Long range absorption and other optical-model effects from strong inelastic coupling

An optical potential component is constructed to represent the effect of a strongly coupled inelastic excitation upon elastic scattering. In the particular case of quadrupole Coulomb excitation a long range imaginary potential component is derived in closed form. The effects of long range absorption upon the elastic scattering are considered in a general way by inserting this potential into a weak absorption model and deriving an elastic scattering cross section in closed form. Below the Coulomb barrier the formula takes a simple form which may be related to the semiclassical theory of Coulomb excitation. The potential component arising from nuclear excitation of an inelastic state may be evaluated numerically on a computer. Two examples computed (50 MeV α-scattering on ¹⁵⁴Sm and 60 MeV ¹⁶O scattering on ⁴⁰Ca) exhibit strong l-dependence in the potential component.     

Excitation of the quasi-bound state in 4He by electron scattering at medium momentum transfer

The cross sections of inelastic electron scattering on ⁴He leading to the quasi-bound state (0⁺, ε = 20.1 MeV) were measured in the momentum transfer range 0.8 fm^?2 < q² < 2.4 fm^?2. We found it to be a pure longitudinal transition and give the form factor of this C0transition.     

Test of vibrational current distributions in transverse electron scattering

The transition current distribution for the low-lying 2⁺ state at 4.085 MeV in ²⁰⁸Pb obtained in a realistic cranked shell-model calculation is used to calculate the transverse PWBA form factor for inelastic electron scattering to this state. By keeping the same density but changing the velocity field to that of irrotational, incompressible flow the sensitivity of the form factor to the dynamics of nuclear vibrational motion is explored. It is found that the two flow patterns lead to significant, observable differences in the form factors.     

High energy elastic and inelastic p-12C scattering and nuclear deformation

Starting from a microscopic deformed picture of the ¹²C nucleus, and using angular momentum projected wave functions, the elastic and inelastic 1 GeV p-scattering differential cross sections have been analysed within the framework of the Glauber theory. From a comparison of the results obtained in a full Glauber calculation and in the optical limit, it has been shown that the elastic scattering and, to a lesser extent, the transition to the 2⁺(4.44 MeV) level are only weakly affected by the long-range correlations. In contrast the scattering to the 4⁺(14.08 MeV) state includes crucial multi-step contributions which affect the differential cross section both in shape and magnitude. Similarly the corrections to the DWIA are essential for the transition to the 3^?(9.64 MeV) state and a satisfactory explanation of the qualitative differences observed between the 2⁺ and 3^? inelastic cross sections in ¹²C is given. The great sensitivity of the cross sections to the nuclear deformation is shown and a generally good agreement with experiment for both the electron form factors and 1 GeV p cross sections has been obtained for the ground-state intraband transitions using a single oblate intrinsic state. The importance of a correct treatment of the rotational motion through the angular momentum projection is underlined and the use of the adiabatic approxmation is critically examined.     

Elastic and inelastic scattering of protons from Li between 25 and 45 MeV

The elastic and inelastic scattering of protons from ⁶Li has been studied at incident energies of 25.9, 29.9, 35.0, 40.1 and 45.4 MeV. The 2.18 MeV (3⁺, T = 0) first excited state of ⁶Li was found to be strongly excited, but the 3.56 MeV (0⁺, T = 1) second excited state was quite weakly excited. Angular distributions for excitation of the 2.18 MeV level were measured at all five energies, while angular distributions for excitation of the 3.56 MeV level were extracted only at 25.9 and 45.4 MeV. To test the applicability of the optical model for the scattering of protons from such a light nucleus the elastic scattering angular distributions have been analyzed using the eleven-parameter search code SEEK. Available polarization angular distributions were included in the analysis. Reasonable fits to the data have been obtained with an average geometry potential. Theoretical estimates of the real part of the optical potential and the inelastic scattering differential cross sections have been made using the microscopic model for proton-nucleus scattering. Both phenomenological and realistic forces have been considered and the necessary nuclear transition densities have been extracted from experimental elastic and inelastic electron scattering data. An estimate of a possible spin-spin term in the optical potential has also been made.     

Large-angle 6Li + 28Si elastic and inelastic scattering at 27 and 34 MeV

Elastic and inelastic scattering data extending to θ_c.m ≈ 175° are reported for ⁶Li + ²⁸Si at 27 and 34 MeV. Optical model analyses of the elastic data were made using a variety of real potential forms. The large-angle data cannot be fitted with a Woods-Saxon real potential, but are well described by Woods-Saxon squared, double-folded or Fourier-Bessel potentials. The real potential is the same at both energies, but the imaginary potential is weaker at 27 MeV. The inelastic data were analyzed using the DWBA and coupled channels techniques with folded real form factors and deformed Woods-Saxon imaginary potentials, with the deformations taken from electron scattering. The 2⁺ state was fitted well at both energies with the DWBA, while the prediction decreased too rapidly at large angles for the 4⁺ state. The large-angle 4⁺ data were better described when two-step excitations were included in the coupled-channels calculations. The forward-angle 2⁺ data are sensitive to the interference between Coulomb and nuclear scattering and show that the nuclear and Coulomb deformation parameters β₂ are equal for this transition.     

