Prospective study on microscopic potential with Gogny interaction
aa r X i v : . [ nu c l - t h ] A p r EPJ manuscript No. (will be inserted by the editor)
Prospective study on microscopic potentialwith Gogny interaction
G. Blanchon , M. Dupuis , and H. F. Arellano CEA,DAM,DIF F-91297 Arpajon, France Department of Physics - FCFM, University of Chile, Av. Blanco Encalada 2008, Santiago, ChileReceived: date / Revised version: date
Abstract.
We present our current studies and our future plans on microscopic potential based on effectivenucleon-nucleon interaction and many-body theory. This framework treats in an unified way nuclear struc-ture and reaction. It offers the opportunity to link the underlying effective interaction to nucleon scatteringobservables. The more consistently connected to a variety of reaction and structure experimental data theframework will be, the more constrained effective interaction will be. As a proof of concept, we present somerecent results for both neutron and proton scattered from spherical target nucleus, namely Ca, usingthe Gogny D1S interaction. Possible fruitful crosstalks between microscopic potential, phenomenologicalpotential and effective interaction are exposed. We then draw some prospective plans for the forthcomingyears including scattering from spherical nuclei experiencing pairing correlations, scattering from axiallydeformed nuclei, and new effective interaction with reaction constraints.
PACS.
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Producing satisfactory data evaluations based solely onmany-body theories and effective nucleon-nucleon (NN)interaction is a long term project. Two keys to success are: (i) robust and well tested nuclear reaction codes, such as
TALYS [1] or
EMPIRE [2], flexible enough to incorpo-rate new microscopic models and (ii) microscopic inputssuch as optical model potentials, nuclear level densities, γ -ray strength functions, and fission properties, all basedon effective NN interaction. Advances made along this lineprovide more opportunities to connect the effective NN in-teraction to a broad body of structure and reaction dataand as a matter of fact to improve its parametrization. Many-body theories based on effective NN interaction suchas Hartree-Fock(-Bogolyubov) (HF(B)) for static proper-ties and (quasiparticle-)random-phase approximation ((Q)RPA)[3] or five-dimension collective Hamiltonian (5DCH) [4]for dynamical properties have proven their ability to de-scribe a wide range of nuclear structure observables, in-cluding binding energy, charge radius, deformation, exci-tation spectrum, density and spectroscopic factors, this fornuclear masses with A &
5. As an illustration of the ex-tended reach of the method, we show in Fig. 1, nuclear a e-mail: [email protected] deformations determined within axially deformed HFBwith Gogny D1S interaction all over the nuclide chart [5].Effective theories are usually based on phenomenologicalparametrizations of the effective NN interaction, such asSkyrme [6,7] or Gogny forces [8,9,10,11]. In the follow-ing we mainly focus on Gogny interaction. The fit of theinteraction mostly relies on connection made with struc-ture data through the effective theory. Those constraintsare then completed by physical filters coming from infi-nite matter calculations. Up to now, two strategies havebeen adopted. The first and original one uses a restrictedHF model where single-particle orbitals are approximatedby harmonic-oscillator wave functions for simplicity. Thismakes possible to determine parameter sets of the inter-action from a limited number of constraints by matrixinversion and from physical filters. This strategy has beenapplied with success to the determination of D1 [9], D1S[10] and D1N [11] versions of Gogny interaction. Recentmultiparticle-multihole configuration mixing studies [12,13] have motivated the elaboration of a generalized Gognyinteraction with finite-range density, spin-orbit and tensorterms. Along that line, Gogny D2 interaction with a finiterange density term has been designed [14]. The secondstrategy is more based on brute force using HFB with aself-consistent account of quadrupole correlations energieswithin the 5DCH approach. This strategy has been usedto develop the D1M parametrization of Gogny interactionwhich reaches a rms deviation with respect to the 2149measured masses of only 798 keV [8]. This improvement G. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction NZ <β> Fig. 1.
Chart of the nuclides showing deformations obtained with axially deformed HFB/D1S [5]. of the parametrization has been done conserving as muchas possible the virtues of former Gogny interactions.As a consequence, the global character of effective theoriesas well as their accuracy and their relatively low compu-tational cost make them well suited to fulfill the needs foraccurate nuclear data files in a reasonable time scale.
Various phenomenological ingredients of reaction codeshave been replaced by their microscopic counter partsbased on effective interaction. Connecting the dots, it willbe possible to obtain satisfactory evaluations on the ba-sis of reliable and accurate microscopic inputs only [15].A good example of this on going work is the use madeof results on
U isotope described within QRPA withGogny force [16,17]. Those results have made possible amicroscopic description of preequilibrium without ad-hoc “pseudo-state” prescription [18]; see also Dupuis et al. inthis issue. Along the same line, nuclear level densities havebeen obtained using a temperature-dependent HFB ap-proach with Gogny interaction [19]. γ − ray strength func-tion studies based on QRPA and Gogny interaction havealso been developed [20,21]. Concerning optical potentials, TALYS uses phenomenological potentials [22,23] in theregions where data are available and a modified version ofthe semi-microscopic Jeukenne-Lejeune-Mahaux [24] else-where. Neither of those methods allows a direct connec-tion with NN interaction. In order to fulfill this lack, weare interested in developing a microscopic potential basedon the effective NN interaction.
