Coupled-channel optical model potential for rare earth nuclei
M. Herman, G. P. A. Nobre, A. Palumbo, F. S. Dietrich, D. Brown, S. Hoblit
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Coupled-channel optical model potential for rare earth nuclei
M. Herman , a , G. P. A. Nobre , b , A. Palumbo , F. S. Dietrich , D. Brown , and S. Hoblit National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA P.O. Box 30423, Walnut Creek, CA, 94598, USA
Abstract.
The global spherical optical model by Koning and Delaroche is generalized toenable its use in coupled-channel calculations on well deformed nuclei in the rare-earthregion. The generalization consists in adding the coupling of the ground state rotationalband, deforming the potential by introducing appropriate quadrupole and hexadecupoledeformations and correcting the optical model potential radius to preserve volume inte-gral of the spherical optical potential. We choose isotopes of three rare-earth elements(W, Ho, Gd), which are known to be nearly perfect rotors, to perform a consistent test ofour conjecture on integrated cross sections as well as on angular distributions for elasticand inelastic neutron scattering. The only additional input are experimentally determineddeformations, which we employ without any adjustments. The results are clearly superiorcompared to the spherical optical model calculations with dramatic improvement at lowincident energies.
The coupled-channel theory is a natural way of accounting for nonelastic channels, in particular thosearising from collective excitations. Proper treatment of such excitations is often essential to the accu-rate description of reaction experimental data. Optical potentials (OP) needed for coupled-channelscalculations are normally obtained through proper parametrization and parameter fitting in order toreproduce experimental data sets for a specific nucleus. Such phenomenological OP’s might achievevery good agreement with experimental data, as they were specifically adjusted to do so, but do notlend themselves to extrapolation to other nuclei, unless there are explicit measurements that allow toreadjust OP individually. Therefore, the predictive power of the coupled-channels method is ham-pered by the lack of reliable OP for the nuclei with no or scarce experimental data. On the other hand,there are quite reliable spherical OP’s applicable to spherical or slightly-deformed nuclei. Therefore,developing a method capable of employing such well-tested global OP’s to the particular case ofdeformed nuclei would be desirable.Inspired by the recent work by Dietrich et al., substantiating the validity of the adiabatic assump-tion in coupled-channel calculations, we explore the possibility of generalizing (deforming) a globalspherical optical model potential to make it usable in coupled-channel calculations on statically de-formed nuclei. a e-mail: [email protected] b e-mail: [email protected] a r X i v : . [ nu c l - t h ] F e b PJ Web of Conferences
Optical potentials (OP) have been widely used to describe nuclear reaction data by implicitlyaccounting for the e ff ects of excitation of internal degrees of freedom and other nonelastic processes.An OP is called global when this fitting process is consistently done for a variety of nuclides.There are global spherical OP’s that have been fit to nuclei below and above the region of stat-ically deformed rare-earth nuclei, but these potentials have been viewed as inappropriate for use incoupled-channels calculations, since they do not account for the loss of flux through the explicitlyincluded inelastic channels. On the other hand, a recent paper [1] shows that scattering from rareearth and actinide nuclei is very near the adiabatic (frozen nucleus) limit, which suggests that the lossof flux to rotational excitations might be unimportant. In this paper we test this idea by performingcoupled channel calculations with a global spherical optical potential by deforming the nuclear radiibut making no further adjustments. We note an alternative approach (Kuneida et al. [2]), which hasattempted to unify scattering from spherical and deformed nuclei by considering all nuclei as staticallydeformed, regardless of their actual deformation.This work corresponds to a preliminary attempt to extend the approach initially presented inRef. [3], focusing on angular distributions for the cases of neutron scattered by Gd, Ho, and W nuclei. Due to the high moment of inertia and consequent low excitation energies of the ground-state bandmembers of the statically deformed nuclei in the rare-earth region, the deformed nuclear configurationmay be regarded as “frozen” during the scattering. This means that all the internal degrees of freedomnot associated with the strong deformation may assumed to be accounted for by a spherical opticalpotential that