Quenching of the electron scattering form factor of the 1+ state at 3.48 MeV in88Sr and the influence of theΔ(1232)-isobar

The form factor for excitation of the 1⁺ state at 3.48 MeV in⁸⁸Sr by inelastic electron scattering has been measured for momentum transfersq=0.24–0.62 fm^?1. Neither its magnitude nor shape can be described employing the best available nuclear wave functions. We demonstrate with a schematic model that the observed reduction of the form factor may be understood by taking into account a renormalization of theM1-operator due to virtualΔ-hole excitations.     

Scattering of polarized protons by iron and nickel isotopes

Elastic and inelastic scattering of polarized protons (E = 17.2, 20.4, 24.6 MeV) by ^{54, 56}Fe and ^{58, 60, 62} Ni have been investigated. Most data can be readily accommodated by standard optical-model and DWBA procedures, including full Thomas coupling. The 2⁺₁ state (1.41 MeV) in ⁵⁴Fe is peculiar because the inelastic scattering data require deformation parameters for the central and spin-orbit parts of the nuclear potential that differ by a factor 2 to 3; moreover, this anomaly shows a marked energy dependence.     

Unelastische Elektronenstreuung am 1,78 MeV- und 11,4 MeV-Niveau des Si28

TheE2 transition at 1.78 MeV and the strongM1 transition at (11.42±0.02) MeV (measured excitation energy) in Si²⁸ have been studied by inelastic electron scattering at the Darmstadt linear accelerator. Primary energies between 30 and 56 MeV, and scattering angles from 104° to 165° were used. In Born approximation, the following radiation widths to the ground state have been deduced: (1.21±0.17) · 10^?3eV (1.78 MeV,E2), and (32.4±4.5)eV (11.4 MeV,M1). Transition radii have been determined from the dependence of transition probability on momentum transfer.     

Spectroscopy of 41Sc by resonant scattering of protons: Observation of an intermediate structure

Excitation functions of elastic and inelastic scattering of protons on ⁴⁰Ca have been measured for incident proton energies between 4.8 and 8.2 MeV. Spin and parity assignments and partial width determinations of ⁴¹Sc states between 5.8 and 8.7 MeV excitation energy have been made on the basis of quantitative analyses of the data for the different inelastic channels. Intermediate structure has been identified in the excitation function for the inelastic scattering to the 3^? (3.73 MeV) state in ⁴⁰Ca. This has been interpreted as being due to the presence of a state in ⁴¹Sc which consists primarily of the core nucleus ⁴⁰Ca in its 3^? state with a 2p particle weakly coupled to it.     

Test of correlation expansions for inelastic scattering of protons from 16O and 40Ca in Glauber theory

Two different versions of a correlation expansion for the A-body nuclear transition density required to evaluate the Glauber amplitude for inelastic proton-nucleus scattering are tested. Antisymmetrized oscillator wave functions, containing only Pauli correlations, are used to calculate the “exact” amplitude as well as various terms in the correlation expansions for the excitation of the 3^? (6.13 MeV) state of ¹⁶O and the 5^? (4.49 MeV) state of ⁴⁰Ca. The leading term in both expansions, which ignores all correlations and corresponds to the Glauber theory version of the DWIA, leads to errors which are larger than present experimental errors, especially at large momentum transfers. In one version of the correlation expansion, due to Alkhazov et al., the first-order correction contains both elastic and inelastic two-body correlations and leads to satisfactory results. In the other version, used by Abgrall et al., the first-order correction contains only inelastic two-body correlations. In this case the first- and second-order corrections are needed to obtain accuracy comparable to that of the latest experiments.     

Elastic and inelastic scattering of 120 MeV 18O from 208Pb and the spin alignment of the 2+ state of 18O

The elastic and inelastic scattering of ¹⁸O ions at 120 MeV from a target of ²⁰⁸Pb have been studied. Cross sections for excitation of the 2⁺ state at 1.982 MeV in ¹⁸O and of the 3^? state at 2.61 MeV in ²⁰⁸Pb were measured. In addition, the populations of the m-substates for the ¹⁸O excitation were deduced from the Doppler-broadened line shapes. The data were subjected to a coupled-channels analysis using either Woods-Saxon or folding-model potentials. In addition, the ¹⁸O excitation was found to be described very well by use of a semi-microscopic model. The analyses consistently indicated the presence of a positive static hexadecapole moment of several e · fm⁴ for the 2⁺ state of ¹⁸O. The m-substate population distributions were found to be better fitted if a vector spin-orbit coupling was introduced for the 2⁺ state of ¹⁸O with a sign opposite to that for the nucleon-nucleus case.     

Inelastic scattering studies on 50Cr

The inelastic scattering of protons, deuterons, τ and α from⁵⁰Cr has been studied. Spins, parities and transition strengths have been inferred for many states. The 1.89 MeV 4⁺state is excited by a double-excitation process, but more weakly than expected for a two-phonon state. A weak 5^? state is found at 4.37 MeV, and a 5⁺ state is inferred at 3.792 MeV.     

