Dispersive Optical Model Analysis Research Articles

Systematic Matter and Binding-Energy Distributions from a Dispersive Optical Model Analysis.

We present the first systematic nonlocal dispersive optical model analysis using both bound-state and scattering data of ^{16,18}O, ^{40,48}Ca, ^{58,64}Ni, ^{112,124}Sn, and ^{208}Pb. In all systems, roughly half the total nuclear binding energy is associated with the most-bound 10% of the total nucleon density. The extracted neutron skins reveal the interplay of asymmetry, Coulomb, and shell effects on the skin thickness. Our results indicate that simultaneous optical model fits of inelastic scattering and structural data on isotopic pairs are effective for constraining asymmetry-dependent nuclear structural quantities otherwise difficult to observe experimentally.

A Dispersive Optical Model Analysis of the Alpha Particles Scattering by Titanium Element Nucleus and Its Natural Isotopes

A dispersive optical model analysis of the alpha particles scattering by titanium element nucleus and its natural isotopes has been applied for a new scattering potential within the energy range (1-100) MeV which has contained the range of the Coulomb barrier, and for constant input values of the parameters of this potential. This potential is extent of the mean field potential and is called by (coulomb-nuclear) interference potential, that contains (spin-orbit) coulomb term. The results according to DOMACNIP program that has been designed for that purpose would contain: continuous energy variation of the depths of the real and imaginary parts of the mean field, which are connected by dispersion relations have been compared with these resulting from global parameterizations of the alpha particles scattering potential. In addition to continuous energy variation of the real radius parameter of the Wood-Saxon approximation to the mean field potential with its Hatree-Fock approximation of the nonlocal potential. Consequently, our results for the continuous energy variations of the predicted total reaction cross section within the energy range (1-100) MeV, and with calculation step of the pervious range whose magnitude (1 MeV), differential cross sections, Ratio of the differential elastic scattering cross section to Rutherford cross section and polarization resulted only from the Coulomb spin-orbit term that has been appeared characteristically for selected energy and for selected center-of-mass scattering angle within the energy range (1-100) MeV, showed the excellent agreement with available experimental data and better than these resulted from global parameterizations of the alpha particles scattering potential.

A Dispersive Optical Model Analysis of the Neutrons Scattering by Titanium Element Nucleus and Its Natural Isotopes

In this paper, a dispersive optical model analysis of the neutrons scattering by titanium element nucleus and its natural isotopes is applied to the construction of the complex single-particle mean field starting from Fermi energy value to the energy value 100 MeV and for constant input values of the parameters of this mean field and the varied input values of Hatree-Fock approximation parameters of the nonlocal potential. The results according to DOMACNIP program that has been designed for that purpose would contain: continuous energy variation of the depths of the real and imaginary parts of the mean field, which are connected by dispersion relations were compared with these resulting from global parameterization of the optical model potential. In addition to continuous energy variation of the real radius parameter of the Wood-Saxon approximation to the mean field potential with its Hatree-Fock approximation of the nonlocal potential. Consequently, our results for the continuous energy variations of the predicted (total, total reaction, elastic) cross sections within the energy range (1-100) MeV, and with calculation step of the pervious range whose magnitude (1 MeV), elastic differential cross section and polarization for selected energy and for selected center-of-mass scattering angle within the energy range (1-100) MeV showed the excellent agreement with available experimental data and better than these resulted from global parameterization of the optical model potential, and thus more reliable for calculation the cross sections of unknown interactions of elements nuclei and their isotopes such neutrons scattering by titanium element nucleus and its natural isotopes.

