Folding Model Calculations Research Articles

Reply to "Comment on 'Elastic scattering of 318 MeV 6Li from 12C and 28Si: Unique phenomenological and folding-model potentials' and the validity of the M3Y effective interaction"

The Comment addresses a long standing problem concerning folding model calculations for the elastic scattering of $^{6}\mathrm{Li}$ ions. We agree that the low-energy form of M3Y potential for the knockon exchange recommended by the authors of the Comment does require a normalization of \ensuremath{\sim}0.7 for the real folded potential. However, the prescription used in our paper, which better reflects the phenomenology of higher-energy nucleon scattering, gives results that are consistent with a normalization of unity.

Transverse and longitudinal polarized-neutron, polarized-93Nb scattering, and the tensor spin-spin potential.

Spin-spin cross sections were measured for 7.5-MeV polarized neutrons incident on cryogenically polarized $^{93}\mathrm{Nb}$. Measurements were made with spin orientations transverse and longitudinal to the beam direction, yielding ${\mathrm{\ensuremath{\sigma}}}_{\mathit{T}}^{\mathit{s}\mathit{s}}$=-5.2\ifmmode\pm\else\textpm\fi{}5.3 mb and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{L}}^{\mathit{s}\mathit{s}}$=-2.2\ifmmode\pm\else\textpm\fi{}6.9 mb, respectively. The results are consistent with recent valence-nucleon folding model calculations. The tensor spin-spin potential is found to be small, ${\mathit{V}}_{12}$=-67\ifmmode\pm\else\textpm\fi{}190 keV. Volume integrals of spin-spin potentials are derived from depolarization and transmission experiments on $^{13}\mathrm{C}$, $^{15}\mathrm{N}$, $^{27}\mathrm{Al}$, and $^{93}\mathrm{Nb}$. The results for the central spin-spin potential are shown to be consistent with the folding model predictions.

A study of 36S by high-resolution inelastic proton and alpha scattering

In an inelastic hadron scattering study of 36S both a high resolution (p, p′) experiment at E p = 28.0 MeV and an (α, α′) experiment at E α = 31.6 MeV were performed. A highly enriched target was used. Angular distributions for levels up to 6.55 MeV have been obtained. Five previously unknown levels have been observed. A macroscopic CC analysis with a phenomenological potential was performed for the inelastic levels, yielding J π assignments for several states. The different angular distributions for the 2 1 + and 2 2 + states could be accounted for by assuming a transition potential that has a contribution in the nuclear interior. Apart from the 3 − state at E x = 4.193 MeV a 4 − state and 5 − state were identified, for which phenomenological octupole band calculations have been performed. Explicit folding-model calculations, essential for a consistent description of 0 + and 1 − states, have been performed for several levels, using the JLM nucleon-nucleon interaction both for (p, p′) and for (α, α′), A rather good description of the differential cross sections of several levels, including the 0 + and 1 − states, is obtained. A direct comparison between the results from a (p, p′) and an (α, α′) folding-model analysis is made, resulting in isoscalar B(E λ) values. Microscopic calculations have been performed for the (p, p′) results. The transition densities are obtained from shell-model (SM) and extended-RPA (E-RPA) calculations. No effective charges were introduced. The E-RPA calculations gave a fair description of the differential cross sections of several levels. In the SM calculations the neutron part of the transition density is severely underestimated.

Spin-spin cross sections of 27Al and 93Nb for neutrons from 20 to 50 MeV.

Spin-spin cross sections of $^{27}\mathrm{Al}$ and $^{93}\mathrm{Nb}$ have been measured for neutrons between 20 and 50 MeV with high precision. Absolute size and energy dependence of the data agree well with existing predictions from folding model calculations. The data can also be reproduced with phenomenological spin-spin potentials, but do not allow to distinguish between spherical and tensor interactions. The deduced spin-spin strengths are close to 1 MeV.

Quadrupole scattering of 135 MeV protons by 9Be.

We have measured cross sections and analyzing powers for $^{9}\mathrm{Be}$(p,p') at ${E}_{p}$=135 MeV, observing the ${(3/2}^{\mathrm{\ensuremath{-}}}$ and ${(5/2}^{\mathrm{\ensuremath{-}}}$ members of the ground-state rotational band. Folding model calculations were performed using a density-dependent effective interaction in the local density approximation. The quadrupole contribution to elastic scattering was evaluated in distorted wave approximation. We find the elastic quadrupole contribution to be important for q>1.5 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$, in good agreement with data. The effect on the elastic analyzing power data is quite large.