Depending on the projectile energy and the target mass,various strategies have been adopted in order to deal withelastic scattering starting from NN interaction. We nowexpose the pros and cons of those different methods. In the following, ab-initio (microscopic) refers to methodsbased on bare (effective) NN interaction.Nuclear matter models [25] provide reasonable descrip-tions of nucleon elastic scattering at incident energies aboveabout 50 MeV [26], even up to ∼ ad-hoc prescription such as mean-field or beyond mean-field approaches. Work toward aconsistent treatment of both density and g-matrix is inprogress [28]. Recent ab-initio calculations address theissue of reactions involving light nuclei and low-energyregime. The resonating group method within the no-coreshell model, has successfully described nucleon scatteringfrom light nuclei [29]. This ab-initio model has recentlybeen extended to include three-nucleon forces for nucleonscattering from He [30,31]. They deal with He, He and Be targets and incident energies below 15 MeV. An-other method, the Green’s function Monte Carlo methodhas been used to describe nucleon scattering from Hein particular the phaseshift of the first partial waves be-low 5 MeV incident energy [32]. Other ab-initio calcula-tions handle magic nuclei. Among them, the self-consistentGreen’s function (SCGF) method has been applied to op-tical potential calculations for , , Ca targets [33]. TheSCGF potential is compared with a phenomenological dis-persive potential [34]. This model underestimates nuclearradii and, as a consequence, is not well suited for scat-tering calculations. Further studies including three-bodyforces may cure this issue. Moreover work on Gorkov-Green’s function theory is in progress to extend SCGFto nuclei around closed-shell nuclei [35,36]. Finally, thecoupled-cluster theory has been applied to proton elasticscattering from Ca [37]. Cross section at 9.6 MeV and12.44 MeV center-of-mass energy are compared with data.They observe a lack of absorption.Although ab-initio methods have made progresses in han-dling light and magic-nuclei, they are still yet suited nei-ther for heavy targets nor for high incident energy projec-tiles. Another option is to build the potential starting from . Blanchon et al.: Prospective study on microscopic potential with Gogny interaction 3 an effective NN interaction. The price to pay is to breakthe explicit link with bare NN interaction. The advantageis once again the extended reach of effective theories andthe wealth of results already available.The so-called nuclear structure method (NSM) for scatter-ing [38,39,40,41] relies on the self-consistent HF and RPAapproximations to the microscopic optical potential [42].The former is a mean-field potential; the latter is a po-larization potential built from target nucleus excitations.This method applies to double-closed shell spherical tar-get nuclei well described with RPA. A strictly equivalentmethod, the continuum particle-vibration coupling usinga Skyrme interaction, has been recently applied to neutronscattering from O [43]. They neglect part of the residualinteraction in the coupling vertices. In addition, they donot address the issue of the double counting of the uncor-related second-order diagram. Other approaches aimingat fitting a Skyrme effective interactions including reac-tion constraints are in progress, where optical potential isapproximated as the HF term and the imaginary part ofthe uncorrelated particle-hole potential neglecting collec-tivity of target excited states [44,45]. A recent applicationof NSM with Gogny interaction is presented in Ref. [46].The same interaction is consistently used to generate themean-field, the excited states and the couplings. In thisstudy, special attention is given to the issue of the doublecounting of the uncorrelated second-order diagram. Thesubtraction of this second-order term is shown not to leadto pathological behaviors when positive incident energyis considered, contrarily to what is expected in Ref. [47].Moreover, the use of the finite-range Gogny interactionprevents from the necessity of ad-hoc momentum cut-offwhen second-order effects are considered.
In the continuity of work presented in Ref. [46], we wish toexplore the possible connections provided by a microscopicoptical potential using NSM. The framework is summa-rized in Fig. 2. In the first line of the diagram, effective
Experimentalcross sectionMicroscopicpotential PhenomenologicalpotentialStructure ExperimentalspectroscopyEffective NNinteraction
Fig. 2.
