Dispersive optical model description of nucleon scattering on Pb and Bi isotopes

PHYSICAL REVIEW C 101, 064618 (2020) Dispersive optical model description of nucleon scattering on Pb and Bi isotopes Xiuniao Zhao ,1Weili Sun,2,*R. Capote ,3,†E. Sh. Soukhovitski˜ ı ,4D. S. Martyanov ,4and J. M. Quesada 5 1Graduate School of China Academy of Engineering Physics, Beijing 100088, China 2Institute of Applied Physics and Computational Mathematics, Beijing 100094, China 3NAPC–Nuclear Data Section, International Atomic Energy Agency, Vienna A-1400, Austria 4Joint Institute for Energy and Nuclear Research, Minsk-Sosny 220109, Belarus 5Departmento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, Sevilla E-41080, Spain (Received 18 February 2020; revised manuscript received 18 April 2020; accepted 22 May 2020; published 18 June 2020) A recently derived dispersive optical model potential (DOMP) for 208Pb is extended to consider the nonlocality in the real potential and the shell gap in the definition of the nuclear imaginary potentials near the Fermi energy. The modified DOMP improves the simultaneous description of nucleon scattering on 208Pb and of the 208Pb particle-hole bound states. This potential is shown to give a very good description of nucleon scattering data on near-magic targets 206,207Pb and 209Bi. DOI: 10.1103/PhysRevC.101.064618 I. INTRODUCTION The nuclear optical model has been comprehensively applied to analyze the elastic scattering of pions, nucleons, and heavier particles by nuclei over a wide range of energies [1–3]. The requirement of causality, namely that the scattering wave is not emitted before the incident wave arrives [4],ledtothe need to consider dispersion effects in the nuclear scattering, and allowed the combination of the optical model potential and the shell model potential into a dispersive optical model potential (DOMP) [5]. The DOMP combined both nuclear reaction (E>0) and nuclear structure (E<0) information to minimize the number of parameters and improve the predictive capabilities of relevant observables. Pioneering work on DOM potentials for strongly deformed nuclei was the contribution of Romain and Delaroche [6], who analyzed the nucleon scattering data on 181Ta and tungsten isotopes. An explicit treatment of the nonlocality of the surface imaginary potential and the Hartree-Fock (HF) potential was introduced following Perey-Buck recipes [7]. Mahaux and Sartor suggested in 1991 [8,9] that the absorptive potential will be asymmetric at large positive and negative energies with respect to the Fermi energy EF.The DOM analysis of neutron scattering on 27Al [10]showed the importance of the asymmetry of the volume absorptive potential and the corresponding dispersive contributions to describe σTdata for energies above 100 MeV. Many studies have also dealt with nucleon scattering on near-magic nuclei. A global spherical potential for nucleoninduced reactions derived by Koning and Delaroche [11]used local dispersive OMPs as a starting point [12]. Recently, *[email protected] †r[email protected]g a global dispersive spherical potential for neutron-induced reactions was derived by Morillon and Romain [13], where an explicit nonlocal HF-like potential was used; bound-state data were also studied [14]. Dispersive optical model has been extensively developed by Washington University (St. Louis) researchers to study nucleon scattering on magic and near-magic nuclei as reviewed recently by Dickhoff and Charity [15]. The need to introduce asymmetric imaginary volume potentials far from the Fermi energy was confirmed in Ref. [16] and led to an improved description of spectroscopic factors of the bound states [17]. An energy gap called EPnear the Fermi energy was introduced in Ref. [17] to describe elastic nucleon scattering data on magic nuclei. Additionally, the importance of the spatial nonlocality in the DOM potential, including both the real and imaginary parts, was highlighted