Elastic and inelastic scattering of 185 MeV protons from 12C

The polarization of 185 MeV protons in elastic scattering and in the excitation of the 2⁺ state at 4.44 MeV and the 0⁺ state at 7.65 MeV in ¹²C has been measured in the angular region 2°–60°. Optical model calculations are performed for the elastic scattering. Angular distributions for the inelastic scattering from the 2⁺ state at 4.44 MeV and the 3^? state at 9.64 MeV are calculated in the distorted-wave impulse approximation (DWIA) as well as in the distorted-wave Born approximation (DWBA).     

Line shape and excitation strength of the first excited state in9Be

The line shape and the excitation strength of the very weak first excited J ^π =1/2⁺ state at E_x=1.684 MeV in Zeitschrift für Physik Zeitschrift für Physik⁹Be has been investigated with high-resolution inelastic electron scattering at E₀=45 and 49 MeV and scattering angles θ=105°, 117°, 129° and 165°, and with high-resolution inelastic proton scattering at E₀=13MeV and θ=15° and 18°. Due to lying just above the neutron threshold the level has a strongly asymmetric line shape which in both experiments can be described consistently with a Breit-Wigner expression modified on the low energy side by the threshold behaviour of the cross section. The resonance energy is E_R=1.684 ± 0.007 MeV and the width T=217± 10 keV in thec.m. system. A single particle potential model calculation reproduces the line shape and the resonance parameters fairly well. In addition, the inelastic electron scattering form factor has been measured. In the range of momentum transfersq =0.24-0.46 fm^?1 it is dominated by a 0p_3/2→ 1s_1/2 particle-hole transition. The transition is mainly longitudinal and of isoscalar nature with a strength of B (E1)↑ =0.027 + 0.002 e² fm², but a small M2 contribution ofB(M2)↑=8.8 ±1.5 μ _N ² fm² has also been detected.     

46.8MeV的p+12C非弹性散射

在Glauber多重散射理论框架下，使用跃迁密度方法和三种N湮没势，计算了46.8MeV的反质子在¹²C上的非弹性散射微分截面. 理论曲线与实验数据符合得较好. 关键词： Glauber理论 反质子 非弹性散射 湮没势 跃迁密度     

Application of a folding-model optical potential to analyzing inelastic pion–nucleus scattering and the in-medium effect on a pion–nucleon amplitude

The folding-model optical potential is generalized in such a way as to apply it to calculating the cross sections for inelastic scattering of π ^±-mesons on ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, and ²⁰⁸Pb nuclei at the energies of 162, 180, 226, and 291 MeV leading to the excitation of the 2⁺ and 3^? collective states. In doing this, use is made of known nucleon-density distributions in nuclei and the pion–nucleon scattering amplitude whose parameters were obtained previously by fitting the elastic scattering cross sections for the same nuclei. Thus, the values of quadrupole (β ₂) and octupole (β ₃) deformations of nuclei appear here as the only adjustable parameters. The scattering cross section is calculated by solving the relativistic wave equation, whereby effects of relativization and distortion in the entrance and exit scattering channels are taken exactly into account. The cross sections calculated in this way for inelastic scattering are in good agreement with respective experimental data. The importance of the inclusion of in-medium effects in choosing parameters of the pion–nucleon amplitude is emphasized.     

Elastische und inelastische Streuung von 104 MeV Alpha-Teilchen an58,60,62,64Ni

Differential cross sections are measured for the elastic and inelastic scattering of 104 MeV α-particles from^{58, 60, 62, 64}Ni. The experimental results are analyzed in terms of coupled channels on the basis of an anharmonic vibrational model and prove to be quite sensitive to the values of the deformation parameters. For forward angles the 4⁺ angular distributions of^58,60Ni are dominated by aL=4 single excitation mechanism. This is in keeping with theE4-transitions observed in (e, e′) scattering. From the 2 ₁ ⁺ - and 3 ₁ ⁺ -results transition rates are derived which can be compared to results of electromagnetic methods and of inelastic proton scattering studies. The comparison indicates that the transition rates differ and are generally higher for the proton scattering. Especially in the case of the 3^? states of^58,60Ni the differences are obvious. The elastic cross sections are analyzed both on the basis of the usual phenomeno-logical model and in terms of a semimicroscopic folding model resulting in values of rms-radii for the nuclear matter distribution.     

Cluster Folding and Coupled-Channels Analysis of 16O + 16O Elastic and Inelastic Scattering

The data of elastic and inelastic ¹⁶O?+?¹⁶O scattering to the lowest 0⁺, 2⁺, and 3^? excitation channels of ¹⁶O in the energy range of 250–1,120 MeV is reanalyzed using the derived double folding cluster (DFC1) potential in the framework of the coupled-channels (CC) mechanism. The potential provides a very good agreement with the experimental data for the elastic and inelastic scattering angular distribution. Furthermore, the deduced deformation length and quadrupole and octupole deformation parameter are quite consistent with the corresponding electromagnetic measurements.