Dispersive spherical optical model of neutron scattering from 27 Al up to 250 MeV

Dispersive spherical optical model of neutron scattering from 27Al up to 250 MeV A. Molina,1,*R. Capote,1,2,† J. M. Quesada,1,‡ and M. Lozano1,§ 1Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Facultad de Fı ´sica, AP 1065, E-41080 Sevilla, Spain 2Centro de Estudios Aplicados al Desarrollo Nuclear, AP 100, Miramar, La Habana, Cuba 共Received 14 November 2001; published 4 March 2002兲 A spherical optical model potential 共OMP兲containing a dispersive term is used to fit the available experimental database of ␴ ( ␪ ) and ␴ Tfor n⫹27Al covering the energy range 0.1–250 MeV using relativistic kinematics and a relativistic extension of the Schro ¨dinger equation. A dispersive OMP with parameters that show a smooth energy dependence and an energy-independent geometry are determined from fits to the entire data set. A very good overall agreement between experimental data and predictions is achieved up to 150 MeV. Inclusion of nonlocality effects in the absorptive volume potential allows one to achieve an excellent agreement up to 250 MeV. DOI: 10.1103/PhysRevC.65.034616 PACS number共s兲: 11.55.Fv, 24.10.Ht I. INTRODUCTION During the last 15 years, a great deal of theoretical attention has been devoted to achieving a proper formulation of the nuclear mean field at positive and negative energies. A significant contribution to the solution of this problem can be considered the work of Mahaux and co-workers on dispersive optical-model analysis 关1–5兴. A unified description of a nuclear mean field in a dispersive optical model is accomplished by using a dispersion relation, which links the real and absorptive terms of the optical model potential 共OMP兲. The dispersive optical model 共DOM兲provides a natural extension of the optical-model-derived data into the boundstate region. In this way a physically self-consistent description of the energy dependence of the OMP is obtained, and a prediction of single-particle, bound-state quantities using the same potential at negative energies becomes possible. Moreover, an additional constraint imposed by dispersion relations helps to reduce the ambiguities in deriving phenomenological OMP parameters from the experimental data. A dispersive OMP analysis was applied to nucleusnucleus systems 关6–9兴, where the energy dependence of the real central potential at low energies near the Coulomb barrier was studied, and contributions of the dispersion terms evaluated. However, for a nucleus-nucleus system, the dispersive OMP analysis is limited to the positive energy region, because it is not yet clear how to deal with particle clusters bound in a nucleus. Some progress has been achieved in applications of the dispersive OMP analysis to the alpha-nucleus scattering, improving our knowledge of the alpha cluster effective interaction inside a nuclear system 关10兴. Pionering works on dispersive OMP analysis for nucleon scattering were made by Passatore 关11兴and Lipperheide and Schmidt 关12兴. A great success was achieved in deriving DOM potentials for nucleon scattering on closedshell nuclei like 40Ca 关4,13–16兴,90Zr 关16–20兴, and 208Pb 关1–4,13,14,21,22兴, for which experimental information on bound states is available. Many studies also dealt with neutron scattering on nonmagic nuclei (39K关23兴,51V关24兴,86Kr 关25兴,89Y关26兴,93Nb 关27兴,113In 关28兴, and 209Bi 关22,29,30兴兲. However, very few studies were devoted to DOM potentials for nuclei with Aⱗ30. Only one preliminary DOM analysis was reported for 27Al(n,n)关31兴. There are two publications making a DOM analysis for proton-induced reactions on aluminum up to 60 MeV 关32,33兴. The main purpose of this contribution is to construct a complex mean field ‘‘felt’’ by neutrons in 27Al theoretically valid from ⫺50 up to 250 MeV energy. There exist two main versions of the dispersion relation approach. In both methods, the real and imaginary parts of the mean field are connected by a dispersion relation; moreover, a mean field is required to reproduce the experimental value of the Fermi energy EFclosely. The main difference between the two methods is the following: 共i兲In the ‘‘variational moment approach’’ 关13,14兴, the parameters of the complex mean field are determined by fitting radial moments of phenomenological optical-model potentials. 