Spin-Spin Potentials inAlpol27+npoland the Nuclear Ramsauer Effect

Spin-spin effects in neutron-nucleus interactions have been studied with a polarized $^{27}\mathrm{Al}$ target and polarized neutrons of energies 5 to 17 MeV. Because of nuclear Ramsauer interference, real and imaginary spin-spin terms in the optical potential give rise to energy dependences of spin-spin effects which are out of phase. We find a central real potential ${V}_{\mathrm{ss}}=750\ifmmode\pm\else\textpm\fi{}440$ keV, consistent with folding-model calculations, and a central volume imaginary potential ${W}_{\mathrm{ss}}=\ensuremath{-}780\ifmmode\pm\else\textpm\fi{}320$ keV, related to compound-nuclear absorption effects.

Optical model and folding model potentials for elastic scattering of 56 MeV polarized deuterons

Differential cross sections σ and all four analyzing powers A y , A xx , A yy and A xz have been measured for deuteron elastic scattering from 12C, 16O, 40Ca, 58Ni, 90Zr, 118Sn, 144Sm and 208Pb at 56 MeV. The tensor analyzing power A xz was measured using horizontally polarized deuterons. These data were analyzed in terms of an optical potential containing central, spin-orbit and T R type tensor terms. A complex T R tensor potential was necessary to explain A xz and the quantity X 2 = (2A xx + A yy) √3 . The optical potentials obtained were compared with exact folding model calculations. The real part of the T R tensor potential is generally smaller than the folding model potential, especially for heavier nuclei. The imaginary T R potential is comparable with the folding model for lighter nuclei and is smaller than the folding model for heavier nuclei. The folding model T R potential overestimates the amplitudes of the A xz and X 2 data. Features of the σ, A y and A yy data at large angles are discussed in connection with nuclear rainbow scattering.

Correlation between the optical model potential and matter distribution parameters in heavy-ion elastic scattering

Starting from the analytical expression for the folding model potential for the heavy-ion scattering, a simple relationship between the potential and the nuclear density parameters is obtained. This relationship is consistent with the analysis of the proximity forces and is confirmed around the strong absorption radius through the results of the folding model calculations for16O projectile incident on a variety of targets ranging from40Ca to208Pb. These relations help to deduceRx the distance where the folding potential has a specific valuex and the densities of the target nuclei. The predictions are found to be in good agreement with the experiment.

Inelastic scattering of 65-MeV polarized protons from 178Hf, 180Hf, 182W, and 184W and multipole moments of the optical potential.

Differential cross sections and analyzing powers of polarized proton elastic and inelastic scattering from $^{178}\mathrm{Hf}$, $^{180}\mathrm{Hf}$, $^{182}\mathrm{W}$, and $^{184}\mathrm{W}$ have been measured at 65 MeV. Analysis has been performed in the framework of the coupled channel formalism of the deformed optical potential for ${J}^{\ensuremath{\pi}}$${=0}^{+}$--${6}^{+}$ members of the ground state rotational band. In this analysis, all the deformation parameters of the real central, volume imaginary, surface imaginary, and spin-orbit parts of the deformed optical potential were searched independently. Up to the ${6}^{+}$ state, excellent fits have been obtained. It is found that the quadrupole moment of the real central part is 6--9 % larger than that of the charge density for all the measured nuclei, but the quadrupole moment of the Woods-Saxon form factor of the spin-orbit part agrees with the charge quadrupole moment within the fitting error. These results are consistent with the difference between the effects of the density dependence of the effective nucleon-nucleon interaction on the real central part and the spin-orbit part of the optical potential. Similar trends are also found for the hexadecapole moments of the real central part and the spin-orbit part. A folding model calculation using a realistic effective interaction has been carried out and its results are compared to those of the coupled channel analysis.

Elastic and inelastic scattering ofBe9+Be9atE(Be9)=35−50MeV

The angular distributions of the elastic and inelastic cross sections for /sup 9/Be+ /sup 9/Be have been measured at laboratory energies of E( /sup 9/Be) = 35, 40, 45, and 50 MeV. Double folding and optical model calculations were carried out in order to fit the elastic scattering while the inelastic scattering cross sections were analyzed using the coupled-channels procedure. The renormalization factor of the real double folded potential was found to decrease as energy increased in order to obtain any successful fit. In the optical model analysis, good fits were obtained using the same set of optical parameters at all energies except at backward angles, where at higher energies better fits were obtained with a shallower attractive potential. In the coupled-channels analysis, good fits were obtained using an optical model with a deep real well. The potential deformation was found to be ..beta.. = 0.36.

