the absence of a field. The kinetic equations are used to analyze the conditions governing accelerated motion of electrons in a dielectric in an external electric field in the continuous-deceleration approximation. Account is taken of fluctuations in the energy loss and of multiple scattering. There are two energy ranges of particles moving in a dielectric in which accelerated motion can occur; in the case of an electron beam with a continuous energy spectrum, this acceleration would be accompanied by monochromatization of the beam.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 7–12, February, 1972.     

Analytic properties of the T matrix for two Yukawa potentials

A study is made of the behavior of the residues at the Regge poles of the partial-wave T matrix on the mass shell as a function of the energy E for nonrelativistic scattering on an exponential potential and two Yukawa potentials. It is shown that the residues do not have left-hand cuts in the E plane.     

Bound and scattering wave functions for a velocity-dependent Kisslinger potential for l > 0

Using formal scattering theory, the scattering wave functions are extrapolated to negative energies corresponding to bound-state poles. It is shown that the ratio of the normalized scattering and the corresponding bound-state wave functions, at a bound-state pole, is uniquely determined by the bound-state binding energy. This simple relation is proved analytically for an arbitrary angular momentum quantum number l > 0, in the presence of a velocity-dependent Kisslinger potential. The extrapolation relation is tested analytically by solving the Schr?dinger equation in the p-wave case exactly for the scattering and the corresponding bound-state wave functions when the Kisslinger potential has the form of a square well. A numerical resolution of the Schr?dinger equation in the p-wave case and of a square-well Kisslinger potential is carried out to investigate the range of validity of the extrapolated connection. It is found that the derived relation is satisfied best at low energies and short distances. Received: 17 October 2001 / Accepted: 4 January 2002     

Continuum states of modified Morse potential

Using a proper approximation scheme to the centrifugal term, we study any l-wave continuum states of the Schrdinger equation for the modified Morse potential. The normalised analytical radial wave functions are presented, and a corresponding calculation formula of phase shifts is derived. It is shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some numerical results are calculated to show the accuracy of our results.     

Peculiarities of the amplitude of elastic subbarrier impurity scattering

A formula for the amplitude of electron elastic subbarrier (in the domain of imaginary momenta) scattering by an impurity modeled as a spherical potential well is derived. An equation is obtained to find the features (poles) of this amplitude in a complex energy plane in the close vicinity of which the energy spectrum of quantum resonant percolation trajectories in M-I-M contacts (M = N (normal metal) or S (superconductor) and I is an insulator) with weak (low impurity concentration) structural disorder in the I layer is concentrated [1–4].