共ii兲In the ‘‘dispersive optical model analysis’’ 关15–17兴, the unknown parameters are derived by performing optical-model fits to experimental scattering cross sections that need to be available over as broad an energy range as possible. In the present work a variation of the dispersive optical model analysis is applied to a determination of the nuclear mean field for a neutron-27Al system. An Ohio University– Los Alamos collaboration published an extensive survey of neutron-nucleus total cross-section measurements up to 600 MeV 关34,35兴. These high precision data, together with earlier neutron differential scattering data available in the interval 1–26 MeV, form the database considered at positive energies. The Fermi energy value derived from nuclear masses is used to constrain the mean-field value at negative energies. Therefore, the energy variation of the model parameters is reasonably defined over a wide range, an extremely important point for a successful dispersive analysis. It is remarkable that our total-cross section database goes up to the region where surface absorption can be safely neglected. Since the employed database extends up to 250 MeV, and since the recent ␴ Tdata are very accurate, i.e., the uncertainty ⌬ ␴ Tis *Email address: [email protected] †Email address: [email protected] ‡Email address: [email protected] §Email address: [email protected] PHYSICAL REVIEW C, VOLUME 65, 034616 0556-2813/2002/65共3兲/034616共13兲/$20.00 ©2002 The American Physical Society65 034616-1 about ⫾1%, we use relativistic kinematics and a relativistic equation equivalent to the Schro ¨dinger equation in all our calculations. Other motivation for our work is that aluminum is an important structural material for accelerator-driven systems, and its cross sections are often used as references to determine other cross sections 关36兴. There exist phenomenological OMP’s 共in the sense that dispersive relations constrain is not used兲describing neutron scattering on aluminum up to high incident energy. The LANL high energy evaluation of Chadwick et al. 关37兴employed the OMP of Petler et al. 关38兴up to 60 MeV and the Madland global OMP 关39兴from 60 up to 150 MeV. Lee et al. 关40兴derived a phenomenological OMP which described neutron scattering from 27Al up to 250 MeV incident energy. Recently a global phenomenological parametrization valid from 1 keV to 200 MeV for A⭓27 nuclei was proposed by Koning and Delaroche 关41兴. Usually in DOM analysis the absorptive potentials are considered symmetric about the Fermi energy EF, and nonzero in the energy gap surrounding EF. However, Mahaux and Sartor 关4兴pointed out that 共i兲due to nonlocality effects, the absorptive potential will be highly asymmetric共with respect to EF兲; and 共ii兲there should be an energy gap centered about EFin which the absorption term drops to zero, at least for energies between the first-hole and first-particle states. A recent DOM analysis of neutron scattering on 208Pb and 209Bi 关22兴failed to describe ␴ Tdata for energies above 40 MeV using an asymmetric version of the absorptive potentials for large positive and large negative energies. We will present strong evidence to favor asymmetric absorptive potentials for a proper description of the neutron-scattering ␴ T data for energies between 150 and 250 MeV. The paper is structured as follows. Section II provides a description of the dispersive optical model formalism, the solved wave equation, and the forms of the energy and radial dependencies of the real, imaginary, and spin-orbit potentials. Section III describes the compound nucleus 共CN兲calculations, the 27Al(n,n) experimental database, our procedure for searching, and the resulting relativistic and nonrelativistic spherical DOM potentials for 27Al(n,n). In the same section we compare derived DOM potentials with phenomenological potentials and experimental data. Finally, Sec. IV contains our conclusions. II. DOM FORMALISM A. Optical-model potential and wave equation The optical-model analysis was carried out with a semirelativistic generalization of the conventional nonrelativistic Schro ¨dinger formulation of the scattering process 关42兴. Relativistic kinematics was used for the projectile, but it was assumed that the target motion in the center-of-mass system could be treated nonrelativistically. A relativistic equivalent to the Schro ¨dinger equation was generated by appropriate reduction of the Dirac equation for a massive, energetic fermion 共mass mand c.m. wave number k) moving in a localized central potential V(r) taken as the time-like component of a Lorentz four-vector. In a reduced two-body problem with a relativistic projectile but a nonrelativistic target 共mass M), the large component of the partial wave function Fl( ␳ ) can be shown to satisfy the radial equation 再 d2 d ␳ 2⫹ 冋 1⫺V共 ␳ 兲 Tc ⫺l共l⫹1兲 ␳ 2 册 冎 Fl共 ␳ 兲⫽0, 共1兲 where ␳ ⫽kr,Tcis the total c.m. kinetic energy, lis the orbital angular momentum, and V( ␳ ) is the renormalized nuclear optical potential: V共 ␳ 兲⫽ ␥ U共r兲, ␥ ⫽1⫹Tc Tc⫹2m.