Microscopic positiveenergy potential based on Gogny interaction
Abstract
We present nucleon elastic scattering calculation based on Green’s function formalism in the RandomPhase Approximation. For the first time, the Gogny effective interaction is used consistently throughout the whole calculation to account for the complex, nonlocal and energydependent optical potential. Effects of intermediate singleparticle resonances are included and found to play a crucial role in the account for measured reaction cross section. Double counting of the particlehole secondorder contribution is carefully addressed. The resulting integrodifferential Schrödinger equation for the scattering process is solved without localization procedures. The method is applied to neutron and proton elastic scattering from Ca. A successful account for differential and integral cross sections, including analyzing powers, is obtained for incident energies up to 30 MeV. Discrepancies at higher energies are related to much too high volume integral of the real potential for large partial waves. Moreover, this works opens the way for future effective interactions suitable simultaneously for both nuclear structure and reaction.
Nuclear structure and nuclear reactions are two aspects of the same manybody problem, although in practice they are often addressed as different phenomena. A consistent, quantitative and predictive account for both is still a challenging open problem in nuclear physics. The description of nucleonnucleus elastic scattering based solely on the nucleonnucleon (NN) interaction is an important step forward toward this unification.
Depending on projectile energy and target mass, various strategies have been adopted in order to treat microscopically elastic scattering. Nuclear matter models Hufner and Mahaux (1972) provide reasonable descriptions of nucleon elastic scattering at incident energies above 50 MeV Dupuis et al. (2006), even up to 1 GeV Arellano and von Geramb (2002). The Resonating Group Method within the NoCore Shell Model, has successfully described nucleon and deuteron scattering from light nuclei Quaglioni and Navrátil (2008). These models have recently been extended to include threenucleon forces for nucleon scattering from He Hupin et al. (2013). The Green’s Function Monte Carlo method has been used to describe elastic scattering from He Nollett et al. (2007). These models yield encouraging results but are still restricted to light targets at low energies. The SelfConsistent Green’s Function (SCGF) method has been applied to microscopic calculation of the optical potentials for proton scattering from O Dussan et al. (2011); Barbieri and Jennings (2005). The coupledcluster theory has been applied to proton elastic scattering from Ca Hagen and Michel (2012). These last two methods are limited to closedshell nuclei. Work on GorkovGreen’s function theory is in progress to extend SCGF to nuclei around closedshell nuclei Somà et al. (2013, 2014). An alternative method consists of using microscopic approaches based on the selfconsistent meanfield theory and its extensions beyond meanfield. In nuclear physics, they are usually based on energy density functionals built from phenomenological parametrizations of the NN effective interaction, such as Skyrme Vautherin and Brink (1972); Hellemans et al. (2013) or Gogny forces Dechargé and Gogny (1980); Berger et al. (1991); Chappert et al. (2008); Goriely et al. (2009). These approaches have successfully predicted a broad body of nuclear structure observables for nuclei ranging from medium to heavy masses. This wealth of developments can be extended to reaction calculations based on NN effective interaction. The socalled Nuclear Structure Method (NSM) for scattering Vinh Mau (1970); Vinh Mau and Bouyssy (1976); Bernard and Van Giai (1979); Bouyssy et al. (1981); Osterfeld et al. (1981) relies on the selfconsistent HartreeFock (HF) and RandomPhase Approximations (RPA) of the microscopic optical potential. The former is a meanfield potential and the latter is a polarization potential built from targetnucleus excitations. A strictly equivalent method, the continuum particlevibration coupling using a Skyrme interaction, has been recently applied to neutron scattering from O Mizuyama and Ogata (2012), but neglecting part of the residual interaction in the coupling vertices. Other approaches are in progress, where optical potential is approximated as the HF term plus the imaginary part of the uncorrelated particlehole potential neglecting the collectivity of target excited states Pilipenko and Kuprikov (2012); Xu et al. (2014).
We report on optical potential calculations using NSM Vinh Mau (1970). Here the optical potential consists of two components,
(1) 
The HF potential, , is the major contribution to the real part of the optical potential. is calculated in coordinate space to ensure the correct asymptotic behavior of singleparticle states. It is non local and energy independent due to the nature of Gogny interaction, which is of finite range and energy independent, respectively. Rearrangement contributions stemming from the densitydependent term of the interaction are also taken into account.
The second component of the potential in Eq. (1) is
(2) 
which is complex, energy dependent and non local. Here and are contributions from particleparticle and particlehole correlations, respectively. The uncorrelated particlehole contribution is accounted for once in , and twice in . As a matter of fact, if twobody correlations are neglected in Eq. (2) for and , then reduces to as expected Vinh Mau (1970).
As mentioned in Ref. Bouyssy et al. (1981), if one works with an NN effective interaction with a densitydependent term, such as Gogny or Skyrme forces, attention must be paid to correlations already accounted for in the interaction. Indeed, part of particleparticle correlations is already contained at the HF level as far as is concerned. We thus use the same prescription as in Ref. Bernard and Van Giai (1979), omitting while is approximated by . Then Eq. (2) becomes
(3) 
From now on, equations are presented omitting spin for simplicity. For nucleons with incident energy E, the RPA potential reads
(4)  
where and are occupation number and energy of the singleparticle state in the HF field, respectively. Subscripts , and refer to the quantum number of particle, hole and the intermediate singleparticle, respectively. and represent the energy and the width of the excited state of the target, respectively. Additionally
(5) 
where and denote the usual RPA amplitudes and
(6) 
where P̂ is a particleexchange operator and is the NN effective interaction. The particlehole contribution reads
(7)  
with , the uncorrelated particlehole energy.