describes well the nuclei in the neighboring region, in an adiabatic approach. Therefore,the only channels that need to be treated explicitly (e. g., through couple-channel methods) are theones arising from the static deformation.The spherical OP that was deformed in our coupled-channel calculations was the global Koning-Delaroche (KD) [4], unmodified except for a small change in the radius parameters to ensure volumeconservation when the nucleus is deformed. Since the KD potential describes scattering from nucleiboth above and below the deformed rare earth region very well, we make the assumption that theimaginary potential adequately describes the internal nuclear excitations in the rare earths also. Thispicture is consistent with the adiabatic approximation. The coupled channel calculations account forthe external (rotational) excitations of the target. These assumptions are tested in the calculationsshown in this paper.The process of deforming a spherical OP to explicitly consider collective excitations within thecouple-channel framework is done in the standard way of replacing the radius parameter R in eachWoods-Saxon form factor by the angle dependent expression: R ( θ ) = R + (cid:88) λ β λ Y λ ( θ ) (1)where R is the undeformed radius of the nucleus, and β λ and Y λ ( θ ) are the deformation parameterand spherical harmonic for the multipole λ , as seen in Ref. [5], for example. The deformed formfactor obtained from Eq. 1 is then expanded in Legendre polynomials numerically.We use in our calculations the E mpire reaction code [6, 7], in which the direct reaction part iscalculated by the code E cis [8, 9]. In order to test our model we perform coupled-channel calcula-tions, coupling the ground state rotational band, for neutron-incident reactions on selected rare-earthnuclei, namely , Sm,
Eu, , , , , Gd,
Tb, , , Dy,
Ho, , , , Er,
Tm, , , , , Yb, , Lu, , , , Hf,
Ta, and , , , W. All those nuclides have at least
NR*13 C r o ss S e c t i on ( ba r n s ) -1 -2 Spherical modelCoupled channelsEXFOR
Figure 1.
Total cross sections for neutron-induced reaction on
Ho. The black curve corresponds to coupled-channel calculations within our model, while the green curve indicates, for comparison purposes, the result froma spherical model calculation. Experimental data taken from EXFOR [11].
90 neutrons, indicating static deformation, therefore making them suitable candidates for interpolationthrough the adiabatic limit. We then compared, as an initial test, the obtained coupled-channel resultsfor total cross sections with plain spherical calculations with the undeformed KD optical potential.In this initial step, only quadrupole deformations were considered, having their values taken from thecompilation of experimental values from Raman et al. [10]. The overall result is a dramatic improve-ment in the agreement with experimental data, in particular in the lower neutron-incident energies.Fig. 1 clearly illustrates the very good description of observed total cross section in the case of
Ho,obtained through our coupled-channel model.In carrying out the calculations, it is important to couple a su ffi cient number of rotational states toachieve convergence, and we have carried out tests to ensure this. Such analysis is shown in Ref. [12],where it is also demonstrated that this convergence can be energy-dependent. After the initial success in describing direct-reaction quantities, such as total cross sections, we an-alyzed the model predictions for observables that depend also on the compound-nucleus decay. Themodels adopted to describe the emissions from the compound nucleus were basically default optionsin E mpire code, which means standard Hauser-Feshbach model with properly parametrized EnhancedGeneralized Superfluid Model (EGSM) level densities [13], modified Lorentzian (version 1) γ -raystrength functions [14–16], width fluctuation correction up to 3 MeV in terms of the HRTW approach[17, 18], and with transmission coe ffi cients for the inelastic outgoing channels also calculated withincoupled-channel approach. Pre-equilibrium was calculated within the exciton model [19], as based onthe solution of the master equation [20] in the form proposed by Cline [21] and Ribansky [22] (usingP cross code [6, 7]) with mean free path multiplier set to 1.5.In Fig. 2, as an example, we compare with experimental data the angle-integrated elastic crosssections for incident neutrons on Ho and
Gd obtained by our coupled-channel calculations. Eventhough there are not as many data available as in the case of total cross sections (Fig. 1), the low-energy
PJ Web of Conferences
Incident Energy (MeV) C r o ss S ec t i on ( b a r n s ) -2 -1 Spherical calculationCoupled channelsEXFOR
Incident Energy (MeV) C r o ss S ec t i on ( b a r n s ) -2 -1 Spherical calculationCoupled channelsEXFOR
Figure 2.
