Multi-step processes in high-energy elastic and inelastic nuclear scattering

Multi-step processes in elastic and inelastic nuclear scattering at intermediate and high energies are investigated using a formulation whereby a finite number of channels are explicitly treated while all the other channels are approximately accounted for through a “second-order potential matrix”. Within the framework of the eikonal approximation the problem reduces to a finite system of first-order coupled integro-differential equations with non-local potentials which depend on the two-body density matrix of the target nucleus. The relationship of the above formulation to the DWIA, the close-coupling method, and the Glauber multiple scattering model is examined. This approach is applied to the elastic and inelastic (2⁺, 4.43 MeV) scattering of 1 GeV nucleons by ¹²C. The corrections to the DWIA are sizeable, and the inelastic scattering appears to be very sensitive to the multi-step contributions and the nuclear structure.     

One- and two-body dissipation in heavy-ion collisions

A microscopic theory has been formulated for one- and two-body dissipation in collisions between two heavy nuclei. With a nucleon-nucleon interaction as the basic perturbation in a density matrix approach with “linear response” approximations, the one- and two-body nuclear friction coefficients for the ⁴⁰Ca + ⁴⁰Ca system have been calculated and their dependence on relative kinetic energy and smearing of nuclear single-particle states was obtained. The results of our calculation show that: (a) the combined one- and two-body friction coefficients compare favourably with phenomenological values, (b) the one-body dissipation is more effective than two-body in kinetic energy damping, while both the mechanisms are comparable for the damping of relative angular momentum, (c) the importance of the two-body friction compared to one-body increases at higher relative kinetic energies and (d) the effect of introducing a smearing in nuclear levels appears as a lowering of nuclear friction.     

Nuclear structure studies on halo nuclei by direct reactions with radioactive beams

The investigation of direct reactions with exotic beams in inverse kinematics gives access to a wide field of nuclear structure studies in the region far off stability. The basic concept and the methods involved are briefly discussed. The present contribution will focus on the investigation of light neutron-rich halo nuclei. Such nuclei reveal a new type of nuclear structure, namely an extended neutron distribution surrounding a nuclear core. An overview on this phenomenon, and on the various methods which gave first evidence and qualitative confirmation of our present picture of halo nuclei, is given. To obtain more quantitative information on the radial shape of halo nuclei, elastic proton scattering on neutron-rich light nuclei at intermediate energies was recently investigated for the first time. This method is demonstrated to be an effective means for studying the nuclear matter distributions of such nuclei. The results on the nuclear matter radii of ⁶He and ⁸He, the deduced nuclear matter density distributions, and the significance of the data on the halo structure is discussed. The present data allow also a sensitive test of theoretical model calculations on the structure of neutron-rich helium isotopes. A few examples are presented. The investigation of few-nucleon transfer reactions in inverse kinematics may provide new and complementary information on nuclear structure, as well as astrophysical questions. The physics motivation and the experimental concept for such experiments, to be performed due to momentum matching reasons at low incident energies around 5–20 MeV/u at the new generation low energy radioactive beam facilities SPIRAL, PIAFE, etc., is briefly discussed.     

The single particle potential in nuclear matter

The energy dependent real part of the optical potential for particles and holes in nuclear matter is calculated from a realistic nuclear hamiltonian that explains the nucleon-nucleon scattering data and equilibrium properties of nuclear matter. The variational method is used with Fermi-hypernetted and single-operator-chain summation techniques. The results are comparable with empirical Woods-Saxon well depths at energies ? 150 MeV. At higher energies the potential has a density dependence suggesting a “wine-bottle” shaped nucleon-nucleus potential.     