A Dispersive Optical Model Analysis of the Protons Scattering by Titanium Element Nucleus and Its Natural Isotopes

A dispersive optical model analysis of the proton scattering by titanium element nucleus and its natural isotopes is applied to the construction of the complex single-particle mean field starting from Fermi energy value to the energy value 100MeV and for constant input values of the parameters of this mean field. This mean field is called (coulomb-nuclear) interference potential, that contains (spin-orbit) coulomb term. The results according to DOMACNIP program that has been designed for that purpose would contain: continuous energy variation of the depths of the real and imaginary parts of the mean field, which are connected by dispersion relations were compared with these resulting from global parameterization of the optical model potential. In addition to continuous energy variation of the real radius parameter of the Wood-Saxon approximation to the mean field potential with its Hatree-Fock approximation of the nonlocal potential. Consequently, our results for the continuous energy variations of the predicted total reaction cross section within the energy range (1-100) MeV, and with calculation step of the pervious range whose magnitude (1 MeV), differential cross sections, Ratio of the differential elastic scattering cross section to Rutherford cross section, polarization for selected energy showed the excellent agreement with available experimental data and with these resulted from global parameterization of the optical model potential.

Neutron Skin Thickness of ^{48}Ca from a Nonlocal Dispersive Optical-Model Analysis.

A nonlocal dispersive optical-model analysis has been carried out for neutrons and protons in ^{48}Ca. Elastic-scattering angular distributions, total and reaction cross sections, single-particle energies, the neutron and proton numbers, and the charge distribution have been fitted to extract the neutron and proton self-energies both above and below the Fermi energy. From the single-particle propagator resulting from these self-energies, we have determined the charge and neutron matter distributions in ^{48}Ca. A best fit neutron skin of 0.249±0.023 fm is deduced, but values up to 0.33fm are still consistent. The energy dependence of the total neutron cross sections is shown to have a strong sensitivity to the skin thickness.

Implications for(d,p)reaction theory from nonlocal dispersive optical model analysis ofCa40(d,p)Ca41

The nonlocal dispersive optical model (NLDOM) nucleon potentials are used for the first time in the adiabatic analysis of a (d,p) reaction to generate distorted waves both in the entrance and exit channels. These potentials were designed and fitted by Mahzoon $et \text{ } al.$ [Phys. Rev. Lett. 112, 162502 (2014)] to constrain relevant single-particle physics in a consistent way by imposing the fundamental properties, such as nonlocality, energy-dependence and dispersive relations, that follow from the complex nature of nuclei. However, the NLDOM prediction for the $^{40}$Ca(d,p)$^{41}$Ca cross sections at low energy, typical for some modern radioactive beam ISOL facilities, is about 70$\%$ higher than the experimental data despite being reduced by the NLDOM spectroscopic factor of 0.73. This overestimation comes most likely either from insufficient absorption or due to constructive interference between ingoing and outgoing waves. This indicates strongly that additional physics arising from many-body effects is missing in the widely used current versions of (d,p) reaction theories.

Neutron scattering from28Si and32S from 8.0 to 18.9 MeV, dispersive optical model analyses, and ground-state correlations

Background: The nucleon-nucleus dispersive optical model (DOM) has been successful in providing good fits to scattering data and in making valuable predictions for bound-state properties in single- and double-closed shell nuclei. However, the generalizability of the DOM remains an ongoing issue.Purpose: We investigate the DOM in the continuum and bound-state regions of the open-shell, self-conjugate nuclei ${}^{28}$Si and ${}^{32}$S. We collect new differential cross section and analyzing power data for elastic scattering at incident neutron energies between 8.0 and 18.9 MeV.Methods: The measurements were conducted using a pulsed deuteron beam, the ${}^{2}$He($d$,$n$)${}^{3}$He source reaction, and time-of-flight techniques. All data were corrected for finite-geometry effects. Phenomenological DOM potentials were tailored to fit the differential and total cross section data, and then extrapolated to the bound-state regions. The DOM bound-state predictions were then compared to experimental data available for single-particle energies, occupation probabilities, root-mean-square radii, and spectroscopic factors.Results: The DOM bound-state predictions are in only fair agreement with experimental data and with USD shell-model predictions. Similar results are found after converting our neutron DOMs into proton DOMs. We investigate the separate effects of the dispersive surface and volume potential components on occupation probability and find that the volume component leads to a uniform depletion of the hole states, while the surface component acts mainly to deplete the valence orbitals. We compare these results to those of a variational multiparticle multihole configuration mixing (mp-mh CM) calculation using the Gogny D1S effective force.Conclusions: We find that the phenomenological DOM, which was originally designed for spherical nuclei, show certain deficiencies when applied to open-shell nuclei and suggest possible avenues of improvement. We also find that the predictions of occupation probability by the DOM using the dispersive surface component are similar to those by the mp-mh CM. This lends support to the interpretation that the surface absorption in the optical model originates from particle-vibration couplings, that is, long-range correlations.

