IWB analysis of scattering and fusion cross sections for the 12C+12C, 13C+16O and 16O+16O reactions for energies near and below the Coulomb barrier

Calculations are presented of the elastic scattering and fusion cross sections for the astrophysically interesting reactions ¹²C+¹²C, ¹²C+¹⁶O and ¹⁶O+¹⁶O. The calculations are performed using the incoming wave boundary condition (IWB) and a real ion-ion interaction potential. The results are compared with the available experimental data for the energy region near and below the Coulomb barrier. With values of two adjustable potential parameters (the radial position of the l = 0 barrier and the diffuseness) determined by fitting elastic scattering data, good agreement is obtained for the average energy dependence of the ¹²C+¹²C and ¹²C+¹⁶O fusion cross sections. In the case of ¹⁶O+¹⁶O, both the calculated absolute magnitude and the energy dependence of the fusion cross section are inconsistent with the data and this discrepancy is discussed.     

Cluster exchange and elastic scattering of heavy ions

The effect of cluster exchange on the nuclear interaction potential is examined. Cluster exchange results in nonlocal interactions that must be added to the direct local potential of the optical model. The dependence of exchange interactions on the parity of relative motion is analyzed. Scattering amplitudes corresponding to nonlocal exchange interactions are calculated in the Born approximation for ¹⁶O + ¹²C scattering.     

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.     

Application of the incoming wave boundary condition to16O+16O and12C+12C elastic scattering

The elastic scattering of¹²C+¹²C and¹⁶O+¹⁶O has been studied in the framework of an incoming wave boundary condition model. Different logarithmic derivatives for the incoming waves have been tested and their effects investigated with the help of Fourier analyses. It is shown that a logarithmic derivative obtained from a JWKB approximation leads to strong absorption of the low partial waves while a logarithmic derivative constant for all partial waves causes reflections. These reflections are necessary to describe the high energy elastic scattering of¹²C+¹²C. The fits thus obtained with shallow real potentials are comparable to those obtained with deep folding potentials. It is shown that not the lowest partial waves, but those within a window just below the grazing angular momentum are most important for the higher energy¹²C+¹²C angular distributions.     

A dependence of inclusive pion double charge exchange

The A dependence of the forward cross section for inclusive pion double charge exchange on nine target nuclei from ⁶Li to ²⁰⁹Bi at T₀ = 0.59 GeV, as well as the cross section for ⁶Li, ⁷Li, and ¹²C nuclei at T₀ = 0.59, 0.75, and 1.1 GeV, was measured with the 3-m magnetic spectrometer of the Institute of Theoretical and Experimental Physics (ITEP, Moscow). The resulting A dependence is well described within the model involving two sequential single charge exchanges and taking into account the renormalization of the amplitude for pion single charge exchange in a nucleus. A relatively weak energy dependence of the cross section for the ⁶Li, ⁷Li, and ¹²C nuclei agrees with the analogous dependence obtained previously for the ¹⁶O nucleus, but it contradicts the predicted sharp decrease in the cross section within the model involving two sequential single charge exchanges. This result provides an additional piece of evidence that the contribution from the mechanism of inelastic Glauber rescattering is significant at T₀ ? 0.6 GeV.     

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.     

Low energy fusion of 12C + 12C and 16O + 16O systems

The adiabatic time-dependent Hartree-Fock method (ATDHF) is applied to the calculation of low energy fusion of ¹²C + ¹²C and ¹⁶O + ¹⁶O systems. The energy dependence of the results is in good agreement with experiment, while the order of magnitude is not correct. It is shown that the dynamical effects included in ATDHF are very important and cannot be neglected at the energies of astrophysical interest.     

Measurement of inclusive pion double charge exchange on 16O at T 0 ≥ 0.5 GeV with the SKS spectrometer at KEK

The inclusive cross section for pion double charge exchange on ¹⁶O at T ₀ = 0.5 and 0.75 GeV was measured with the superconducting kaon spectrometer (SKS) at KEK in a joint ITEP/KEK experiment. The result shows a relatively weak energy dependence of the measured cross section, which is in contradiction with its rapid drop predicted within the conventional model of two sequential single charge exchanges. The data of this experiment agree with the results that were obtained previously from similar measurements at ITEP and which are indicative of a significant contribution from the mechanism of inelastic Glauber rescatterings for T ₀ ? 0.6 GeV.     

Studies of multi-nucleon transfer reactions in <Superscript>90</Superscript>Zr(<Superscript>18</Superscript>O,X) and <Superscript>90</Superscript>Zr(<Superscript>16</Superscript>O,X)

