On-shell Approximation Research Articles

On the damping of giant resonances and the independent propagation of particles and holes

Abstract In the literature, damping of nuclear collective motion has been attributed to the imaginary part of the self-energies of particles and holes, when assumed to move independently. We examine this picture within a schematic model adapted to the case of giant resonances. In this model we take degenerate 1p-1h states and dress them by a simple analytical form for the self-energies. We are able to calculate the intrinsic response function analytically and thus to study various approximations. We demonstrate that the on-shell approximation appreciably overestimates the widths of the collective vibrations.

Effects of a refined Coulomb treatment on π±-deuteron elastic scattering observables

The sensitivity of π ±d differential cross section, vector analyzing power, tensor polarization and charge asymmetry parameters to different Coulomb contributions to the total scattering amplitudes is investigated at eleven energies in the energy range T π lab = 47 to 360 MeV. For that purpose we derive a simple on-shell approximation for the Coulomb-corrected hadronic T-matrix, which is superior to widely used semiclassical descriptions of the Coulomb corrections. In addition we use relativistic Coulomb amplitudes and phase shifts, which consider also the pion and deuteron charge extension. It is shown that the vector analyzing power it 11 is more sensitive to different Coulomb prescriptions than the differential cross section and tensor polarization t 20. We have also succeeded in describing the charge asymmetry parameter of differential cross section A π satisfactorily and find that the charge asymmetry parameter of vector analyzing power A it 11 is even more sensitive than A π to different Coulomb treatments. For a search of charge-symmetry-breaking effects in terms of the πd charge asymmetry parameter, the necessity of including our careful treatment of Coulomb contributions to the total scattering amplitude is emphasized.

Photoelectron motion in the presence of a probe electromagnetic wave

Photoionization in the presence of a probe electromagnetic wave is studied. A continuum wave function is obtained in a weak-field, low-frequency limit for low-energy photoelectrons. When the electron and photon energies are comparable, the absorption and stimulated-emission angular distributions show a significant departure from results based on a much-used on-shell approximation of Kroll and Watson.

Calculation of free-free radiative absorption cross sections

One-photon free-free absorption cross sections are evaluated by means of an integral relationship involving electron-atom off-shell scattering amplitudes. Off-shell effects are explicitly calculated and compared to various on-shell approximations. This calculation shows that off-shell effects become small with increasing energy as expected. One of the on-shell approximations agrees with the exact result over the entire energy range under consideration to within a few percent. The only integrals which appear in the calculations are absolutely convergent in contrast to methods used in previous calculations.

Excitation of atomic hydrogen by electron impact in an on-shell Faddeev-Watson approximation

The second-order Faddeev-Watson approximation is applied to the excitation of atomic hydrogen by electron impact. By restricting the total kinetic energy of the electrons in the intermediate state to the three-body energy shell a simplification is effected when approximates the dynamics to a double Coulomb binary encounter. The resulting transition matrix elements involve multi-dimensional integrals which are evaluated numerically. Although the model fails for low momentum transfers it predicts differential cross sections and angular correlation parameters for electron-hydrogen excitation which are in reasonable agreement with experiment at high momentum transfers.

Approximate Coulomb effects in the three-body scattering problem

From the momentum space Faddeev equations we derive approximate expressions which describe the Coulomb-nuclear interference in the three-body elastic scattering, rearrangement, and breakup problems and apply the formalism to $p\ensuremath{-}d$ elastic scattering. The approximations treat the Coulomb interference as mainly a two-body effect, but we allow for the charge distribution of the deuteron in the $p\ensuremath{-}d$ calculations. Real and imaginary parts of the Coulomb correction to the elastic scattering phase shifts are described in terms of on-shell quantities only. In the case of pure Coulomb breakup we recover the distorted-wave Born approximation result. Comparing the derived approximation with the full Faddeev $p\ensuremath{-}d$ elastic scattering calculation, which includes the Coulomb force, we obtain good qualitative agreement in $S$ and $P$ waves, but disagreement in repulsive higher partial waves. The on-shell approximation investigated is found to be superior to other current approximations. The calculated differential cross sections at 10 MeV raise the question of whether there is a significant Coulomb-nuclear interference at backward angles.NUCLEAR STRUCTURE Coulomb effects, approximation formulas; three-body elastic scattering, rearrangement, breakup; applied to $p\ensuremath{-}d$ scattering.

