Microscopic Optical Potential Research Articles

Scattering of polarized protons from 6Li at 200 MeV.

New cross section and analyzing power data are presented for elastic and inelastic scattering of 200 MeV protons from $^{6}\mathrm{Li}$. The elastic data are well described by a standard, spherical, 12-parameter phenomenological Woods-Saxon (WS) optical potential. Microscopic folding-model optical potentials, obtained by convoluting free and Pauli-corrected effective nucleon-nucleon (NN) interactions with $^{6}\mathrm{Li}$ ground-state densities constrained by electromagnetic data, produce satisfactory descriptions of the elastic data. The effects of the spin-spin optical potential, estimated via the distorted wave approximation (DWA), are small but not negligible. The inelastic transitions leading to the ${3}^{+}$,T=0 state at 2.18 MeV and the ${0}^{+}$,T=1 state at 3.56 MeV are also examined within the framework of the microscopic folding model using the DWA. Transition potentials are generated by convoluting the two effective NN interactions considered with target transition densities constrained by weak and electromagnetic data. These are employed in DWA calculations using both folded optical potentials consistent with the transition potentials and the WS optical potential. The self-consistent calculation with the Pauli-corrected interaction provides a good description of the data for the ${3}^{+}$ excitation. The results for the ${0}^{+}$ excitation provide a clear indication that shell model configurations outside the p shell are important for a complete description of this transition.

Effect of 3α-like states in 12C to 12C- 12C scattering

The polarization potential arising from the excitation of 0 2 + (7.66 MeV), 2 2 + (10.3 MeV) and 3α breakup states in 12C are calculated using the transition densities to these states which are obtained by Kamimura in the 3α resonating-group method. Adding this to the microscopic optical potential of Faessler et al. in the nuclear matter approach, the elastic and inelastic cross sections for the 12C- 12C scattering at low ( E lab = 360 MeV) to intermediate ( E lab = 1016 MeV) energies are calculated including the 2 1 + (4.44 MeV) and 3 1 − (9.64 MeV) states in the coupled channel formalism. We found that at low energy the polarization potentials cure the overestimation of the cross section seen in the calculation of Faessler et al. and improve the agreement with the experimental data while at intermediate energy they affect little the cross section and do not spoil the agreement with the experimental data obtained without the polarization potential.

Optical potentials in relativistic meson-nucleon model

The relativistic microscopic optical potentials (RMOP) at E < 300 MeV have been derived and investigated based on Walecka's meson-nucleon model. An effective lagrangian including nucleon, σ- and ω-mesons, which is required to reproduce the nuclear matter saturation properties, has been introduced and used to calculate the self-energy of a nucleon in the nuclear medium. Systematical analyses of the scattering data are performed with the RMOP. Finally, several effects, such as the meson-nucleon vertex form factor, isovector meson exchanges, non-linear σ-model are studied.

Complex microscopic optical potential between two nuclei based on Dirac-Brueckner theory for nuclear matter

The real and imaginary parts of the optical-model potential between two nuclei are calculated in the energy density formalism. The energy density is derived from the Dirac-Brueckner approach to nuclear matter. In this approach, both free NN scattering and the saturation properties of nuclear matter can be explained starting from a realistic NN interaction. The relativistic features incorporated in the Dirac-Brueckner approach make the real part of the optical potential less attractive than that obtained in a non-relativistic calculation while the imaginary part is enhanced. The comparison of the calculated differential cross section for elastic 12C- 12C scattering with the experimental data suggests that the enhancement of the imaginary part due to the relativistic treatment is favourable while its repulsive contribution to the real part is unfavourable.

Analysis of 40Ca(p,p′) at 500 MeV using tϱ optical potentials in a collective framework

Inelastic scattering to the low-lying collective states in 40Ca excited by 500 MeV protons is examined using both relativistic and nonrelativistic microscopic optical model potentials in the Tassie model. The non-standard shapes produced by these microscopic models does not produce different deformation lengths than those obtained using standard Woods-Saxon shapes. For comparison, calculations using the distorted wave impulse approximation are presented.

