Double-folding Model Research Articles

Double-folding model analysis of deuteron elastic scattering

The applicability of the Daehnick et al. global deuteron potential and the validity of treating the deuteron as a spin- 1 2 particle has been investigated. A wide range of deuteron elastic scattering data has been fitted with a double folding model utilizing the M3Y interaction. An average normalization of = 0.87 ± 0.05 was found for the interaction. Some evidence was found for a possible energy and mass dependence of N.

Optical model and coupled-channels analysis ofLi7+Si28andLi7+Ca40scattering

Angular distributions for elastic scattering and for excitation of the 1.78 MeV state in $^{28}\mathrm{Si}$ and for elastic scattering and excitation of the 3.73 and 3.90 states in $^{40}\mathrm{Ca}$ by $^{7}\mathrm{Li}$ scattering at a bombarding energy of 45 MeV have been measured and analyzed. Double folding model calculations using a realistic effective nucleon-nucleon interaction similar to that performed for $^{9}\mathrm{Be}$ + $^{28}\mathrm{Si}$ and $^{9}\mathrm{Be}$ + $^{40}\mathrm{Ca}$ scattering have been carried out for the elastic angular distributions and the real potential must be renormalized to yield agreement with the measured cross sections. Coupled channels calculations using a Woods-Saxon potential were performed in an effort to describe the inelastic angular distributions. The extracted deformation parameters are in reasonable agreement with those obtained from light and heavier ion scattering from the same target nuclei. The effect of strong excitation of the 0.48 MeV state in $^{7}\mathrm{Li}$ and of mutual excitation of target and projectile is considered in a qualitative manner.NUCLEAR REACTIONS $^{28}\mathrm{Si}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$)$^{28}\mathrm{Si}$, $^{28}\mathrm{Si}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$*)$^{28}\mathrm{Si}$* $Q=\ensuremath{-}0.48, \ensuremath{-}1.78, \mathrm{and} \ensuremath{-}2.26$ MeV, $E(^{7}\mathrm{Li})=45$ MeV; $^{40}\mathrm{Ca}$ ($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$)$^{40}\mathrm{Ca}$, $^{40}\mathrm{Ca}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$*)$^{40}\mathrm{Ca}$* $Q=\ensuremath{-}0.48, \ensuremath{-}3.73, \mathrm{and} \ensuremath{-}3.90$ MeV, $E(^{7}\mathrm{Li})=45$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, ${\ensuremath{\theta}}_{\mathrm{lab}}=10\ensuremath{-}70\ifmmode^\circ\else\textdegree\fi{}$;; performed optical model calculations using Woods-Saxon potential and double folding procedure, deduced optical model parameters; performed coupled-channels calculations, deduced deformation parameters.

Potential model for fusion and angular momentum localisation

The elastic scattering and fusion cross sections for ten different heavy-ion systems with (A1+A2)<100 have been successfully described with an interaction potential obtained from the double-folding model. Similarly, the (9Be, 8Be) reaction as a representative example for direct processes in the system 9Be+28Si is well reproduced by a DWBA calculation which in turn leads to a localisation of fusion and direct processes in mutually exclusive regions of the angular momentum space.

Folding model analysis of triton and 3He elastic scattering

A double folding model using the M3Y interaction has been successfully applied to a wide range of triton and 3He data. A projectile energy and mass dependence in the exchange potential proportional to E/ A p is shown to be adequate between 11 and 27 MeV/nucleon. Normalization factors for triton and 3He real potentials of ∼ 0.6 and ∼ 0.8, respectively, are in accordance with those of other light projectiles subject to break-up.