in Refs. [17–22] to describe both the nucleon scattering as well as bound-state data. Nonlocality in the DOM was also shown to have a large impact on calculated (p,d) transfer cross sections [23]. Phenomenological local DOM potentials following the Lane formulation [24,25] have been developed by authors [26–32] and mostly applied to describe nucleon scattering on well-deformed target nuclei using a coupled-channel formalism. Calculated scattering cross sections included quasielastic (p,n) scattering data; e.g., see Ref. [33]. Those potentials very accurately describe available experimental data of nucleon scattering from keV up to 150–200 MeV of incident nucleon energy. However, deformed nuclei do not have bound-state experimental data available as the bound states are very fragmented due to the deformation. The analysis of nucleon scattering of 208Pb by DOMP was recently undertaken [32]. The DOMP from Ref. [32] was also used to test the derived DOMP at negative energies using our methodology [34]. Calculated DOMP energies of the particle-hole bound states were compared to other calculated 2469-9985/2020/101(6)/064618(9) 064618-1 ©2020 American Physical Society XIUNIAO ZHAO et al. PHYSICAL REVIEW C 101, 064618 (2020) TABLE I. Dispersive optical-model potential parameters for nucleon induced reactions on lead and bismuth isotopes. Volume Surface Spin-orbit Coulomb V0=81.5+0.0292(A−208) MeV VSO =7.61 MeV CCoul =1.288 MeV Real β=0.912 fm Dispersive λso =0.006 MeV−1 Potential Cviso =29.35 MeV (Vs)+dispersive (Vso) +dispersive (Vv) Av=12.81 MeV W0=19.66 MeV WSO =−3.1MeV Imaginary Bv=65.56 MeV Bs=8.99 MeV Bso =160 MeV Potential Ea=56 MeV Cs=0.025 MeV−1 α=0.12 MeV1/2Cwiso =50.71 MeV Potential rHF =1.226 −0.00176(A−208) rs=1.1858 +0.03418(A−208) rso =1.194 rc=1.27 Geometry aHF =0.647 +0.002417(A−208) as=0.6195 aso =0.6426 ac=0.671 (fm) rv=1.321 av=0.6267 −0.00658(A−208) values [13,14] as well as to the existing experimental data [35]. Some inconsistencies in the data description were found in Ref. [34] including problems to describe accurately the total cross sections in the region from 5 up to 10 MeV and, at the same time, achieve a nice description of the bound-state data. A very recent publication from the Washington University (St. Louis) group also studied the nucleon scattering on 208Pb by DOMP using their own methodology [36]. In this work, some of the physical ideas advanced by Mahaux and Sartor [8], the CEA Bruyères-le-Châtel group [6,13,14], and the Washington University (St. Louis) group [16,17] will be tested using our phenomenological DOMP framework to study the impact on calculated observables. Our main goal is to derive a Lane consistent potential for lead and bismuth isotopes that reproduces very well both scattering and bound-state data. II. DISPERSIVE SPHERICAL OPTICAL MODEL POTENTIAL A dispersive optical model is defined by energy-dependent real Vi(i=HF, v, s, C, so) and imaginary Wi(i=v, s, so) functionals for the so-called Hartree-Fock (HF), volume (v), surface (s), Coulomb (C) and spin-orbit (so) potentials, respectively and also by the corresponding dispersive contributions to the real potential Vv,Vs, and Vso, which are calculated analytically from the corresponding imaginary potentials [31,37,38]. The general formulation of the Laneconsistent spherical dispersive optical potential has been published previously (e.g., see Eqs. (1)–(3) in Ref. [34]) and is not repeated here. Note that our formulation considers the Coulomb corrections in all orders through an effective TABLE II. The average particle (hole) single-particle energies EP (for neutrons and protons) in MeV for nucleon-induced reaction on selected targets. 