Off-energy Shell Effects Research Articles

Off-energy-shellp−pscattering at sub-Coulomb energies via the Trojan horse method

Two-proton scattering at sub-Coulomb energies has been measured indirectly via the Trojan horse method applied to the p + d{yields}p + p + n reaction to investigate off-energy shell effects for scattering processes. The three-body experiment was performed at 5 and 4.7 MeV corresponding to a p-p relative energy ranging from 80 to 670 keV. The free p-p cross section exhibits a deep minimum right within this relative energy region due to Coulomb plus nuclear destructive interference. No minimum occurs instead in the Trojan horse p-p cross section, which was extracted by employing a simple plane-wave impulse approximation. A detailed formalism was developed to build up the expression of the theoretical half-off-shell p-p cross section. Its behavior agrees with the Trojan horse data and in turn formally fits the n-n, n-p, and nuclear p-p cross sections given the fact that in its expression the Coulomb amplitude is negligible with respect to the nuclear one. These results confirm the Trojan horse suppression of the Coulomb amplitude for scattering due to the off-shell character of the process.

Deuteron-proton elastic scattering at intermediate energies

The deuteron-proton elastic scattering has been studied in the multiple scattering expansion formalism. The essential attention has been given to such relativistic problem as a deuteron wave function in a moving frame and transformation of spin states due to Wigner rotation. Parameterization of the nucleon-nucleon $t$-matrix has been used to take the off-energy shell effects into account. The vector, $A_y,$ and tensor, $A_{yy}$, analyzing powers of the deuteron have been calculated at two deuteron kinetic energies: 395 MeV and 1200 MeV. The obtained results are compared with the experimental data.

Relativistic structure of heavy quarkonia. Application to their leptonic decays

We investigate the relativistic structure of J=1, L=0 heavy quarkonia in the framework of a covariant formulation of light-front dynamics. This formalism enables us to treat in a very transparent way off-energy shell effects as well as relativistic kinematical effects coming from the finite momentum between the quark and antiquark. We apply our formalism to the leptonic decay width of J=1 charmonium and bottomonium S-states. Relativistic corrections are found to be very large in the charm sector (up to about 60 % reduction for ψ ′ leptonic decay width) and smaller in the bottomonium sector, but in any case not negligible.

Collisional scaling within a multichannel square representation

A new approach is examined for the state-to-state scaling of collision problems. It derives from multichannel scattering at a square interaction and is referred to as mutlichannel square (MS) scaling. Since it retains the full internal energy spectrum while approximating the radial coupling elements, it may be viewed as complementing the conventional energy sudden (ES) scaling. Some general comments are made with regard to the structure of the MS relations and we show how several attractive features of ES scaling can be matched by the new scaling. Application is made to the analysis of rotationally inelastic integral cross sections for the scattering of He from p-H2. We compare MS scaled results with both ES and modified ES results. It is found that for the range of total energies examined, the MS scaling yields results which are in good agreement with exact ones and are dramatically improved over the basic ES ones. The modified ES procedure depends upon incorporating off-energy shell effects into the ES scaling and like the MS scheme is here set up to contain a single free parameter. We find that the MS results are roughly twice as accurate as the modified ES ones. A number of avenues for further development and application of the MS scaling are discussed.

Polarization potentials in heavy-ion scattering

A polarization potential is defined in terms of the Feshbach projection operator formalism to represent the effect upon the elastic channel of the coupling to non-elastic channels in heavy-ion scattering. The polarization potential represents coupling to specific surface degrees of freedom of the particular reaction considered and it is contrasted to the complementary global approaches for the volume potential such as the folding model and the proximity potential. The coupled channels method is used both as a source of exact model solutions for comparison with the various approximate potential forms and also as a numerical means of constructing trivially equivalent local potentials. The imaginary Coulomb polarization potential is due in lowest order to quadrupole coupling to the lowest collective 2 + state of a nucleus. It is considered in detail since it provides the insight of closed analytical forms in various approximations. Multistep coupling to higher states, energy loss and off-energy shell effects are also considered analytically. The real Coulomb polarization potential due to the virtual excitation of multipole giant resonances, and the polarization potential arising from relativistic corrections, are investigated in detail. Polarization potential components due to nuclear coupling are investigated numerically. Analytical cross section approaches are contrasted with the polarization potential approach and with coupled channels.