Exploring Correlations in Exotic Nuclei

The difference in neutron total cross section between 40 Ca and 48 Ca is sensitive to the n/p asymmetry dependence of the surface imaginary potential in a dispersive optical model analysis, from which spectroscopic information can be extracted. The measured cross sections imply that the strength of neutron correlations changes little as neutron number increases in neutron-rich isotopes. In contrast, for the light-nucleus-induced knockout of the valence neutron in 36 Ca, the small experimental knockout cross section (as compared to an eikonal reaction theory) implies a very small spectroscopic factor (and a strong trend in correlations with asymmetry). If this type of knockout analysis is correct, it implies that standard shellmodel calculations miss spectroscopic strength as the Fermi energy nears the continuum. If this is not the case, it may be that there is a problem with the application of the eikonal theory to very deeply bound nucleons at these intermediate energies. The term correlations refers to interactions between nucleons that go beyond the mean field considered in an independent-particle-model (IPM) approach. The effect of these correlations is that in real nuclei, the spectroscopic strength of a singleparticle orbital is fragmented over energy. The spectroscopic factor (SF) quantifies the strength found at a discrete energy, and the sum of spectroscopic strength over all energies below the Fermi energy EF gives the occupation of the orbital. Correlations reduce the occupation of a single-particle state relative to its IPM value, since some of the fragmented spectroscopic strength is shifted above EF . The nuclear shell model (SM) can account for some of these correlations by including mixing between single-particle states (within a finite basis set). But due to the finite model space, the presence of the hard-core of the N-N interaction, and the tensor interaction (which further reduce the occupancies by shifting strength to high-momentum states), standard SM calculations still overestimate the occupancies of bound states. The reduction in occupancy due to correlations is not a direct experimental observable, but it will affect particle removal cross sections. By comparing experimental cross sections to calculated single-particle cross sections, one can learn about the strength of correlations. Electron-induced proton knockout (e,e’p) experiments have indicated a roughly constant 30% reduction in spectroscopic strength for valence protons in beta-stable nuclei. For unstable nuclei, light-nucleus-induced knockout experiments in inverse kinematics suggest that the affect of correlations is weaker for weakly-bound nucleons and much stronger for very deeply-bound nucleons (e.g.

Neutron spectroscopic factors ofAr34andAr46from(p,d)transfer reactions

Single-neutron-transfer measurements using ($p$,$d$) reactions have been performed at 33 MeV per nucleon with proton-rich $^{34}\mathrm{Ar}$ and neutron-rich $^{46}\mathrm{Ar}$ beams in inverse kinematics. The extracted spectroscopic factors are compared to the large-basis shell-model calculations. Relatively weak quenching of the spectroscopic factors is observed between $^{34}\mathrm{Ar}$ and $^{46}\mathrm{Ar}$. The experimental results suggest that neutron correlations have a weak dependence on the asymmetry of the nucleus over this isotopic region. The present results are consistent with the systematics established from extensive studies of spectroscopic factors and dispersive optical-model analyses of $^{40\ensuremath{-}49}\mathrm{Ca}$ isotopes. They are, however, inconsistent with the trends obtained in knockout-reaction measurements.

