Localization of peripheral reactions and sensitivity to the imaginary potential
LLocalization of peripheral reactions and sensitivity to the imaginary potential
Imane Moumene , , Angela Bonaccorso High Energy and Astrophysics Laboratory Department of Physics, Cadi Ayyad University, Marrakesh, Morocco. Department of Physics, University of Pisa, 56127 Pisa, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy.
Abstract
The aim of the present study is to make for the first time in the literature a systematic and quantitative assessment of theevaluation of the imaginary part of the optical potential calculated within the folding model and its consequences on the localizationof surface reactions. Comparing theoretical and experimental reaction cross sections, for some light projectiles on a Be, it hasrecently been shown that a single-folded (s.f.) (light-) nucleus- Be imaginary optical potential is more accurate than a double-folded (d.f) optical potential. Within the eikonal formalism for the cross sections and phase shifts, the single-folded potential wasobtained using a n- Be phenomenological optical potential and microscopic projectile densities. This paper is a follow-up in whichwe systematically study a series of di ff erent light and medium-mass projectile induced reactions on Be. Our results confirm thatthe (s.f.) cross sections are larger than the (d.f.) cross sections and the e ff ect increases with the projectile mass. Furthermore thestrong absorption radius parameter extracted from the S matrices calculated with the (s.f.) has a stable value r s = Keywords:
Exotic nuclei, optical potentials, folding models.
1. Introduction
In a seminal paper, about forty years ago, De Vries and Peng[1] shown that the energy dependence of reaction cross sectionsfor medium to heavy mass nuclei could be reasonably well re-produced by using the eikonal formalism and a double foldingmodel for the optical potential and the phase shift. The fold-ing model had been discussed just one year before by Satch-ler and Love [2] which however warned that while the doublefolding model was well justified for the real potential, it wasless for the imaginary potential because the latter must be allorder in the nucleon-nucleon (nn) interaction [3]. The foldingmodel was also used with good success by Kox et al. [4] ina systematic study of reaction cross sections in the intermedi-ate energy range. From then on the folding model has beenone of the most used methods to generate optical potentialsand there is a huge literature on the subject, see for exampleRefs.[5, 6, 7, 8, 9, 10, 11, 12] and references therein. Because ofthe incertitude on the method for the imaginary potential, in cal-culations of elastic scattering, transfer, partial fusion and fusionwhile the real potential is often obtained by double folding theprojectile and target densities with an e ff ective nn interaction,the imaginary part is treated phenomenologically by a Woods-Saxon potential or by renormalizing the real folded potential.However since the advent of radioactive beams, measured to-tal reaction cross sections have been often studied by the theGlauber model and the double folding model for the imaginarypotential [14, 15, 17, 18, 16, 13]. Total reaction cross sections are relatively easy to measure and using the folding model onemight hope to obtain information on the density distribution ofthe exotic nucleus projectile. On the other hand there would stillbe some sensitivity to the target density and the nn interaction. Be has been one of the most used targets in reactions withradioactive beams. It is very deformed [2] and it has strongbreakup channels. There exist an almost continuous series ofneutron- Be data as a function of the neutron incident energy.The optical potentials of Ref.[19] were able to reproduce at thesame time all those data, namely the total, elastic and reactioncross sections and all available elastic scattering angular distri-butions. Using such potential it has recently been shown thata single-folded (s.f.) (light)-nucleus- Be imaginary optical po-tential is more accurate than a double-folded (d.f) optical po-tential. Within the eikonal formalism for the cross sections andphase shifts, the single-folded potential was obtained using then- Be phenomenological optical potential [19] and various mi-croscopic projectile densities[20, 21] for light projectiles suchas C, Li and B. This paper is a follow-up in which we sys-tematically study a series of di ff erent light and medium massprojectile induced reactions on Be.We hope to clarify the sensitivity of the results for reac-tion cross sections, S matrices and strong absorption radii tothe method used to obtain the optical potential and the phaseshifts. For all these reactions resumed in Table I, our interest isto assess the interaction of the projectile with the target. In par-ticular we wish to understand whether there is a clear and con-sistent way to determine geometrical parameters that can help Preprint submitted to Elsevier July 13, 2020 a r X i v : . [ nu c l - t h ] J u l etermine the range of impact