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate. Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio V E . Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.

The 6Li(p, 2p) 5He reaction at 100 MeV

The 6Li(p, 2p) 5He reaction has been studied at a proton bombarding energy of 100 MeV with good energy and angular resolution for proton separation energies up to 50 MeV. Data were taken for 17 coplanar symmetric angle pairs yielding both energy sharing spectra and coplanar symmetric angular correlations for various regions of separation energy. The data for the 5He ground state ( 3 2 − resonance ) does not show a significant minimum at zero recoil momentum, suggesting a contribution from an s-state knockout leading to this continuum state. DWIA calculations are presented which reproduce most of the important features of the data. These calculations indicate the importance of both the distortion effects and the offenergy-shell effects at this energy.

Optical-model calculations with a realistic nucleon-nucleon interaction: (I). Theory

The real parts of the optical potentials for the elastic scattering of 65 MeV protons on 16O, 17O and 18O are calculated in an impulse approximation. The prescription used is derived from three-body theory, considering the system as consisting of the incident nucleon, an active bound nucleon, and an inert core. The effective interaction resulting is a fully off-shell T-matrix and is both non-local and energy dependent. Reid's soft-core potential is used to generate the requisite T-matrices. Antisymmetry between the incident and active nucleon is included and all relative two-nucleon partial waves with l ≦ 3 are retained. The optical potential is constructed by assuming a pure shell-model target and summing the state of the active nucleon through the single-particle states. The resulting non-locality is investigated explicitly and is found to be negligible. A coordinate space potential is constructed. When compared to the real parts of phenomenologically generated optical potentials, our potentials are found to be similar in the surface ( r ≈ 2.5 fm) but deeper in the interior. When our parameter-free real well is combined with phenomenological imaginary and spin-orbit wells, high-quality fits to the experimental data are obtained, comparable to those obtained when all three wells are chosen phenomenologically. Off-energy shell effects and the energy dependence are investigated.

Three-particles final state reactions

The first iteration solution of the Faddeev-equation is used to evaluate the transition amplitude for the (α, 2α) reactions. Reasonable approximations are adopted to calculate the off-energy shell effects. Good agreement with the experimental results is obtained.

Elastic Nucleon-Deuteron Scattering in a Soluble Model as a Test of the Impulse Approximation

A comparison of the results of the Born and impulse approximations for a well-defined situation is made by calculating the nucleon-deuteron differential elastic cross sections in both approximations at incident nucleon laboratory energies of 32, 94, and 146 Mev. In order to carry out the impulse approximation computation in complete detail, including, in particular, contributions from off- energy-shell two-particle matrix elements, the assumption is made that the two- particle scattering is completely described by effective range theory. This assumption, though incorrect, nevertheless retains some of the features of the actuai physical situation and provides a means of comparing the results of calculations using the impulse approximation and the Born approximation as well as a way of studying the effect of including two-particle matrix elements off the energy shell. Exact solutions to the two-particle model are written down with the aid of the Gel'fand-Levitan theory and are used in a precise numerical evaluation of the impulse approximation expressions. The results of the Born and impulse approximations differ considerably. Also, indications are that, as expected, the pickup term (which dominates at large angles) is not adequately treated by either of the above approximations. When this term is not considered, a study ofmore » the remaining expressions shows that off-energy-shell effects are significant for the deuteron wave functions employed. In view of the rather restrictive assumptions made on the twoparticle data, a detailed comparison of the results with experiment is not possible. Nevertheless, there is a suggestion that the impulse approximation, including offenergy-shell effects, describes the experimental results best. (auth)« less