206Pb 207Pb 208Pb 209Bi EP(n)−6.75 −6.74 −3.95 −4.62 EP(p)−3.57 −3.72 −3.81 −3.81 energy shift in the potential definition; i.e., the effect of Coulomb interaction on the nuclear interaction is not an averaged energy-independent constant as usually done (e.g., see Koning-Delaroche potential definition [11]). It is well known (see, e.g., Ref. [15]) that the real meanfield potential VHF (r,r) is nonlocal and energy independent. A parametrization of such nonlocal potential was postulated by Perey and Buck to be of Gaussian type [7]: VHF (r,r)=V(r)exp(−|r−r|2/β2),(1) where the parameter βis a nonlocality range given in fermi. The local energy approximation of such nonlocal potential [7] then results in the following implicit equation: VHF (E)=AHF exp −μβ2 (¯hc)2[E+VHF (E)].(2) Note that both AHF and the potential VHF (E)inEq.(2)are assumed to be positive. To obtain the potential depth VHF(E) at a given energy Eit is necessary to solve the Eq. (2)by iterations.1Note that both AHF and βare independent of iterations on VHF for a given energy E. The reduced mass μ in the formula is calculated using relativistic kinematics and, therefore, is also a function of the incident nucleon energy E. The isospin dependence of the potential (the Lane term [24,25]) was considered in real VHF (E) and imaginary surface Ws(E) potentials as follows: AHF =V01+(−1)Z+1Cviso V0 N−Z A,(3) As=W01+(−1)Z+1Cwiso W0 N−Z A,(4) where V0,Cviso,W0, and Cwiso are undetermined constants. Many authors found that the imaginary volume potential does not depend on the isospin. The isospin constants Cviso and Cwiso should be determined mainly using quasielastic (p,n) scattering data. 1Solution of Eq. (2) can be expressed explicitly through the special function Lambert W(as known as the product logarithm) as VHF (E)=W[AHFλexp (−λE)] λ,whereλ≡μβ2 (¯hc)2. 064618-2 DISPERSIVE OPTICAL MODEL DESCRIPTION OF … PHYSICAL REVIEW C 101, 064618 (2020) FIG. 1. DOMP depths and dispersive contributions as a function of Efor the n+208Pb reaction between −50 and 200 MeV. (a) Potential depths for real spin-orbit Vso (top panel), imaginary spin-orbit Wso, volume Wv, and surface Wspotentials (middle panel), and corresponding dispersive contributions Vso,Vv,andVs(bottom panel) and (b) “Hartree-Fock” potential depth from Ref. [34] (previous, solid) compared to the present one (dashed) given by Eq. (2). The energy dependencies for the imaginary volume term Wv, the imaginary surface term Ws, and the spin-orbit imaginary term Wso are taken as the ones suggested by Brown and Rho [39], Delaroche et al. [40], and Koning et al. [11], respectively. The imaginary potentials used in all our studies so far are local ones. Some groups advocate the need to consider nonlocal imaginary potentials [6,19,20], but this is deferred to future works. In this work, following Mahaux et al. [8] and Molina et al. [10], a modified definition for the imaginary part of the OMP is taken as follows: Wv(E)=0EF<E<EP Av(E−EP)2 (E−EP)2+(Bv)2E>EP ,(5) Ws(E)=⎧ ⎨ ⎩ 0EF<E<EP As(E−EP)2 (E−EP)2+(Bs)2 ×exp (−Cs|E−EP|)E>EP ,(6) Wso(E)=0EF<E<EP WSO (E−EP)2 (E−EP)2+(Bso )2E>EP .(7) The imaginary part of the DOM potential is assumed to be zero inside the shell gap , which is related to the average energy of the single-particle (single-hole) states EPas = 2(EP−EF). Obviously, there are no states in the shell gap, and therefore we have to set the absorption to zero. A similar definition of the shell gap was employed in Refs. [16,17]. Both EPand EFare different for neutron- and proton-induced reactions. For nuclei far from magic EPis approximately equal EF, and therefore the shell gap is zero and can be neglected. The symmetry condition W(2EF−E)=W(E)isusedto extend the imaginary part of the OMPs for energies below the Fermi energy. This analytical extension is needed for the calculation of the dispersive corrections. Asymmetric absorptive potentials were used in many analysis of DOMPs derived on different targets [26–32]. Following Mahaux and Sartor [9], the assumption that the imaginary potential Wv(E) is symmetric about E=EF[according to equation W(2EF−E)=W(E)] is modified above some fixed energy Ea, which is expected to be close to 60 MeV, but it is treated as a parameter. FIG. 2. Energy dependence of the imaginary potential depths (solid curve) and corresponding dispersion correction terms (dashed curve) near the Fermi energy EFcalculated for the n+208Pb reaction. The effect of the assumed shell gap =2(EP−EF)=3.4MeV on the imaginary potentials is clearly seen. (a) Volume imaginary potential (left Y-axis) and corresponding dispersive correction (right Y-axis) and (b) Surface imaginary potential (left Y-axis) and corresponding dispersive correction (right Y-axis). 