The microscopic feshbach optical potential for a schematic coupled channel example

By means of a method of expanding the solutions of a system of coupled equations into a set of positive-energy sturmian basis functions, a separable approximation to the microscopic (Feshbach) elastic optical potential is obtained. The properties of this nonlocal potential are analysed in terms of the moments of the potential and of the wronskian of two independent solutions of the single-channel nonlocal Schroedinger equation. A numerical application is given for a schematic coupled channel example, representing a 60 MeV neutron incident on a schematic nucleus with up to 10 channels, coupled by phenomenological surface-peaked potentials. The nature of the nonlocality of this schematic potential is very different from the nonlocality due to exchange effects in that the wronskian is strongly angular momentum dependent and can be larger as well as smaller than unity. The range of the nonlocality is found to be larger than the wavelength of the elastic projectile.

Complex microscopic optical potential and medium energy elastic scattering of 12C on 12C and 208Pb

Differential elastic scattering cross sections of 12C on 12C and on 208Pb at laboratory energies 1440 and 2400 MeV are calculated with microscopically determined complex optical potentials. The complex optical potentials are derived in the energy density formalism from the potential energy density generated by solving the Bethe-Goldstone equation with the Reid soft core NN interaction. Effects of the excitation of the low-lying collective states are taken into account explicitly by the coupled channel formalism. Considering that there is no adjustable parameter to fit the cross section in the calculations, the results are in reasonable agreement with recent experimental data.

Nonlocality in medium energy proton scattering.

The microscopic optical model potential is investigated in its full form starting from the nonlocal two-body effective force. It is found that the nonlocality causes an effective repulsion in the nuclear interior which produces a so called wine-bottle shape which characterizes the optical model potential at intermediate energy. Its relationship to the relativistic approaches based on the Dirac equation is discussed.

Real and imaginary parts of the microscopic optical potential between nuclei in the sudden and adiabatic approximation and its application to medium energy 12C- 12C scattering

Real and imaginary parts of the optical potential between two nuclei have been calculated from the realistic two-nucleon interaction of Reid using the approach of Faessler and collaborators but with the following modifications: (a) A surface curvature term proportional to the square of the gradient of the density has been added to the energy density functional and the parameters are determined so as to reproduce the binding energies and rms radii of nuclei considered and (b) in addition to the sudden approximation, an adiabatic situation is also considered. Whereas, in the sudden approximation the density of the composite system is obtained by adding the densities of the two colliding nuclei, in the adiabatic approximation, the maximum density in the composite system is limited to the saturation value while conserving the nucleon number. Although the real part of the potential is in the center quite different in these two approximations, the cross sections are not. The real and imaginary parts of the optical potential are calculated for the nuclear pairs 12C+ 12C, 16O+ 16O, 40Ca+ 40Ca and 208Pb+ 208Pb for relative momenta per nucleon k r = 0 and 1 fm −1. Elastic, inelastic and reaction cross sections are calculated for 12C+ 12C scattering at E lab = 300, 360 and 1016 MeV using the calculated potential and coupling the elastic channel to the 2 + at 4.44 MeV and 3 − at 9.64 MeV, and compared with the experimental data. The agreement with the data at 1016 MeV is satisfactory and the theory can also reproduce the general trend of the angular distributions at 300 and 360 MeV but the calculated values at larger angles overestimate the data. The energy dependence and the magnitude of the observed reaction cross section are well reproduced by the theory.

Microscopic relativistic description of optical model potential

The microscopic relativistic optical potentials in energy region of E < 300MeV have been studied and calculated. An effective Lagrangian density with two adjustable coupling constants (gσ, gω) is chosen by reproducing the bulk properties of nuclear matter in the Dirac-Hartree-Fock (DHF) approximation. It is found that the optical potential calculated in this model is in good agreement with those obtained by phenomenological fitting up to the energy of 300MeV.

Measurements and calculations of neutron scattering in the actinide region.

Neutron scattering differential cross sections were measured for $^{232}\mathrm{Th}$, $^{238}\mathrm{U}$, and $^{239}\mathrm{Pu}$ at 14.1 MeV. Optical model calculations were performed using the coupled-channel formalism for deformed nuclei as given by Tamura. Several microscopic and phenomenological optical model potentials were tested for these heavy nuclei: (i) a folding optical model potential using the microscopic nucleon-nucleon effective interaction of Jeukenne, Lejeune, and Mahaux with deformed nuclear density distributions (these calculations are addressed in the text as semimicroscopic optical model potentials), (ii) global phenomenological optical model potentials optimized for the actinide region, and (iii) neutron potentials derived from 35 MeV proton optical potentials using the Lane formalism. The total cross sections calculated from these optical model potentials at 14.1 MeV are also compared with measurements in the literature. The semimicroscopic calculations reproduce reasonably well the measured angular distributions with only two free parameters used to optimize fits to the data. The calculations carried out with optical potentials inferred from proton data give good agreement with the neutron data, an agreement which is similar to that between calculational results using neutron global potentials optimized for this mass region and the measured data.