Connections. Dashed arrows emphasize connectionsdiscussed in the following. interaction is used within structure models to make spec-troscopic predictions. Feedback from experiment provides constraints on the interaction whenever a reliable struc-ture model is used. Effective interaction and structure cal-culation can then be used to define the microscopic po-tential through NSM. Once the corresponding scatteringproblem is solved, feedbacks are made possible from crosssection data providing reaction constraints to the fit ofthe interaction. Microscopic potentials are nonlocal, com-plex and energy dependent. They can provide prescrip-tions for future phenomenological potentials, in particularconcerning the shape of the nonlocality, the energy de-pendence. One can as well investigate the origin of thevolume and the surface part of the potential in terms oftarget excitations. Reciprocally, when data are available,phenomenological potentials can help identifying contri-butions missing in microscopic potentials. Moreover when-ever phenomenological potentials obey dispersion relation,a connection is also made with spectroscopy. In Fig. 2, themain connections we wish to investigate in the followingare highlighted.In the following, the NSM formalism for spherical-target nuclei and the integro-differential Schr¨odinger equa-tion are briefly exposed in Sec. 2.1 and Sec. 2.2, respec-tively. In Sec. 2.3, we emphasize the importance of theexact treatment of the intermediate HF propagator andmore precisely the account of single-particle resonances.As a proof of concept in Sec. 3, we apply NSM to nucleonelastic scattering from Ca. Some possible crosstalks be-tween phenomenological potentials and their microscopiccounter part are discussed in Sec. 4. In Sec. 5, we showhow phenomenological nonlocal potential can relate to theeffective NN interaction through volume integrals. Finallyin Sec. 6, we draw plans for the decade to come. In par-ticular, we mention the issue of spherical target nuclei ex-periencing pairing correlations and the one of deformedtarget nuclei.
The NSM formalism is presented in detail in Ref. [42]. Webriefly introduce the key points of the formalism. Equa-tions are presented omitting spin for simplicity. The po-tential, V , consists of two components, V = V HF + ∆V. (1)This potential will be referred to as the NSM potentialin the following. The former is a mean-field potential; thelatter is a polarization potential built from target nucleusexcitations. The HF potential, V HF , is the major contri-bution to the real part of the optical potential. The polar-ization potential, ∆V , brings only a correction to the realpart of V and entirely generates its imaginary contribu-tion. The HF potential is obtained in the Green’s functionformalism neglecting two-body correlations [48]. It reads, V HF ( r , r’ ) = Z d r v ( r , r ) ρ ( r ) δ ( r − r’ ) − v ( r , r’ ) ρ ( r , r’ ) , (2) G. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction where v is the effective NN interaction and ρ ( r ) = X i | φ i ( r ) | , (3) ρ ( r , r’ ) = X i φ ∗ i ( r ) φ i ( r’ ) , (4)are the local and nonlocal densities with i running over oc-cupied states. V HF is made of a local direct term and anexchange term which is nonlocal because of the finite rangeof Gogny interaction. It is energy independent. Rearrange-ment contributions stemming from the density-dependentterm of the interaction are also taken into account. To-gether with Schr¨odinger equation, Eq. (2) defines a self-consistent scheme as shown in Fig. 3. Schr¨odinger equation V HF ( ρ ) ρ NN interaction
Fig. 3.
Self-consistent Hartree-Fock.
The polarization potential, ∆V in Eq. (1), is built cou-pling the elastic channel to the intermediate excited statesof the target nucleus. Those excited states are describedwithin the RPA formalism. Both excited states and cou-plings are generated using Gogny interaction. Togetherwith Schr¨odinger equation, Eq. (1) now defines a new self-consistent scheme, illustrated in Fig. 4 when consideringboth full-line and dashed-line arrows. In practice, as de- Schr¨odinger equation V HF ( ρ ) + ∆V ( ρ ) ρ NN interaction
Fig. 4.
Self-consistent RPA (full-line and dashed-lined ar-rows). Consistent RPA on top of a self-consistent HF (full-linearrow only). The HF propagator is dressed only once. scribed in Fig. 4, we first converge the HF scheme, asshown in Fig. 3, then we dress only once the HF prop-agator by coupling to the excitations of the target. Thismakes the scheme only consistent at that stage. Going intomore details, the polarization contribution to the potentialreads, ∆V = V P P + V RP A − V (2) , (5)where V P P and V RP A are contributions from particle-particle and particle-hole correlations, respectively. Theuncorrelated particle-hole contribution, V (2) , is accounted for once in V P P and twice in V RP A . When two-body cor-relations are neglected in Eq. (5), one expects ∆V to re-duce to V (2) . As a matter of fact V (2) shall be subtractedtwice [42]. This formalism takes into account all the cor-relations explicitly. Although it is well suited for ab-initio developments, we wish to make the connection with effec-tive NN interactions. In practice, if one uses an effectiveinteraction with a density-dependent term, such as Gognyor Skyrme forces, attention must be paid to correlationsalready accounted for in the interaction [40]. Indeed, insuch a case, part of particle-particle correlations is alreadycontained at the HF level. We thus use the same prescrip-tion as in Ref. [39], omitting the real part of V P P whileapproximating the imaginary part of V P P by Im (cid:2) V (2) (cid:3) .Then Eq. (5) reduces to ∆V = Im h V (2) i + V RP A − V (2) . (6)Both ingredients of Eq. (6), V RP A and V (2) , can be ex-pressed in terms of diagrams. In Fig. 5, wavy lines standfor the effective NN interaction and up (down) arrowsstand for HF particle (hole) propagators. Subscript p ( h )refers to the quantum numbers of the single-particle (hole)HF states used to build target excitations. The subscript λ refers to the quantum numbers of the intermediate single-particle state of the scattered nucleon. Both discrete andcontinuum spectra of the intermediate single-particle stateare accounted for. Label (a) refers to V RP A built with anunoccupied intermediate state. Label (a’) refers to V RP A built with an occupied intermediate state. Labels (b) and(b’) refer to the corresponding uncorrelated particle-holecontributions. Exchange diagrams are taken into account. λ p h ( a ) RP A λ p h ( a ′ ) RP A λ p h ( b ) λ p h ( b ′ ) Fig. 5.