共2兲 Equation 共1兲is formally identical to the radial equation for the solution of the nonrelativistic Schro ¨dinger equation for an analogous scattering problem with a nuclear potential renormalized by a factor ␥ . This factor becomes increasingly important as the projectile kinetic energy increases 关see Eq. 共2兲兴, leading to an effective increase of the potential depth. The spin-orbit term in V(r) employed in this analysis is a purely phenomenological one, since the intrinsic spin-orbit term in the Dirac equation is negligibly small in the above limits. Equation 共1兲was used in all calculations. In a nonrelativistic case, we set the factor ␥ equal to 1 and nonrelativistic kinematics was employed; otherwise relativistic kinematics and the factor ␥ according to Eq. 共2兲were used. Our analysis spans an energy range from 0.1 to 250 MeV. Both direct and statistical processes contribute to nucleonnucleus elastic scattering at these energies. According to our estimation, the statistical processes are important up to 12 MeV in aluminum. A compound nucleus calculation will be described in Sec. IIIC. The direct processes, increasingly dominant at higher energies, can be described by the optical model. Although the 27Al nucleus is deformed, the spherical OMP was applied successfully 关38,40,43兴.Ana posteriori analysis of the impact of this approximation on the calculated observables will be discussed below. The optical model potential may be written as U共r,E兲⫽⫺关Vv共E兲⫹iWv共E兲兴fWS共r,Rv,av兲⫺关Vs共E兲 ⫹iWs共E兲兴gWS共r,Rs,as兲⫹ 冉 ប m ␲ c 冊 2 关Vso共E兲 ⫹iWso共E兲兴1 r d dr fWS共r,Rso ,aso兲共l ជ • ␴ ជ 兲,共3兲 where the successive complex-valued terms are the volume central, surface central, and spin-orbit potentials. The volume shape fWS(r,Rv,av) is a standard Woods-Saxon form factor specified by a potential radius Rvand diffuseness av. The surface shape is the first derivative of the Woods-Saxon form specified by a potential radius Rsand diffuseness as: gWS共r,Rs,as兲⫽⫺4as d dr fWS共r,Rs,as兲.共4兲 A. MOLINA, R. CAPOTE, J. M. QUESADA, AND M. LOZANO PHYSICAL REVIEW C 65 034616 034616-2 The reduced radius parameter riis introduced as usual by the relation Ri⫽riA1/3. In our formulation of the OMP in Eq. 共3兲, the real and imaginary central volume terms share the same geometry parameters rvand av, and likewise the real and imaginary central surface terms share the same rsand as. This assumption 关3兴can be seen as a consequence of the dispersive relations, allowing us to reduce the number of geometrical parameters in the OMP. For the spin-orbit potential we adopt the parameters obtained by Koning et al. 关36兴, namely, Vso共E兲⫽6.0exp共⫺0.005E兲MeV, Wso共E兲⫽0.2⫺0.011EMeV, rso⫽1.017 fm, aso⫽0.6 fm. 共5兲 In a dispersion relation treatment, the real central potential strength consists of a term which varies slowly with energy, the so called Hartree-Fock 共HF兲term VHF(E), plus a correction term 䉭V(E) which is calculated using a dispersion relation. The depth of the dispersive term of the potential 䉭V(E) can be written in the subtracted form 䉭V共E兲⫽ P ␲ 冕 ⫺⬁ ⬁W共E⬘兲 冉 1 E⬘⫺E⫺1 E⬘⫺EF 冊 dE⬘.共6兲 With the assumption that W(E) be symmetric with respect to the Fermi energy EF, Eq. 共6兲can be expressed in a form which is stable under numerical treatment 关17兴, namely, 䉭V共E兲⫽2 ␲ 共E⫺EF兲 冕 EF ⬁W共E⬘兲⫺W共E兲 共E⬘⫺EF兲2⫺共E⫺EF兲2dE⬘, 共7兲 where W(E) is the imaginary part of the OMP. The dispersive term 䉭V(E) is divided into two terms 䉭Vv(E) and 䉭Vs(E), which arise through dispersion relations 共7兲from the volume Wv(E) and surface Ws(E) imaginary potentials, respectively. If the imaginary potential geometry is energy dependent, then the radial dependence of the dispersive correction cannot be expressed using a Wood-Saxon form factor, i.e., 䉭Vv(r,E)⫽䉭V(E)f(r,R,a). However, to simplify the problem, the OMP geometry parameters used in this work are energy independent. In this case, using the definitions of Eq. 共3兲, the real volume Vv(E) and surface Vs(E) central part of the DOM potential are given by Vv共E兲⫽VHF共E兲⫹䉭Vv共E兲, Vs共E兲⫽䉭Vs共E兲.共8兲 It is known that the energy dependence of the depth 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 E, and is an exponential for large positive E. Following Mahaux and Sartor 关4兴, the energy dependence of the Hartree-Fock part of the nuclear mean field is taken as that found by Lipperheide 关44兴, VHF共E兲⫽V0exp关⫺ ␣ HF共E⫺EF兲兴,共9兲 where the parameters V0and ␣ HF are undetermined constants. Equation 共9兲can be used to describe HF potential in the scattering regime 关4兴. It is useful to represent the variation of surface Ws(E) and volume absorption potential Wv(E) depth with energy in functional forms suitable for the dispersive optical model analysis. An energy dependence for the imaginary volume term has been suggested in studies of nuclear matter theory 关45兴, Wv共E兲⫽Av 共E⫺EF兲n 共E⫺EF兲n⫹共Bv兲n,共10兲 where Avand Bvare undetermined constants. Following Mahaux and Sartor 关2兴, we adopt n⫽4. An energy dependence for the imaginary-surface term was suggested by Delaroche et al. 