Proton-nucleus elastic scattering at 156 MeV

Cross sections for the elastic scattering of 156 MeV protons on eleven targets ranging from ¹²C to ²⁰⁹Bi were measured and an optical model analysis has been performed. The effect of different optical potentials in DWIA inelastic scattering calculations is shown by some examples.     

High energy proton-carbon scattering in the alpha-particle model

We investigate elastic and inelastic 0⁺–2⁺ high energy proton-¹²C scattering in the alpha-particle model. We use a rigid equilateral triangle nuclear wave function with a Gaussian dispersion function allowing theα-particles of the¹²C-nucleus to deviate from their most probable positions at the triangle vertices. Expressions for the differential scattering cross sections are deduced using Glauber multiple diffraction theory. Thus we need thep?α-particle scattering amplitude, which is calculated from a Gaussian nucleon-nucleon profile function. Numerical calculations show that the model reproduces the experimental results onp?α andp-¹²C scattering.     

The He scattering and reactions on C and cluster states of C

The ⁶He+¹²C elastic and inelastic scattering and the ⁶He+¹²C→α+¹⁴C reaction have been measured using a 18.0 MeV ⁶He beam. Experimental results for the elastic scattering are in fair agreement with optical model predictions, using the potentials found in the analysis of ⁶Li scattering on ¹²C at similar energies. In triple coincidences, the ⁶He+¹²C→¹⁰Be+2α reactions were clearly seen, with the ¹⁰Be nucleus left in ground and several excited states. The dominant mechanism of this reaction is sequential decay through cluster states of ¹⁴C.     

Scattering and reactions of 14C + 14C and 14C + 14C

We have studied elastic scattering, inelastic scattering and several transfer channels of the systems ¹⁴C + ¹⁴C and ¹⁴C + ¹²C over a wide range of energies up to E_c.m. = 35 eMeV and 32 MeV, respectively. The reaction channels were identified by means of kinematic coincidences between solid-state detectors, γγ coincidences were measured to determine cross sections for mutual inelastic scattering of ¹⁴C + ¹⁴C.Pronounced regular gross structures, similar to those found for ¹⁶O + ¹⁶O, are observed in the elastic excitation function of ¹⁴C + ¹⁴C at θ_c.m. = 90°, The angular distributions measured at the energies of the maxima and an optical-model analysis suggest that one or a few surface partial waves dominate the scattering behaviour. Correlated structure of narrower width is found in the inelastic channels and, to a lesser degree, in the transfer channels which appear with rather small cross sections.In ¹⁴C + ¹²C elastic scattering the gross structures are strongly fragmented, in contrast to ¹⁴C + ¹⁴C but similar to ¹²C + ¹²C. While the ¹²C(2⁺) excitation is very weak, the observed strengths of the ¹⁴C(3^?) excitation and of neutron transfer point to a substantial role of these channels as coupling partners to the elastic configuration and to their influence on the elastic scattering behaviour. A correlated intermediate structure is observed near 23.5 MeV, where a dominance of l = 18 is suggested by the elastic scattering angular distribution. This unexpectedly high l-value exceeds l_graz at this energy by at least two units of ?.     

Breakup effects of 6,7li on elastic and inelastic scattering from 12C at 18–28 MeV/nucleon

The differential cross sections for elastic scattering of ^6,7Li from ¹²C and inelastic one from the lowest three excited states of ¹²C have been measured at bombarding energies of 18–28 MeV/nucleon. Theoretical analyses of the data have been performed in which consistent treatments of density distributions for the ground and excited states of both projectile and target nuclei are made in the framework of microscopic cluster models for ^6,7Li and ¹²C and projectile-target interactions are generated by the double folding of the M3Y effective nucleon-nucleon interaction. About 25% reduction of the real part of folded potentials is required both in the analyses of elastic scattering with the single-channel calculation and in those of inelastic scattering with the coupled-channel calculation including the excited states of ¹²C. This reduction can be explained as a projectile breakup effect on elastic and inelastic scattering in comparison with a coupled-discretized-continuum-channels (CDCC) calculation and an extended CDCC one which allows mutual excitations of both projectile and target nuclei for ⁶Li case, respectively. It is also seen that an effect due to the target excitation on elastic scattering is of less importance than that of the ^6,7Li projectile breakup processes even fora deformed nucleus like ¹²C. Discrepancy between the extended CDCC calculation and inelastic data for the 0⁺₂ state of ¹²C suggests a strong influence from the ¹²C → 3α breakup channels in the ⁶Li case.     