064618-3 XIUNIAO ZHAO et al. PHYSICAL REVIEW C 101, 064618 (2020) FIG. 3. Comparison of the calculated total cross section for the n+208Pb reaction with measurements. Calculations using the Koning-Delaroche potential [11], the DOMP from our previous work [34], and the current DOMP are shown. Experimental data are taken from EXFOR [45]andRefs.[46–49]. Optical model code OPTMAN [41–43] that includes the calculation of (p,n) quasielastic scattering [33]was used for cross-section calculations for positive energies. The parameters of the dispersive optical model potential were searched for by minimizing the quantity χ2in the usual way [44]. All experimental data used in the fitting process are taken from the EXFOR database [45] which is exactly the same database used to derive the DOMPs describing scattering on 208Pb target and published in Refs. [32,34]. Additionally, the calculation of 208Pb bound states that depends on the real potential [34]isalsousedinthe DOMP optimization using the experimental data quoted in Ref. [35]. Newly derived DOMP parameters are listed in Table Iand corresponding average particle (hole) energies EP(n) and EP(p) that define the imaginary potentials are listed in Table II. Figure 1(a) shows the obtained energy dependence of the real spin-orbit potential, of the imaginary (absorptive) potentials, and of the corresponding dispersive correction terms near the Fermi energy for the n+208Pb reaction. A comparison of the energy dependence of the Hartree-Fock VHF potential is shown in Fig. 1(b). “Previous” refers to the VHF potential from Ref. [34] which is compared to the local approximation of the nonlocal potential used in this work [see Eq. (2) labeled as “Present”]. The depth of Hartree-Fock VHF potential given by Eq. (2) is lower below the Fermi energy, falls more slowly up to 100 MeV and decreases faster above that energy as compared FIG. 4. Comparison of calculated total cross section for (a) n+206Pb, (b) n+207Pb, (c) n+208Pb, and (d) n+209Bi reactions with measurements, as well as the results of Koning-Delaroche calculations [11]. Experimental data are taken from Refs. [46–57]. 064618-4 DISPERSIVE OPTICAL MODEL DESCRIPTION OF … PHYSICAL REVIEW C 101, 064618 (2020) FIG. 5. Neutron single-particle (hole) energies in 208Pb, the first and second columns display the results from Ref. [14], the third column shows those of Ref. [34], and the fifth column contains those of the current work. In the fourth column, the experimental values taken from Ref. [35] are shown. Note that EF≈−5.6 MeV and it defines the N=126 shell. Levels below −5 MeV are hole levels; above that are particle levels. to the exponentially decreasing potential used in Ref. [34]. A shallower potential well at negative energies given by the Perey-Buck nonlocal approximation [7] improved the description of the bound states as well as scattering data, as will be shown below. Figure 2shows the energy dependence of the imaginary (absorptive) potentials and corresponding dispersivecorrection terms near the Fermi energy for the n+208Pb system. The figure clearly shows that the imaginary potentials vanish from the energy (2EF−EP) up to the energy EPreflecting the shell gap. However, the dispersive correction remains nonzero in that region, as discussed in Refs. [37,38]. III. RESULTS AND DISCUSSION The calculation of neutron total cross section for the 208Pb target using our DOMP is compared with the results of Koning-Delaroche [11] and our