Reaction between twoLi6at rest to give threeαparticles

For $^{6}\mathrm{Li}$ beam energies of 2 MeV, $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ coincidence measurements show a large, narrow peak due to the direct reaction $2^{6}\mathrm{Li}\ensuremath{\rightarrow}3\ensuremath{\alpha}$, where both $^{6}\mathrm{Li}$ nuclei contribute one deuteron to make the third $\ensuremath{\alpha}$ particle. The large volume of three-body phase space allows the various parts of the reaction process to be separated from each other. Excellent fits to the data were obtained using a three-body, extended-range, distorted-wave Born-approximation model. Elastic scattering potentials for the calculation were generated by the double-folding technique in combination with $^{6}\mathrm{Li}$ + $^{6}\mathrm{Li}$ elastic scattering measurements for $^{6}\mathrm{Li}$ beam energies from 2 to 5.5 MeV.NUCLEAR REACTIONS $^{6}\mathrm{Li}$($^{6}\mathrm{Li}$, $^{4}\mathrm{He}$)$^{4}\mathrm{He}$$^{4}\mathrm{He}$, $E=2$ MeV; DWBA analysis. $^{6}\mathrm{Li}$($^{6}\mathrm{Li}$, $^{6}\mathrm{Li}$), $E=2.0\ensuremath{-}5.5$ MeV; fitted $\ensuremath{\sigma}(\ensuremath{\theta},E)$ with double-folding model potentials.

Double folding with a density-dependent effective interaction and its analytical approximation

The real part of the optical potential for heavy ion elastic scattering is obtained by double folding of the nuclear densities with a density-dependent nucleon-nucleon effective interaction which was successful in describing the binding, size, and nucleon separation energies in spherical nuclei. A simple analytical form is found to differ from the resulting potential considerably less than 1% all through the important region. This analytical potential is used so that only few points of the folding need to be computed. With an imaginary part of the Woods-Saxon type, this potential predicts the elastic scattering angular distribution in very good agreement with experimental data, and little renormalization (unity in most cases) is needed.NUCLEAR REACTIONS Optical model for nucleus-nucleus, double folding model, nucleon-nucleon effective interaction.

Inelastic scattering ofB11fromCa40

Angular distributions have been measured for inelastic scattering of $^{11}\mathrm{B}$ + $^{40}\mathrm{Ca}$ at ${E}_{\mathrm{lab}}=51.5$ MeV leading to the first ${1/2}^{\ensuremath{-}}$ state in $^{11}\mathrm{B}$ and the first ${2}^{+}$, ${3}^{\ensuremath{-}}$, and ${5}^{\ensuremath{-}}$ states in $^{40}\mathrm{Ca}$. Distorted-wave Born approximation calculations reproduce the data quite well and are of quality equal to that of coupled channels calculations. Along with the conventional, deformed Woods-Saxon potential, the analysis employed the double-folding model to fit the inelastic data. The nuclear deformation lengths of the double-folding model agree well with the experimental $B(\mathrm{EL})$ values.NUCLEAR REACTIONS $E=51.5$ MeV, inelastic ($^{11}\mathrm{B}$, ${\mathrm{\textonehalf{}}}^{\ensuremath{-}}$, 2.12 MeV; $^{40}\mathrm{Ca}$, ${2}^{+}$, 3.90 MeV, ${3}^{\ensuremath{-}}$, 3.74 MeV, and ${5}^{\ensuremath{-}}$, 4.49 MeV). Measured $\ensuremath{\sigma}(\ensuremath{\theta})$, deduced nuclear deformation lengths. DWBA, CC, double-folding model analysis.

A woods-saxon and double-folding optical-model description of 14N elastic and inelastic scattering from 24Mg, 27Al and 28Si

Angular distributions of the elastic and inelastic scattering of 14N from 24Mg, 27Al and 28Si have been measured at energies near 53 MeV. The elastic data were fitted using both a Woods-Saxon and a G-matrix double-folding optical model. The potentials determined from these analyses were used in DWBA calculations of the cross sections for the 1.37 MeV 24Mg and 1.78 MeV 28Si levels. For the double-folding model calculations, matter and transition densities for the targets and projectile were taken from electron scattering and a “realistic” nucleon-nucleon interaction was used. A good description of the elastic scattering data was obtained by both models. The normalization of the real double-folded potential was close to unity and the inelastic scattering data were well reproduced with transition densities close to those obtained from electron scattering.

