Calculation of the imaginary part of the nucleon-nucleus optical-model potential

The imaginary part of the optical-model potential for the scattering of nucleons by nuclei is studied in the frame of the shell-model approach to nuclear reactions. Special attention is paid to the one-hole target nuclei. The imaginary part of the optical-model potential in the second order in the nucleon-nucleus interaction is divided into two parts. The first corresponds to the average resonant scattering. The second corresponds to the inelastic scattering leading to the non-collective states of the target nuclei. A local potential equivalent to the non-local theoretical one is constructed in order to facilitate comparison with experiment. Numerical calculations concern the scattering of 14.5 MeV protons by ³⁹K. It is found that the imaginary part depends upon the angular momentum and that its radial variation is governed by strong shell effects. The predicted absorption is approximately 60% of the experimental one. The average resonant scattering contributes to the imaginary part of the optical-model potential as much as the inelastic non-collective excitations of the target.     

Effect of ground state correlations on the imaginary part of the optical-model potential

The effect of the ground state correlations on the imaginary part of the nucleon-nucleus optical-model potential is qualitatively discussed within the random phase approximation (RPA). We also investigate the effect of terms of order higher than two in the microscopic nucleon-nucleus interaction. A rough estimate of both effects is given for the case K³⁹+p.     

Effective interactions in the proton optical-model potential

A six-parameter optical model is developed in which the real central part is calculated by folding several effective nucleon-nucleon interactions into matter distributions which reproduce single-particle binding energies and electron scattering data. A simple local approximation is made to take into account the exchange term. It is concluded that the density-dependent effective interactions derived from nuclear-structure calculations are also appropriate for nucleon-nucleus scattering. Off-shell effects are apparent from the worsening of the quality of fit for light nuclei.     

The real part of the nucleon-nucleus optical potential

The nucleus²⁴Na has been investigated by studying the gamma-rays emitted following thermal neutron capture in²³Na, with curved crystal and Ge(Li) spectrometers. Of the 277 transitions assigned to²⁴Na, 216 were placed in the²⁴Na level scheme containing 45 levels, of which six (1,961, 1,977, 3,866, 5,810, 5,918, and 6,222 keV) are reported for the first time. An average gamma-ray multiplicity of 3.3 gammas per neutron capture was observed. The neutron binding energy was determined to be 6,959.73 (14) keV. The resulting level scheme is compared to shell and rotational model predictions.     

Low density expansion of the nucleon-nucleus optical-model potential

The optical-model potential for nucleon-nucleus scattering is studied within the framework of the Green function approach to the many-body problem. The optical potential is identified with the self-energy, for which an expansion in terms of irreducible graphs exists. We propose to group the diagrams of this expansion according to the number of independent hole lines and to sum the graphs within each class. This procedure essentially amounts to an expansion in powers of the density and is closely related to the Bethe-Brueckner expansion for the binding energy of nuclear matter. We show that the same convergence parameter appears in both expansions. The one-and two-hole line contributions are studied in detail, numerical estimates are provided and compared with experiment. At low energy, our expansion can be related to the calculation of the optical potential within the framework of nuclear reactions (e.g., using doorway states). At high energy one is led in a natural way to the expressions derived from multiple scattering theory. Thus the hole-line expansion ties together the low and high energy domains of the optical potential. Hole line expansions for the momentum distribution and the total energy are derived from the expansion of the self-energy. The self-consistency requirement is discussed. The present study is restricted to nuclear matter but most results apply to finite nuclei as well.     

The proton optical-model potential near the Coulomb barrier

Proton elastic scattering data from ¹⁹⁷Au, ²⁰⁸Pb and ²⁰⁹Bi at energies near the Coulomb barrier are analyzed. The energy dependences of the real volume and imaginary surface-derivative potential depths V_R and W_SF of a local optical-model potential with fixed geometric parameters are found to be much more rapid than at higher energies. The strong energy dependence of V_Rnear the Coulomb barrier is explained in terms of the non-locality of the nucleon-nucleus interaction.     

Energy dependence of the isovector imaginary nucleon optical-model potential

The differences between the imaginary depths of the optical-model potential s felt by neutrons and protons on ²⁰⁸Pb show evidence for an energy decrease of the isovector imaginary term of the nucleon-nucleus optical-model potential, analogous to that of the real part.     

Ambiguities in the scattering tomography for central potentials

Invisibility devices exploit ambiguities in the inverse scattering problem of light in media. Scattering also serves as an important general tool to infer information about the structure of matter. We elucidate the nature of scattering ambiguities that arise in central potentials. We show that scattering is a tomographic projection: The integrated scattering angle is a projection of a scattering function onto the impact parameter. This function depends on the potential but may be multivalued, allowing for ambiguities where several potentials share the same scattering data. In addition, multivalued scattering angles also lead to ambiguities. We apply our theory to show that it is, in principle, possible to construct an invisibility device without infinite phase velocity of light.     