Angle-integrated elastic cross sections for the case of
Ho (left panel) and
Gd (right panel) targets.Black curves correspond to predictions by our coupled-channel model while green curves were obtained byspherical model calculations. Experimental data taken from EXFOR [11].
Incident Energy (MeV) C r o ss S ec t i on ( b a r n s ) -1 -1 Spherical modelCoupled channelsEXFOR
Figure 3.
Angle-integrated inelastic cross sections for
Ho. Black curves correspond to predictions by ourcoupled-channel model while green curves were obtained by spherical model calculations. Experimental datataken from EXFOR [11]. point ( ∼ Ho (Fig. 2, left panel) and the few high energy points ( (cid:38)
Gd(Fig. 2, right panel) indicate again a very good agreement between our couple-channel model and theexperimental data (in contrast with the spherical-model calculations).Fig. 3 shows the total inelastic cross section in the case of neutrons scattered by a
Ho target.Again, our coupled-channel model describes well the observed experimental data.We also obtain a very good agreement with experimental data for inelastic cross sections forindividual excited states. As an example, we show in Fig. 4 the predictions of our model for theangle-integrated inelastic cross section of the first inelastic channel of the target
W, which is a 2 + state at excitation energy of 111.2 keV, as a function of the neutron incident energy. We observe againthat we are able to achieve a very good description of measured data within our couple-channel model. NR*13 C r o ss S e c t i on ( ba r n s ) -1 Spherical modelCoupled channelsEXFOR
Figure 4.
Angle-integrated inelastic cross sections for the 2 + state (excitation energy of 111.2 keV) of thetarget W, excited on a neutron-induced reaction. The black curve corresponds to coupled-channel calculationswithin our model, while the green curve indicates, for comparison purposes, the result from a spherical modelcalculation, using Hauser-Feshbach model to describe the excitation. Experimental data taken from EXFOR [11].
A more careful analysis of di ff erential cross-section experimental data proved necessary due to thelarge amount of angular distribution data available in the literature, and also because some measure-ments do not contain pure elastic isotopic data. It is quite common for experiments measuring elasticangular distributions for rare-earth nuclei to be unable to separate inelastic contributions due to thelow-lying excitation energies of their rotational states. In such cases, measured data correspond actu-ally to “quasi-elastic” angular distributions, and the calculated elastic and inelastic di ff erential crosssections have to added up together accordingly for appropriate comparison. In addition, some experi-ments were done using the natural form of the element, rather than the isotope-specific one.For these reasons, application of the coupled-channel model for angular distributions was focusedon three elements only: Gadolinium, Holmium, and Tungsten. Those three elements were chosenbecause the lighter and heavier ones are close to the border of the rare-earth region, while the other isroughly in the middle. When an originally spherical configuration assumes a deformed shape, defined by quadrupole andhexadecupole deformation parameters β and β , respectively, the volume and densities are not con-served. In Ref. [23], a method to ensure volume conservation was proposed, corresponding to apply-ing a correction to the reduced radius R , of the form: R (cid:48) = R − (cid:88) λ β λ / π , (2)where terms of the order of β λ and higher have been discarded. Ref. [3] tested the e ff ects of such cor-rection, showing that it is not negligible and seems to bring the integral and di ff erential cross-section PJ Web of Conferences calculations to a slightly better agreement with the experimental data. Therefore, in the followingcalculations of angular distributions, we decided to implement the radial corrections calculated fromEq. 2, as it should correspond to a more realistic modeling of the deformed nuclei.
In this work we present preliminary results for quasi-elastic di ff erential cross sections for the Hotarget. In Ref. [12] one can also find preliminary results within the same coupled-channel model ofangular distributions for
Gd and
W.The ground state of
Ho has spin and parity corresponding to 7 / − . As an odd nucleus, itsrotational band does not follow the standard 0 + , 2 + , 4 + , etc., scheme. For coupling purposes, it wasconsidered to belong to the ground state rotational band the successive negative-parity states with adi ff erence of spin equal to 1 relative to the ground state, i. e., 7 / − , 9 / − , 11 / − , 13 / − , etc. Couple-channel calculations were performed coupling up to the 23 / − state. The values adopted for thedeformation parameters were β = . β = − .