Diagrammatic contributions of V RPA (a) and V (2) (b).Indices p , h and λ refer to particle, hole and the intermediatestate in the HF field, respectively. Wavy lines stand for theeffective NN interaction.. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction 5 For nucleons with incident energy E , the RPA potential,corresponding to Figs. 5a and 5a’, reads, V RP A ( r , r ′ , E ) = X N =0 ZX λ (cid:20) n λ E − ε λ + E N − iΓ ( E N )+ 1 − n λ E − ε λ − E N + iΓ ( E N ) (cid:21) × Ω Nλ ( r ) Ω Nλ ( r’ ) , (7)where n i and ε i are occupation number and energy of thesingle-particle state φ i in the HF field, respectively. E N and Γ ( E N ) represent the energy and the width of the N th excited state of the target, respectively. Additionally, Ω Nλ ( r ) = X ( p,h ) h X N, ( p,h ) F phλ ( r ) + Y N, ( p,h ) F hpλ ( r ) i , (8)where X and Y denote the usual RPA amplitudes and F ijλ ( r ) = Z d r φ ∗ i ( r ) v ( r , r ) h − ˆP i φ λ ( r ) φ j ( r ) , (9)where ˆP is a particle-exchange operator and v is the sameeffective NN interaction as in Eq. (2). The uncorrelatedparticle-hole contribution, corresponding to Figs. 5b and5b’, reads V (2) ( r , r ′ , E ) = 12 X ij ZX λ (cid:20) n i (1 − n j ) n λ E − ε λ + E ij − iΓ ( E ij )+ n j (1 − n i )(1 − n λ ) E − ε λ − E ij + iΓ ( E ij ) (cid:21) × F ijλ ( r ) F ∗ ijλ ( r ′ ) , (10)with E ij = ε i − ε j , the uncorrelated particle-hole energy.In practice, V HF is determined in coordinate space toensure the correct asymptotic behavior of single-particlestates. This nonlocal potential is then used to build thesingle-particle intermediate state used to determine thepolarization potential, ∆V . It is worth mentioning thatthe HF potential in coordinate space reproduces boundstate energies obtained with the HF/D1S code on oscilla-tor basis which makes our result reliable.The description of target excitations is obtained solvingRPA equations in a harmonic oscillator basis, includingfifteen major shells [49] and using the Gogny D1S inter-action [10]. We account for RPA excited states with spinup to J = 8, including both parities, in order to achievethe convergence of cross section calculations. The first J π = 1 − excited state given by RPA, containing the spu-rious translational mode, is removed from the calculation.Moreover, in order to avoid spurious modes in the uncor-related particle-hole term, we approximate the J π = 1 − contribution in V (2) by half that of the J π = 1 − contri-bution in V RP A .Even though RPA/D1S method provides a good over-all description of the spectroscopic properties of double-closed shell nuclei, still some contributions are left out. First, the projection on an oscillator basis discretizes theRPA continuum. As a consequence, the escape width ismissing from the structure calculation. Second, couplingsto two or more particle-hole states are excluded from themodel space even though they may play a significant role.The impact of these couplings is a strength redistributionthrough a damping width as well as a shift in energy of ex-cited states. Third, the optical potential is, by definition,built to provide the energy-averaged S -matrix. Hence, therapid fluctuations, the potential exhibits at low energydue to compound-elastic contribution, shall be averagedbefore identifying the result of Eq.(1) with an optical po-tential [50].In the present work, we simulate those three differentwidths assigning a single phenomenological width, Γ ( E N ),to each RPA state. Γ ( E N ) takes the value of 2, 5, 15and 50 MeV, for excitation energies of 20, 50, 100 and200 MeV, respectively. Those values have not been fittedin order to better reproduce cross section data. Longerterm solutions are planned in order to provide more mi-croscopic prescriptions for those widths. The escape widthcan be obtained using continuum RPA [51,43]. We alsoplan to determine the damping width and the energy shiftusing the multiparticle-multihole configuration mixing method[52]. The integro-differential Schr¨odinger equation for boundstates and scattering is solved without localization proce-dures. The radial Schr¨odinger equation reads, − ~ µ (cid:20) d dr − l ( l + 1) r (cid:21) f lj ( r )+ r Z ν lj ( r, r ′ ; E ) f lj ( r ′ ) r ′ dr ′ = Ef lj ( r ) , (11)where f lj ( r ) = rφ lj ( r ) is the partial wave for the projectile-target relative motion, E is the incident nucleon energyand ν lj ( r, r ′ ; E ) is defined from the multipole expansionof the nonlocal potential V ( r σ, r’ σ ′ ; E ) = X ljm Y ljm ( ˆr σ ) ν lj ( r, r ′ ; E ) Y † ljm ( ˆr ′ σ ′ ) , (12)with Y ljm ( ˆr σ ) ≡ [ Y l ( ˆr ) ⊗ χ / ( σ )] jm . (13)The potential, V , is complex and energy dependent for E >
0, and real and energy independent for
E <
0. Dis-crete solutions are obtained by expressing Eq. (11) on amesh in coordinate space and performing the correspond-ing matrix diagonalization [53]. For positive energies, thescattering problem with the correct asymptotic conditionsturns into a matrix inversion following J. Raynal’s methodfor scattering exposed in the
DWBA code explanatoryleaflet [54].
G. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction
We now would like to emphasize the impact of an ex-act treatment of φ λ on the second-order terms of thepotential, V RP A and V (2) (see Eqs. (7) through (10));especially, the role of single-particle resonances alreadydiscussed by Rao et al. [55] within a phenomenologicalapproach. In previous works, φ λ has often been approxi-mated by a plane wave for neutron and a Coulomb wavefor proton [40] or discretized [39]. In this work, we includeboth discrete and continuum spectra of φ λ determined inthe HF field with the correct asymptotic solutions. Phase-shifts for neutron and proton scattering from Ca in theHF field are shown in Fig. 6. We observe single-particleresonances for several partial waves each time phaseshiftincreases rapidly through an odd multiple of π/
2. Reso-nance energies are summarized in Table 1. Those reso- δ (r ad / π ) E (MeV) (a) j=1/2 l=0j=1/2 l=1j=3/2 l=1j=3/2 l=2j=5/2 l=2 δ (r ad / π ) E (MeV) (c) j=1/2 l=0j=1/2 l=1j=3/2 l=1j=3/2 l=2j=5/2 l=2 δ (r ad / π ) E (MeV) (b) j=5/2 l=3j=7/2 l=3j=7/2 l=4j=9/2 l=4j=9/2 l=5 δ (r ad / π ) E (MeV) (d) j=5/2 l=3j=7/2 l=3j=7/2 l=4j=9/2 l=4j=9/2 l=5
Fig. 6.
Phaseshift in the HF field for neutron (panel (a) and(b)) and proton (panel (c) and (d)) scattering from Ca as afunction of incident energy for the first ten partial waves.
Table 1.
HF single-particle resonance energies (in MeV) forneutron (n) and proton (p) scattering from Ca.n p12.18 (j=7/2, l=4) (j=1/2, l=1) (j=9/2, l=4) (j=3/1, l=1) (j=11/2, l=5) (j=5/2, l=3) (j=13/2, l=6) (j=7/2, l=4) (j=9/2, l=4) (j=11/2, l=5) (j=13/2, l=6) nances will result in fluctuations of the imaginary part of V RP A and V (2) (Eqs. (7) and (10)) whenever the energy E − E N matches a resonance energy ε λ of the intermediatesingle-particle state. As a consequence, those resonanceswill strongly influence the corresponding cross section. As
10 12 14 16 18 20
E (MeV) σ R ( m b ) DWCoul ε λ = 2.15 MeVj=3/2 l=1 ε λ = 5.65 MeVj=5/2 l=3 ε λ = 9.55 MeVj=9/2 l=4 ε λ = 3.70 MeVj=1/2 l=1 Fig. 7.
Reaction cross section vs. proton incident energy with V = V HF + Im[ V RPA /
2] coupling only to the first 1 − excitedstate in Ca ( E x = 9 . φ λ treatedas a Coulomb wave (dashed line) or as a single-particle statein the HF field (solid line). an illustration of resonance effects, we show, in Fig. 7,the reaction cross section for proton scattering with a po-tential including the HF potential as real part and theimaginary part of the RPA potential generated restrictingcouplings to the second J π = 1 − excited state of Cawith E x = 9 . Γ ( E N ) = 0 MeV in Eq. (7) in order to emphasizethe effect of resonances. This result is compared with acalculation using a Coulomb wave as intermediate state.The exact treatment of the intermediate state leads to aglobal enhancement of the reaction cross section. Moreover E (MeV) σ R ( m b ) V HF + Im [V RPA ] Γ = 0 MeVV HF + Im [V (2) ] Γ = 0 MeV Fig. 8.