The description of target excitations has been obtained by solving the RPA/D1S equations in a harmonic oscillator basis, including fifteen major shells Blaizot and Gogny (1977) and using the D1S Gogny interaction Berger et al. (1991). We account for RPA excited states with spin up to , including both parities in order to achieve convergence of the cross section. The first 1 state given by RPA, containing the translational spurious mode, is removed. In order to avoid spurious modes in the uncorrelated particlehole term, we approximate the 1 contribution in by half that of the 1 contribution in . Coupling to excited states results in a number of poles in Eqs. (4) and (7). Moreover, fluctuations appear in the imaginary part of the potential whenever the energy matches a resonance energy of the intermediate singleparticle state . The leading inelastic doorways are those containing singleparticle resonances. Although the RPA/D1S method provides a good overall description of the spectroscopic properties of doubleclosed shell nuclei, couplings to two or more particlehole states and to continuum states are neglected. The impact of these couplings is a strength redistribution that can be handled assigning a finite width to each RPA state. It has the effect of averaging in energy and smoothing the potential. The resulting potential can then be identified with an optical model Feshbach (1958). A microscopic calculation of these widths is beyond the scope of the present study. We include them phenomenologically as an interpolation between reasonable values. takes the value of 2, 5, 15 and 50 MeV, for excitation energies of 20, 50, 100 and 200 MeV, respectively. The integrodifferential Schrödinger equation for elastic scattering is solved retaining the nonlocal structure of the potential Raynal (). Moreover optical potential calculations yield shape elastic, reaction and total cross sections Feshbach (1958). The compoundelastic cross section has to be added to shape elastic cross section and subtracted from reaction cross section before comparison with data Feshbach (1958). In a first attempt, we use the compoundelastic contribution from HauserFeshbach calculations with TALYS code Koning et al. (2008) using KoningDelaroche global potential Koning and Delaroche (2003). These considerations are particularly relevant for neutron scattering below 10 MeV.
In Fig. 1, we present results for the calculated differential cross sections based on NSM for both neutron and proton scattering from Ca. References to data are given in Ref. Koning and Delaroche (2003). Error bars are smaller than the size symbols. NSM results compare very well to experiment and those based on KoningDelaroche potential up to about 30 MeV incident energy. Beyond 30 MeV, backwardangle cross sections are overestimated. Discrepancies at 16.9 MeV (23.5 MeV) for neutron (proton) scattering are related to resonances in the intermediate singleparticle state when not completely averaged. A detailed treatment of the width might cure this issue.
In Fig. 2 we show calculated analyzing powers for neutron and proton scattering at several energies, in good agreement with measurements. Moreover, agreement with the data is comparable to that obtained from KoningDelaroche potential. These results suggest that NSM potential retains the correct spinorbit behavior.
In Fig. 3 we show reaction cross section for proton scattering (a) and total cross section for neutron scattering (b). Calculated reaction cross sections are in good agreement with experiments. For neutrons, however, we underestimate the total cross section below 10 MeV. Considering that the differential elastic cross section is well reproduced, this underestimate suggests that part of the absorption mechanism is not accounted for, as targetexcited states beyond RPA or doublecharge exchange process.
To understand the limited energy range of applicability of the NSM approach, we compare in Fig. 4 the volume integral, , of the central HF potential with the one obtained from the real part of the PereyBuck nonlocal potential Perey and Buck (1962). Black segments denote the strongest partialwave contributions accounting for 80% of the reaction cross section at the selected incident energies. Keep in mind that the HF potential is the leading contribution to the real part of in Eq. (1). Its contribution to is similar to that from PereyBuck up to about the twelfth partial wave (17 MeV). Beyond this point HF saturates, following the trend of the Hartree potential which is local and thus partialwave independent. This departure from PereyBuck explains why increasing incident energy (partial wave) yields much too high for HF, with the subsequent overestimate of the differential cross section at backward angles. It would be interesting to investigate to what extent the effective interaction has incidence on this behavior at high partial wave.
We now address the subtraction of the uncorrelated secondorder term in Eq. (2). As pointed out in Ref. Barbieri and Dickhoff (2001), this subtraction can lead to unphysical solutions with spurious poles and negative occupation numbers. The smooth and averaged potential obtained from Eq. (1) no longer suffers these pathologies. Indeed if one approximates , then Eq. (2) reduces to
(8) 
This approximation has the drawback of neglecting the real part of as well as part of the collectivity of the excited states. However, it has the advantage of avoiding secondorder double counting.
As seen in Fig. 5, both approximations in Eqs. (3) and (8) yield very similar shapes for each partialwave contribution of the diagonal imaginary part of for neutron scattering from Ca at 9.91 MeV. This trend remains true for higher partial waves and incident energies, confirming the good behavior of . In Fig. 6 we present the differential cross section for proton scattering from Ca based on these two approximations. The diffractive minima obtained with agree better with experiment than those obtained from . This result emphasizes the important role played by the real part of .
The work presented here constitutes a promising step forward aimed at a model keeping at the same footing both reaction and structure
aspects of the manynucleon system. Within the optical model potential, NSM is able to account reasonably well for
low energy scattering data. An important feature of the approach is the extraction of the imaginary part of the potential by means
of intermediate excitations of the system. It has been based on Gogny effective interaction, although it can be applied to any interaction
of similar nature. The study has been restricted to closedshell targets but can be extended to deformed
nuclei described with Quasiparticle RPA. Those results also open the way to new parametrizations of NN effective interactions including reaction
phenomena. A comprehensive work on the formalism and applications shall be presented elsewhere.
H. F. A. acknowledges partial funding from FONDECYT under Grant No 1120396.
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