Mean field for the p+90Zr system in the energy range −60 MeV<E<+65 MeV and single-particle features of proton states in 90Zr from a dispersive optical-model analysis

Data on the scattering of protons with energies 5 MeV<E<65 MeV by 90Zr nuclei and data on the energies of proton particle and hole levels in the A+1 and A−1 systems with A=90 are analyzed within the dispersive optical model. The parameters of the mean proton field for 90Zr are determined in the energy range −60 MeV<E<+65 MeV. The predicted single-particle features of the levels (root-mean-square radii of orbits, occupation numbers, spread widths, spectroscopic factors, and spectral functions) comply well with experimental data obtained in (d, 3He), (3He, d), (n, d), and (d, n) reactions for levels near the Fermi surface and in (e, e′p) and (p, 2p) reactions for deep levels.

Analyzing power measurements for 209Bi(n,n) at 6 and 9 MeV and consistent dispersive optical-model analyses for n+209Bi and n+208Pb from -20 to +80 MeV.

High-accuracy measurements of ${\mathit{A}}_{\mathit{y}}$(\ensuremath{\theta}) data for elastic scattering for n${+}^{209}$Bi have been performed at 6 and 9 MeV. The data are incorporated into a large database of \ensuremath{\sigma}(\ensuremath{\theta}), ${\mathit{A}}_{\mathit{y}}$(\ensuremath{\theta}), and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{T}}$ for n${+}^{209}$Bi covering the energy range 1.0--80 MeV. A complementary database is constructed for n${+}^{208}$Pb and a dispersive optical-model analysis is performed for both scattering systems while constraining many of the parameters to be identical for both systems. A good representation of both databases is obtained with conventional geometry and spin-orbit parameters. The $^{208}\mathrm{Pb}$ model predicts quite well the measured energies of valence single-particle and single-hole bound states. Occupation probabilities and spectroscopic factors for the same bound states are also calculated. Finally, a fully constrained model is presented in which the only differences between the n${+}^{208}$Pb and the n${+}^{209}$Bi systems are the Fermi energy and the isospin dependence in the real volume potential. \textcopyright{} 1996 The American Physical Society.

Proton-90Zr mean field between -60 and +185 MeV from a dispersive optical model analysis.

Elastic scattering of protons from $^{90}\mathrm{Zr}$ in the energy range ${\mathit{E}}_{\mathit{p}}$=9.8 to 135 MeV is analyzed using a dispersive optical model potential (OMP). In this analysis, a dispersion relation connects the volume integrals of the imaginary and the real parts of the OMP. Best-fit dispersive OMP parameters are obtained from fits to experimental cross section and analyzing power data at each energy, while the volume integrals of the imaginary potential and dispersive correction terms are fixed at the empirical values obtained in the individual proton elastic scattering analyses from 9.8 to 135 MeV. Predictions of the cross section and analyzing power angular distributions from the best-fit dispersive OMP and conventional OMP are obtained, and give similar quality fits to data. A dispersive OMP with parameters that show a smooth energy dependence are determined from fits to the entire data set. Comparison of cross sections and analyzing powers calculated by the dispersive OMP with experimental data at 160 and 185 MeV is also presented. The dispersive OMP with a smooth energy dependence is extended to the negative energy region with the guidance of the known first single-particle and single-hole state energies near the Fermi energy, ${\mathit{E}}_{\mathit{F}}$=-6.8 MeV, to provide parameters for the shell model potential. This analysis also provides estimates for single-particle and hole energies ${\mathit{E}}_{\mathit{n}\mathit{l}\mathit{j}}$, root-mean-square radii ${\mathit{R}}_{\mathit{n}\mathit{l}\mathit{j}}$, expectat ion values of the effective mass 〈${\mathit{m}}^{\mathrm{*}}$/m${\mathrm{〉}}_{\mathit{n}\mathit{l}\mathit{j}}$, occupation probabilities ${\mathit{N}}_{\mathit{n}\mathit{l}\mathit{j}}$, absolute spectroscopic factors ${\mathit{S}}_{\mathit{n}\mathit{l}\mathit{j}}$, and spectral functions ${\mathrm{\ensuremath{\zeta}}}_{\mathit{n}\mathit{l}\mathit{j}}$(${\mathit{E}}_{\mathit{x}}$) for proton single-particle and single-hole orbits in $^{90}\mathrm{Zr}$. These estimates are compared with available experimental information.

