Dispersive coupled-channels optical-model potential with soft-rotator couplings for Cr, Fe, and Ni isotopes

PHYSICAL REVIEW C 87, 054611 (2013) Dispersive coupled-channels optical-model potential with soft-rotator couplings for Cr, Fe, and Ni isotopes Rui Li,1Weili Sun,2,*E. Sh. Soukhovitskiı˜,3J. M. Quesada,4and R. Capote5 1Graduate School, China Academy of Engineering Physics, Beijing 100088, China 2Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 3Joint Institute for Energy and Nuclear Research, Minsk-Sosny 220109, Belarus 4Departmento de F´ ısica At´ ømica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, 41080 Sevilla, Spain 5NAPC–Nuclear Data Section, International Atomic Energy Agency, Vienna A-1400, Austria (Received 20 March 2013; revised manuscript received 16 April 2013; published 17 May 2013) An approximate Lane-consistent dispersive coupled-channels optical potential is derived that describes nucleon-induced reactions on even iron isotopes. Realistic saturated couplings for 54,56,58Fe nuclei are built using nuclear wave functions of the soft-rotator model with the Hamiltonian parameters adjusted to reproduce the energy of the low-lying collective levels of these isotopes. E2and E3-transition probabilities between low-lying collective levels are well reproduced. The comprehensive experimental database used in the fitting process includes all scattering data for neutron and proton scattering up to 200 MeV on iron nuclei. The derived potential is shown to be applicable to Ni and Cr isotopes, assuming the applicability of the soft-rotator model to these nuclei and to the odd 57Fe nucleus within the rigid-rotor model. The approximate Lane consistency of the derived potential is validated by describing the quasielastic (p, n) scattering with excitation of isobaric analog states. Elastic and inelastic analyzing powers for both neutronand proton-induced reactions are shown to be in good agreement with experimental data, demonstrating the reliability of the derived dispersive spin-orbit potential. DOI: 10.1103/PhysRevC.87.054611 PACS number(s): 24.10.Eq, 24.70.+s, 25.40.Kv, 27.40.+z I. INTRODUCTION Being a major component of steel, iron is the most important structural element in nuclear technology. Chromium (18–20%) and nickel (8–10%) are the other major components of austenitic steels that are widely employed in the nuclear industry. All three materials are therefore important structural materials for nuclear applications. A reduction of the uncertainty of elastic and inelastic scattering data on iron, chromium, and nickel isotopes in the fast neutron region has been requested, therefore motivating additional work on the improvement of phenomenological optical-model potentials for nuclear data modeling and evaluation. Phenomenological nucleon optical-model potentials (OMPs) that describe successfully the nucleon scattering data have been extensively studied in the past [1]. A widely used one is the global spherical optical potential by Koning and Delaroche [2] for nucleon-induced reactions up to 200 MeV. Recently, more emphasis has been put on the development of the dispersive OMPs proposed and extensively developed by Mahaux and Ngo [3] and Mahaux and Sartor [4,5]. The unified description of the nuclear mean field in the dispersive optical model (DOM) is accomplished by using a dispersion relation, which links the imaginary OMP parts to the corresponding dynamic real parts without requiring additional parameters. Furthermore the dispersive constraint helps to reduce the ambiguities and the number of OMP parameters and also eliminates the need for energy-dependent geometry. Extensive works on extending the DOM approach into the domain below the Fermi energy by employing additional *[email protected] experimental data to constrain the potentials have been recently carried out by Morillon and Romain [6], Bespalova and co-workers [7,8], and Dickhoff and co-workers (Ref. [9] and references therein). Progress has been also achieved in the development of the dispersive OMPs for the positive-energy domain that describe nucleon scattering using both spherical [10] and coupled-channels formalism [11–14]. An additional constraint to derived OMP parameters arises from the use of quasielastic (p, n) scattering data involving the excitation of isobaric analog states. Such calculations are a good test of the isovector part of the OMP and check for the OMP Lane consistency [15,16]. It has been previously shown by some of the authors [17] that the dispersive coupled-channels optical-model (DCCOM) potential for nucleon-induced reactions on actinides was approximately Lane consistent, but no such a study has been previously undertaken for iron isotopes. On