Folding Model Potentials Research Articles

Detailed study and mean field interpretation of16O+12Celastic scattering at seven medium energies

Detailed measurements of the elastic scattering of ${}^{16}\mathrm{O}$ ions from ${}^{12}\mathrm{C}$ have been carried out at seven energies from 62 to 124 MeV, at center-of-mass angles from about $10\ifmmode^\circ\else\textdegree\fi{}$ to about $145\ifmmode^\circ\else\textdegree\fi{}.$ A coherent optical model analysis of these data has been performed using both the Woods-Saxon and the folding-model potentials. The extracted results are consistent with analyses of data at higher energies for this and similar light heavy-ion systems. Some model-independent spline forms for the real potentials were also investigated.

Elastic scattering of16O+16Oat energiesE/Abetween 5 and 8 MeV

The elastic scattering of ${}^{16}\mathrm{O}{+}^{16}\mathrm{O}\mathrm{}$ has been measured at nine energies between ${E}_{\mathrm{lab}}=75$ and 124 MeV. The data cover up to $100\ifmmode^\circ\else\textdegree\fi{}$ in the c.m. and can be described in terms of phenomenological and folding model potentials which reproduce the main features observed. In agreement with studies at higher energies in this and similar systems, refractive effects are present in the angular distributions at all energies. In particular, the passage of Airy minima through $90\ifmmode^\circ\else\textdegree\fi{}$ at ${E}_{\mathrm{c}.\mathrm{m}.}=40,$ 47.5, and 62 MeV explains the deep minima observed in the excitation function. The real part of the optical potential is found to vary very little with energy over the studied interval, but the imaginary part shows a rapid change in its shape at incident energy about 90 MeV. Nonetheless, the energy dependence of the volume integral of the real and imaginary parts is in agreement with dispersion relation predictions.

Dynamic polarization potential of 12C+12C system at molecular-resonance energies

The nuclear reaction dynamics leading to the formation of recently discovered resonance in the mutual-02+ channel of the 12C+12C inelastic scattering around Ec.m.≃ 32 MeV is studied in terms of the dynamic polarization potential (DPP) induced by the channel coupling among various excited states in 12C. The microscopic 3α cluster-model wave functions are used to generate the 12C−12C diagonal and coupling potentials in the double-folding model. It is found that DPP for the 02++ 02+ channel is an unusually strong attractive potential which even exceeds the zeroth-order folding-model potential of this channel around the nuclear surface region and that the strong coupling between the 02+ and 22+ states is predominantly responsible for the unusual DPP in this channel. The effective potential, the sum of the original folding-model potential and the attractive DPP, is found to generates resonance states in the same energy region as that of the resonance states generated by the original folding-model potential but the former states are found to be higher-nodal states having four additional radial nodes. Similar but more moderate property of DPP is also found in the entrance (elastic) channel. These results suggest that the reaction dynamics of generating the resonance in the 12C(02+) +12C(02+) channel may rather differ from that of the simple crossing of the zeroth-order molecular band generated by the potentials in the entrance and exit channels suggested by the standard band-crossing model.

16O + 16O elastic scattering at 350 MeV studied by inversion

We have analysed 16O + 16O elastic scattering at 350 MeV using a two-step method (ifS(l) search followed by ifS(t)-to-V(r) inversion. We have constrained the procedure so that only rather smooth potentials are determined. The fits are nevertheless better than others involving direct fits of parameterized V(r) to data. We present the results of two independent analyses which both yield V(r) closely resembling the type A potential of Kondo et al. and also folding model V(r) for r ⩾ 4 fm. The potentials differ in their range of sensitivity; for one potential changes in V(r) for r < 3.8 fm have no influence whatever on the fit in strong contrast to certain folding model potentials. The general properties of the real components for both fits closely resemble folding model predictions. For both potentials, the imaginary component that emerges is remarkably similar to that obtained from a simplified Glauber model. We compare with results of previous two-step fit and discuss strengths and dangers of two-step phenomenology.

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.

Local representation of the exchange nonlocality in n-16O scattering.