Neutron scattering from 12C between 15.6 and 17.3 MeV

The differential cross section sigma ( theta ) for neutron elastic scattering from 12C and for inelastic scattering from the 4.44 MeV state was measured at 15.57, 16.75 and 17.29 MeV. The sigma ( theta ) data, together with published analysing power Ay( theta ) data, were analysed in the framework of the spherical optical model and in the coupled-channels formalism. It was concluded that the present 12C(n,n)12C data and published data at higher energies appear to be well suited for determining properties of valence single-particle excitations in 11C and 13C via an iterative-moment approach or a dispersive optical-model analysis.

Dispersion relation approach to the mean field and spectral functions of nucleons in 40Ca

The variational moment approach is applied to the construction of the complex single-particle mean field felt by protons and neutrons in 40Ca, at negative as well as at positive energies. The results are compared with those obtained from recent dispersive optical-model analyses. At positive energies, the reliability of the mean field is checked by comparing predicted with experimental differential and polarization cross sections. At negative energies, the model predictions are compared with empirical values of the single-particle energies, spectroscopic factors, rms radii of single-particle orbits and energy distributions of the spectroscopic factors associated with spread quasiparticle excitations. In particular, the predicted spectral functions are in close agreement with empirical strength distributions recently measured by means of the 40 Ca( d , p), 40 Ca p , d) and 40 Ca( e, e′ p) reactions.

N+209Bi mean field between -20 and 60 MeV.

New measurements of differential elastic neutron scattering for {sup 209}Bi at energies between 7.5 and 24.0 MeV are presented along with new measurements of {sigma}{sub {ital T}} up to 60 MeV. These data, taken together with earlier measurements at lower energy, provide a very large data set for testing and extending the dispersive optical-model analysis. The dispersion correction to the optical model has been obtained from the scattering and total cross-section data. The potential is extrapolated to negative energy for comparison with bound-state properties. A very good description of all of the data is obtained from {minus}20 to +60 MeV. The present analysis suggests somewhat less depletion of the Fermi sea in this mass region than has been obtained from electron-scattering data and from other recent treatments of the dispersion correction to the optical model.

Proton mean field in 40Ca between -60 MeV and +200 MeV deduced from a dispersive optical-model analysis.

The p${\mathrm{\ensuremath{-}}}^{40}$Ca mean field is derived from an optical-model (OM) analysis that explicitly incorporates the dispersion relation connecting the real and imaginary parts of the mean field. This analysis is based on differential cross-section, analyzing power, and reaction cross-section data available in the energy range between 20 and 180 MeV. The extrapolation of the OM potential from positive to negative energies provides the shell-model potential. This extrapolation is guided by known single-particle energies. The deeply bound 1p and 1s orbits clearly indicate the need for a linear rather than an exponential energy dependence of the Hartree-Fock potential at large negative energies. The analysis also provides root-mean-square radii, occupation probabilities, spectral functions, and absolute spectroscopic factors for proton single-particle orbits in $^{40}\mathrm{Ca}$. Our calculated 15% depletion of the hole states in $^{40}\mathrm{Ca}$ is lower than that suggested for $^{208}\mathrm{Pb}$ from theoretical and experimental studies. We also find that a substantial amount of single-particle strength in $^{40}\mathrm{Ca}$ is located at rather high excitation energy.

From scattering to very deeply bound neutrons in 208Pb: Extended and improved moment approaches