关17兴to be Ws共E兲⫽As 共E⫺EF兲m 共E⫺EF兲m⫹共Bs兲mexp共⫺Cs 兩 E⫺EF 兩 兲, 共11兲 where m⫽4 and As,Bs, and Csare undetermined constants. According to Eqs. 共10兲and 共11兲, the imaginary part of the OMP is assumed to be zero at E⫽EF, and nonzero everywhere else. A more realistic parametrization of Wv(E) and Ws(E) forces these terms to be zero in some region around the Fermi energy. A physically reasonable energy for defining such a region is the average energy of the single-particle states Ep关4兴. For aluminum we used a value Ep ⫽⫺5.66 MeV, obtained by averaging the first three particle states reported in the microscopical single-particle level calculation by Moller and Nix 关46兴. The experimental value of the Fermi energy EF, derived from mass differences, is equal to ⫺10.392 MeV. Therefore, a definition for imaginary part of the OMP can be written as Wv共E兲⫽ 再 0 for EF⬍E⬍Ep Av 共E⫺Ep兲n 共E⫺Ep兲n⫹共Bv兲nfor Ep⬍E, 共12兲 and likewise for surface absorption: DISPERSIVE SPHERICAL OPTICAL MODEL OF... PHYSICAL REVIEW C 65 034616 034616-3 Ws共E兲⫽ 再 0 for EF⬍E⬍Ep As 共E⫺Ep兲m 共E⫺Ep兲m⫹共Bs兲mexp共⫺Cs 兩 E⫺EP 兩 兲for Ep⬍E.共13兲 The symmetry condition W共2EF⫺E兲⫽W共E兲共14兲 is used to define imaginary part of the OMP for energies below the Fermi energy. Equations 共12兲and 共13兲are used to describe the imaginary absorptive potential in this contribution. B. High-energy behavior of the volume absorption The assumption that the imaginary potential Wv(E)is symmetric about E⬘⫽EF关according to Eq. 共14兲兴 is plausible for small values of 兩 E⬘⫺EF 兩 ; however, as pointed out by Mahaux and Sartor 关4兴this approximate symmetry no longer holds for large values of 兩 E⬘⫺EF 兩 . In fact the influence of the nonlocality of the imaginary part of the microscopic mean field will produce an increase of the empirical imaginary part W(r,E⬘) at large positive E⬘, and approaches zero at large negative E⬘关1,47兴. Following Mahaux and Sartor 关4兴, we assume that the absorption strengths are not modified below some fixed energy Ea. They used Ea⫽60 MeV; however, this value is fairly arbitrary 关4兴. Let us assume the nonlocal imaginary potential to be used in the dispersive integral is denoted by W ˜ v(E); then we can write 关5兴 W ˜ v共E兲⫽Wv共E兲 冋 1⫺共EF⫺E⫺Ea兲2 共EF⫺E⫺Ea兲2⫹Ea 2 册 , for E⬍EF⫺Ea共15兲 and W ˜ v共E兲⫽Wv共E兲⫹ ␣ 冋 冑 E⫹共EF⫹Ea兲3/2 2E ⫺3 2 冑 共EF⫹Ea兲 册 , for E⬎EF⫹Ea.共16兲 These functional forms are chosen in such a way that the function and its first derivative are continuous. At large positive energies nucleons ‘‘sense’’ the ‘‘hard core’’ repulsive region of the nucleon-nucleon interaction, and W ˜ v(E) diverges like ␣ 冑 E. Using a model of a dilute Fermi gas hard sphere, the coefficient ␣ can be estimated to be equal to 1.65 MeV1/2 关47兴, assuming that the Fermi impulse kFis equal to 1.36 fm⫺1and the radius of the repulsive hard core is equal to 0.4 fm. Conversely, at large negative energies the volume absorption decreases and goes asymptotically to zero. The nonlocal imaginary absorption potential W ˜ v(E) and the symmetric imaginary absorption potential W(E) are represented by solid and dotted lines, respectively, in the lower panel of Fig. 1. The asymmetric form of the volume imaginary potential of Eqs. 共15兲and 共16兲results in a dispersion relation that must be calculated directly from Eq. 共6兲, and separates into three additive terms 关48兴. Therefore, we write the dispersive correction in the form 䉭V ˜ v共E兲⫽䉭Vv共E兲⫹䉭V⬍共E兲⫹䉭V⬎共E兲,共17兲 where 䉭Vv(E) is the dispersive correction due to the symmetric imaginary potential of Eq. 共12兲, and the terms 䉭V⬍(E) and 䉭V⬎(E) are dispersive corrections due to the asymmetric terms of Eqs. 共15兲and 共16兲, respectively. The resulting energy dependence of the dispersive integrals 䉭V ˜ v(E) and 䉭Vv(E) for both the nonlocal imaginary absorption potential W ˜ v(E) and the symmetric imaginary absorption potential W(E) is represented by solid and dotted lines, respectively, in the upper panel of Fig. 1. While the symmetric case features equal contributions coming from negative and positive energies, in the asymmetric case the negative-energy contribution to the dispersive integral is very different from the positive-energy value. The resulting dispersive correction for the asymmetric case starts to increase for energies above 50 MeV, making a significant contribution to the real part of the OMP. FIG. 1. Dependence upon energy of the dispersive volume contribution of the real central potential of the n⫹27Al mean field. The dotted curve corresponds to Eq. 