previously derived DOMP [34]inFig.3in the energy range from 6 up to 16 MeV. The potential from Ref. [34] was worse than Koning-Delaroche description [11] in this region. Results from the current work shows a clear improvement over our previous work, and these DOMP results are in good agreement with data as well as with Koning-Delaroche potential calculations in this energy region. The calculation of neutron total cross sections for 206Pb, 207Pb, 208Pb, and 209Bi are compared in Fig. 4with the results TABLE III. Spectroscopic factors of valence-neutron-particle states (denoted by quantum numbers in bold) and valence-neutron-hole states in 208Pb. 3d3/22g7/24s1/23d5/21j15/21i11/22g9/23p1/22f5/23p3/21i13/22f7/21h9/2 This work 0.90 0.89 0.92 0.90 0.73 0.74 0.78 0.83 0.77 0.86 0.70 0.93 0.97 Ref. [35] 0.90 0.86 0.91 0.88 0.82 0.82 0.81 0.80 0.81 0.81 0.81 0.85 0.85 Ref. [61] 0.72 0.80 0.61 0.77 0.55 0.73 0.67 0.79 0.73 0.71 0.62 0.53 0.51 Ref. [62] 0.79 0.79 0.81 0.78 0.79 0.79 0.88 0.78 0.74 0.82 0.83 0.70 0.81 Ref. [63] 0.80 0.82 0.86 0.83 0.71 0.75 0.86 0.73 0.82 0.76 0.72 0.59 0.44 Ref. [64] 0.72 0.76 0.80 0.78 0.66 0.86 0.94 0.94 0.91 0.94 0.92 0.75 0.82 Ref. [65] 0.88 0.92 0.83 0.90 0.62 0.89 0.97 0.95 0.92 0.94 0.87 0.70 0.84 064618-5 XIUNIAO ZHAO et al. PHYSICAL REVIEW C 101, 064618 (2020) FIG. 6. Comparison of nucleon elastic scattering angular distributions with measurements at different incident nucleon energies. (a) 206Pb (n,n) angular distributions, (b) 207Pb (n,n) angular distributions, (c) 209Bi (n,n) angular distributions, and (d) 209Bi (p,p) angular distributions. of Koning-Delaroche potential [11] from 500 keV up to 200 MeV of incident neutron energy. The calculated total cross section using this DOMP is in fair agreement with Koning-Delaroche results above 5 MeV, but reproduces better the experimental data below that energy for all targets. The real part of our derived DOM potential is the shell model potential, which can be used to calculate the energies of the bound single-particle states of the magic nuclei 208Pb. This potential includes the sum of the Hartree-Fork term VHF(Enlj ), the real spin-orbit term Vso(Enlj), and all dispersive correction terms Vv(Enlj), Vs(Enlj), and Vso(Enlj) with the corresponding geometry form factors. The experimental values of the neutron single-particle energies of the various single-particle and hole states for 208Pb were taken from Ref. [35]. The predicted singleparticle (single-hole) energies are compared with the experimental data in Fig. 5. Results labeled “DOM(MR)” and “DOM(MR+35%)” represent the Morillon and Romain calculations from Ref. [14]; the label “DOMprevious” corresponds to calculations from our previous publication [34], and the label “DOMpresent” represents the current work. The description of the single-particle bound states is significantly improved compared to Ref. [14], and slight improvement can be seen relative to our previous work. The order of both particle and hole states agrees with the experimental one; the particle energies agree well for the five single-particle levels near the Fermi energy; the agreement deteriorates for more unbound states. A similar situation is observed for hole states: better agreement near the Fermi energy, worse for deeper hole states. The neutron single-particle energies for last single-particle state and first single-hole state were calculated for the 208Pb target; the absolute values of these two energies define the neutron separation energies Sn(A) and Sn(A+1). The calculated values of Sn(A) and Sn(A+1) are 7.47 and 3.85 MeV, respectively. These results are in excellent agreement with the corresponding experimental data 7.37 and 3.94 MeV [58,59]. The root mean square (rms) radii for each orbit and single-particle densities were also calculated and the agreement with results from Ref. [35] is similar to what we already published for 208Pb [34]. The spectroscopic factor is given by the following expression: Snlj =¯u2 nlj(r)[m/¯m(r;Enlj)]dr =u2 nlj(r)[m∗ H(r;Enlj)/m] [¯m(r;Enlj)/m]dr =u2 nlj(r)1−d dEVHF(r;E)E=Enlj  1−d dE V(r;E)E=Enlj dr.