the other hand, a DCCOM potential for actinides [11,12] used an imaginary spin-orbit potential found by Koning and Delaroche [2], but assumed a dispersive contribution in the real spin-orbit potential that has never been checked by using polarization data (e.g., the analyzing power Ay). Therefore, the aim of the present work is to construct a DCCOM potential for chromium, iron, and nickel isotopes and then perform a joint analysis using almost all types of experimental nucleon-nucleus scattering data, including cross sections, elastic and inelastic angular distributions, quasielastic (p, n) data, and analyzing powers. For the latter, a computational method based on the Tamura formalism [18]is given. Such a joint analysis is a good validation for the Lane consistency of the derived DCCOM potential. This contribution is structured as follows. Section II describes the soft-rotator model analysis. The coupled-channels 054611-1 0556-2813/2013/87(5)/054611(11) ©2013 American Physical Society LI, SUN, SOUKHOVITSKI˜ I, QUESADA, AND CAPOTE PHYSICAL REVIEW C 87, 054611 (2013) TABLE I. Assignments of the SRM quantum numbers for the lowlying collective levels of 54,56,58Fe. Quantum numbers nγ,nβ2,andnβ3 correspond to the number of γ, quadrupole, and octupole phonons, respectively. Kis the projection of the total angular momentum of the vibrational bandhead (approximate quantum number because strictly speaking Kis not conserved in SRM). 54Fe 56Fe 58Fe (i) K≃00 +(g.s.) 0+(g.s.) 0+(g.s.) nγ=02 +(1.408 MeV) 2+(0.847 MeV) 2+(0.811 MeV) nβ2=04 +(2.538 MeV) 4+(2.085 MeV) 4+(2.077 MeV) nβ3=0 (ii) K≃22 +(2.959 MeV) 2+(2.658 MeV) 2+(1.675 MeV) nγ=03 +(3.345 MeV) 3+(3.445 MeV) 3+(2.134 MeV) nβ2=04 +(3.835 MeV) 4+(2.600 MeV) nβ3=0 (iii) K≃00 +(2.561 MeV) 0+(2.942 MeV) 0+(2.258 MeV) nγ=02 +(3.166 MeV) 2+(3.748 MeV) nβ3=0 nβ2=1 (iv) K≃03 −(4.782 MeV) 3−(4.510 MeV) 3−(3.660 MeV) nγ=0 nβ2=0 nβ3=0 (iv) K≃23 −(6.400 MeV) nγ=0 nβ2=0 nβ3=0 optical-model analysis is presented in Sec. III. Results and discussions are presented in Sec. V. Finally, the summary and conclusions are given. II. SOFT-ROTATOR MODEL ANALYSIS A coupled-channels optical model using matrix elements derived by the soft-rotator model (SRM) [19] is employed to simultaneously describe the low-lying collective level nuclear structure and the nucleon scattering data at positive energies. This approach has been successfully applied to a number of vibrational nuclei, including 58Ni [19], 56Fe [20,21], and 52Cr [22], which are studied in this work. We first adjusted the SRM Hamiltonian parameters, with the consideration of both quadrupole and octupole dynamic vibrations, to fit the experimentally observed collective levels of 54,56,58Fe isotopes. The assignments of the SRM quantum numbers for these isotopes using a standard approach [21]are shown in Table I. Such assignments allow us to search for the Hamiltonian parameters; derived parameters for iron isotopes are listed in Table II. Figure 1displays a comparison of the calculated and experimental levels of 56Fe as an example. We show only experimental levels used or predicted by the SRM, while noncollective levels are not shown. One can see that the SRM reproduces well the experimental levels with deviations lower than 10%. The same is true for 54,58Fe nuclei. TABLE II. SRM Hamiltonian parameters used to reproduce the experimental level schemes of 54,56,58Fe. γ0is the quadrupole nonaxiality parameter; 0=β30/β20 is the equilibrium octupole deformation. Quantum numbers μx(x=β20,γ 0,) denote the nuclear softness parameter for quadrupole, γ, and octupole phonons, respectively. a32 and η(a42,δ4,andγ4) determine the nonaxiality of octupole (hexadecapole) deformations; 2δnis the energy splitting of a doubly degenerated level in octupole vibrations due to the tunneling effect; and the ¯hω0parameter normalizes the overall scale of the predicted levels. 54Fe ¯hω0=1.6630 μβ20 =0.9700 μγ0=0.7600 γ0=0.66998 a32 =0.004873 a42 =0.0053 δ4=0.69718 γ4=0.093896 μ=0.65298 η=0.0802 δn=7.6615 0=0.70991 56Fe ¯hω0=2.6016 μβ20 =0.48922 μγ0=0.1260 γ0=0.32552 a32 =0.004873 a42 =0.0δ4=0.69718 γ4=0.093896 μ=0.3500 η=0.14557 δn=7.6791 0=0.85582 58Fe ¯hω0=1.8935 μβ20 =0.5439 μγ0=0.1260 γ0=0.42385 a32 =0.004873 a42 =0.0δ4=0.69718 γ4=0.093896 μ=0.29933 η=0.14557 δn=7.3601 0=0.64596 Optimal SRM Hamiltonian parameters for Ni isotopes and 52Cr are listed in Tables III and IV, respectively. A similar level of agreement between SRM calculations and experimental energies is observed for 58,60,62Ni and 52Cr nuclei. Derived deformation parameters β20,β30, and β4for eveneven Fe isotopes are listed in Table V. Note that only the β4 parameter can be directly related to the similar parameter in the rigid-rotor model. β20 and β30 cannot be related, because they are the equilibrium quadrupole and octupole deformations (with respect to dynamical soft vibrations), respectively. Therefore, we used two approaches to calculate the effective 0 1 2 3 4 5 Excitation Energy (MeV) 0+ 2+ 4+ 2+ 0+ 3+ 2+ 30+ 2+ 4+ (ii) 2+ 3+ (iii) 0+ 2+ (iv) 3- (i) Experiment Calculation FIG. 1. Comparison of the experimental and predicted level schemes for the 56Fe nucleus. 