Density matrix expansion for microscopic optical model potential

An approximation scheme for evaluating the folding optical model potential for nucleon-nucleus scattering is developed. The microscopic optical model potential is expressed as a simple algebraic function of nuclear densities. Nonlocality and finite-range effects are properly incorporated. A few numerical examples are given.

Quasi-free nucleon scattering on 16O

The 16O(p, 2p) 15N and 16O(p, pn) 15O reactions have been observed using the MRS facility and a 505 MeV proton beam at TRIUMF. Data were acquired for sixteen pairs of angles of the emitted nucleons leading to the mirror 1p 1 2 ground and 1p 3 2 excited states of the residual nuclei over a broad range of recoil momenta. Cross sections and cross-section ratios were extracted from the data. The results are compared with DWIA calculations in which microscopic optical potentials, with medium effects included, are used in the calculation of the scattering wave functions. The amplitudefactorized form of the cross section is used rather than the usual cross-section factorized form. Comparisons of the (p, 2p)/(p, pn) cross-section ratios are made for calculations with and without the spin-orbit optical potential, and with the free nucleon-nucleon elastic scattering cross-section ratios. The experimental cross sections and ratios are generally smaller than any of the calculations, and these differences remain to be clarified.

Measurements and calculations of neutron scattering angular distributions over a wide mass and energy range

Neutron elastic and inelastic differential cross sections for targets between /sup 9/Be and /sup 239/Pu at energies, E > 14 MeV have been measured using the Livermore and Ohio University neutron time-of-flight facilities. We review here the data and the analyses based on two local microscopic optical potentials: that of Jeukenne, Lejeune and Mahaux, and that of Brieva and Rook. The results are also compared with calculations using global potentials. Coupled channel formalism has been used in the analysis of targets with strong deformations, such as Be, C, Ta, and actinides. The value of the microscopic optical potentials as a tool to predict elastic and inelastic neutron cross sections over a wide mass and energy range is discussed. The need for neutron measurements up to higher energies and their analysis in conjunction with (p,p) and charge exchange (p,n) data is addressed. 17 refs.

Microscopic optical model analysis of nucleon scattering from light nuclei.

Differential cross sections for neutron elastic scattering from $^{12}\mathrm{C}$, $^{14}\mathrm{N}$, $^{16}\mathrm{O}$, and $^{27}\mathrm{Al}$ have been measured in the energy range 18--26 MeV. These data, together with previously published neutron and proton data for these nuclei and for $^{12}\mathrm{C}$ in the energy region 20 to 65 MeV, have been analyzed in terms of two microscopic optical potentials. The spherical potential derived from the work of Jeukenne, Lejeune, and Mahaux provides a consistent representation of the differential data, provided the imaginary central potential is renormalized. Predictions from the interaction of Brieva and Rook give less satisfactory agreement. In particular, there is a significant overprediction of the forward angle data below \ensuremath{\sim}25 MeV and an underprediction above \ensuremath{\sim}35 MeV.

Potential sensitivity in projectile excitation for 7Li

Detailed angular distributions have been measured for /sup 7/Li elastic and inelastic scattering to the first 1/2/sup -/ state in /sup 7/Li (projectile excitation) and the first 2/sup +/, 3/sup -/, and 5/sup -/ states in /sup 40/Ca at E( /sup 7/Li) = 34 MeV. Distorted-wave Born approximation and coupled-channels calculations of the inelastic processes were carried out with both Woods-Saxon and microscopic double-folded real optical potentials. No renormalization of the real double-folded potential is needed to fit the elastic data when the contribution from the ground state quadrupole moment of /sup 7/Li is accounted for in coupled-channels calculations, otherwise a renormalization of 0.6 is required. Moreover, these coupled-channels calculations matched the phase of the oscillations in the projectile excitation data, while calculations employing a Woods-Saxon real potential did not. The calculations for the excitation of states in /sup 40/Ca were insensitive to the form of real potential chosen.