(8) where ¯unlj is the eigenstate of the full microscopic mean field, and unlj is the eigenstate of its local equivalent that we use. Normalized eigenstates were used in spectroscopic factor calculations; details of the definition can be found in Refs. [35,60]. The spectroscopic factors of valence neutron particle and hole states in 208Pb are compared in Table III with previous calculations from Refs. [35,61–65] (values were taken from Ref. [8], except Johnson et al. values [35]). A reasonable agreement is observed. Figure 6shows calculated elastic scattering angular distributions of neutrons and protons incident on 206Pb, 207Pb, and 209Bi for different incident nucleon energies. Results for 208Pb are similar to those presented in Ref. [34] and are not shown in this paper. The results for both neutron and proton elastic scattering describe the experimental data rather well over the entire energy and angular range. Slight underestimation of data below 5 MeV of neutron incident energy is probably associated with the missing compound-elastic contribution. Figure 7shows elastic scattering analyzing powers of neutrons and protons incident on 208Pb and 209Bi for different incident nucleon energies. Results for 208Pb are similar to those presented in Ref. [34] in the regions where data are available; the agreement is reasonable but not perfect. A 064618-6 DISPERSIVE OPTICAL MODEL DESCRIPTION OF … PHYSICAL REVIEW C 101, 064618 (2020) FIG. 7. Nucleon elastic scattering analyzing powers compared with the experimental data at different incident nucleon energies. (a) 208Pb (n,n) analyzing powers, (b) 208Pb (p,p) analyzing powers, (c) 209Bi (n,n) analyzing powers, and (d) 209Bi (p,p) analyzing powers. similar level of agreement is observed for nucleon scattering on 209Bi. The newly fitted DCCOM potential has not been tested on quasielastic (p,n) scattering to the isobaric analog states (IAS) of the target nucleus, which represent the best test of the isovector part of the optical potential. Figures 8and 9show the calculated quasielastic (p,n) angular distribution for scattering on 208Pb, and on 206Pb and 209Bi targets, respectively. Reasonable agreement with data is achieved showing the Lane consistency of the derived DOMP, i.e., the same potential describes both neutron and proton scattering indistinctly, including the quasielastic (p,n) scattering which is defined by the isovector potential. However, additional work is needed to clarify a potential improvement of the quasielastic data description by introducing a shift of the isovector and isoscalar geometries as recently proposed by Danielewicz et al. [66]. In fact, our calcula- FIG. 8. Calculated angular distributions of the quasielastic (p,n) scattering on 208Pb target. tions underestimate the oscillations in data as observed in Ref. [66]. IV. CONCLUSIONS In order to improve the description of the nucleon scattering data and the bound-state energies using a dispersive potential, this work considered the nonlocality of the real potential as suggested by Perey and Buck [7] and extensively used in papers by CEA Bruyères-le-Châtel and Washington University (St. Louis) groups, and the impact of the large shell gap in magic nuclei on the definition of the imaginary potential [8,10]. The improved physical model allowed to derive a Lane-consistent dispersive optical model potential that accurately describes scattering data for nucleon-induced reactions on the double-magic target 208Pb. The real part of the same DOMP, which corresponds to the shell potential, gives FIG. 9. Calculated angular distributions of the quasielastic (p,n) scattering on 206Pb and 209Bi targets. 064618-7 XIUNIAO ZHAO et al. PHYSICAL REVIEW C 101, 064618 (2020) a good description of the bound-state data. 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