parameters for which surface re-actions dominate the projectile-target interaction from regionsin which the strong absorption regime applies. For example inthe case of fusion and incomplete fusion on heavy targets theexperimental localization of varius reactions has been of fun-damental importance in disentangling the reaction mechanisms[22].In the following from the calculated S -matrices we obtainthe values of the strong absorption radii and then the value ofthe strong absorption radius parameter. The latter needs to be analmost constant quantity in order to consider reliable a geomet-rical model such as the eikonal model. The eikonal approach[23] is used in this paper to obtain the phase shifts, S matri-ces and reaction cross sections. Most of the reactions discussedhere are calculated in the incident energy range 40-100 A. MeVand just a few at smaller or larger energies. At small energy ourcalculations for the phase-shift are performed by substitutingthe impact parameter with the classical distance of closest ap-proach [24, 1]
2. Reminder of eikonal formulae
As in Ref.[1] we calculate the eikonal reaction cross sectionaccording to σ R = π (cid:90) ∞ b db (1 − | S PT ( b ) | ) (1)where | S PT ( b ) | = e χ I ( b ) (2)is the probability that the projectile-target (PT) scattering iselastic for a given impact parameter b .The imaginary part of the eikonal phase shift is given by χ I ( b ) = (cid:126) v (cid:90) dz W PT ( b , z ) = (cid:126) v (cid:90) dz (cid:90) d r W nT ( r − r ) ρ P ( r ) (3)where W PT is negative defined as W PT ( r ) = (cid:90) d b W nT ( b − b , z ) (cid:90) dz ρ P ( b , z ) . (4)This quantity is the imaginary part of the single-folded opticalpotential given in terms of a nucleon-target (nT) optical poten-tial W nT ( r ) and the matter density ρ P ( b , z ) of the projectile.In the single-folding method used in this paper, W nT ( r ) is theimaginary part of the n + Be phenomenological nucleon-targetpotential (AB) of Ref. [19]. In the double-folding method, W PT is obtained from Hartree-Fock microscopic densities ρ P , T ( r ) forthe projectile and target respectively and an energy-dependentnucleon-nucleon (nn) cross section σ nn , i.e., W PT ( r ) = − (cid:126) v σ nn (cid:90) d b ρ T ( b − b , z ) (cid:90) dz ρ P ( b , z ) . (5) Also W nT ( r ) = − (cid:126) v σ nn ρ T ( r ) (6)is a single-folded zero-range n -target imaginary potential and v is the nucleon-target velocity of relative motion. The W nT potential of Eq.(6) has the same range and profile as the targetdensity because σ nn is a simple scaling factor. On the otherhand the phenomenological potential of Eq.(4) has a range andin particular a profile which represents the localization of thevarious n-target reactions (c.f. figures in Ref.[20, 21]). Withthe potential of Eq.(5), the phase shift becomes: χ I ( b ) = − σ nn (cid:90) d b (cid:90) dz ρ T ( b − b , z ) (cid:90) dz ρ P ( b , z ) . (7)At low energy (E inc < b with the classical dis-tance of closest approach d = b + (cid:112) b + a c where a c is theCoulomb length parameter.A finite-range potential can also be defined as: W PTI ( r ) = − (cid:126) v (cid:90) d r d r ρ P ( r ) ρ T ( r ) v nn ( r + r − r ) (8)where v nn can be a zero-range or a finite-range nucleon-nucleoninteraction such as Gogny [26] or M3Y [27] or a phenomeno-logical form. However, the imaginary parts obtained in this wayneed to be renormalized most of the time. For this reason wedo not use such a method here.The previous equations can be generalized in a obvious wayin order to distinguish between the proton and neutron densitiesand the proton-neutron and proton-proton cross sections, using: ρ P = ρ nP + ρ pP , and W nT ( r ) = − (cid:126) v ( σ np ρ pT ( r ) + σ pp ρ nT ( r )).This is the formalism followed in the present work.The strong-absorption radius R s [25, 28] is obtained fromthe S-matrices calculated according to Eq.(2) as the radius where | S PT ( R s ) | = , and a ”strong absorption radius parameter” r s can be extracted from R s = r s ( E inc )( A / P + A / T ) . (9)We will in the following refer to the previous formulae, thisis why, although they are well known, we resume them here. Inthis way we have a set of quantities which define the geometryof the reactions in a transparent way and allow comparisons be-tween di ff erent projectile-target combinations and incident en-ergies.In Ref.[20, 21] we compared results obtained using di ff erentmicroscopic densities. We found that the Hartree-Fock densi-ties described better the reaction cross section values. For thisreason we will present here only results obtained using HF den-sities calculated with the code HFBTHO [29]. We use for σ pp , np the parametrization of Ref. [30]. In this paper we compare re-sults obtained with the potential defined in Eq.(4) and Eq.(5).2
10 20 30 40 50 60Mass Number800100012001400160018002000 [ m b ] R s s.f.d.f. Figure 1: (Color online) Reaction cross sections from Eq. (1). These in-clude the double-folded (d.f.), red triangles and the single-folded (s.f.), bluecrosses results. Mass number refers to the A P nuclei of Table I.