共12兲, in which it is assumed that the imaginary part is symmetric about the Fermi energy. The thick solid curves correspond to the asymmetric model, considering the nonlocal behavior of the imaginary volume absorption above certain energy Eafollowing Eqs. 共15兲and 共16兲. The thin dashed line corresponds to the Fermi energy. A. MOLINA, R. CAPOTE, J. M. QUESADA, AND M. LOZANO PHYSICAL REVIEW C 65 034616 034616-4 It should be noted that nonlocality corrections 关Eqs. 共15兲 and 共16兲兴 can be used either for the volume or surface imaginary potential; however, Mahaux and Sartor 关4兴showed that the nonlocality consideration for the surface imaginary potential has a very small effect on the calculated cross sections. Therefore, in this work we followed Ref. 关5兴and only considered the effects of nonlocality in the volume absorption. III. DISPERSIVE OPTICAL MODEL ANALYSIS A. DOM software Search optical model codes ECIS95 in the external input mode 关49,50兴and COH v 2.2 关51兴were used for DOM analyses using relativistic and nonrelativistic kinematics respectively. A modification was introduced into the latter code to force equality of the real and imaginary surface 共volume兲 geometry parameters Rs,as(Rv,av) during the search procedure, as is implicit in Eq. 共3兲. The code does not include the dispersion relations; therefore, the dispersion integrals 共7兲of the symmetric forms 共12兲and 共13兲of the imaginary potential were calculated numerically using a Gauss quadrature method 关52兴, while the asymmetric contribution was calculated analytically 关see Eqs. 共16兲–共19兲of Ref. 关48兴兴. An auxiliary code system was developed to produce proper input data sets for both optical model codes, and to calculate for each data set 共i.e., for each energy兲the ␹ 2quantity according to ␹ 2共E兲⫽兺 i⫽1 N ␴ 冋 ␴ expt共E, ␪ i兲⫺ ␴ calc共E, ␪ i兲 ⌬ ␴ expt共E, ␪ i兲 册 2 ⫹ 冋 ␴ expt tot 共E兲⫺ ␴ calc tot 共E兲 ⌬ ␴ expt tot 共E兲 册 2 .共18兲 Here, ␴ calc(E, ␪ i)关 ␴ calc tot (E)兴and ␴ expt(E, ␪ i)关 ␴ expt tot (E)兴, are the differential 共total兲cross sections from the optical model calculations and experiments for a given laboratory energy E, respectively, and ⌬ ␴ expt(E, ␪ i)关⌬ ␴ expt tot (E)兴is the experimental uncertainty reported. The N ␴ is the number of data points for ␴ expt(E, ␪ i). Our code system allows us to fine tune the OMP parameters of interest to minimize the total search ␹ 2of the entire data set. B. Summary of the experimental databases A survey of the experimental data spanning from 0.1 to 250 MeV used in the DOM analyses is presented in this section. The 27Al(n,n) ␴ ( ␪ ) data were obtained from Towle and Gilboy 关53兴at 1, 2, 3, and 4 MeV; Tanaka et al. 关54兴at 4.8, 6, 7, and 8 MeV; Kinney and Perey 关55兴at 5.4, 6.4, 7.5, and 8.6 MeV; Dagge et al. 关56兴at 7.62 MeV; Velkley et al. 关57兴at 9 MeV; Boerker et al. 关58兴at 10.2 MeV; Whisnant et al. 关43兴at 11, 14, and 17 MeV; Petler et al. 关38兴at 18, 20, 22, 25, and 26 MeV; Bratenahl et al. 关59兴at 84 MeV; Salmon 关60兴at 96 MeV; and Van Zyl et al. 关61兴at 136 MeV. The 27Al(n,n)Ay( ␪ ) data were obtained from Dagge et al. 关56兴 at 7.62 MeV and Martin and Walter 关62兴at 14 and 17 MeV. These polarization data were used only for testing spin-orbit interaction. Energy-averaged total cross sections ␴ Tfor 27Al were obtained from Finlay and co-workers 关34,35兴from 5.3 to 250 MeV. Additional energy-averaged ␴ Tdata were taken from Refs. 关63–74兴to be used for comparing predictions of the model. We selected measurements containing several points in energy, specially, all with data above 20 MeV. In examining all the available experimental total cross-section data, the high-resolution cross-section data of Ref. 关75兴were found to be inconsistent with the rest of the dataset, and were ignored in our analysis. C. Compound-nucleus corrections The statistical model of nuclear reaction according to the Hauser-Feshbach theory 关76兴, with width fluctuation corrections as modified by Moldauer 关77兴, is used to compute the CN contributions to the elastic channel. When the cross section is averaged over many CN resonances, the shape elastic differential cross section can be incoherently added to the FIG. 2. Cumulative number of levels as a function of the excitation energy for the three residual nuclei considered in the CN cross-section calculations. The discrete level data are from the RIPL 关46兴and are well represented by the ‘‘constant temperature’’ level density formula of Ref. 