Two new methods are developed which improve and extend the iterative moment approach to the extrapolation of the nuclear mean field from positive towards negative energy and to the prediction of various single-particle properties. These two methods still use as sole input a set of phenomenological optical-model potentials. They are improvements of the original approach because they yield more accurate predictions. They are extensions of the original approach because they provide the imaginary part of the mean field, in addition to its real part; this enables one to evaluate scattering cross sections, spectral functions and occupation probabilities, which was not possible in the previous version. These extended approaches are used to construct the neutron- 208Pb mean field from +40 MeV down to −60 MeV. They yield practically identical results. These results are moreover extremely close to those recently obtained from a dispersive optical-model analysis of the experimental n− 208Pb scattering cross sections. It is shown that the radial shape of the real part of the full mean field depends upon energy but remains very close to a Woods-Saxon. One of the two new methods, dubbed the variational moment approach, is well suited for the evaluation of the accuracy of the calculated Woods-Saxon parameters. If the diffuseness is set equal to 0.70 fm, the potential radius at the Fermi energy ( E F = −5.65 MeV) is found equal to (1.238±0.015)A 1 3 fm , and its volume integral per nucleon at E F to −401 ±6 MeV · fm 3. The energy dependence of the calculated real part of the full mean field is characterized by an effective mass m∗(r; E). The effective mass at the nuclear centre and at the Fermi energy, m∗(0; E F ) , must always be larger than the value that it takes in the Hartree-Fock approximation; this property was violated in the original iterative moment approach, but is fulfilled in both of the new methods developed here. One obtains m∗(0; E F )/m = 0.82 , in close agreement with the value found in a recent dispersive optical model analysis; in the latter, however, the quantity m∗(0; E) was infinite at several energies, while here m∗(0; E) is a smooth function of energy. The complex mean field constructed from the extended iterative moment approaches predicts n- 208Pb cross sections which are in quite good agreement with the experimental values in an energy domain which extends up to 40 MeV. The following properties are calculated for the very deeply, deeply, weakly bound and quasibound single-particle states: energies, spreading widths, spectral functions, spectroscopic factors, occupation probabilities and root-mean-square radii. The calculated energies of the valence subshells are in close agreement with experiment. Right below the Fermi energy, the calculated occupation probability is equal to 0.85, and the spectroscopic factor to 0.73. At the bottom of the Fermi sea, the calculated occupation probability is close to 0.95. The predicted energy distributions of the strengths of the 1h 11 2 , 1g 7 2 and 1 g 9 2 deeply bound states and of the 2 h 11 2 , 1 k 17 2 and 1 j 13 2 quas good agreement with experimental evidence.

Neutron-40Ca mean field between -80 and +80 MeV from a dispersive optical-model analysis.

The n${\mathrm{\ensuremath{-}}}^{40}$Ca complex mean field is derived from a dispersive optical-model analysis of the available experimental cross sections. In this analysis the real part of the mean field contains dispersive contributions which are derived from the imaginary part by means of a dispersion relation. These dispersive contributions must be added to the Hartree-Fock potential which is assumed to have a Woods-Saxon shape, with a depth ${V}_{H}$(E) that depends exponentially upon energy. The input experimental data are 14 differential cross sections in the energy domain (5.3, 40.0 MeV), five polarization cross sections in the domain (9.9, 16.9 MeV), and the total cross section in the domain (2.5, 80 MeV).The resulting optical-model potential is an analytic function of energy. It can thus be extrapolated towards negative energies, where it should be identified with the shell-model potential. This extrapolation yields good agreement with the experimental single-particle energies in the two valence shells of $^{40}\mathrm{Ca}$. The model also predicts the radial shape and the occupation probabilities of the single-particle orbits and the spectroscopic factors of the single-particle excitations. In order to reproduce the experimental energies of the deeply bound 1p and 1s orbits, one must use a linear rather than an exponential energy dependence of ${V}_{H}$(E) at large negative E. It is shown that this is precisely the behavior expected from the fact that the energy dependence of ${V}_{H}$(E) actually represents the nonlocality of the original microscopic Hartree-Fock field. The model also correctly predicts the distribution of the single-particle strength of the 1d(5/2 excitation in $^{39}\mathrm{Ca}$. The calculated distributions of the 1p strength in $^{39}\mathrm{Ca}$ and of the 1f(5/2 strength in $^{41}\mathrm{Ca}$ show that the available experimental information extends over less than half the expected peak, whose energy is thus poorly known experimentally. In the energy domain (2.5, 9 MeV) the predicted total cross section deviates from the experimental data; this reflects the fact that at low energy the calculated cross section is very sensitive to small modifications of the mean field.