020 [24].As an example, Fig. 5 presents the predictions of our model when attempting to describe the elasticangular distribution data for
Ho, at the neutron-incident energy of 11 MeV, as measured by Ferrer etal. [25]. Actually, a careful analysis of Ref. [25] indicates that in that experiment it was not possible toseparate the elastic channel from the inelastic ones. Therefore, the data points in Fig. 5 should containinelastic contributions. It is seen Fig. 5 that the predictions of our couple-channel model for the elasticangular distribution (green curve) lies consistently below the experimental data. However, when thecontribution from the first inelastic state, which is a 9 / − state (excitation energy of 94.7 keV) is added(blue curve), the couple-channel prediction approaches the observed cross sections. When the secondinelastic state (11 / − state lying at 209.8 keV) is further added (black curve), we achieve a very gooddescription of the observed quasi-elastic angular distribution. For comparison purposes we plot on thesame figure the result obtained from spherical-model calculations, as the dashed red curve. In this work, we demonstrated that we can use the spherical Koning-Delaroche optical potential incoupled channels calculations by simply deforming it and making no further modification. We foundthat we achieved encouraging results in the description of neutron-induced reactions on the rare-earthsdespite the fact that this potential was not designed to describe reactions on deformed nuclei. We stud-ied the e ff ect of reducing the radius to ensure volume conservation when deforming an the originalspherical configuration. This correction was found to produce small but significant e ff ects which im-proved the agreement with experimental data for the cases we tested. With our approach, we describedexperimental data not only for optical-model observables (such as total cross sections, elastic and in-elastic angular distributions), but also for those obtained through compound-nucleus formation (suchas total elastic and inelastic, capture cross sections). Our results are consistent with the insight thatthe scattering is very close to the adiabatic limit as shown in Ref. [1]. Although imperfect, this simplemethod is a consistent and general first step towards an optical potential capable of fully describingthe rare-earth region and fills the need of an optical model potential in this important region. Acknowledgments
The work at Brookhaven National Laboratory was sponsored by the O ffi ce of Nuclear Physics, Of-fice of Science of the U.S. Department of Energy under Contract No. DE-AC02-98CH10886 withBrookhaven Science Associates, LLC. NR*13 d σ / d Ω ( m b / s r) θ C.M. (deg.)0 30 60 90 120 150 180
J.C. Ferrer et al.
Elastic onlyElastic + 9/2 - Elastic + 9/2 - + 11/2 - Spherical model
Ho(n,n+n') E inc = 11.0 MeV Figure 5.
Quasi-elastic angular distribution for the neutron-induced reaction on
Ho. The green curve cor-responds to the results for the elastic channel only, obtained within our couple-channel model. The blue andblack curves contain, in addition to the elastic di ff erential cross section, contributions from the first, and first andsecond inelastic channels, respectively. For comparison purposes, we also plot the result from a spherical modelcalculation, as the dashed-red curve. Experimental data taken from Ref. [11], corresponding to measurementsfrom Ref. [25]. References [1] F.S. Dietrich, I.J. Thompson, T. Kawano, Phys. Rev. C , 044611 (2012)[2] S. Kunieda, S. Chiba, K. Shibata, A. Ichihara, E.S. Sukhovit-ski˜ı, Journal of Nuclear Science and Technology , 838 (2007), [3] G.P.A. Nobre, A. Palumbo, D. Brown, M. Herman, S. Hoblit, F.S. Dietrich, to be published on Nuclear Data Sheets (2014), http://arxiv.org/abs/1311.0426 [4] A. Koning, J. Delaroche, Nuclear Physics A , 231 (2003)[5] H. Krappe, Annals of Physics , 142 (1976)[6] M. Herman, R. Capote, B. Carlson, P. Obložinský, M. Sin, A. Trkov, H. Wienke, V. Zerkin,Nucl. Data Sheets , 2655 (2007)[7] M. Herman, R. Capote, M. Sin, A. Trkov, B.V. Carlson, C.M.M. P Obložinský, H. Wienke,S. Hoblit, Y.S. Cho, G.P.A. Nobre et al., Tech. Rep. INDC(NDS)-0642, BNL-101378-2013(2013)[8] J. Raynal, Tech. Rep. SMR-9 /
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