Multipole moments ofEr166,Er168,Yb174, andYb176from 65 MeV polarized proton inelastic scattering and density dependence of the effective interaction

Differential cross sections and analyzing powers of polarized proton elastic and inelastic scattering from $^{166}\mathrm{Er}$, $^{168}\mathrm{Er}$, $^{174}\mathrm{Yb}$, and $^{176}\mathrm{Yb}$ have been measured at 65 MeV. Analysis for ${J}^{\ensuremath{\pi}}={0}^{+}\ensuremath{-}{6}^{+}$ members of the ground state rotational band has been performed using coupled-channel calculations for scattering from deformed optical potentials. Excellent fits have been obtained for both cross sections and analyzing powers for ${0}^{+}$, ${2}^{+}$, and ${4}^{+}$ states and fairly good fits for the ${6}^{+}$ state. In the coupled-channel calculations, the multipole moments of each part of the deformed optical potentials were set to be equal. The quadrupole moments of the deformed optical potentials for these nuclei were found to be 4-6% larger than those of charge densities obtained by electron scattering and Coulomb excitation. A folding model calculation shows that the main part of this difference is attributed to the density dependence of the effective interaction. Mass number dependence of the phenomenological range of the effective interaction can be also reproduced from the folding calculation using the density-dependent effective interaction.

Elastic scattering of polarized tritons fromC12,V15, andNi58at 9 and 11 MeV

Angular distributions of the differential cross section and of the vector analyzing power were measured over a wide range of angles at 9 MeV for $^{12}\mathrm{C}$ and $^{58}\mathrm{Ni}$ and at 11 MeV for $^{12}\mathrm{C}$, $^{51}\mathrm{V}$, and $^{58}\mathrm{Ni}$. Optical-model parameters which could serve as input into $^{7}\mathrm{Li}$ folding model calculations were determined from the data.[NUCLEAR REACTIONS $^{12}\mathrm{C}, ^{58}\mathrm{Ni}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{t}},\mathrm{t})$, $E=9$ MeV, $^{12}\mathrm{C}, ^{51}\mathrm{V}, ^{58}\mathrm{Ni}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{t}},\mathrm{t})$, $E=11$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, ${A}_{y}(\ensuremath{\theta})$; optical-model analysis.]

Dynamically induced spin-dependent interaction in the elastic scattering of heavy ions

Dynamical polarization effects in heavy-ion elastic scattering are investigated in the framework of the coupled-reaction channel theory. By using the adiabatic approximation at low incident energies, these effects are expressed as a spin-orbit ( L· S ) interaction with a L - and S independent radial function. The strength of the ( L· S ) interaction calculated for the 12C + 13C system is in the same order of magnitude as deduced from experiments and is about two orders of magnitude larger than that obtained from the folding model calculation.

Elastic and inelastic scattering of neutrons from 54, 56Fe and 63, 65Cu : (II). Scattering and nuclear structure effects from coupled channels calculations

The differential cross sections for neutron scattering from 54, 56Fe and 63, 65Cu at energies between 8 and 14 MeV, which were reported in a companion paper, have been analyzed in the framework of the coupled channels formalism. Neutron potential deformations were extracted and compared with corresponding deformations deduced from the reanalysis of proton scattering measurements and Coulomb excitation results. The size of the Coulomb correction term in the absorptive proton potential is discussed. Matter root-mean-square radii are estimated from our potential and compared with results from α-particle folding model calculations.

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.

Giant multipole resonances from inelastic scattering of 152-MeV alpha particles

Giant resonance spectra have been measured for $^{208}\mathrm{Pb}$, $^{120}\mathrm{Sn}$, $^{90}\mathrm{Zr}$, $^{58}\mathrm{Ni}$, and $^{46}\mathrm{Ti}$, using inelastic scattering of 152-MeV alpha particles. In addition to some low-lying states, the spectra were analyzed to yield separate peaks for the giant quadrupole resonance and giant monopole resonance, except in $^{46}\mathrm{Ti}$, where no evidence was found for a giant monopole resonance. The results for the monopole resonance when compared to distorted-wave Born-approximation calculations show that \ensuremath{\sim}100% (\ifmmode\pm\else\textpm\fi{}20%) of the $T=0$, $L=0$ energy weighted sum rule is depleted for nuclei with $A>90$. For $^{58}\mathrm{Ni}$ only 40% of the monopole sum rule is found. The excitation energy of the giant monopole resonance follows the systematic trend $\ensuremath{\sim}80{A}^{\ensuremath{-}\frac{1}{3}}$ MeV. The data are also compared with folding model calculations using Tassie transition densities and an effective interaction derived from elastic data. Good agreement was obtained for low-lying quadrupole and octopole excitations, but this model yields considerably too little cross section for the monopole resonance. This probably indicates a deficiency in the Tassie model breathing mode transition density.NUCLEAR REACTIONS $^{208}\mathrm{Pb}$, $^{120}\mathrm{Sn}$, $^{90}\mathrm{Zr}$, $^{58}\mathrm{Ni}$, $^{46}\mathrm{Ti}(\ensuremath{\alpha}, {\ensuremath{\alpha}}^{\ensuremath{'}})$, ${E}_{\ensuremath{\alpha}}=152$ MeV, ${E}_{\ensuremath{\chi}}=0\ensuremath{-}40$ MeV; measured $\ensuremath{\sigma}({E}_{\ensuremath{\chi}}, \ensuremath{\theta})$; discuss giant resonances, folding model calculations, deduced $L$, ${\ensuremath{\beta}}_{L}$.