The nonlocal Schr\odinger equation is solved rigorously in a microscopic folding model, incorporating both direct and knock-on exchange potentials, for n${\mathrm{\ensuremath{-}}}^{16}$O scattering at laboratory energies of 20 and 50 MeV. The model uses the complex and density dependent n-n interaction of N. Yamaguchi et al. uses harmonic oscillator wave functions for the bound nucleons, and calculates the scattering wave function for this nonlocal problem using a Bessel-Sturmian expansion method incorporating correct boundary conditions. All spins are neglected. The local phase-equivalent potential is obtained from the scattering matrix elements at a given energy by using the iterative perturbative inversion method. This representation allows comparison between the microscopic model and a phenomenological potential, showing good agreement for the local real part of the potential at 20 MeV. From the ratio of the wave functions for the nonlocal potential and for the potential calculated by inversion, a Perey damping factor (PDF) is obtained which is of similar form to the well-known Perey-Buck prescription for the PDF for a Gaussian nonlocality of the conventional range of 0.85 fm. The significance of these results for distorted wave Born approximation calculations is discussed.

Elastic scattering of 58Ni+27Al at near-barrier energies.

The elastic angular distributions for [sup 58]Ni+[sup 27]Al have been measured at five energies from [ital E][sub c.m.]=48.8 to 69.5 MeV. The optical model analysis assumes both phenomenological and folding model potentials. Regions of sensitivity, where the potential is well determined, are found to be different for the real and the imaginary parts, and to vary with energy. The values of the real potential in the surface are weakly dependent on energy.

Elastic-scattering phenomenology by inversion: (I). 12C + 12C from 140 to 2400 MeV

We present a two-step procedure for determining the local internuclear potential which fits elastic-scattering data. The first step is the determination from the angular-distribution data of a complex S-matrix S( l) using a “parameterized improved phase-shift” procedure, PIPS. The second stage is the determination by inversion of the corresponding complex potential. The method is used to determine the 12C + 12C potential from 139.5 to 2400 MeV. We present much better fits than have previously been published; the corresponding potentials are compared with both Woods-Saxon potentials and DDM3Y folding-model potentials. We discuss certain deficiencies of Woods-Saxon potentials and inversion models based on unmodified McIntyre-Wang-Becker parameterizations. The method we present can exploit elastic-scattering data of greater precision and angular range than generally exists, accurately representing such data by a local potential. It is less computer intensive than model-independent fitting for heavy ions.

Nuclear refraction in 14N+12C elastic scattering at E/A=20 MeV.

The elastic scattering of 280-MeV $^{14}\mathrm{N}$ off $^{12}\mathrm{C}$ has been measured. The angular distribution, extending up to 38\ifmmode^\circ\else\textdegree\fi{} in the c.m. system, displays a Fraunhofer crossover followed by a structureless falloff, indicative of the influence of nuclear refraction. Results of an optical-model analysis of the data, using phenomenological and folding model potentials, are presented.

Elastic scattering of alpha-particles from Si isotopes at 26.6 MeV

Experimental elastic-scattering angular distributions of 26.6 MeV α-particles scattered from29Si and30Si nuclei have been measured. These data together with previously measured α+28Si elastic angular distribution have been analysed using the optical model. Both the phenomenological (Woods-Saxon type with four and six parameters) and the double folding model (density-independent and density-dependent real part with squared Woods-Saxon imaginary form factor) potentials have been employed in fitting the data. The best agreement of theory with experiment has been obtained when the real part of the optical potential has been calculated with density-independent M3Y interaction of Bertschet al.

12C+12C elastic scattering analysis above E/A=6 MeV using deep real potentials.

We present phenomenological and folding optical-model descriptions of {sup 12}C+{sup 12}C elastic scattering between 6 and 120 MeV/nucleon. The potentials have a deep real part, a weakly absorbing imaginary part, and are energy dependent. The folding-model real potential is calculated from the energy- and density-dependent effective interaction DDM3Y. It requires renormalizations of about 1.1 below 10 MeV/nucleon, and close to unity above. The phenomenological real potential is about 300 MeV at the center for the lower energies, slowly reducing to 120 MeV for the highest energies. The discrete ambiguities for the real part found in the analysis at energies below 10 MeV/nucleon are resolved by demanding consistency with the higher energy results. The volume integrals for the folding-model and phenomenological potentials agree and display a smooth and slow energy dependence that is consistent with simple dispersion-relation predictions. The values for the total reaction cross sections show the transparency effect expected from microscopic calculations. A possible extension of this description towards lower energies is discussed.