Microscopic analysis of antiproton scattering from carbon

The antiproton-12C scattering data at 300 MeV/c are analysed with a microscopic optical potential. The Paris or the Bryan-Phillips NN potential and the12C ground-state density are the sole inputs.

Test of microscopic optical model potentials for neutron elastic scattering at 14.6 MeV over a wide mass range.

Differential cross sections for the elastic scattering of 14.6 MeV neutrons from $^{9}\mathrm{Be}$, C, $^{27}\mathrm{Al}$, Fe, $^{59}\mathrm{Co}$, $^{89}\mathrm{Y}$, $^{93}\mathrm{Nb}$, In, $^{140}\mathrm{Ce}$, $^{181}\mathrm{Ta}$, $^{197}\mathrm{Au}$, $^{208}\mathrm{Pb}$, and $^{209}\mathrm{Bi}$ have been measured using a time-of-flight facility. The measured cross sections cover the angular range from 9.2\ifmmode^\circ\else\textdegree\fi{} to 159\ifmmode^\circ\else\textdegree\fi{}. Calculations have been carried out using two local microscopic optical potentials, based on the work of Jeukenne, Lejeune, and Mahaux, and of Brieva and Rook. Reasonably good agreement with the measurements is found over the whole mass range, with the Jeukenne-Lejeune-Mahaux potential giving systematically better agreement with the data. The quality of the results with the Jeukenne-Lejeune-Mahaux potential compares well with that obtained from phenomenological optical model calculations using global parameters.

Boltzmann equations and hydrodynamics for nuclear systems

The Keldysh-Schwinger formalism for causal Green's functions gives an-in principle-self-contained coupled system of (functional) equations for transport processes, which is used for the microscopic investigation of nuclear non-equilibrium systems. First, the general statistical equations are obtained by means of the energy moments of the GF-equations. The essential quantity is the causal mass operator (effective single-particle) potential, which determines the generalized densities. The transition to nuclear physics is achieved by assuming a weak energy dependence of the mass operator. With this simplification one obtains — as desired — a generalization of the time-dependent Hartree-Fock-approach, in which the single-particle propagation is determined by the real part of the total microscopic optical potential. The damping is caused by the imaginary part of the optical potential, where two-particle collisions and excitations contribute. The Vlasov-Uehling-Uhlenbeck-equation — recently applied in nuclear physics — is a special approximation, in which for the single-particle description only the Hartree-term enters. The dissipation term is simplified by approximating the ladder approximation via the optical theorem. Nuclear hydrodynamics emerges by means of the momentum-moments of the Wigner transformed statistical equations. Collision contributions — in the ladder approximation — in the “Navier-Stokes” equations demand at least the inclusion of second order moments. Furthermore one can derive equations for macroscopic variables. As example we give the equation of motion for the centre-of-mass coordinate for two colliding nuclei.

Momentum space approach to microscopic effects in elastic proton scattering

The microscopic nonrelativistic first-order optical potential for proton-nucleus scattering is studied in some detail. Momentum-space calculations have been performed for a number of different target nuclei at proton energies above \ensuremath{\sim}100 MeV and these microscopic predictions are compared with experimental cross section, analyzing power, and spin-rotation function data. The input to these calculations consists of the free on-shell nucleon-nucleon $t$ matrix, its nonlocal and off-shell structure, the treatment of the full-folding integral, and target densities obtained from electron scattering. Off-shell and nonlocal effects, as well as various factorization approximations, are studied. The sensitivity to uncertainties in the off-shell extension of the $t$ matrix, within the context of the Love-Franey model, is explicity displayed. Similarly, uncertainties due to nonlocalities and incomplete knowledge of nuclear densities are shown. Explicit calculations using the $t$ matrix of Love and Franey indicate that these effects play significant roles only for relatively large angles ($\ensuremath{\theta}\ensuremath{\lesssim}60\ifmmode^\circ\else\textdegree\fi{}$) and/or lower energies (\ensuremath{\sim}150 MeV). These studies reinforce the conclusion that the lack of agreement between such first-order predictions and the data for spin observables at small angles arises from a physical effect not included in the nonrelativistic first-order theory, rather than from any uncertainty in the calculation or in its input.