Double-folding model potentials forLi6elastic scattering at 99 MeV

Double-folding model potentials generated from a realistic nucleon-nucleon interaction and densities that are consistent with electron scattering are used to generate angular distributions for $^{6}\mathrm{Li}$ elastic scattering at 99 MeV. When used to calculate observed angular distributions it is found that the double-folding model potential is roughly twice the potential required to generate a fit to the data. This is consistent with observations at both lower and higher $^{6}\mathrm{Li}$ energies.NUCLEAR REACTIONS $^{12}\mathrm{C}$, $^{28}\mathrm{Si}$, $^{40}\mathrm{Ca}$, $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, $^{208}\mathrm{Pb}$($^{6}\mathrm{Li}$, $^{6}\mathrm{Li}$), $E=99$ MeV; calculated $\ensuremath{\sigma}(\ensuremath{\theta})$ with double-folding model potentials.

A test of the fish bone optical model on?-16O scattering

The recently proposed fish bone optical model is tested on0α-16 O scattering. Using a microscopic interaction similar to that of resonating group calculations the characteristic rotational bands in20Ne as well as the gross structure of the elastic differential cross section are reproduced. It is shown that the fish bone optical model can be extended by inclusion of a phenomenological imaginary part. A numerical comparison with the corresponding double folding model and with the Saito model confirms the theoretical prediction that the fish bone optical model is a further step towards inclusion of the effect of the Pauli principle in the optical model.

Double folding model analysis of 7Li scattering

Elastic scattering angular distributions have been measured for the scattering of 7Li by 44Ca, 56Fe and 58, 60Ni at E( 7 Li) = 34 MeV . These data are well described by the optical model using volume Woods-Saxon potentials. These and previously measured 7Li scattering data also have been analyzed with a double-folding real potential which employs the nucleon-nucleon potential of Bertsch et al. The overall best fit to the experimental data down to σ/σ R ~ 10 −3 is obtained when the DF potential is multiplied by approximately 0.6. However, the best fit for the peripheral data alone ( σ/ σ R ∼ 10 −1 is obtained without any renonnalization.

Phenomenological and microscopic optical potentials for 88 MeV7Li scattering

The elastic scattering cross sections for 88 MeV 7Li ions have been measured for targets of 24,26Mg and 40,48Ca. Analyses using both phenomenological and microscopic optical potentials provide information on the energy dependence of optical parameters, and the extent to which the potentials are determined for these light ions. The use of a double-folding microscopic model demonstrates the need for normalisation of the real potential by a factor of 0.5 in contrast to measurements at lower energies. The contribution of exchange effects density dependence and break-up are discussed.

Folding model analysis of 10, 11B, 12C + 27Al, 39K and 40Ca

Elastic scattering angular distributions for 10, 11B + 40Ca at E lab = 46.6 and 51.5 MeV and 12C+ 39K at E lab = 54 and 63 MeV have been measured and compared with Woods-Saxon and double-folding optical models. The oscillatory structure observed previously for 12C + 40Ca disappears when the projectile is changed to 10,11B or the target is changed to 39K. The angular distributions are adequately reproduced by a double-folding analysis, which employs the nucleon-nucleon potential of Bertsch et al., with a range of real normalizations N R = 1.0–1.38. This same range of real normalizations was also able to describe previously measured 10,11B, 12C + 27A1 data. The double-folding analysis of 12C + 40Ca scattering indicates that this system behaves differently from neighboring systems.