054611-2 DISPERSIVE COUPLED-CHANNELS OPTICAL-MODEL ... PHYSICAL REVIEW C 87, 054611 (2013) TABLE III. SRM Hamiltonian parameters used to reproduce the experimental level schemes of 58,60,62Ni. Notation is the same as in Table II. 58Ni ¯hω0=1.7482 μβ20 =1.0239 μγ0=0.3900 γ0=0.6515 a32 =0.0001 a42 =0.014857 δ4=0.69719 γ4=0.1141 μ=0.50193 η=0.14194 δn=5.9953 0=1.0901 60Ni ¯hω0=1.3062 μβ20 =1.887 μγ0=0.42 γ0=0.49172 a32 =0.0001 a42 =0.011933 δ4=0.6974 γ4=0.11365 μ=0.43003 η=0.10964 δn=6.1169 0=0.95215 62Ni ¯hω0=1.1794 μβ20 =2.7235 μγ0=0.63 γ0=0.29742 a32 =0.0001 a42 =0.011933 δ4=0.6974 γ4=0.11365 μ=0.31179 η=0.1044 δn=5.9375 0=0.69 deformation parameters, equivalent to those deformation parameters β2and β3in the rigid-rotor model. The first approach is to average the equilibrium deformation parameters over the nuclear wave functions of the initial |iand final |f states, i.e., β2(β3)=i|β20(β30)|f. The wave functions are the eigenfunctions of the SRM Hamiltonian. Such averaging, taking μλsoftness to β2and β3into account, demonstrates an enhancement of the coupling strength between different channels [19,23]. The second approach is to derive the β2(β3) value from B(E2) [B(E3)] given by the rotational-vibrational model [24,25]. The results are shown in Table VI, together with β2and β3determined from experimental B(E2) and B(E3). The effective β2values calculated by our two approaches is within 5% of those determined from B(E2) recommended by Raman et al. [26]. The effective β3calculated by B(E3) is consistent with those recommended by Spear [27], with the largest deviation being lower than 10% for 54,58Fe nuclei. However, for the case of 56Fe, the effective β3, calculated by 0+ g.s.|β30|3−, is 0.1537, which is 30% lower than Spear’s recommended β3=0.2145. A similar situation has been observed for the 28Si nucleus in previous work [28], where a possible reason for this discrepancy is discussed. III. COUPLED-CHANNELS OPTICAL MODEL ANALYSIS Coupled-channels calculations were performed with a coupling built on SRM nuclear wave functions with the above SRM Hamiltonian parameters; e.g., eight collective levels were coupled for 56Fe as shown in Fig. 2. It was verified that the coupling of additional levels changes calculated cross TABLE IV. SRM Hamiltonian parameters used to reproduce the experimental level scheme of 52Cr. Notation is the same as in Table II. 52Cr ¯hω0=1.4752 μβ20 =1.2124 μγ0=0.175 γ0=0.33586 a32 =0.0 a42 =0.0 δ4=0.69718 γ4=0.093896 μ=0.20677 η=0.000024 δn=7.3544 0=0.45 TABLE V. Static deformation parameters of Fe isotopes. 54Fe 56Fe 58Fe β20 0.131 0.232 0.232 β30 0.710 0.856 0.646 β40.109 0.046 0.046 sections within quoted experimental uncertainties. Therefore, we call such a coupling scheme a saturated coupling. Two additional levels were coupled for (p, n) quasielastic scattering calculations as shown in the top of this figure: The first level is the ground state isobaric analog state (IAS), and the second level is a 2+excited analog state (EAS). The energy difference between the ground state of 56Fe and its corresponding IAS state is just the Coulomb displacement, which can be obtained by the empirical relation [29]C= 1.444Z/A1/3−1.13 MeV, with Zbeing the average charge of the target and residual nuclei in the reaction. Note that the energy difference between the analog states in the residual nucleus is kept the same as that between the ground-state rotational band levels in the target nucleus. The empirical value of Cis 8.8719 MeV for 56Fe. However Ccan also be given approximately by C= Eth +Eex, where Eth is the (p, n) reaction threshold energy and Eex is the measured excited energy of the residual nucleus. For the case of the 56Fe(p, n)56Co reaction, Eth is 5.444 MeV, while the measured Eex of 56Co is 3.5 MeV [30]. Therefore the sum given in such a way is 8.944 MeV, quite close to the empirical one. Similarly, the coupled schemes with the consideration of (p, n) transitions for 54,58Fe, 58,60,62Ni, and 52Cr can be built. A dispersive coupled-channels OMP, taking into account the deformed nuclear shapes, is used in the coupled-channels optical-model calculations. The general formulation with conventional definition of the symbols is given by V[r, R(θ,ϕ),E] =−[VHF(E∗)]fWS [r, RHF(θ,ϕ)] −[Vv(E∗)+iWv(E)]fWS [r, Rv(θ,ϕ)] −[Vs(E∗)+iWs(E)]gWS [r, Rs(θ,ϕ)] TABLE VI. The effective deformation parameters β2and β3 calculated by two approaches as described in the text, together with β2and β3, determined from experimental B(E2) and B(E3), respectively. β2β2β2 by i|β20|fby B(E2) by B(E2) [26] 54Fe 0.1955 0.1755 0.1958 56Fe 0.2461 0.2645 0.2393 58Fe 0.2543 0.2696 0.2587 β3β3β3 by i|β30|fby B(E3) by B(E3) [27] 54Fe 0.1193 0.0930 0.1144 56Fe 0.1537 0.1891 0.2145 58Fe 0.1566 0.1807 0.1716 054611-3 LI, SUN, SOUKHOVITSKI˜ I, QUESADA, AND CAPOTE PHYSICAL REVIEW C 87, 054611 (2013) 0+(G.S.) 2+ 4+ 0+ (IAS) 2+ (EAS) 2+ 3+ 30+ 2+ 56Fe FIG. 2. Coupling scheme used in the coupled-channels calculation for the 56Fe nucleus. +¯h mπc2 [Vso(E)+Vso(E)+iWso(E)] ×1 r d drfWS [r, Rso]( ˆσ·ˆ L)+VCoul[r, Rc(θ,ϕ)],(1) with the geometric form factors given as fWS [r, Ri(θ,ϕ)] =[1 +exp[r−Ri(θ,ϕ)]/ai]−1, i=HF,v fWS [r, Rso]=[1 +exp[r−Rso]/aso]−1, gWS =−4as d drf[r, Rs(θ,ϕ)],(2) where the “effective” nucleon energy E∗=Enfor neutrons and E∗=Ep−ECoul for protons, which accounts for the Coulomb shift due to the repulsion by the target nucleus. The effective energy E∗for protons is discussed in more detail later. The expressions of the Hartree-Fock (HF) potential, VHF(E), the imaginary and corresponding dispersive potentials Wi(E) and Vi(E),(i=v,s, and so), and the Coulomb potential VCoul[r, Rc(θ,ϕ)] as well as the parameters therein have been described in detail in a previous paper [11]. For the sake of clarity and completeness, we give a brief description below. The employed HF potential, VHF(E), is due to the replacement of a microscopic nonlocal HF potential by a local equivalent. For a Gaussian nonlocality, VHF(E) is a linear function of Efor large negative Eand is an exponential for large positive E. Following Mahaux and Sartor [31], the energy dependence of the Hartree-Fock part of the nuclear mean field is taken as that found by Lipperheide [32]: VHF(E)=AHF exp[−λHF(E−EF)],(3) where the parameters AHF and λHF are undetermined constants and EFis the Fermi energy. The energy dependencies for the imaginary volume term Wv(E) and the imaginary surface term Ws(E)are taken as the ones suggested by Brown and Rho [33] and Delaroche et al. [34], respectively, as follows: Wv(E)=Av (E−EF)2 (E−EF)2+(Bv)2, Ws(E)=As (E−EF)2 (E−EF)2+(Bs)2exp(−Cs|E−EF|),(4) where Av,Bv,As,Bs, and Csare undetermined constants. The isospin dependence of the potential (the Lane term [15,16]) was considered in real VHF(E) and imaginary surface Ws(E) potentials as follows: AHF =V01+(−1)Z+1Cviso V0 N−Z A,(5) As=W01+(−1)Z+1Cwiso W0 N−Z A,(6) where V0,Cviso,W0, and Cwiso are undetermined constants, and Nand Zare the neutron and atomic numbers of the target nucleus. The spin-orbit potential is taken in the same functional form used by Koning and Delaroche [2], but contains a dispersive contribution, Vso(E)[10,35]: Vso(E)=Vso exp[−λso(E−EF)] +Vso(E),(7) Wso(E)=Wso (E−EF)2 (E−EF)2+(Bso)2.(8) The calculations for all dispersive contributions are performed using a dispersion relation as follows: V (r, E)=P π+∞ −∞ W(r, E) E−EdE,(9) where symbol Pdenotes that the principal value of the integral should be taken. Analytical dispersive integrals are calculated following Ref. [36]. In Eq. (4), the imaginary volume potential is assumed to be symmetric about E=EF. Such a symmetry assumption holds for small values of |E−EF|⩽Ea, but it is no longer valid for large values of |E−EF|>E a[31]. Here Eais a constant energy suggested by Mahaux and Sartor to be approximately equal to the depth of the nuclear potential [31]. In that energy region, we adopted a modified (asymmetric) imaginary volume potential as given in Ref. [11] that leads to an additional dispersive volume contribution. Eais used as an additional fitting parameter in this work. The derived value of Ea=52 MeV is close to Mahaux and Sartor’s suggestion of 60 MeV [31]. The Coulomb potential VCoul[r, Rc(θ,ϕ)] is calculated using a multipole expansion of a charged ellipsoid with uniform charge density with the Coulomb radius Rcand zero outside as suggested by Satchler et al. [37]. The spherical term of the Coulomb potential was calculated by taking into account the diffuseness of the charge density distribution of the form fc=[1 +exp(r−Rc/ac)]−1. A similar form of the potential as Eq. (1) was used by Capote et al. [14]. However, an “effective” proton energy E∗=Ep− 054611-4 DISPERSIVE COUPLED-CHANNELS OPTICAL-MODEL ... PHYSICAL REVIEW C 87, 054611 (2013) TABLE VII. Dispersive coupled-channels OMP parameters for 54,56,58Fe, with the SRM Hamiltonian parameters given in Table II and the deformation parameters given in Table V. Volume Surface Real depth (MeV) V0=52.2848 +0.0292ADispersive λHF =0.008 Cviso =20.00 +dispersive