10 20 30 40 50 Mass Number11.11.21.31.41.5 [f m ] s r s.f.d.f. HF r Figure 2: (Color online) Values of the r s parameters as a function of theprojectile-mass. They correspond to the strong absorption radii of Eq.(9)for the S matrices. Red triangles from the the double-folded potentials, andblue crosses from the single-folded potentials.
3. Results
Fig.1 and Table 1 show the calculated reaction cross sec-tions as a function of the projectile mass. In Fig. 1 the results ofEqs.(1,4), blue crosses, obtained by single-folding of the (AB)-potential [19] with the HF projectile density are almost alwayslarger than the double - folding cross sections from Eqs.(1,5),red triangles, again in agreement with what found in Ref.[21].Notice that the systematics presented here is di ff erent than inRef.[21]. There we studied the energy dependence of C + Bescattering while here we are studying various systems A p + Bein the range of incident energy 40-100AMeV. In Table I we givethe reaction cross sections on Be (fourth column) of the nucleiindicated in the first column, incident energies on the secondcolumn and strong absorption radius parameter (fifth column)from Eq.(9). Results are given for the single folding and doublefolding methods as indicated in the third column.The sixth andseven columns contain the HF radii of the projectiles and HFradius parameter for the sake of completeness.From the obtained S matrices we extract the strong absorp-tion radii according to | S PT ( R s ) | = . Then using Eq.(9) we obtain the values of the r s parameter given in the fifth columnof Table I. They are also shown in Fig. 2. Red triangles areobtained from the calculations with the double-folded poten-tials while blue crosses are from the single-folded potentials asa function of the projectile-mass. It is very interesting to notethat almost all (s.f.) results are concentrated in the range r s = S matrix[13, 28] of the type | S PT | = exp ( − ln e ( R s − b ) / a ) . (10)The results from (d.f.) potentials are more scattered. This isdue in part to the sensitivity to the incident energy which istreated in a more approximate way, partially to a less accuratelocalisation of the n-target scattering. The two red trianglescorresponding to r s > Li and C scattering around 30A.MeV, which is the small-est energy considered in this paper. The e ff ect is less dramaticwhen using the (AB) potential and the single folding methodfor the target. Thus we note that in almost all cases the use of3he phenomenological n-target potential produces larger crosssections and a localization of the projectile-target scattering atlarger impact parameters than the double folding, as seen inFig.2. This is in agreement with what found in Ref.[21] wherewe studied the C + Be scattering. It is due to the fact that the(AB) potential contains correctly all surface e ff ects and energydependence of the n + Be scattering. Note that the Be target isitself a weakly bound, strongly deformed nucleus. If the reac-tion cross sections are larger with the (s.f) model it means that | S PT ( b ) | is localized at larger impact parameters and thus willbe the elastic scattering.In Fig.2 and Table I we provide also for comparison andcompleteness the values of the radius-parameters of the HF den-sities, green points, for the given projectile nuclei, defined as R HF = r HF A / . They are scattered around with respect to theprojectile mass and then we cannot extract from them a uniformr s ( HF ) radius parameter. This is due to the fact that the HF ra-dius is obtained from the total density, sum of the proton andneutron densities which are very di ff erent for the nuclei studiedhere.
4. Conclusions
In this paper we have made for the first time in the literaturea systematic comparison of calculated reaction cross sectionson a Be target. Calculations are made via a (s.f) vs. (d.f.)optical potential considering the strong absorption radius as asignificative parameter to extract. From the results presented,in particular Fig. 2 and Table I of this work it appears evidentthat the (d.f.) method, used to calculate optical potentials, phaseshifts and S matrices, localises the overlap and elastic scatter-ing between exotic projectiles and the Be target at smaller dis-tances than the (s.f.) method and with no consistent distinctionbetween the surface region and the region of strong absorption.This produces smaller reaction cross sections on a Be target.Also it appears that the radius parameter of the (HF) densitiesshows strong variations as a function of projectile mass andthe di ff erence in the number of neutrons and protons. In ouropinion, this could make it a doubtful quantity for systematicreaction studies. Furthermore, because of the not consistent lo-calization of surface reactions, results might suggest unrealisticand unphysical correlations in the analysis of experimental data.On the other hand the (s.f.) model has provided very sta-ble values of the strong absorption radius parameter r s = ff ects. Smallvariations between the two values are due to the energy depen-dence of the cross sections. Finally we suggest that the value r s = Acknowledgements
This work was done while one of us (I.M.) was visiting theINFN, Sezione di Pisa and Department of Physics, Universityof Pisa. I.M. acknowledges the full financial support from theUniversity of Pisa under scheme ERASMUS + KA107 Interna-tional Mobility. She is also grateful to Profs. M. Gaidarov, M.V. Stoitsov and colleagues for allowing her to run and use re-sults from the code HFBTHO[29].
