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.     

Refractive scattering and reactions,comparison of two systems:16O+16O and20Ne+12C

Elastic, inelastic scattering as well as one-neutron transfer channels have been measured over a wide angular range for systems¹⁶O+¹⁶O at the incident energy of 350 MeV and²⁰Ne+¹²C at 390 MeV, respectively, using the Q3D magnetic spectrometer. In both cases differential cross sections have been measured down to about 50 nb/sr (or d/d _R10^–4) at large angles. For the¹⁶O+¹⁶O system refractive contributions are found at the level of these cross sections, whereas in the²⁰Ne+¹²C case a steeper decrease of the differential cross section with the angle is observed and the refractive contribution can not be determined. The elastic scattering data have been analyzed using standard Woods-Saxon potentials and potentials calculated in different versions of the double-folding model. Some properties of these potentials are tested in the calculations for inelastic scattering and one-neutron transfer within the DWBA. With the refractive pattern observed for the¹⁶O+¹⁶O system, the scattering and transfer data are found to be sensitive to the interaction potential at small internuclear distances down to about 2.5 fm.It should be acknowledged that part of the folding analysis reported here was done while one of the authors (D.T.K.) was staying at the Institute for Theoretical Physics, University of Tübingen. We also thank Prof. H. Clement and H. Abele for numerous discussions and contributions and Prof. G.R. Satchler for helpful comments on the use of the DWBA code PTOLEMY.     

Study of the diffraction scattering 12C + 12C with the excitation of the 12C exotic state 0 2 + (the Hoyle state)

New results from a series of experiments dedicated to the study of the ¹²C exotic state (the so-called Hoyle state) are presented. In spite of the many investigations that have been carried out, the structure of this state (which lies above the threshold for breaking up into three alpha particles) is still unknown. The different models assume that the nucleus has an abnormally large size in this excited state. However, until recently, methods for measuring the radii of unbound states have not been suggested. The best way to solve this problem seems to be by measuring the angular distributions of elastic and inelastic scattering of ¹²C on different target nuclei, and the determination of the radii is based on the fact that, at small scattering angles, the cross sections for direct reactions at high enough energies behave like Frauenhofer diffraction on a black ball. Accordingly, an experiment was performed aimed at measuring the elastic and inelastic angular distributions of ¹²C with an energy of (121.5 ± 0.5) MeV on a ¹²C target. The elastic scattering was measured in the angular range from 18° to 50° in the c.m. system with uncertainty in the angle of measurement equal to Δθ = ± 0.6°. The inelastic cross section was measured for the ¹²C excited state 2⁺ (4.44 MeV) and 0⁺ (7.65 MeV). Estimates were made for the diffraction radii for the ground and excited states. An increase was observed in the radius of the state at 7.65 MeV compared to those of the ground and first excited states.     

Microscopic interpretation of 7Li+24Mg inelastic scattering at 34 MeV

Theoretical differential cross sections for ⁷Li+²⁴Mg inelastic scattering have been calculated in the DWA using a double folding model. The projectile and target transition densities are matched to electron scattering data and the nucleon-nucleon interaction is chosen to reproduce the real part of the optical potential at the distance $\text{D}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ where there is 50 % transmission in the elastic channel. There are no free parameters in the calculations. The agreement between the theoretical results and the available experimental data is good which serves to establish a consistency between elastic and inelastic scattering. The cross section for a mutual transition which leaves both the projectile and target in their first excited states is found to come mainly from relative L_r = 4 transfer. It is speculated that a quadrupole term in the optical potential might make important contributions at backward angles in the elastic cross section.     

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.     

Kernradien von12C,13C,14N und16O aus Elektronenstreuung zwischen 30 und 60 MeV

Elastic electron scattering cross sections of¹²C,¹⁴N and¹⁶O relative to the proton, and of¹³C relative to¹²C have been measured. Using harmonic oscillator wave functions the followingrms charge radiiR _m were deduced by phaseshift calculations: 2.395 (28) fm for¹²C, 2.384 (47) fm for¹³C, 2.492 (33) fm for¹⁵N, 2.666 (33) fm for¹⁶O. The ratioR _m (¹³C)/R _m (¹²C) is 0.995±0.008. The errors given do not include uncertainties from the model dependence of the evaluation which may be of the same magnitude.     

Detailed energy dependence of the elastic and inelastic12C +12C scattering between 5 and 10 MeV/nucleon

Excitation functions for¹²C+¹²C elastic and inelastic scattering to 2⁺ level have been measured over the energy range 30–60 MeV (cm) by 250 keV steps using the kinematical coincidence method. The intermediate structure resonances disappear aboveE _cm=35 MeV while the broad and irregular structure becomes a general feature of the interaction at higher energies.     

Elastic and inelastic scattering of 178 MeV protons from 58Ni and 60Ni

The elastic and inelastic scattering of 178 MeV protons from ⁵⁸Ni and ⁶⁰Ni has been studied. Angular distributions were measured for the differential cross sections for elastic scattering as well as inelastic scattering from excited states below about 5 MeV, all with natural parity. For the elastic and for the inelastic scattering from the first excited state (2⁺ in both nuclei, the angular distributions for the polarization were also measured. The measurements extend out to c.m. angles of about 60°, corresponding to a momentum transfer of about 600 MeV/c.The elastic and inelastic scattering data were compared to the results of coupled-channel calculations in the vibrational model using a deformed spin-orbit interaction of the full Thomas form. Good agreement was found in general showing that the main features of the experimental results are well described in this model.     