Channel coupling and exchange of an alpha-particle cluster in deuteron scattering on 6Li nuclei

Existing experimental data on elastic and inelastic deuteron scattering on ⁶Li nuclei in the energy range from 8 to 50 MeV were analyzed within the approach of coupled reaction channels. The coupling of elastic scattering and inelastic scattering accompanied by the transition to the 3⁺ state at E _x = 2.186 MeV and the mechanism involving the exchange of an alpha-particle cluster were taken into account in respective calculations. The phenomenological potentials obtained from the present analysis describe well experimental angular distributions at all energies and in full angular ranges. The depths of the real and imaginary parts of the potentials in question depend smoothly on energy at fixed values of the remaining parameters. The energy dependence of relevant volume integrals agrees well with similar data for the p + ⁶Li, ?? + ⁶Li, and ¹²C + ¹²C systems and with the predictions of a microscopic theory.     

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.     

Elastic and inelastic scattering of nucleons from 16O

Measurements of differential elastic and inelastic cross sections for neutron scattering from ¹⁶O at incident energies 18 to 26 MeV are presented. In addition to cross sections for neutron scattering differential cross sections for proton scattering up to 66 MeV are described in terms of phenomenological optical model potentials. At 24.5 MeV incident energy inelastic scattering up to 11.5 MeV excitation was measured. The elastic and inelastic compound nucleus contributions were examined. Direct inelastic scattering from the normal parity states was calculated using the DWBA and coupled-channel formalisms. The inelastic scattering cross section from non-normal parity state 2⁻ was calculated using the coupled-channel formalism via multi-step processes. Cross sections due to inelastic scattering from some of the states, which are thought to be members of an excited state rotational band were calculated using both vibrational and rotational approaches and were compared.     

Folding model analysis of pion elastic and inelastic scattering from 6Li and 12C

π ^±-Nucleus scattering cross sections are calculated applying the Watanabe superposition model with a phenomenological Woods-Saxon potential. The phenomenological potential parameters are searched for π ^± scattering from ⁶Li and ¹²C to reproduce not only differential elastic cross sections but also inelastic and total and reaction cross sections at pion kinetic energies from 50 to 672 MeV. The optical potentials of ⁶Li and ¹²C are calculated in terms of the alpha particle and deuteron optical potentials. Inelastic scattering has been analyzed using the distorted waves from elastic-scattering data. The values of deformation lengths thus obtained compare very well with the ones reported earlier.     

Observation of the nuclear rainbow scattering for12C+12C atE Lab =300 MeV

The elastic and inelastic scattering of¹²C on¹²C has been measured in the angular range between 2.8° and 70.4° in the c.m. system atE _Lab =300 MeV. Optical model calculations have been performed with Woods-Saxon and folded potentials, the ground state and the first 2⁺-state were coupled in the calculations. The large cross sections of the elastic scattering at large angles is related to the nuclear rainbow scattering, which is centered at about 56°. This requires a potential depth of 100 MeV at a distance of 3 fm, the fit to the data is sensitive down to this region. The calculations with the folded potential show a better agreement with the data than those with the Woods-Saxon shape. The total reaction cross section of 1,420 mb, obtained from the optical model analysis, corresponds to the geometrical value; no transparency is observed.     

Phenomenological local potential analysis of π+ scattering

π⁺-nucleus scattering cross section are calculated by solving a Schrödinger equation reduced from the Klein-Gordon equation. Local potentials are assumed, and phenomenological potential parameters are searched energy dependently for π⁺ scattering from ¹²C, ⁴⁰Ca, and ²⁰⁸Pb to reproduce not only differential elastic cross sections but also inelastic and total and reaction cross sections at 800 MeV/c pion laboratory momentum. The collective model is used to calculate the angular distributions of differential inelastic cross sections for pions leading to the lowest 2⁺ and 3^? states of ¹²C. The deformation parameters and lengths are extracted and compared to the corresponding ones from other works. Local potentials well describe the scattering of pions from nuclei.     

Coupled-channels analysis of K<Superscript>+</Superscript> scattering from <Superscript>6</Superscript>Li and <Superscript>12</Superscript>C using the Watanabe superposition model

Coupled-channels calculations for the elastic and inelastic scattering of K⁺ at 715 MeV/c by ⁶Li and ¹²C at 635, 715 and 800 kaon Lab momenta have been analysed. The optical potentials of ¹²C and ⁶Li are calculated in terms of the alpha-particle and deuteron optical potentials. Good fits to the experimental data and phenomenological calculations are obtained for ⁶Li and ¹²C nuclei.     