关81兴using parameters from Table I. The cutoff energy is indicated by the vertical dashed line. Above the cutoff energy the ‘‘constant temperature’’ level density formula was used. DISPERSIVE SPHERICAL OPTICAL MODEL OF... PHYSICAL REVIEW C 65 034616 034616-5 compound elastic contribution to compare with the experimentally observed elastic-scattering cross section. For neutron energies larger than 12 MeV, the compound-elastic contribution can be neglected. The CN cross-section calculation is built-in inside the search code COH 关51兴. Three reaction channels are considered in the statistical-model calculations of the 28Al CN decay: (n,n), (n,p), and (n, ␣ ). Transmission coefficients for proton and alpha emission in the exit channels were calculated from the spherical OMP parameters by Perey 关78兴and Arthur and Young 关79兴共a modification of Lemos OMP 关80兴兲, respectively. The transmission coefficients in the entrance and inelastic channels were calculated using the DOM potential of the present work. Discrete level information is used to represent low-lying states, and the Gilbert-Cameron level density formula 关81兴is used to represent the high-lying continuum of states. Figure 2 shows the cumulative number of levels as a function of excitation energy for the residual nuclei of the three reaction channels. The discrete state data are taken from the Belgya compilation contained in RIPL 关46兴. The vertical lines indicate the cutoff energy between the discrete states and the continuum. It is well known that a CN calculation is highly sensitive to the level density parameters modeling the continuum of the excited states. We used a ‘‘constant temperature’’ formula 关81兴to estimate the total number of excited states available at excitation energy E,N(E)⫽exp关(E ⫺E0)/T兴, where Tis the ‘‘nuclear temperature’’ and E0is the energy shift. These two parameters are determined by fitting the cumulative number of available experimental states up to some cutoff energy. The level density parameters for all three residual nuclei involved in CN cross-section calculations are listed in Table I. A cumulative number of levels, as calculated by the ‘‘constant temperature’’ model using these parameters, is shown by solid lines in Fig. 2. D. Search procedure It is well known that the search routine does not always converge to the optimum solution, especially when we are dealing with strongly correlated OMP parameters. In our DOM analysis we performed a global ␹ 2optimization combined with a grid search using a ␹ 2fit in a limited energy region, using a maximum number of two fitting parameters simultaneously. Our search procedure can be divided in four main steps. 共1兲The search for an imaginary Wv emp(E) empirical potential depth using total cross-section data between 70 and 150 MeV, neglecting real and imaginary surface contributions. This energy range is selected in order to neglect the surface absorptive potential in the first iteration. Once empirical values Wv emp(E) are obtained, a fit of the absorptive volume potential Wv(E) using Eq. 共12兲is carried out. In this way volume absorption is fixed, as well as the dispersive volume contribution 䉭Vv(E) to the central real potential, which is calculated by integration. The empirical values of the real volume potential depth Vv emp(E), combined with 䉭Vv(E), are used to obtain a set of empirical points corresponding to VHF emp(E). A typical set of empirical values derived in the above described way can be seen in Fig. 3, as obtained with the search code COH. Finally Eq. 共9兲is used to obtain the V0and ␣ HF parameters that offer a best fit to the TABLE I. Constant temperature level density parameters for residual nuclei in an n⫹27Al reaction. Residual nucleus Ecut (MeV) T(MeV) E0(MeV) 27Al 11.2 2.071 ⫺0.678 24Na 5.2 1.875 ⫺2.046 27Mg 6.0 2.113 ⫺1.2157 FIG. 3. Empirical real volume 共solid circles兲and imaginary volume potential depth 共empty circles兲of the OMP for n⫹27Al as determined from individual best ␹ 2fit searches using ␴ tot data in the interval 70⬍E⬍150 MeV after the first iteration. Upper panel: the solid line for the Hartree-Fock potential is the functional representation defined in Eq. 共9兲. The dashed line denotes the starting guess values calculated using the OMP of Ref. 关40兴. The crosses represent the empirical values of the Hartree-Fock-type potential obtained after the dispersive contribution coming from the volume imaginary part of the OMP was subtracted from the real volume empirical values. Lower panel: the solid line for the absorptive potential is the functional representation defined in Eq. 共12兲. The dashed line denotes the starting guess values calculated using the OMP of Ref. 关40兴. A. MOLINA, R. CAPOTE, J. M. QUESADA, AND M. LOZANO PHYSICAL REVIEW C 65 034616 034616-6 empirical real potential data. In the fitting process the strength V0was constrained for the DOM predicted firstparticle and first-hole states to be centered around the experimental value of the Fermi energy. 