Imaginary (MeV) Av=12.36 W0=12.03 Bv=77.4Bs=12.64 Ea=52 Cs=0.01355 Cwiso =14.00 Geometry (fm) rHF =1.2873 −0.0016Ar s=1.0277 +0.0039A aHF =0.504 +0.00225Aa s=0.503 rv=1.0551 av=0.92676 −0.00021A Spin-orbit Coulomb Real depth (MeV) Vso =6.98 CCoul =1.0 λso =0.005 +dispersive Imaginary (MeV) Aso =−3.1 Bso =160.00 Geometry (fm) rso =1.0388 rc=1.188 aso =0.59 ac=0.32 CCoul Z A1/3is used here for the real potentials. The constant CCoul is an adjustable parameter accounting for the effective radius of proton interaction in the nucleus. Note that the constant e2in our definition is included in the constant CCoul.Atthis effective energy E∗, the optical potential can be expanded in a Taylor series as V(E∗)=VE−CCoul Z A1/3 =V(E)−CCoul Z A1/3 d dE[V(E)] +··· (10) Equation (10) holds for VHF(E∗), Vv(E∗), and Vs(E∗). One can see that the first derivative term −CCoul Z A1/3 d dE [V(E)] is just the Coulomb correction term used in previous works [11,14,38]. In fact, Eq. (10) is a generalization of previously used Coulomb correction with the consideration of the full Coulomb correction in all orders. This energy shift has been applied to the real potential only (denoted by E∗); therefore the potential is not fully charge independent. We can speak of the approximate Lane consistency as previously discussed for actinides in Ref. [17]. For the sake of simplicity, the energy-independent potential geometry is used, but small Adependence of the geometry parameters was introduced to derive a regional potential. The DCCOM potential parameters were searched to fit the experimental proton and neutron scattering data by minimizing the quantity χ2inausualway[19]. All experimental data used in the fitting process are taken from the EXFOR database [39]. In total we adjusted 15 parameters from the real volume potential, the surface and volume absorptive potential, the spin-orbit potential, and the Coulomb interaction, namely, V0, λHF ,Cviso,Av,Bv,Ea,W0,Bs,Cs,Cwiso,Vso,λso,Aso,Bso, and CCoul, and 10 energy-independent geometric parameters, namely, rHF ,aHF ,rv,av,rs,as,rso,aso,rc, and ac. They allow the best fit to the experimental data and are summarized in Table VII. We assumed that parameters V0,rHF ,aHF ,av, and rs are linearly dependent on mass number A. However this dependence is found to be quite weak, which implies that the derived regional potential for Fe isotopes can be extended to neighboring nuclei. The energy and mass number dependen- -8 -6 -4 -2 0 2 4 0 50 100 150 200 -50 -40 -30 -20 -10 ΔVs and ΔVv (MeV) VHF (MeV) E (MeV) 56Fe 96Mo 28Si ΔVs VHF ΔVv FIG. 3. Dependencies on energy and mass number for the real volume VHF(E), the real dispersive-volume contribution Vv,and the real dispersive-surface potential Vsfor 28Si, 56Fe, and 96Mo nuclei. VHF(E) is scaled on the right, while Vsand Vsare scaled on the left. 054611-5 LI, SUN, SOUKHOVITSKI˜ I, QUESADA, AND CAPOTE PHYSICAL REVIEW C 87, 054611 (2013) -10 -5 0 0 50 100 150 200 Wv and Ws (MeV) E (MeV) 56Fe 96Mo 28Si Ws Wv FIG. 4. Volume and surface imaginary potential depths for 28Si, 56Fe, and 96Mo. cies of the real volume potential VHF(E) and the corresponding dispersive contribution Vv(E), as well as the surface dispersive contribution Vs, are shown in Fig. 3for the cases of 28Si, 56Fe, and 96Mo to study the mass dependence of derived OMPs. A similar comparison is given in Fig. 4for imaginary potentials. It can be seen from Figs. 3and 4that the dependencies of VHF,Wv, and Vvon mass number are weak within a wide mass range. However, for the imaginary surface potential Wsand the corresponding real dispersive-surface contribution Vs, the dependencies are stronger, deviating from 56Fe by less than 10%. Calculated DCCOM results for the iron group nuclei are presented under Results and Discussion. 100 102 104 106 108 10-1 100101102 Total cross section (b) Neutron energy (MeV) 62Ni ( x 100) 60Ni ( x 101) 58Ni ( x 102) 58Fe ( x 103) 56Fe ( x 104) 54Fe ( x 105) 52Cr ( x 106) Present FIG. 5. Comparison of neutron total cross sections with measurements. IV. A COMPUTATIONAL METHOD FOR ANALYZING POWER In this work a computational method to derive the analyzing power in the actual coupled-channels calculations has been given following the formalism for a polarization vector developed by Tamura in his canonical work [18]. It is assumed that initially both the projectile and the target are in some polarized states described by amplitudes a(i) msand b(i) M1, respectively. Here the superscripts iand i specify different spin ensembles. Then, the polarization vector Pn(θ,φ), parallel to a given unit vector n(the normal to the scattering plane) of the particle scattered in the direction specified by the polar angles (θ,φ), is given in Eq. (50) in