Multi-nucleon transfer reactions in ¹⁸O + ⁹⁰Zr and ¹⁶O + ⁹⁰Zr have been studied at an incident energy of 90 MeV. The energy spectra and angular distributions are measured. The data have been analyzed to obtain cross-section dependence on the number of nucleons transferred and on the ground-state Q-values. In the ⁹⁰Zr(¹⁸O, X); X = ¹⁶O, ^17,16,15,14N, ^14,13,12C, ^12,11,10B, ^10,9,7Be and ^7,6Li reactions, 2n and 2n-correlated transfer cross-sections are observed to be enhanced as compared to the ¹⁶O + ⁹⁰Zr reaction. A detailed comparison in the multi-particle stripping and elastic-scattering cross-section between these two systems are made in order to investigate the possible influence of the two valence neutrons in ¹⁸O nucleus. Diffractional model DWBA calculations, based on the direct surface transfer model, have been performed to understand the reaction mechanism of multi-nucleon transfer to continuum.Received: 12 September 2002, Revised: 24 September 2003, Published online: 27 January 2004PACS: 25.70.Hi Transfer reactions - 25.70.Bc Elastic and quasielastic scattering     

Consequences of weak absorption in light heavy-ion reactions: Molecular resonances and nuclear rainbows

In the elastic angular distributions of the systems composed of tightly bound α-nuclei, ¹²C+¹²C, ¹⁶O+¹⁶O and ¹⁶O+¹²C, the higher-order Airy minima are observed in the energy region between 5 and 10 MeV per nucleon. The obtained optical potentials are deeply attractive and with incomplete absorption. The same potentials successfully describe the lower-energy (molecular resonances) and the higher-energy (first Airy minimum) data.     

Manifestation of exchange effects in heavy-ion interactions

Three different approaches to taking into account exchange effects in heavy-ion collisions are studied. Within the first of them, the lowest eigenstates of the Hamiltonian are treated as forbidden states. In the second approach, the eigenstates of the normalization kernel of the resonating-group model that correspond to zero eigenvalues are treated as forbidden states. The third approach takes additionally into account semiforbidden states. The ¹⁶O + ¹⁶O system is considered. A hybrid approach that combines the methods of discrete and continuous mathematics is developed for calculating the widths of narrow resonance states. The resonance width calculated within the approach that takes into account semiforbidden states proves to be sharply different from the widths obtained within traditional approaches.     

Nuclear rainbow in scattering and reactions and nucleus-nucleus interaction at small distances

A review of recent experiments on the study of the nuclear rainbow phenomenon in scattering and some reactions induced by light heavy ions is given. It includes (i) the study of the differential cross sections of the ¹⁶O + ¹²C elastic scattering at seven ¹⁶O energies between 130 and 281 MeV; (ii) finding of the phenomenological potential deeper than that of folding model; (iii) the first data on ¹⁶O + ¹⁴C elastic scattering; (iv) dispersion relation analysis of the obtained data and observation of abnormal nuclear dispersion; and (v) use of the charge-exchange ¹⁴C(⁶Li,⁶He)¹⁴N reaction to search for pion-condensation effects.     

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.     

Effective potentials and threshold anomaly

The strongE andL dependence of the effective elastic channel potentials is shown to be an implicit radial kinetic energy (ε) dependence. It is also shown that this effective potential satisfies the dispersion relation inε variable at the strong absorption radius. Further, the experimental data for both elastic and fusion channels are consistent with thisL-dependence of the corresponding effective potentials. The effective transfer channel potentials derived using CRC code FRESCO are shown to exhibit strong energy dependence as a result of couplings. The energy dependence of effective transfer strength for¹⁶O+²⁰⁸Pb and¹⁶O+²³²Th systems is determined using the experimental transfer angular distributions.     

Comparison of cross sections for C + O reactions in the second regime of complete fusion

Kinetic energy spectra, angular distributions, and elemental yield distributions have been measured for the ¹² C + ¹⁶ O, ¹² C + ¹⁸ O and ¹³ C + ¹⁷ O reaction products over an energy range from 2 to 7 times the Coulomb barrier energy. A careful kinematic analysis of the evaporation residues and comparisons with statistical-model calculations show that fusion proceeds with full momentum transfer followed by a statistical decay of the compound nucleus. The competition between complete fusion process and peripheral reactions in the ¹² C + ¹⁶ O system is less important than for the ¹² C + ¹⁸ O and ¹³ C + ¹⁷ O reactions. The unexpectedly high ¹² C + ¹⁶ O complete fusion cross sections are related to the possible occurrence of a superdeformation of the ²⁸ Si compound nucleus.     

Static versus energy-dependent nucleus–nucleus potential for description of sub-barrier fusion dynamics of _8~(16)O+_(50)~(112,116,120)Sn reactions

The static and energy-dependent nucleus–nucleus potentials are simultaneously used along with the Wong formula for exploration of fusion dynamics of168O+112,116,12050 Sn reactions. The role of internal structure degrees of freedom of colliding pairs, such as inelastic surface vibrations, are examined within the context of coupled channel calculations performed using the code CCFULL. Theoretical calculations based on the static Woods–Saxon potential along with the one-dimensional Wong formula fail to address the fusion data of168O+112,116,12050 Sn reactions.Such discrepancies can be removed if one uses couplings to internal structure degrees of freedom of colliding nuclei.However, the energy-dependent Woods–Saxon potential model(EDWSP model) accurately describes the sub-barrier fusion enhancement of168O+112,116,12050 Sn reactions. Therefore, in sub-barrier fusion dynamics, energy dependence in the nucleus–nucleus potential governs barrier modification effects in a closely similar way to that of the coupled channel approach.