Quadrupole contributions toB10,11+O16elastic scattering

Elastic scattering angular distributions have been measured for the systems $^{10}\mathrm{B}$ + $^{16}\mathrm{O}$ at ${E}_{L}=33.7, 41.6, \mathrm{and} 49.5$ MeV and $^{11}\mathrm{B}$ + $^{16}\mathrm{O}$ at ${E}_{L}=41.6 \mathrm{and} 49.5$ MeV. The $^{10}\mathrm{B}$ and $^{11}\mathrm{B}$ data are oscillatory and similar for angles less than 50\ifmmode^\circ\else\textdegree\fi{} c.m. For angles greater than 50\ifmmode^\circ\else\textdegree\fi{} c.m., the oscillations in the $^{10}\mathrm{B}$ data are damped and the data stop decreasing with increasing angle. Elastic scattering angular distributions have been calculated for $^{10,11}\mathrm{B}$ + $^{16}\mathrm{O}$ with the coupling to the ground-state quadrupole moment included explicitly. Potentials which describe the scattering are extracted with the double folding model. It is found that the quadrupole coupling is necessary to explain the difference in $^{10}\mathrm{B}$ and $^{11}\mathrm{B}$ scattering from $^{16}\mathrm{O}$.NUCLEAR REACTIONS $^{16}\mathrm{O}$ ($^{10}\mathrm{B}$, $^{10}\mathrm{B}$) $^{16}\mathrm{O}$, ${E}_{L}=33.7, 41.6, \mathrm{and} 49.5$ MeV, $^{16}\mathrm{O}$ ($^{11}\mathrm{B}$, $^{11}\mathrm{B}$) $^{16}\mathrm{O}$, ${E}_{L}=41.6 \mathrm{and} 49.5$ MeV. Measured $\ensuremath{\sigma}(\ensuremath{\theta})$, deduced Woods-Saxon, and double-folding model parameters. Coupled channels calculations for the ground-state quadrupole moment contributions.

Elastic and inelastic scattering ofC12byC12fromEc.m.=35−63MeV

Differential cross sections for the elastic scattering, single excitation ($Q=\ensuremath{-}4.43$ MeV), and mutual excitation ($Q=\ensuremath{-}8.86$ MeV) of $^{12}\mathrm{C}$ by $^{12}\mathrm{C}$ have been measured at 14 bombarding energies in the range ${E}_{\mathrm{c}.\mathrm{m}.}=35\ensuremath{-}63$ MeV. The angular distributions extend typically from 16\ifmmode^\circ\else\textdegree\fi{} to 90\ifmmode^\circ\else\textdegree\fi{} c.m. The results have been analyzed with the optical model, the distorted-wave approximation, and the method of coupled equations. The optical model analysis of the elastic scattering reveals a sensitivity of the predicted cross sections at large angles to the strength and shape of the real part of the nuclear potential in the region 4 to 6 fm. The potentials predicted by the double-folding model fit the data remarkably well over the entire angular range, whereas the shallow potentials which we have explored generally fit the data only in the forwardangle region. The results thus indicate that the real potential for the $^{12}\mathrm{C}$ + $^{12}\mathrm{C}$ system is more attractive in the region of 4 to 6 fm than would have been expected on the basis of previous analyses of comparable systems at lower bombarding energies.NUCLEAR REACTIONS $^{12}\mathrm{C}$($^{12}\mathrm{C}$, $^{12}\mathrm{C}$), ${E}_{\mathrm{c}.\mathrm{m}.}=35\ensuremath{-}63$ MeV; measured $\ensuremath{\sigma}(E,\ensuremath{\theta})$ for elastic, single, and mutual excitation of ${2}^{+}$. Optical model, DWBA analyses.

Coupled-channel analysis with double-folding potential of 16O and 18O scattering on Ni and Ge isotopes

The elastic and inelastic scattering of 16O and 18O from 72,74,76Ge and 16O from 64Ni are reanalyzed by performing coupled-channel calculations with a double-folding potential. The double-folding potential model explains the experimental data with fewer adjustable parameters than the usual Woods-Saxon potential.