Ref. [18]as σ(s) n(θ,φ) Pn(θ,φ) = ii (m) X∗ msM1;m sMn(θ,φ)X¯ ms¯ M1;¯ ms¯ Mn(θ,φ) ×m s|(σ·n)|¯ msa(i)∗ msb(i)∗ M1a(i) ¯ msb(i) ¯ M1,(11) where the summation (m)is over all projections (magnetic quantum numbers). Note that the subsequent explanations follow the same symbol definitions used by Tamura, except for new quantities. σ(s) n(θ,φ) is the shape differential cross section with the target in its nth state, and σis the Pauli spin matrix. The amplitude XmsM1;m sMn(θ,φ) given by Eq. (48) in Ref. [18] describes the scattering probability from the initial state with msand M1to the final state with m sand Mnquantum numbers. For the case of unpolarized projectile and unpolarized target, the amplitudes a(i) msand b(i) M1 can be simplified as a(1) 1 2 =a(2) −1 2 =(1 2)1 2,a (1) −1 2 =a(2) 1 2 =0, and b(N1) M1=(1 ˆ I1)δM1,N1,M1,N 1=−I1,...,I 1. Correspondingly, Eq. (11) is considerably simplified, but it is still hard to use such an equation in actual coupled-channels calculations. We restrict our discussion to the calculation of the y component Py n(θ,φ) of the polarization vector  Pn(θ,φ), because the ycomponent Ay(θ) of the analyzing power vector  Ais usually measured. Here, the yaxis is chosen as the normal to the scattering plane. Correspondingly, the matrix elements m s|(σ·n)|¯ mscan be simplified as m s|σy|¯ ms, where σy is the ycomponent of the Pauli matrix. These y-component matrix elements can be evaluated by 2 ×2 combinations of two spin functions: |¯ msand |¯ ms. With such a treatment and the consideration of symmetry in the azimuthal angle φ, Eq. (11) leads to σ(s) n(θ)Py n(θ) =1 2ˆ I1 2 msM1Mn 2ImX∗ msM1;1 2Mn(θ)XmsM1;−1 2Mn(θ),(12) where the differential cross sections σ(s) n(θ) are reduced from Eq. (49) in Ref. [18]to σ(s) n(θ)=1 2ˆ I1 2 msM1m sMnXmsM1;m sMn(θ) 2.(13) 054611-6 DISPERSIVE COUPLED-CHANNELS OPTICAL-MODEL ... PHYSICAL REVIEW C 87, 054611 (2013) In our coupled-channels calculations, the Coulomb amplitude fc(θ) and C-matrix elements included in the expression of the amplitude XmsM1;m sMn(θ) are evaluated by OPTMAN [25] code. Meanwhile, σ(s) n(θ) are just the elastic and inelastic angular distributions. Equation (12) can be used to calculate both elastic and inelastic y-component polarizations. For the case of elastic scattering, it is well known that the asymmetry (analyzing power) of the reaction induced by the completely polarized particle is equal to the polarization produced in the (time- )reversed reaction initiated by unpolarized particles [40]. Therefore, Py n(θ)(n=1) is equivalent to the analyzing power Ay(θ). For the inelastic case both formulations are not equal in the general case [41]. However, it has been shown [41,42] that the polarization and the asymmetry (analyzing power) are equal in distorted-wave Born approximation (DWBA) calculations within an adiabatic approximation (i.e., when the projectile energy is much higher than the level energy, leading to a weak coupling limit). We found a similar behavior for the strong coupling case in coupled-channels calculations. Numerical values calculated with the ECIS code [43]using both formulations display very good agreement, with only minor differences observed at backward angles. Therefore, it is reasonable to assume that Py n(θ) can be used for comparison with experimental Ay(θ) data. Further investigations of the inelastic analyzing power are warranted. V. RESULTS AND DISCUSSION This section shows the results of total cross sections, proton reaction cross sections, nucleon elastic/inelastic angular distributions, (p, n) data, and analyzing powers, mainly for 100 101 102 100101102 Nonelastic cross section (b) Proton energy (MeV) 62Ni ( x 21) 60Ni ( x 22) 58Ni ( x 23) 58Fe ( x 24) 56Fe ( x 25) 54Fe ( x 26) 52Cr ( x 27) Present FIG. 6. Comparison of proton nonelastic cross sections with measurements. 10-3 100 103 106 109 1012 1015 1018 1021 1024 1027 0 30 60 90 120 150 26.00 24.00 22.00 20.00 16.93 14.70 13.92 11.93 11.00 9.94 8.50 7.96 7.00 ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) ( x 1012) ( x 1014) ( x 1016) (a) 54Fe(n,n) ELASTIC ( x 1018) ( x 100) ( x 1020) ( x 1022) ( x 1024) 100 103 106 109 1012 1015 1018 1021 1024 1027 0 30 60 90 120 150 180 96.00 65.00 55.00 26.00 20.00 14.70 14.00 ( x 106) 11.00 9.94 7.96 ( x 108) ( x 1010) ( x 1012) ( x 1014) ( x 1016) ( x 1018) (b) 56Fe(n,n) ELASTIC ( x 1020) 11.93 ( x 1022) ( x 1024) ( x 1026) 10-3 100 103 106 109 0 30 60 90 120 150 26.00 24.00 22.00 20.00 16.93 13.92 ( x 100) ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) (c) 54Fe(n,n′)2 + 0 30 60 90 120 150 180 10-3 100 103 106 109 1012 1015 26.00 21.10 15.20 13.92 11.93 11.00 9.94 7.96 ( x 100) ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) ( x 