Reaction cross section vs. neutron incident energy. Neu-tron scattering from Ca with V HF + Im [ V RPA ] (solid line)and V HF + Im [ V (2) ] (dashed line) coupling to all availableopen channels.. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction 7 we notice that coupling to only one excited state of thetarget already leads to four resonances between 10 and 20MeV. One expects to get a large number of resonant con-tributions once coupling to the thousand target excitedstates. As an example, we show in Fig. 8 the same cal-culation as in Fig. 7 but including all the open channelsfor a given incident energy. Once again as discussed inSec. 2.1, one needs to average fluctuating contributionsbefore comparing scattering observables with experiment.This shows the importance of a complete treatment of theintermediate wave. As a first application, NSM has been applied to neutronand proton scattering from Ca using Gogny D1S inter-action. The corresponding differential cross sections forincident energies below 40 MeV are presented in Fig. 9.Compound-elastic corrections furnished by the Hauser-Feshbach formalism using Koning-Delaroche potential with
TALYS are applied to cross sections obtained from NSMand Koning-Delaroche potential, respectively. It is mostlyrelevant below 10 MeV for neutron projectile while it givesa smaller contribution for proton. NSM results comparevery well to experiment and those based on Koning-Delarochepotential up to about 30 MeV incident energy. Referencesto data are given in Ref. [22]. Error bars are smaller thanthe size symbols. Beyond 30 MeV, backward-angle crosssections are overestimated. Discrepancies at 16.9 MeV (23.5 MeV)for neutron (proton) scattering are related to resonancesin the intermediate single-particle state when not com-pletely averaged. A detailed treatment of the width mightcure this issue. In Fig. 10, we show calculated analyz-ing powers for neutron and proton scattering at severalenergies, in good agreement with measurements. More-over, agreement with data is comparable to that obtainedfrom Koning-Delaroche potential. These results suggestthat NSM potential retains the correct spin-orbit behav-ior. In Fig. 11, we show reaction cross section for protonscattering and total cross section for neutron scattering.Calculated reaction cross sections are in good agreementwith experiments. For neutrons, however, we underesti-mate the total cross section below 10 MeV. Consider-ing that the differential elastic cross sections are well re-produced (see Fig. a), this underestimation suggests thatpart of absorption mechanisms is not accounted for, suchas target-excited states beyond RPA, double-charge ex-change process or an intermediate deuteron formation.This microscopic potential makes the bridge between crosssection data and effective interaction. It is worth mention-ing that we already get nice agreement with data withoutany adjustable parameter and using an effective interac-tion, the Gogny D1S interaction, originally tailored forstructure purposes. This framework opens the way for neweffective interactions based both on structure and reactionconstraints. θ c.m. (deg.) -4 -2 d σ / d Ω ( m b / s r) (a) θ c.m. (deg) -10 -8 -6 -4 -2 σ ( θ ) / σ R u t h (b) Fig. 9.
Differential cross sections for neutron (a) and proton(b) scattering from Ca. Comparison between data (symbols), V HF + ∆V results (solid curves) and Koning-Delaroche poten-tial results (dashed curves). G. Blanchon et al.: Prospective study on microscopic potential with Gogny interaction -1-0.500.51-1-0.500.51-1-0.500.510 20 40 60 80 100 120 140 160 180 θ c.m. (deg) -1-0.500.51 9.9111.13.916.9 A y ( θ ) (a) -1-0.500.51-1-0.500.51-1-0.500.510 20 40 60 80 100 120 140 160 180 θ c.m. (deg) -1-0.500.51 14.5115.9718.5740. A y ( θ ) (b) Fig. 10.
Same as Fig. 9 for analyzing powers.
We shall now highlight the possible crosstalk between mi-croscopic and phenomenological potentials depicted in Fig. 2.A large variety of local potentials have been developedin order to describe reaction data. Mahaux and Sartorhave then demonstrated the need for the potential to sat-isfy a dispersion relation, connecting its imaginary partto its real part, which provides a link with shell model[56]. Along that line, local dispersive potentials have beendeveloped [57,58]. One issue in those local approaches isthe spurious energy dependence of the potential comingfrom the use of a local ersatz to represent a nonlocal ob-
E (MeV) σ R ( m b ) Exp.KDV HF + ∆ V (a) E (MeV) σ T o t ( m b ) Exp.KDV HF + ∆ V (b) Fig. 11.
Reaction cross section for proton (a) and total crosssection for neutron (b) scattering from Ca. Comparison be-tween data (symbols), V HF + ∆V results (solid curve) andKoning-Delaroche potential (dashed curve). ject. This issue has been overcome building a dispersivepotential with a nonlocal static real component [59]. Arecent version of this dispersive potential is fully nonlocalboth in its real and imaginary parts [60]. It is parametrizedonly for Ca but using all the structure and reaction dataavailable for this nucleus. Another nonlocal dispersive po-tential is currently being developed for a broader range ofnuclei [61]. It is interesting to compare such phenomeno-logical potentials with microscopic and ab-initio ones [33,62]. This connection can help identifying missing compo-nents in microscopic potentials. Reciprocally microscopicpotential can provide some guidance for next-to-come po-tential parametrizations regarding for example the shapeand the range of the nonlocality or the incident energydependence.As an illustration, we now consider the case of neutronscattering from Ca at E = 9 .
91 MeV. In Fig. 12, wecompare the NSM potential with the nonlocal dispersive(NLD) potential from Ref. [60]. We focus on the multipole . Blanchon et al.: Prospective study on microscopic potential with Gogny interaction 9 expansion (see Eq. (12)) of the imaginary part of bothnonlocal potentials and depict their diagonal contribu-tions. NSM and NLD potentials compare very well around r = 4 . r I m [ ν (r , r) ] ( M e V /f m ) NLDV HF + ∆ Vr (fm)PW r I m [ ν (r , r) ] ( M e V /f m ) Fig. 12.