共2兲At each energy for which neutron elastic differential cross-section data and neutron total cross-section data are available from 1 to 26 MeV, we have conducted a best ␹ 2fit by searching in volume real Vv emp(E) and surface imaginary Ws emp(E) empirical potential depths. In the first iteration the corresponding dispersive surface contribution 䉭Vs(E) to the central real potential was calculated by integration from the starting OMP parameters. CN contributions and width fluctuation corrections were considered in all calculations for an incident energy below 12 MeV. Once empirical values Ws emp(E) are obtained, a fit of the absorptive surface-peaked potential Ws(E) using Eq. 共13兲is carried out. The dispersive surface contribution 䉭Vs(E) to the central real potential is re-evaluated by integration. The empirical values of the real volume potential depth Vv emp(E), combined with the 䉭Vv(E) calculated for these energies, are used to increase the set of empirical points corresponding to VHF emp(E). Equation 共9兲is used to refine the fitting of the V0and ␣ HF parameters, derived in point 共1兲, using the whole empirical set of potential values obtained in steps 共1兲and 共2兲. We iterate over steps 共1兲and 共2兲until the empirical potential strengths are consistent with our predefined energy functional 关see Eqs. 共9兲,共13兲, and 共12兲兴 over the whole energy range. 共3兲After fixing potential strengths, the optimum geometry parameters were searched for, iterating over steps 共1兲and 共2兲 to redefine the potential strengths corresponding to the optimized geometry parameters. 共4兲Finally, a global ␹ 2optimization using the whole experimental database was carried out to obtain the minimum in the ␹ 2multiparameter surface. E. 27Al„n,n…DOM analysis We started our analysis by using a nonrelativistic formulation to fit the experimental data. Initial values for geometrical parameters were provided by the energy-independent geometry deduced by Whisnant et al. 关43兴and used by Petler et al. 关38兴for a phenomenological analysis of the data up to 26 MeV. They found rv⫽1.18 fm, av⫽0.64 fm, rs ⫽1.26 fm, and as⫽0.58 fm. Because the general form of the energy dependence of the imaginary potential used in the present model is similar to the 27Al(n,n) phenomenological OMP of Lee et al. 关40兴, we used their real volume and imaginary potential parameters as a starting point for our analysis. We used symmetric imaginary absorptive potentials according to Eqs. 共13兲and 共12兲; therefore, we adjusted seven parameters, namely, (V0, ␣ HF), which define the smooth energy dependence of the real volume potential, and (Av,Bv) and (As,Bs,Cs) defining the volume and surface absorptive potentials, respectively. After proper values were obtained by this global minimization, the energy-independent geometry parameters were also optimized. The derived nonrelativistic DOM potential parameters are listed in Table II. The final ␴ TDOM fits using a nonrelativistic potential are compared to 27Al(n,n) data in Fig. 4. It should be stressed that the experimental total cross-section data 共except those TABLE II. Optical model parameters for the nonrelativistic dispersive potential for an n⫹27Al reaction up to 150 MeV. Parameter 共Unit兲Value V0(MeV) 52.24 ␣ HF (MeV⫺1) 0.0071 Av(MeV) 12.5 Bv(MeV) 58.8 rv(fm) 1.20 av(fm) 0.65 As(MeV) 12.6 Bs(MeV) 3.25 Cs(MeV⫺1) 0.0395 rs(fm) 1.11 as(fm) 0.64 EF(MeV) ⫺10.392 Ep(MeV) ⫺5.66 FIG. 4. Energy dependence of the n⫹27Al total cross section from 10 up to 150 MeV. The curve has been calculated using the nonrelativistic 共solid line兲DOM potential of the present work. Empty circles correspond to the experimental data of Refs. 关34,35兴used in the fitting procedure. The diamonds, crosses, and triangles are obtained from the measurements of Refs. 关64兴,关66兴, and 关67兴. DISPERSIVE SPHERICAL OPTICAL MODEL OF... PHYSICAL REVIEW C 65 034616 034616-7 represented by the empty circles兲shown in this figure were not used in the DOM parameter search. We can observe that the experimental total cross section at energies above 130 MeV was always underestimated by our nonrelativistic calculations. We cannot change the real volume potential depth 共or the so-called Hartree-Fock potential兲without spoiling the fits to the differential cross section. One solution could be to consider an increase of the radius of the real part of the OMP. However, this approach would obscure our treatment with an energy-independent geometry. Furthermore, it is theoretically obvious that relativistic effects and nonlocality should show up at this energy regime. Therefore, we decided to carry out a fully relativistic treatment, including nonlocal contributions to the absorptive potential, which will be reflected on the dispersive contribution to the real potential. The starting point in this second stage was the nonrelativistic DOM potential. We took into account the nonlocal contribution to the volume absorptive potential according to Eqs. 