1012) ( x 1014) (d) 56Fe(n,n′)2 + dσ/dΩ (b/sr) θc.m. (deg) FIG. 7. Comparison of neutron elastic and inelastic scattering angular distributions with measurements for Fe isotopes at different incident energies. Fe isotopes. Selected results for Ni isotopes and 52Cr are also presented showing the predictive power of the derived regional DCCOM potential. Figures 5and 6present predictions of neutron total cross sections and proton nonelastic cross sections for Fe-group nuclei up to 200 MeV, respectively. It is seen that the neutron total cross sections are in good agreement with experimental data from 1 to 200 MeV. The deviation of our predictions from the experiments is on average within 5% above 1 MeV. Only at energies below 1 MeV does the prediction clearly deviate from the experimental data. However, it is not expected that the present potential will reproduce the total cross sections at low energies very precisely, because resonance structures often appear in the low-energy ranges, which are beyond the average description assumed by the optical model. Similarly, good agreement with measurement is seen for proton nonelastic cross sections. On average, the difference between the calculation and the measurement is within 5%. The nucleon elastic and inelastic scattering angular distributions for Fe isotopes are shown in Figs. 7and 8, respectively. 054611-7 LI, SUN, SOUKHOVITSKI˜ I, QUESADA, AND CAPOTE PHYSICAL REVIEW C 87, 054611 (2013) 100 103 106 109 1012 1015 1018 1021 0 30 60 90 120 150 σ(θ)/σRuth 65.00 61.50 40.00 39.70 38.80 35.20 30.40 28.80 24.60 20.40 ( x 100) ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) ( x 1012) ( x 1014) ( x 1016) (a) 54Fe(p,p) ( x 1018) 19.60 ( x 1020) 0 30 60 90 120 150 156.00 65.00 61.50 39.80 35.20 30.30 24.60 20.40 19.10 17.20 ( x 100) 7.74 ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) ( x 1012) ( x 1014) ( x 1016) (b) 56Fe(p,p) ( x 1018) ( x 1020) 10.93 ( x 1022) 10-6 10-3 100 103 106 109 0 30 60 90 120 150 dσ/dΩ (b/sr) 65.00 61.50 40.00 39.70 30.40 18.60 17.90 ( x 100) ( x 102) ( x 104) ( x 106) ( x 108) ( x 1010) ( x 1012) (d) 54Fe(p,p′)30 30 60 90 120 150 61.50 19.10 65.00 65.00 22 + ( x 106) 3- ( x 108) 17.20 17.20 (e) 56Fe(p,p′) 22 + ( x 104) 3- ( x 101) 3- ( x 109) 4+ ( x 1012) 100 103 106 109 0 30 60 90 120 150 180 11.66 22.20 35.20 ( x 104) 10.93 6.00 (c) 58Fe(p,p) ( x 102) ( x 100) ( x 106) ( x 108) 0 30 60 90 120 150 180 10-3 100 103 106 109 1012 1015 49.35 49.35 49.35 21 + ( x 104) 17.50 17.50 17.50 11.66 10.93 (f) 58Fe(p,p′) 22 + ( x 103) 3- ( x 100) 3- ( x 106) 22 + ( x 108) 21 + ( x 109) 21 + ( x 1011) 21 + ( x 1013) θc.m. (deg) FIG. 8. Same as Fig. 7, but for protons. Obviously, the predictions for both neutron and proton elastic scattering describe the experimental data rather well over the entire energy and angular range. The same is generally true for neutron and proton inelastic scattering on 3−,4 +, and 2+excited levels. The exception is an observed slight underestimation for the 2+level of the 56Fe(n, n) reaction at 7.96 MeV in Fig. 7, perhaps due to the missing compoundinelastic contribution. A slight underestimation for the 3−level of the 54Fe(p, p) reaction at 40 MeV is also seen in Fig. 8. The reason might be due to some problems with experimental data (for instance, the subtraction of contributions from other levels, because the 3−level contribution is relatively smaller than the others and such subtraction may give large errors). Our calculations show good agreement for scattering data on the 3−level at all other energies. The results of the (p, n) transition exciting the ground state IAS and 2+EAS are shown in Fig. 9for Fe isotopes, 52Cr, and 58,62 Ni. An overall agreement with measurements is seen for all energies and the whole angular region, with the exceptions of forward angles for the 2+state of the 58Fe(p, n) reaction at 120 MeV. Note that experimental (p, n) data were not used in the fitting of our optical potential parameters. Even so, it is very satisfactory that the DCCOM potential can describe these data rather well to such an extent. This demonstrates the approximate Lane consistency of the derived potential. Figure 10 shows the neutron and proton elastic analyzing powers Aycalculated for Fe isotopes, along with the experimental data and those corresponding to the spherical optical model calculated by TALYS code [44] with the KoningDelaroche potential [2]. Our calculations for neutrons are in excellent agreement with data, while the results for proton elastic Ayare also satisfactory, except for a slightly poor description at backward angles for 54Fe at 10 MeV. It is observed that our calculations are, in general, consistent with TALYS spherical optical-model calculations. However, a slight shift between both calculations can be seen at some energies. 