Optical potential for nucleus-nucleus systems

The electrostatic and nuclear parts of the nucleus-nucleus optical potential are constructed using the double folding model by taking into account nuclear charge and matter distributions, along with a suitably chosen effective nucleon-nucleon interaction. For nuclei with uniform charge distributions, a closed form expression for the electrostatic potential has been obtained earlier. This is found to be a very good approximation to the electrostatic potential corresponding to commonly known realistic charge distributions. Optical model calculations for the $^{16}\mathrm{O}$-$^{16}\mathrm{O}$ system using this improved electrostatic potential are reported which provide a slightly better fit to elastic scattering data. The six dimensional integral for the nuclear part of the optical potential is reduced to a three dimensional integral for the case of spherically sysmmetric density distributions and this leads to substantial reduction in computations. In contrast with the electrostatic part, it is found that the nuclear part of the optical potential is quite sensitive to the shape of the density distributions. The optical potential constructed by the "double folding" method with only a single adjustable parameter corresponding to the strength of the imaginary part, is used to fit successfully $^{12}\mathrm{C}$-$^{12}\mathrm{C}$ scattering data at ${E}_{\mathrm{lab}}=102.1$ MeV.NUCLEAR REACTIONS Optical model for nucleus-nucleus system, double folding model, realistic electrostatic potentials.

Functional integral representation of Glauber theory for high-energy heavy-ion scattering

A functional integral representation is presented for the Glauber theory amplitude for high- energy heavy-ion scattering with arbitrary nucleon-nucleon interactions, closely related to that previously obtained by Winbow using Reggeon field theory. As in Winbow's paper the amplitude can be evaluated in the large- A limit using a saddle point technique. The evaluation is carried out in a simple case: it predicts less absorption than does the double-folding model.

Co59(O16,O16)Co59elastic scattering

Differential cross sections for the $^{59}\mathrm{Co}$($^{16}\mathrm{O}$, $^{16}\mathrm{O}$)$^{59}\mathrm{Co}$ elastic scattering have been measured at laboratory energies 36, 40, 45.5, 49, 52, and 56 MeV. The angular range was 15\ifmmode^\circ\else\textdegree\fi{}-170\ifmmode^\circ\else\textdegree\fi{} (lab) in 5\ifmmode^\circ\else\textdegree\fi{} steps at lower energies. At higher energies detailed angular distributions were carried out in 1\ifmmode^\circ\else\textdegree\fi{} steps until the ratio to Rutherford was ${10}^{\ensuremath{-}3}$ with sample measurements at larger angles to assure that the differential cross sections were less than ${10}^{\ensuremath{-}4}$ ratio to Rutherford. The data have been analyzed with a four parameter optical model and the continuous ambiguities in this model are discussed. Strong absorption radii and the values of the nuclear potentials at these points are determined. Good fits to the data have been obtained using a $G$-matrix double-folding model with double-folded or Woods-Saxon forms for the imaginary potential. The required normalization of the real double-folded potential is near unity for all energies. Total reaction cross sections have been obtained by the optical model, quarter-point, "half partial-wave," and the sum of differences methods. The sum of differences method gives slightly larger total reaction cross sections than do the other methods by an amount consistent with estimates of the Coulomb excitation cross section. Estimates of the errors in the sum of differences values are computed by applying the optical theorem and are shown to be negligibly small.NUCLEAR REACTIONS: $^{59}\mathrm{Co}$($^{16}\mathrm{O}$, $^{16}\mathrm{O}$) $^{59}\mathrm{Co}$; measured $\ensuremath{\sigma}(\ensuremath{\theta}, E)$, ${E}_{\mathrm{lab}}=36, 40, 45.5, 49, 52, \mathrm{and} 56$ MeV; measured total reaction cross sections.

Inferences concerning the real part of the heavy-ion optical potential through folding techniques

Analytic solutions are given for the real part of the heavy-ion optical potential through the use of single and double folding models. The effective potential between the nucleons is in the form of a sum of Yukawa terms and the distributions of the densities and the nucleon optical potential are taken to be of Woods-Saxon form. Consideration is given to the influence of the tails of the density distribution as well as that of the effective potential. Calculations suggest there is little need to delineate effective interactions with momentum components much beyond the Fermi level. This provides a rationale for realistic treatments of the effective interaction. A crucial parameter specifying the density distribution is found to be the rms radius.