共15兲and 共16兲. Only one additional parameter was included, TABLE III. Optical model parameters for the relativistic dispersive potential for an n⫹27Al reaction up to 250 MeV. Parameter 共Unit兲Value V0(MeV) 53.5 ␣ HF (MeV⫺1) 0.0087 Av(MeV) 7 Bv(MeV) 75 rv(fm) 1.20 av(fm) 0.63 As(MeV) 12.5 Bs(MeV) 5 Cs(MeV⫺1) 0.034 rs(fm) 1.11 as(fm) 0.64 EF(MeV) ⫺10.392 Ep(MeV) ⫺5.66 Ea(MeV) 90.0 FIG. 5. Relativistic and nonlocality contribution to the total cross section. The total crosssection curves were calculated using the relativistic 共solid line兲and nonrelativistic 共dotted line兲 DOM potentials of the present work. The dashed line denotes relativistic DOM potential results without nonlocality correction. FIG. 6. Comparison between the neutron elastic differential cross-section experimental data and our DOM calculations 共solid line兲. CN contributions were added to the direct reaction predictions for incident energies up to 12 MeV. The ␴ ( ␪ ) data were obtained from Ref. 关53兴at 1, 2, 3, and 4 MeV; Ref. 关54兴at 4.8, 6, 7, and 8 MeV; Ref. 关55兴at 5.4, 6.4, 7.5, and 8.6 MeV; Ref. 关56兴 at 7.62 MeV; Ref. 关57兴at 9 MeV; Ref. 关58兴at 10.2 MeV; Ref. 关43兴at 11, 14, and 17 MeV; and Ref. 关38兴at 18, 20, 22, 25, and 26 MeV. It should be noted that data above 26 MeV was not used in the fitting process. The neutron incident energy is quoted above each calculated curve. A. MOLINA, R. CAPOTE, J. M. QUESADA, AND M. LOZANO PHYSICAL REVIEW C 65 034616 034616-8 namely, the energy Ea, above which the nonlocal behavior of the volume absorptive potential is considered. In this latter ␹ 2minimization the total cross-section data up to 250 MeV were included in the experimental database. All potential parameters changed because of the sizable contribution of the nonlocal absorption for energies above 40 MeV, as can be seen from Fig. 5. In the same figure the total cross-section calculated with the nonrelativistic DOM potential is shown for comparison. It is interesting to remark that the relativistic correction alone is clearly not enough for a correct description of the total cross section from 130 to 250 MeV. The final set of parameters of our dispersive relativistic optical model potential is summarized in Table III. F. Comparison with the experimental cross section in the energy domain 0.1ËEË250 MeV We now compare the experimental cross sections with those calculated from our DOM potentials. The geometrical parameters of the model and the strengths of the various components are specified in Tables II and III. The dispersion relations fully determine the dispersive contribution once the imaginary part of the mean field is specified. The ␴ ( ␪ ) relativistic DOM fits are compared to 27Al(n,n) data in Fig. 6. In general, the fits to ␴ ( ␪ ) are of high quality. A very good agreement between experimental data and calculations is observed in the energy region below 12 MeV, where the CN contribution is important. The highest deviation is observed for energies 25–26 MeV located near the diffraction maximum. In this energy region a difficulty was encountered during the fit process, evidenced by the fact that a common set of surface absorptive potential parameters giving acceptable fits to each type of data 共differential and total cross section兲could not be found. The fits to ␴ ( ␪ ) indicate smaller values of the imaginary surface potential depth Asparameter, while fits to total cross section point to values larger by about 2 MeV. Experimental ␴ ( ␪ ) data for energies higher than 26 MeV were not included in the fitting procedure, but our relativistic DOM potential displays an FIG. 7. Energy dependence of the n⫹27Al total cross section above 10 MeV. The curves were calculated using the relativistic 共solid line兲DOM potential of the present work. The dotted, dotdashed, and dashed lines were obtained from the phenomenological OMP of Refs. 关39兴,关40兴, and 关41兴, respectively, in their range of validity. Empty circles correspond to experimental data of Refs. 关34,35兴used in the fitting procedure. The diamonds, crosses, triangles, solid circles, and solid squares were from the measurements of Refs. 关64兴,关66兴,关67兴,关72兴, and 关63兴. FIG. 8. Low-energy dependence of the n ⫹27Al total cross section from 0.1 up to 10 MeV. The curves have been calculated using the relativistic DOM potential without 共solid line兲and with 共dashed line兲reorientation effects. The circles, up triangles, and down triangles were obtained from the phenomenological OMP by Harper in Refs. 关82兴,关38兴, and 关41兴. The highresolution experimental data were obtained from measurements in Refs. 关73兴and 关74兴. DISPERSIVE SPHERICAL OPTICAL MODEL OF... PHYSICAL REVIEW C 65 034616 034616-9