Multipole expansion of the imaginary part of V HF + ∆V (solid curve) and NLD potential (dashed curve) for r = r ′ , as a function of radius and partial waves for n+ Caat 9.91 MeV. compare the nonlocality of both imaginary components atthe surface of the nucleus ( r = 4 . s = | r − r’ | . We get a good agreementbetween the microscopic approach and the phenomenolog-ical one. Even though small emissive contributions appearin some of the multipoles, NSM validates the choice of aGaussian nonlocality as originally proposed by Perey andBuck [63] and used as well for NLD potential. NSM also re-produces the range of the nonlocality of the NLD potentialwhich corresponds to a nonlocality parameter β = 0 .
94 fmat the surface of the nucleus.We have presented here results only for a given projec-tile at a given incident energy. A more exhaustive andhopefully conclusive study is in progress. Along the sameline, it will be interesting to look at the nucleon asym-metry dependence of the NSM potential for example go-ing toward neutron-rich Ca isotopes and comparing withthe dispersive potential obtained by Charity et al. for , , , , , Ca [34]. -20246 l=0 j=1/2 l=1 j=1/2 l=1 j=3/2-20246 I m [ r r ’ ν ( s ) ] ( M e V /f m ) l=2 j=3/2 l=2 j=5/2 l=3 j=5/2-20246 -3 -2 -1 0 1 2 3l=3 j=7/2 -3 -2 -1 0 1 2 3 s (fm) l=4 j=7/2 -3 -2 -1 0 1 2 3l=4 j=9/2 Fig. 13.
Same as Fig. 12 for nonlocality at r = 4 . We now consider the connection between phenomenologi-cal potential and effective NN interaction through the mi-croscopic potential and the structure calculation as shownin Fig. 2. The link between microscopic potential and phe-nomenological one is done using volume integrals. Volumeintegrals are useful means of comparison between poten-tials as they are well constrained by scattering data. Herewe focus on the real part of the potential. The volumeintegral for a given multipole ( l, j ) of the real part of thenonlocal potential is defined as, J ljV ( E ) = − πA Z dr r Z dr ′ r ′ Re[ ν lj ( r, r ′ , E )] , (14)where A is the nucleon number of the target. The HF po-tential is the leading contribution to the real part of NSMpotential in Eq. (1). In Fig. 14, we present the volumeintegral of the multipole expansion of the HF potential,Eq. (2), as a function of the partial wave. We compare itto the same quantity obtained from Perey-Buck (PB) non-local potential [63]. HF potential gives results similar toPB potential up to about the twelfth partial wave. Blacksegments denote the strongest partial-wave contributionsaccounting for 80% of the reaction cross section at the se-lected incident energies. Hence taking the PB potential asa reference, the HF potential has a reasonable behaviorup to about 17 MeV incident energy. Beyond this energyHF saturates, following the trend of the Hartree poten-tial which is local and thus partial-wave independent. Asa result increasing incident energy, HF yields a much toolarge volume integral reflected in an overestimate of thedifferential cross section at backward angles as shown inFig. 9. We present here results obtained with the D1Sparametrization of the Gogny force, but we came to thesame conclusions using the D1M parametrization [8]. Thebehavior of the volume integral as a function of the partialwave is dictated by the shape and the range of the non-locality. PB potential is built with a Gaussian nonlocalitywhereas the HF potential is made of a local Hartree term J V ( M e V f m ) PW 40 MeV17 MeV10 MeV5 MeV0.5 MeV V HF HartreePB
Fig. 14.
Volume integral as a function of partial waves for neu-tron scattering from Ca: HF potential (solid curve), Hartreepotential (dash-dotted curve) and Perey-Buck potential (dot-ted curve). Horizontal segments denote the partial-wave inter-val to sum up 80% of the reaction cross section at selectedincident energies. and a nonlocal Fock term as shown in Eq. (2). Those twocontributions can be related to the different terms of theeffective NN interaction. Gogny interaction is built withtwo Gaussian ranges in its central part. The contributionsto the Hartree potential of those two Gaussians as wellas the contribution of the density-dependent term of theinteraction are shown in panel (a) of Fig. 15. In panel (b),we show the contributions of the two central terms withfinite range to the nonlocality of the first partial waveof the Fock term. The summation of those two Gaussiannonlocalities with opposite sign yields the ”W” shape ofthe total nonlocality. We then present in panel (c) thevolume integral corresponding to the nonlocal Fock po-tential. The good behavior of the HF volume integral forlow partial waves is related to the downward slope of theFock volume integral and its change of sign around theseventh partial wave. Then for higher partial waves, theFock volume integral converges to the zero limit. The cor-responding behavior of the Fock term as a function of thepartial wave is depicted in panel (d).We plan to investigate to what extent the effective interac-tion could be improved in order to get a better behavior ofthe HF potential above 30 MeV. This issue emerges fromthe use of a local and energy-independent ersatz for the ef-fective NN interaction instead of the actual g-matrix withits full complexity. One way to get a fully nonlocal HF po-tential would be to use a nonlocal effective NN interactionas proposed for example by Tabakin [64].