Nuclear-rainbow scattering and nucleus-nucleus potentials at short distances

The elastic scattering of strongly bound nuclei at energies of 10 to 70 MeV per nucleon shows the phenomenon of “rainbow scattering.” A nuclear rainbow appears because of deflection to negative angles. This process involves a strong overlap of nuclear densities, with values of up to twice the saturation density of nuclear matter. The ¹⁶O+¹⁶O system is studied with a high precision over a wide energy range from 7 to 70 MeV per nucleon in several laboratories. Primary Airy maxima and higher order Airy structures are observed. At all energies, excellent fits are obtained with deep potentials as deduced from the double-folding model involving a nucleon-nucleon interaction weakly dependent on the density. It is shown that Pauli blocking expected at low energies is strongly reduced if the local momenta are calculated self-consistently. Systematics confirms a refractive origin of large-angle scattering, at low energies inclusive. Thus, nuclear-rainbow scattering yields unique information about the properties of cold nuclear matter at higher densities.     

Kinematically correct optical potentials

The Watson multiple scattering formalism for the optical potential describing elastic nucleon-nucleus scattering is developed while paying careful attention to the kinematics. No approximations are made in handling the recoil of the target nucleus so that internal and external energies can be uniquely separated. The resultant formalism allows a careful examination of the impulse, form-factor, and closure approximations without the confusion of the admixture of recoil energies; it also allows an explicit display of the nuclear correlations due to target recoil. With the many necessary (but incompletely understood) approximations explicity displayed, it appears that their confluence makes it unlikely that meaningful information about dynamical correlations in light nuclei can be obtained from elastic double-scattering experiments.     

Закон подобия для контура спектральной линии и сечения неупругого перехода

Connection is made between the velocity dependence of the inelastic cross section and the profile of the broadened spectral line. The transition between the Born and adiabatic limits in the scattering problem corresponds to the transition between the impact and static limits in the broadening problem. Both the cases with term crossing and without term crossing are considered, corresponding to “positive” and “negative” domains of the line profile. The results obtained lead to the conclusion that line-broadening theory is useful in the calculations of inelastic cross sections, where the region of intermediate energies of the perturbing particles is included. Detailed calculations of the inelastic cross section have been made for the dipole interaction because exact solutions of the broadening problem are available for comparison in this case. Comparison is made with the results of other authors. The possibility of calculating the effect of field rotation is discussed.     

Diffractive,refractive and interference effects in heavy ion elastic scattering

The elastic scattering amplitude is divided into contributions arising from the “near” and “far” sides of the interaction region. Each of these contributions is further divided into a diffractive component arising from strong absorption and refractive components arising from the repulsive Coulomb and attractive nuclear fields. Damped orbiting of the projectile around the target is also allowed for. Simple closed form expressions are derived for the various contributions by assuming specific forms for the elastic scattering matrix. The results are discussed by analysing several typical elastic scattering data.     

Pion scattering from a bound nucleon at medium energies

In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable πN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for ⁴He ～ ¹⁶O. The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 ～ 70%) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.     

Explanation of the backangle anomaly and the isotope effect within a microscopic study of elastic α-scattering on the even Ca isotopes

Elastic α-scattering on the isotopes ^40,42,44,48Ca has been studied microscopically within the framework of the generator coordinate method. For energies above the Coulomb barrier the backangle enhancement of the cross section in elastic α-⁴⁰Ca scattering is explained by several overlapping barrier resonances, while for energies below the barrier the backangle rise is caused by individual long-living resonances which — in agreement with recent experimental data — confirm the concept of an “α-⁴⁰Ca quasimolecule”. The isotope effect showing up in a different behaviour of the cross section at backangles is due to a different absorptive strength of the isotopes. “Neutron blocking” has been ruled out as a possible explanation of the isotope effect.     

Properties of nuclear S-wave matter

Comparative Brueckner and Jastrow studies of the properties of nuclear matter are somewhat hampered by the complexities of realistic nucleon-nucleon potentials. To avoid some of these difficulties we investigate a nuclear matter model called S-wave matter, which consists of nucleons interacting via various of the Aviles S-wave delta function potentials. These potentials all fit the two-body S-wave scattering data for energies up to 500 meV. By using Jastrow methods we find two-body contributions to the ground state energy ranging from 18 to 29 MeV, depending on the particular potential used; the results for a given potential are in good agreement with the Brueckner results of Haftel. In addition, there are significant Jastrow three-body contributions, indicating that the equivalent three-body Brueckner contributions should be evaluated.     