054611-8 DISPERSIVE COUPLED-CHANNELS OPTICAL-MODEL ... PHYSICAL REVIEW C 87, 054611 (2013) 10-12 10-6 100 106 0 30 60 90 120 150 135.00 35.00 23.00 0+ ( x 10-4) 35.00 0+ ( x 10-1) 23.00 0+ ( x 104) (a) 54Fe(p,n) 2+ ( x 102) 2+ ( x 107) 10-12 10-8 10-4 100 104 35.20 23.00 17.00 35.20 0+ ( x 10-5) 23.00 0+ ( x 10-1) 17.00 0+ ( x 104) (c) 56Fe(p,n) 2+ ( x 10-2) 2+ ( x 102) 2+ ( x 106) 10-12 10-8 10-4 100 104 0 30 60 90 120 150 120.00 23.00 120.00 0+ ( x 10-2) 23.00 0+ ( x 102) (e) 58Fe(p,n) 2+ ( x 102) 2+ ( x 105) 10-14 10-10 10-6 10-2 102 0 30 60 90 120 150 180 17.30 18.00 120.0 0+ ( x 100) 0+ ( x 10-3) 0+ ( x 10-6) (b) 52Cr(p,n) 10-12 10-8 10-4 100 23.00 32.00 35.00 35.00 2+ ( x 100) 2+ ( x 10-3) 0+ ( x 10-7) 2+ ( x 10-9) (d) 58Ni(p,n) 0 30 60 90 120 150 180 10-12 10-8 10-4 100 16.00 16.00 22.80 35.00 35.00 22.80 0+ ( x 100) 2+ ( x 10-3) 0+ ( x 10-6) 0+ ( x 10-12) 2+ ( x 10-15) 2+ ( x 10-9) (f) 62Ni(p,n) dσ/dΩ (b/sr) θc.m. (deg) FIG. 9. Predictions for (p, n) transition exciting IAS and EAS states for Fe isotopes (left) and three other nuclei (right) at different incident energies and levels. TALYS calculations show an irregular smooth pattern 65◦at 65 MeV, contrary to our oscillation pattern. The predictions for neutron and proton inelastic Ayfor Fe isotopes are presented in Fig. 11, compared with DWBA results. TALYS drives the ECIS code [43] to perform such DWBA calculations. The results for neutrons are rather good, while the results for protons are also good including the inelastic polarization for 3−and 2+ 2states. However, they are of somewhat lower quality than those for neutrons, with slight underestimation seen near extrema for the cases of the 2+ 1 state of 54Fe(p, p) at 18.6 MeV, as well as for the 2+ 1state of 56Fe(p, p)at65MeV. It is expected that the DCCOM is a better approximation (due to the strong coupling of collective levels) than DWBA (weak coupling) at lower excitation energies. Indeed, DCCOM results are slightly better than DWBA ones, although both results describe relatively well the experimental data. A similar situation is seen as shown in Fig. 10; that is, a shift between both results at some energies is observed, and the DWBA results display an irregular smooth pattern above 65◦at 65 MeV. The new option implemented in the OPTMAN code [25] allows calculating elastic and inelastic Ayfor both even-even and odd nuclei. We also used our derived DCCOM potential 0 4 8 12 16 20 0 30 60 90 120 150 10.00 14.00 (+3) 18.60 (+6) 30.40 (+9) 40.00 (+12) 65.00 (+15) 54Fe(p,p) (a) Present Spherical 0 4 8 12 16 20 24 0 30 60 90 120 150 180 5.77 14.00 (+3) 18.60 (+6) 24.60 (+9) 30.30 (+12) 65.00 (+15) 179.0 (+18) 56Fe(p,p) (b) Present Spherical 0 4 8 12 0 30 60 90 120 150 5.85 6.51 (+3) 14.00 (+6) 14.50 (+9) 58Fe(p,p) (c) Present Spherical 0 30 60 90 120 150 180 0 4 8 12 16 20 9.941 13.90 (+3) 13.94 (+6) 16.93 (+9) 9.993 (+14) Present Spherical 0 4 8 12 16 20 9.941 13.90 (+3) 13.94 (+6) 16.93 (+9) 9.993 (+14) 54Fe(n,n) 56Fe(n,n) (d) Analyzing power Ay(θ) θc.m. (deg) FIG. 10. Neutron and proton elastic analyzing powers for Fe isotopes, compared with the experimental data and TALYS calculations using a Koning-Delaroche potential. to predict the measured proton elastic Aydata, as shown in Fig. 12, for the odd nucleus 57Fe. Note that, for the case of the odd-Anucleus 57Fe, the coupled-channels calculation is based on the rigid-rotor model with the deformation parameter β2being 0.2017, β4being 0.0158, and β6being −0.0071. The agreement of our calculations with the measurements is satisfactory. This OMP extension to odd nuclei is important for applications, because the same OMP can be used for all iron isotopes, eliminating inconsistencies in calculated reaction cross sections between odd and even isotopes. Unfortunately, few experimental inelastic data for odd-Anuclei are available, so it is hard to make a further test of the applicability of this approach. Finally, we present some additional results of the elastic and inelastic analyzing powers for 58Ni in Fig. 13 to show the predictive power of our potential. As before, one can see that calculated results for elastic Ayreproduce well the experimental data at all energies. The description for inelastic Ayis also good and reproduces the experimental data for all states, except the poor agreement with data measured at 178 MeV. Our description reproduces well the main tendencies of experimental data, but predicts narrow oscillations and deeper minima at higher energies. Those predicted minima are not observed in the experiments, but such narrow oscillations might be difficult to measure due to degraded experimental 054611-9