On the spin-dependence of the 6Li- 120Sn interaction

Tensor analyzing powers of first and second rank determined for elastic 6Li- 120Sn scattering at 44 MeV can be understood quite satisfactorily using a d- α cluster folding model to generate the 6Li- 120Sn potentials. The first-rank tensor analyzing power is mainly due to coupling of the ground state of 6Li with excited states even though the static folding spin-orbit potential is of some importance. The vanishing second-rank tensor analyzing powers are caused by a vanishing second-rank tensor deuteron-target potential which enters the cluster folding model potentials.

Optical model analysis of90, 92, 94, 96Zr+4He elastic scattering at 35.4 MeV

The elastic scattering data of 35.4 MeV alpha particles from Zr-isotopes have been analysed using the optical model. Both the phenomenological and the folding model potentials have been employed in fitting the data with the optical model. The resultant systematics. concerning the volume integrals and the root mean square radii are summarized.

Tensor interaction in elastic scattering of polarized deuterons from medium-weight nuclei near Ed = 22 MeV

Differential cross section σ(θ), vector analyzing power iT 11( θ) and all three tensor analyzing powers T 20( θ), T 21( θ), and T 22( θ) have been measured for elastic scattering of deuterons from twelve medium-weight nuclei 40Ca, 58,60,62,64Ni, 90Zr, 92,100Mo, and 104, 106, 108, 110 near E d = 22 MeV. Scattering angles were covered over a wide range from θ = 30° to 170°. The data were analyzed in terms of an optical model potential including a complex tensor interaction of the type T r. The tensor analyzing power T 21( θ) and the quantity X 2 T 22(θ)-√ 3 2 T 20(θ) play an essential role in extracting the tensor interaction and its strength. The obtained T r tensor potential has the real (imaginary) part with well depth, radius, and diffuseness parameters of 1.0 ~2.5 MeV (1.0~ 2.5 MeV), 1.0 ~1.5 fm (0.9~1.2 fm), and 0.4~0.6 fm (0.8 ~ 1.7 fm), respectively. The imaginary T r potential has larger diffuseness than the real one. In extracting these parameter values the T 21( θ) data of the backward angular region (100° ≲ θ ≲ 170°) are indispensable. The optical potential thus obtained is compared with a folding model potential. The effect of the tensor potentials of the types T P and T L was investigated.

Antisymmetrization corrections in deuteron elastic scattering and deuteron-induced transfer reactions

Antisymmetrization corrections to deuteron elastic scattering and (d, p) and (p, d) transfer reactions are studied using an approximate three-body model in which the D-state component of the deuteron is included. We find that Pauli exclusion principle modifications of the Watanabe folding model potential for deuteron elastic scattering are small for incident deuteron energies above the target Coulomb barrier where the present model is applicable. A novel spin-dependence is predicted for the effective nucleon-nucleon interaction which should be used to take account of nuclear medium corrections to transfer reaction transition matrix elements.

On the optical-potential deformation in heavy-ion inelastic scattering

The descriptions of heavy-ion inelastic scattering given by the folding model and phenomenological potentials require different degrees of deformation of the optical potential. This is because the rule of equal normalised multipole moments of the optical potential and the nucleon density distribution, imposed by the folding model, conflicts with the rule of equal deformation lengths borne out by analyses with Woods-Saxon potentials. It is shown, in an example of inelastic scattering of heavy ions from deformed nuclei in the region of strong Coulomb-nuclear interference, that the double-folding model description of the experimental data is as good as any description with deformed Woods-Saxon potentials. Thus neither of the two different rules for deforming the optical potential is a universal prescription since they only apply to specific classes of potentials.

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.

Some optical-model analyses of the elastic scattering of 40Ar at 1760 MeV

The elastic scattering cross sections for 40Ar on 60Ni, 120Sn and 208Pb at 1760 MeV were reanalyzed using both Woods-Saxon and folding-model potentials. The latter required renormalizing by factors N≲0.7. This provides further evidence that the real potentials at this energy are substantially weaker than those obtained for other systems at lower energies. This was confirmed by comparison with data for 40Ar + 208Pb at 236 MeV. The 1760 MeV data appear to determine the values of both real and imaginary potentials in the vicinity of a strong-absorption region, nearly 1 fm inside the rainbow radius.

On a Calculational Method of Folding Model Potential with Density-Dependent Effective Interactions

A new calculational method of folding model potentials is proposed. This method facilitates the application of the density-dependent effective interaction to the folding model without the short range approximation. An effective interaction which corresponds to the exchange term of the two-particle ladder correlation is newly introduced.