Coulomb correction due to the true nonlocality of the optical-model potential

We show that the value currently used for the Coulomb correction is approximately 30 percent too small, mainly because it includes a spurious contribution from the dynamical energy dependence of the optical potential. This observation brings in good agreement the empirical values of the symmetry potential U₁ (≈ 16 MeV) determined respectively from neutron and from proton scattering data.     

On the energy dependence of the real part of the heavy-ion optical potential

The energy dependence of the real part of the heavy-ion optical potential is attributed to the J(J + 1) dependence of the-potential which is caused by the rotation induced during the collision. The magnitude and the form of the energy dependence are calculated for several light nucleus-nucleus systems with the use of the distortion potential which is defined by the diagonalization of the coupling interaction. It is shown that the heavy-ion scatterings with the inelastic channels of high and low excitation energies lead to weak and strong energy dependence of the potential, respectively.     

Ambiguities in chiral-symmetry breaking using the effective potential approach

Using a form of the effective potential for composite operators with a variational approach we show that it is possible to get different directions of the chiral phase transition in QCD. Which one occurs depends on the way the Schwinger-Dyson equation for the fermion self-energy is used in the 2-loop term of the effective potential. We must choose the 2-loop term which agrees with phenomenology in each form of the effective potential.     

Inferences concerning the real part of the heavy-ion optical potential through folding techniques

Analytic solutions are given for the real part of the heavy-ion optical potential through the use of single and double folding models. The effective potential between the nucleons is in the form of a sum of Yukawa terms and the distributions of the densities and the nucleon optical potential are taken to be of Woods-Saxon form. Consideration is given to the influence of the tails of the density distribution as well as that of the effective potential. Calculations suggest there is little need to delineate effective interactions with momentum components much beyond the Fermi level. This provides a rationale for realistic treatments of the effective interaction. A crucial parameter specifying the density distribution is found to be the rms radius.     

On the energy dependence of the real central optical potential for proton-nucleus scattering

The energy dependence of the real central optical potential in the energy range 25–1000 MeV has been determined from optical model analyses of p+¹²C, p+¹⁶O, p+²⁷Al, p+⁴⁰Ca and p+²⁰⁸Pb elastic scattering data. The volume integral and the strength can be represented by a relation linear in the incident energy only if a limited energy range is chosen. When the energy range 25–1000 MeV is considered a logarithmic energy dependence gives a better representation of the phenomenological results, especially for the light nuclei.     

Calculation of the real part of the interaction potential between two heavy ions in the sudden approximation

We have calculated the interaction potential between two heavy ions using the energy density formalism and Fermi distributions for the nuclear densities. The experimental fusion barriers are rather well reproduced. The conditions for the observation of fusion between two heavy ions are discussed. As far as the nuclear part of the interaction potential is concerned, the proximity scaling is investigated in detail. It is found that the proximity theorem is satisfied to a good extent. However, as far as the neutron excess is concerned, disagreement with the proximity potential is observed.     

Pion-nucleus scattering and systematics of the delta-nucleus potential

A new calculational technique for the isobar-hole model is used to study pion elastic scattering from ¹⁶O, ²⁸Si, and ⁴⁰Ca. The central value of the Δ-nucleus interaction potential is found to be independent of nuclear mass number.     

Properties of the quasiparticle excitations in 207Pb and 209Pb from an extrapolation of the optical-model potential

A previously developed dispersion relation approach is used to calculate the shell-model potential in the case of neutrons in ²⁰⁸Pb, in the energy domain (-50 MeV, 0). This potential contains a dispersive contribution besides a Hartree-Fock type component, and thereby includes correlation and polarization effects. The shell-model and the Hartree-Fock type potentials are assumed to have Woods-Saxon shapes with diffuseness a_v = 0.70 fm; the energy dependence of their depths and radii is calculated. The energy dependence of the shell-model potential is characterized by the effective mass, whose dependence upon radial distance and neutron energy is determined. The effective mass is a sensitive function of energy, in contrast to its Hartree-Fock type component which is nearly independent of energy. Attention is drawn to the fact that the effective mass in nuclear matter cannot be straightforwardly identified with the effective mass at the nuclear centre. The effective mass presents a sharp peak at the nuclear surface near the Fermi energy and a dip at the surface for energies 10 to 20 MeV away from the Fermi energy. The spectroscopic factors of single-particle excitations in ²⁰⁷Pb and ²⁰⁹Pb are calculated from the difference between the effective mass and its Hartree-Fock type component. The predicted values of the valence single-particle wave functions at large radial distances are in fair agreement with experimental values deduced from analyses of sub-Coulomb pickup reactions. It is shown that the dispersive contribution increases the level density parameter by about 25%, in agreement with previous microscopic or semi-phenomenological models; the calculated level density parameter is in good agreement with the empirical value.     

Geometrical scaling and the real part of the Pomeron

Consequences of the hyptohesis of geometrical scaling of the inelastic overlap function applied to the Pomeron amplitude are discussed. From analyticity and crossing symmetry some predictions are given for the asymptotic real part of the Pomeron.Presented at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.