References [1] R.M. De Vries, J.C. Peng, Phys. Rev. C 22 (1980) 1055.[2] G.R. Satchler and W.G. Love, Phys. Rep. 55 183 (1979).[3] H. Feshbach, Ann. Physics 5 (1958) 357.[4] S. Kox et al., Phys. Rev. C35,1678 (1987)[5] Chamon, L.C., Pereira, D., Hussein, M.S., Cˆandido Ribeiro, M.A.,Galetti, D., Phys. Rev. Lett. 79, 5218 (1997).[6] L. F. Canto et al., Phys. Rep. 596, 1 (2015).[7] M. Dupuis, E. Bauge, S. Hilaire, F. Lechaftois, S. P´eru, N. Pillet and C.Robin Eur. Phys. J. A (2015) 51: 168[8] V. Lapoux and N. Alamanos, Eur. Phys. J. A (2015)51:91.[9] Dao T. Khoa, Nguyen Hoang Phuc, Doan Thi Loan,and Bui Minh Loc,Phys. Rev. C94, 034612 (2016)[10] B. Mukeru, M. L. Lekala, J. Lubian and L. Tomio, Nucl. Phys. A 996,121700 (2020).[11] N.Keeley, R.Raabe, N.Alamanos, J.L.Sidac, Progr. Part. and Nucl. Phys.59, 579 (2007). Prog.Part.Nucl.Phys.59:579-630,2007[12] J. Lei and A. M. Moro, Phys. Rev. Lett. 122 042503 (2019).[13] Angela Bonaccorso,
Direct Reaction Theory for Exotic Nuclei: An intro-duction via semiclassical methods ,Progress in Particle and Nuclear Physics, 101(2018) 1-54, and referencestherein.[14] Isao Tanihata, Herve Savajols, Rituparna Kanungo, Prog. Part. Nucl.Phys. 68 (2013) 215.[15] A. Ozawa et al., Nucl. Phys. A 691, 599 (2001). A. Ozawa, AIP Conf.Proc. 865, 57 (2006); http: // dx.doi.org / / Nuclear Reactions with Heavy Ions , Springer-Verlag, Berlin,Heidelberg, New York, 1980, Sec. 3.3.[26] D. Gogny, Proc. Int conf. Nucl. Phys, Munich 1973, eds J. de Boer andH. J. Mang, p. 48.[27] N. Anantaraman, H. Toki, and G.F. Bertsch, Nucl. Phys. A 398, 269(1983).[28] A. Bonaccorso, D. M. Brink and L. Lo Monaco, J. Phys. G 13 1407(1987) .
29] Stoitsov, M. V.; Schunck, N. ; Kortelainen, M. ; Michel, N ; Nam, H. ;Olsen, E. ; Sarich, J. ; Wild, S.; Comp. Phys. Comm. 184, 1592, 2013,DOI: 10.1016 / j.cpc.2013.01.013[30] C. A. Bertulani, and C. De Conti, Phys. Rev. C 81, 064603 (2010).[31] C. Mahaux and R. Sartor,Advances in Nuclear Physics, editedby J. W.Negele and Erich Vogt, Vol. 20 (Kluwer AcademicPublishers, New York,1991), p. 1.[32] A. J. Koning , J. P. Delaroche, Nucl. Phys. A 713 231 (2003), able 1: Reaction cross sections on Be (fourth column) of the nuclei shownin the first column, incident energies on the second column and strong absorp-tion radius parameter (fifth column) from Eq.(9). As for column fourth resultsare given for the single folding and double folding methods. For the sake ofcompleteness the sixth and seven columns contain the HF radii and HF radiusparameter.
Projectile E(MeV) σ (mb) r s (fm) R HF (fm) r HF (fm) Li 20 s.f.
892 1.34 d.f. s.f.
922 1.37 d.f.
998 1.4560 s.f.
855 1.31 d.f.
878 1.34 2.59 1.51 Be 65.2 s.f.
830 1.30 d.f.
903 1.33 2.24 1.23 C 63.8 s.f.
989 1.34 d.f.
978 1.32 2.57 1.28 Be 80 s.f.
943 1.29 d.f.
905 1.25 2.35 1.12 C 20 s.f. d.f. s.f. d.f. s.f. d.f. s.f.
949 1.29 d.f.
936 1.27120 s.f.
862 1.25 d.f.
849 1.20 O 28.5 s.f. d.f. O 53 s.f. d.f. C 103 s.f. d.f.
935 1.18 2.59 1.07 C 75 s.f. d.f. Si 85.3 s.f. d.f. S 80.7 s.f. d.f. Ar 65.1 s.f. d.f. S 62.8 s.f. d.f. Cl 66.4 s.f. d.f. Ar 70 s.f. d.f. Si 73.4 s.f. d.f. Ca 70 s.f. d.f. Ar 70 s.f. d.f. Ni 73 s.f. d.f.d.f.