The deuteron breakup reaction induced by protons of 65, 85 and 100 MeV

Coplanar energy sharing spectra for p + d breakup at 65, 85 and 100 MeV proton bombarding energies were measured using the University of Maryland sectored isochronous cyclotron, by measuring the energies of either two protons or one proton and one neutron in coincidence. The detector angles were chosen to enhance either the p-p or p-n quasifree scattering, or the p-n final state interaction. The energy dependence of the peak cross section at equal symmetric quasifree scattering angle pairs was extracted for the ²H(p, 2p)n and ²H(p, pn)p reactions. Quasifree angular distributions were obtained for the reaction ²H(p, 2p)n at 65 MeV and for the reaction ²H(p, pn)p at 65, 85 and 100 MeV. The plane wave impulse approximation theory can only qualitatively reproduce the shape of the quasifree scattering peak in the energy sharing spectra and the shape of the p-p quasifree angular distribution. The discrepancies observed between the plane wave impulse approximation theory and the experimental data imply that the presence of the spectator particle (i.e., the multiple scattering effects) has a strong influence on the magnitude and the shape of the experimental results. Multiple scattering calculations were carried out in the three-body model of Aaron, Amado and Yam except that the S-wave separable two-body amplitudes were modified to fit two-nucleon elastic scattering data at higher energies. Comparisons of the results of these multiple scattering calculations to the experimental data show excellent quantitative agreement throughout the energy range and the angular region of this experiment, except for a few cases in which this model is inherently insufficient; namely, regions in which the Coulomb interaction is important, or regions for which a Hulthén wave function is inaccurate and the off-shell effects are not properly taken into account.     

Proximity forces

We have generalized a theorem according to which the force between two gently curved objects in close proximity is proportional to the interaction potential per unit area between two flat surfaces made of the same material, the constant of proportionality being a measure of the mean curvature of the two objects. This theorem leads to a formula for the interaction pontential between curved objects (e.g., two smooth cylinders of mica or two atomic nuclei) which is a product of a simple geometrical factor and a universal function of separation, characteristic of the material of which the objects are made, and intimately related to the surface energy coefficient. We have calculated and tabulated this universal function for nuclear surfaces, using the nuclear Thomas-Fermi approximation. The results can be expressed by a simple cubic-exponential formula which gives the potential between any two nuclei in the separation degree of freedom. Even simpler expressions are found for the interaction energy associated with the “crevice” or neck in the nuclear configuration that would be expected immediately after contact of two nuclei. These “proximity energies” are used to supplement the usual expansion of the energy of a thin-skinned system into volume, surface, curvature, and higher-order terms. The resulting elementary formulas are tested against explicit models of interacting nuclei and against elastic scattering data, and are found to be useful for even quite small mass numbers.     

Elastic scattering of heavy ions — The simplest interpretation?

Elastic scattering of heavy ions with incident energies well above the Coulomb barrier is predicted from a simple extension of the Glauber theory for high energy particle-nucleus scattering. The only parameters needed are the “measured” nuclear density distribution function and the measured forward amplitude for free nucleon-nucleon scattering. An extensive comparison between theory and experiment is presented. It is demonstrated that the nuclear absorption needed to reproduce the measured differential cross section for elastic scattering can be well reproduced by means of the measured free nucleon-nucleon total cross section and the nuclear density distribution as “measured” for instance by electron scattering. Remarkable agreement between theory and experiment is demonstrated.     

The influence of overlapping potentials and short range correlations on high-energy scattering

High-energy nucleon scattering from light nuclei is investigated using several different scattering formalisms. Off-energy-shell effects due to overlapping target particle potentials and short range correlations are included in some of the calculations. The results indicate that the uncertainties in the computed cross section due to choice of scattering formalism, and due to neglect or inclusion of off-energy-shell effects, or two-body correlations are all of comparable size.