Magnetic excitations in12C by inelastic electron scattering at 180°

Nuclear excitations in¹²C have been investigated by inelastic scattering of 50, 65 and 70 MeV primary electrons at 180°. Cross sections were measured of the inelastic peaks observed between 15 and 20 MeV. The behaviour of the corresponding transverse form factor as a function of momentum transfer is compared with theoretical predictions and other experimental¹²C (e, e′) results.     

The 12C+24Mg elastic scattering: An example of anomalous transparency at coulomb barrier energies

Fifteen complete angular distributions of the elastic scattering of ¹²C+²⁴Mg were measured at energies around the Coulomb barrier (E_cm = 10.67–16 MeV). The angular distributions are strongly oscillating and could be well described by an optical potential family, whose real part was determined without continuous ambiguity. The imaginary part of this optical potential is very shallow. At four energies the inelastic scattering angular distributions leading to the 2⁺ state of the ²⁴Mg were also measured and analysed with coupled-channels calculations. The volume integrals of the optical potentials used in the coupled-channels calculations present the threshold anomaly in their energy dependence, with a clear Q-value dependence.     

π+ and π− scattering from 12C at 150 MeV

High-resolution elastic and inelastic data were obtained for π⁺ and π^? scattering ¹²C. Spectra were taken with the SIN pion spectrometer at 148 MeV over a wide angular range.     

The 19.7 MeV resonance in the system 16O + 12C

The 19.7 MeV structure in the system ¹²C + ¹⁶O is investigated by measuring angular distributions and excitation functions for elastic and inelastic scattering with very high resolution. From a quasi phase shift analysis, it is deduced that the structure is a true resonance with J^π = 14⁺. The deduced small elastic partial width indicates that the structure of this state is intermediate between a molecular and a compound state.     

Interpretation of airy minima in 16O+16O and 16O+12C elastic scattering in terms of a barrier-wave/internal-wave decomposition

The observation of refractive effects in ¹⁶O+¹⁶O and ¹⁶O+¹²C elastic scattering data has definitively established the fact that the optical potential for some light heavy-ion systems is relatively transparent and that its real part is deep. Most of the interpretations of the rainbow features of these data rely on the so-called nearside-farside decomposition of the scattering amplitude. Starting from recent optical model analyses of ¹⁶O+¹⁶O and ¹⁶O+¹²C elastic scattering around 100 MeV incident energy as an example, we present an alternative interpretation based on the barrier-wave/internal-wave decomposition first proposed by Brink and Takigawa. This method, which complements the nearside-farside approach, demonstrates clearly the exceptional transparency of the ¹⁶O+¹⁶O, and to a lesser extent ¹⁶O+¹²C, interactions at the investigated energies and makes possible the extraction of the two contributions whose interference explains the Airy oscillations seen in the farside amplitude.     

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).     

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.     

Spins of resonances in the 12C + 16O system

The spins of resonances appearing at 22 MeV c.m. entrance channel energy in the ¹²C + ¹⁶O system are determined. Several inelastic transitions are used and a value of J^π = 15^? is deduced. This value disagrees with a previous J^π = 14⁺ assignment based on elastic scattering, but agrees with the J^π = 15^? value predicted at this energy by a recent microscopic calculation of ¹²C + ¹⁶O scattering.     

Coupling to the elastic channel in electron impact spectroscopy: A simple second born model and its application to excitation of the 61P1 state of mercury and to the excitation of molecules

The matrix element in the infinite channel close coupling approximation responsible for coupling to the elastic channel in electron impact inelastic encounters is investigated. The contribution from the imaginary part of the energy denominator in the elastic coupling matrix element for dipole allowed transitions is shown to yield large angle differential cross sections in good agreement with experiment. This coupling mechanism predicts that the shape of the inelastic differential cross section will be dominated by the shape of the elastic cross section in the large angle high energy limit. In fact the coupling matrix element exhibits a dependence on incident energy, k², and momentum transfer, K, of the form 1/kK² which is in agreement with the theoretical predictions of Huo and means that in the limit of high incident energy the non-first-Born elastic coupling will dominate the angular dependence of the inelastic differential cross section at large scattering angles. In the case of molecular electron impact spectra it is shown that the analog of the Massey—Moore coherence features depending on the symmetry of the states involved in the excitation process will also occur in the coupling contribution. It is suggested that this new mechanism for producing coherent features in inelastic differential cross sections may be the explanation of the coherent features observed experimentally by Karle and Swick.It can be concluded on the basis of the results obtained here that the coupling to the elastic cross section provided by the imaginary contribution from the second Born energy denominator is sufficient to explain presently available experimental data on the large angle differential cross section and spin polarization. The simple coupling model was found to be inadequate to explain the small angle differential cross section in the range 10° < θ < 30° even at incident energies as high as 400 eV. The calculations also showed significant differences between first and second Born calculations at zero scattering angle. No conclusion can be drawn about this observation as all the omitted terms should make significant contributions in the small angle region. It is important to again emphasize that the large angle scattering even in the limit of high incident electron energy will be completely dominated by the coupling to the elastic channel^{7, 11}. On the basis of this work it appears that the coherent structure in the large angle inelastic differential cross section observed by Swick and Karle^{12, 13} at incident electron energies in the keV region may well be due to coupling to the elastic channel.