Global optical model potential for elastic deuteron scattering from 12 to 90 MeV

A set of about 4000 data points has been used in a global search for a general optical model parameter prescription. This potential for deuteron-nucleus scattering was confined to remain within the range given by folding model calculations. The data set spans the energy range from 11.8 to 90 MeV and includes many targets ranging in mass from $^{27}\mathrm{Al}$ to $^{238}\mathrm{Th}$. The set includes 50 reaction cross sections, 13 angular distributions for vector polarization at and above 15 MeV, and 103 angular distributions for elastic scattering. A successful 11-parameter potential of the conventional form fits all data analyzed to about 18% rms. Two of these parameters, ${V}_{R}$ and ${a}_{I}$, are functions of proton number and target mass, while five (${V}_{R}$, ${a}_{o}$, ${W}_{S}$, ${W}_{D}$, ${V}_{\mathrm{LS}}$) are smooth functions of bombarding energy. The remainder (${r}_{o}$, ${r}_{I}$, ${r}_{c}$, ${r}_{\mathrm{LS}}$, ${a}_{\mathrm{LS}}$) are constants. A simple functional dependence for ${a}_{I}(A,Z)$ approximately reproduces the rapid changes in $\ensuremath{\sigma}(\ensuremath{\theta})$ found experimentally for nuclei at or near neutron shell closure. A comparison with earlier work suggests that the optical model analyses furnish the rms radii for the real and imaginary potentials with little ambiguity. General agreement but with a somewhat larger scatter (10%) is also found for the real and the imaginary volume integrals deduced. However, little consistency is seen for the imaginary volume integral if the bombarding energy is below 15 MeV. The rms radii and volume integrals have a small but distinct energy dependence. It was found that the 11-parameter potential occasionally has difficulty delivering good simultaneous fits for cross sections and vector polarizations. This situation was improved by the addition of a small, three-parameter imaginary spin-orbit term, which in our global prescription required short range and very small diffuseness.

Antisymmetrization effects in alpha-nucleus scattering

Antisymmetrization effects in alpha-nucleus scattering are analyzed with respect to the number of nucleons exchanged between the alpha-particle and the nucleus. It turns out that all exchange contributions are essential to reproduce the results of a completely antisymmetrized calculation. Folding model calculations should, therefore, be viewed with caution at low energies. As an example, the backangle anomaly in elastic alpha-nucleus scattering is discussed.

Nuclear shape measurement and the validity of the collective model: results for N not=Z nuclei

Previous work which examined the collective-model transition rates in the light of projected wavefunction calculations has been extended to nuclei with N not=Z. Some interesting quantitative effects have been found which have implications for folding-model calculations and ultimately for experiments. A relationship with shell-model calculations is suggested and comments on the inadequacies of certain Skyrme-type interactions for light nuclei are noted. It would appear that 26Mg is a much less 'rotational' nucleus than 24Mg but 22Ne might be a better rotational nucleus than 20Ne. The author suggests that how 'rotational' a nucleus is may depend on the probe it is examined with.

Neutron component deformations, folding models and Satchler's theorem

The identity, proved by Satchler, between a folded quadrupole moment and the underlying nuclear quadrupole moment has been exploited in order to survey deformed optical model and folding model calculations. We find reasonable agreement for light nuclei between e.m. and nuclear deformations, and for heavy nuclei the following alternative holds: either the folding model for nucleons breaks down in a manner not understood, or, heavy nuclei have much greater deformations of the proton component than of the neutron component for the hexadecapole degree of freedom. In addition we show that α-particle folding models appear to break down completely for heavier nuclei.