Neutron-proton bremsstrahlung at 4.8 MeV

The differential cross section for neutron-proton bremsstrahlung has been measured at an incident energy of 4.8 MeV and outgoing nucleon angles of θ n = 35°, θ p = 25°. An upper limit of d 2σ/ dΩ n dΩ p = 210 μ b · sr −2 was obtained. The experimental method was unusual in that the hydrogen target was a stilbene scintillation crystal, which also served as recoil proton detector, and the bremsstrahlung photon was detected in addition to the two nucleons. A theoretical estimate of the cross section corresponding to that measured in this experiment was made using an on-shell approximation and gave a result of 89 μb · sr −2. The measured upper limit exceeds this and other theoretical predictions but nevertheless represents a marked improvement over previous measurements of n-p bremsstrahlung at energies below 100 MeV.

Simple Coulomb correction for phase-shifts and application to the p-p system

The Coulomb contributions to the phase-shifts of two-particle systems are described within non-relativistic scattering theory to order e2 exactly. A simple on-shell approximation is then derived and applied to p-p scattering. Both the exact and the approximate version are tested for the particular case of a separable strong interaction model. It is found that this on-shell formula is-in contrast to the WKB approach-reliable also for low partial waves, except for energies Elab below 20 MeV in the 1S0, where a low-energy off-shell approximation is effective.

Coulomb corrections in non-relativistic π+-p scattering

Separable potentials are used to calculate non-relativistic first-order Coulomb corrections to pi +-p phase-shifts. The off-shell sensitivity of the Coulomb corrections is studied and found to be small in S and P waves if the finite extent of the charged particles is taken into account. A simple on-shell approximation provides good agreement with the model residual phase-shifts in the S31 and P31 partial wave, but fails in the resonant P33.

Coulomb corrections in NN coupled states

Within nonrelativistic scattering theory we derive an on-shell approximation for the Coulomb corrections to elastic p-p scattering in coupled states. The effect of the Coulomb-nuclear interference to the mixing parameters is calculated using both point-like charge and finite charge distribution of the protons. Comparison with available model calculations demonstrates fair agreement, specifically up to E lab = 100 MeV.

Off-energy-shell behavior of theTmatrix for realistic localN−Npotentials with soft or hard cores

Off-energy-shell T matrix elements for any partial wave are calculated using realistic local two-nucleon potentials with soft or hard cores. The off- shell behaviors are discussed in relation to on-shell values. Off-shell cross sections involved in the impulse approximation of nuclear reactions are analyzed. The validity of some usual on-shell approximations is examined.

Retardation in the resonant interaction of two identical atoms

The problem considered is that of two identical two-level atoms a fixed distance $r$ apart, one of which is excited at $t=0$. A simple improvement on the usual on-shell approximation (which in the single-atom case is part of the Weisskopf-Wigner approximation) yields new solutions for the various probability amplitudes in the form of infinite series involving all the retardation times $\frac{\mathrm{nr}}{c}$. The truncated solutions involving only the single retardation time $\frac{r}{c}$ are compared with previously published results both when all photon modes are allowed and when only photons propagating along the interatomic axis are allowed. When the retardation times are neglected, the series are summed to give the well-known results of Stephen and others.

The off-shell spectator model for d(p, 2p)n

The quasi-free peak for the reaction d(p, 2p)n is examined in the off-shell spectator approximation at incident energies from 14.4 MeV to 100 MeV using the Hamada-Johnstpn potential. At all these energies it is essentially indistinguishable from the on-shell (impulse) approximation, which gives correct shapes but overestimates the magnitudes of differential cross sections due to lack of unitarity. Theoretical interest in the approximation lies in its possible use as a starting point for improved approximations. The energy at which partial waves higher than the s-wave in the pp system are important is investigated. The Coulomb interference is shown to result in the amplitude in this approximation for the (n, 2n) quasi-free peak being larger than that for the (p, 2p) peak at the same energy by an approximate scale factor, thus simplifying the possibility of using these reactions to investigate charge symmetry.

Multiple Scattering of Pions by Deuterons

The general theory of multiple scattering of pions from nuclei is expressed in a way which does not require the use of a series expansion. In an on-shell approximation this theory may be reduced to the solution of a set of $A$ coupled integral equations. As a demonstration of the method the equations are specialized to the case of the deuteron and solved to give comparison with the experimental data. The deuteron tensor ${T}_{20}$ is shown to be sensitive to the percentage of $D$ state for deuterons scattered at 0\ifmmode^\circ\else\textdegree\fi{}.

Proton-proton bremsstrahlung at 20 MeV, 35°: Theory

Proton-proton bremsstrahlung predictions from the Hamada-Johnston and Tabakin potentials are found to be in close agreement with each other, with the on-energy-shell approximation, and with the experimental value at 20 MeV, 30°. The Coulomb interaction is included in the on-shell approximation. The Low approximation is also examined.