The present work constitutes a promising step forwardaimed to a model keeping at the same footing both re- -2000-1500-1000-500 0 500 1000 1500 0 1 2 3 4 5 6 7 8 V H ( M e V ) r (fm) (a) TotalRange 1Range 2Density -150-100-50 0 50 100 150 200 0 5 10 15 20 25 30 J V ( M e V f m ) PW (c) l = 0 -> 8l = 9 -> 15 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 V F ( M e V /f m ) s (fm) (b) TotalRange 1Range 2 -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 V F ( M e V /f m ) s (fm) (d) l = 0 -> 8l = 9 -> 15 Fig. 15.
Contributions for n + Ca to: (a) to the Hartree localpotential ( V H ): Total (solid line), first range of D1S (dashedline), second range of D1S (dash-dotted line) and density term(dotted line). (b) First partial wave of the nonlocal Fock termat r = r ′ = 4 . r = r ′ = 4 . action and structure aspects of the many-nucleon sys-tem. Starting from an effective NN interaction, NSM ac-counts reasonably well for low-energy scattering data. Weuse consistently the Gogny D1S interaction, although thisscheme can be applied to any interaction of similar nature.An important feature of the approach is the extraction ofthe imaginary part of the potential by means of intermedi-ate excitations of the target. The study has been restrictedto closed-shell target but can be extended to account forpairing correlations as well as axial deformation. We nowexpose in more detail our plans for the forthcoming years. In the short-term future, we wish to investigate the NSMscheme for spherical targets according to the followingplan, • NSM will be applied to a broader range of target nucleiwell described within RPA including Ca, Zr,
Snand
Pb. • The link between NSM potential and phenomenologycan be carried on exploring energy, multipole and massdependences of the potential. The NSM potential canalso provide some trends for the shape and the rangeof the nonlocality. We plan to investigate the buildingof surface and volume contributions as a function oftarget excitations. . Blanchon et al.: Prospective study on microscopic potential with Gogny interaction 11 • Above about 50 MeV incident energy, a connectioncan be established between the NSM potential and thefolding potential relying on g-matrix and thus with thebare NN interaction. This can be a fertile ground fornew effective interactions. • At low incident energy, NSM provides a volume partof the imaginary potential larger than phenomenology.The small volume contribution in phenomenologicalpotential is often justified by the fact that the pro-jectile does not have sufficient energy to knock out atarget nucleon. This discrepancy is possibly due to thefact that at low energy the volume part of the imagi-nary potential is not well constrained because the pro-jectile does not explore the interior of the potential. Weplan to use the NSM potential in inelastic scatteringcalculations in order to disentangle this issue. • In its present version, NSM requires a phenomeno-logical width. This width has several microscopic ori-gins, as discussed in Sec. 2.1. A microscopic accountof those widths is planed using continuum RPA [51]and multihole-multiparticle configuration mixing [52]for the escape and the damping widths, respectively. • The NSM potential will be used to provide transmis-sion coefficients for compound-elastic calculations. More-over, we plan to develop a compound-nucleus formal-ism based only on NSM. Indeed, NSM gives access tothe fluctuating contribution of the S-matrix and as aconsequence to the compound-elastic contribution. • The study of the volume integral of the real part of thepotential has exhibited the possible crosstalk betweenphenomenological potentials and effective NN interac-tions. In particular, the interaction is not well suitedfor the description of partial waves with more thanabout ℓ = 7. Those reaction constraints will be usedfor new parametrizations of the interaction. Moreover,a nonlocal version of the effective NN interaction couldtackle the issue of the saturation of the HF volume in-tegral to the Hartree one. The main next step will be to take into account pairingcorrelations in spherical nuclei. • We plan to develop a HFB potential in coordinatespace. The mean-field and the pairing field have al-ready been studied in coordinate space in a previ-ous work on Cooper’s pairs [65]. The goal is then todeal with quasiparticle scattering with a special careof resonances in both mean-field and pairing channelas shown in Ref. [66,67]. • In the present approach, the intermediate particle hasthe same nucleonic nature than the incident and theoutgoing particle. Previous studies by Osterfeld et al. [41] have shown the importance of double charge-exchange.This process can be accounted for in a consistent wayusing HFB [68]. • The target excited states will then be described withinQRPA [3].
In the midterm future, we plan to deal with axially-deformedtargets according to the following plan, • This will require the development of an axially-deformedHFB potential in coordinate space. The correspond-ing mean-field and the pairing field have already beenstudied in coordinate space in a previous work on Cooper’spairs [69]. • QRPA [3] will be used to generate excited states in theintrinsic frame of the target. • A projection an good angular momentum, using the ro-tational approximation [70], will provide the monopoleand different coupling potentials to model nucleon elas-tic scattering from axially deformed target. • The problem of solving coupled equations with nonlo-cal potentials will have to be addressed.
Acknowledgments
H. F. A. acknowledges partial funding from FONDECYTunder Grant No 1120396.
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