Analysis of inelastic scattering of3He on14C at 37.9 MeV

The differential cross sections of elastic and inelastic scattering of³He ions on the¹⁴C nucleus have been measured at an energy of 37.9 MeV. By fitting the shape of the measured angular distribution of the elastic scattering the parameters of the optical model have been found. These parameters have been used for the standard DWBA calculations of angular distributions corresponding to excitations of the¹⁴C levels 6.73(3^?), 7.01(2⁺) and 8.32(2⁺) MeV and for coupled channels calculations of the level 8.32(2⁺) MeV. The vibration parametersβ _L of the¹⁴C nucleus have been deduced.     

The contribution of nuclear inelastic excitation to the optical potential for heavy ions

Using the plane-wave approximation we derive analytical expressions for both the real and imaginary parts of the polarization potential arising from nuclear inelastic scattering. These potentials and the resulting elastic and inelastic cross sections are compared with exact coupledchannel calculations for ¹³C on ⁴⁰Ca at 68 MeV. The agreement, for the most part, is good. We also briefly discuss the numerical non-local potentials for this system and the imaginary polarization potential for ¹⁶O on ²⁰⁸Pb at 104 MeV.     

Quadrupole moments and nucleon excitation strengths in even-A Sm isotopes

We present a coherent coupled-channel analysis of 7 MeV neutron and 16 MeV proton elastic and inelastic scattering from ^{148, 152, 154}Sm. The optical potential and nuclear deformation parameters are determined so as to fit not only these elastic and inelastic scattering data but also the low-energy neutron scattering properties and the total cross sections over a wide energy range. This analysis provides evidence of the same excitation strengths for both projectiles in the case of ^{152, 154}Sm, and of a smaller excitation strength for the proton than for the neutron in case of ¹⁴⁸Sm. Moreover the quadrupole moments of these deformed optical potentials are in good agreement with those extracted from Coulomb excitation measurements and from nuclear matter distribution calculations.     

The elastic and inelastic scattering of 33 MeV polarized 3He from oxygen and iron isotopes

Angular distributions of differential cross sections and analysing powers have been measured for elastic and inelastic scattering of polarized 33 MeV ³He from ¹⁷O, ¹⁸O, ⁵⁴Fe and ⁵⁶Fe targets. The elastic data have been analysed to deduce the optical model parameters. The inelastic scattering together with the elastic scattering have been interpreted in terms of a macroscopic model within a coupled-channels framework. From this analysis the deformation parameters, β, were deduced.     

Elastic and inelastic scattering of 58Ni +90,94Zr

Pure elastic and inelastic scattering cross sections have been measured for the systems ⁵⁸Ni +^90,94Zr at energies near the Coulomb barrier where not only quasi-elastic and fusion but also deep-inelastic process come into play. Coupled channels calculations including both projectile and target inelastic excitations can successfully explain the elastic and inelastic scattering angular distributions with an energy-independent semi-empirical bare potential. The calculation reproduces also the sum of the total quasi-elastic, fusion and deep-inelastic cross sections. Received: 14 September 1998 / Revised version: 21 October 1998     

Folding model potentials from realistic interactions for heavy-ion scattering

The double-folding model, with “realistic” nucleon-nucleon interactions based upon a G-matrix constructed from the Reid potential, is used to calculate the real part of the optical potential for heavy-ion scattering. The resulting potentials are shown to reproduce the observed elastic scattering for a large number of systems with bombarding energies from 5 to 20 MeV per nucleon. Some representative inelastic transitions are also reproduced. Exceptions are the elastic scattering of ⁶Li and ⁹Be for which the folded potentials must be reduced in strength by a factor of about two.The same effective interactions are shown to give a good account of two particular cases of alpha scattering as well as some cases of nucleon-nucleus scattering. Some typical examples of inelastic heavy-ion scattering are also predicted successfully.Some general properties of the folding model are reviewed and its theoretical basis is discussed. An explicit density-dependence is examined for one particular realistic interaction and found not to change the results. Single nucleon exchange is included in an approximate way and its importance is studied.In addition to being a study of the folding model, this work also provides a systematic and comprehensive optical model analysis of heavy-ion elastic scattering in this energy range.     

Elastic and the 2+ inelastic scattering of high energy protons on12C and the alpha-particle model

The alpha-particle model has been applied to calculate the elastic and the 2⁺ inelastic angular distribution of 1 GeV protons on¹²C within the framework of Glauber multiple scattering theory. The results are compared with the recent experimental data and with some previous calculations. The model gives a fairly good account of the inelastic data and its predictions for the elastic scattering are essentially the same as those of the shell model.