Excitation energy spectra and absolute cross-section angular distributions were measured for the 13C(18O,16O)15C two-neutron transfer reaction at 84 MeV incident energy. Exact finite-range coupled reaction channel calculations are used to analyse the data considering both the direct two-neutron transfer and the two-step sequential mechanism. For the direct calculations, two approaches are discussed: The extreme cluster and the newly introduced microscopic cluster. The latter makes use of spectroscopic amplitudes in the centre-of-mass reference frame, derived from shell-model calculations. The results describe well the experimental cross sections

Agodi, C.

Cappuzzello, F.

Carbone, D.

Cavallaro, M.

Ferreira, J. L.

Foti, A.

Gargano, A.

Lenzi, S. M.

Linares, R.

Lubian, J.

Santagati, G.

Acta Physica Polonica B

English

Padua Research Archive

Microscopic cluster model for the description of (18O,16O) two-neutron transfer reactions

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Vol. 49 (2018) ACTA PHYSICA POLONICA B No 3MICROSCOPIC CLUSTER MODELFOR THE DESCRIPTION OF (18O,16O)TWO-NEUTRON TRANSFER REACTIONS∗D. Carbonea, J.L. Ferreirab, F. Cappuzzelloa,c, J. LubianbC. Agodia, M. Cavallaroa, A. Fotic,d, A. Garganoe, S.M. Lenzif,gR. Linaresb, G. SantagatiaaINFN — Laboratori Nazionali del Sud, Catania, ItalybInstituto de Fisica, Universidade Federal Fluminense, Niteroi, RJ, BrazilcDipartimento di Fisica e Astronomia, Università degli Studi di Catania, ItalydINFN — Sezione di Catania, Catania, ItalyeINFN — Sezione di Napoli, Napoli, ItalyfDipartimento di Fisica e Astronomia, Università di Padova, Padova, ItalygINFN — Sezione di Padova, Padova, Italy(Received December 28, 2017)Excitation energy spectra and absolute cross-section angular distribu-tions were measured for the 13C(18O,16O)15C two-neutron transfer reac-tion at 84 MeV incident energy. Exact finite-range coupled reaction chan-nel calculations are used to analyse the data considering both the directtwo-neutron transfer and the two-step sequential mechanism. For the di-rect calculations, two approaches are discussed: the extreme cluster andthe newly introduced microscopic cluster. The latter makes use of spec-troscopic amplitudes in the centre-of-mass reference frame, derived fromshell-model calculations. The results describe well the experimental crosssections.DOI:10.5506/APhysPolB.49.3731. IntroductionDirect two-nucleon transfer reactions play an important role in the studyof specific features of the atomic nucleus and indeed they were extensivelyexplored [1–5] to study, for example, pairing correlations. Among these,heavy-ion direct transfer reactions at energies close to the Coulomb bar-rier are useful tools to obtain precise spectroscopic information. In thiscontext, a wide range of systems was explored by heavy-ion induced one-and two-neutron transfer reactions at INFN-LNS (Italy), specifically using∗ Presented at the XXXV Mazurian Lakes Conference on Physics, Piaski, Poland,September 3–9, 2017.(373)374 D. Carbone et al.the (18O,17O) and (18O,16O) reactions. The MAGNEX spectrometer [6]was used to detect the ejectiles. Its large acceptance and high resolutionallowed to obtain high-quality energy spectra up to the region above thetwo-neutron separation energy in the residual nucleus [7–9]. New phenom-ena were observed, e.g. the dominance of the direct one-step transfer of thetwo neutrons [10] and the presence of the first experimental signature of theGiant Pairing Vibration [11, 12] at high excitation energy in the 14C and 15Cspectra. The analysis of the broad structures in the 15C spectrum at highexcitation energy was presented in Ref. [13] and the neutron decay of thesestructures was investigated in Ref. [14], exploiting the use of the EDEN ar-ray coupled to the spectrometer [15, 16]. Moreover, it was demonstrated inseveral works that the (18O,16O) two-neutron transfer reaction can be usedfor quantitative spectroscopic studies of pair configurations in nuclear states[19–23]. A good description of the reaction mechanisms involved in multi-nucleon transfer reactions is important also to go deep inside the dynamicsof more complex processes such as double charge-exchange reactions, whichare of interest for obtaining information about the nuclear matrix elementsinvolved in neutrino-less double beta decay [24, 25].A complete theoretical treatment of the transfer process should containa description of three terms: the one-step channel with the inclusion of allpossible inelastic excitations of the target, projectile, ejectile or residual nu-cleus; the sequential channel, with the inclusion of intermediate partitionsand the non-orthogonal term derived from the limited model space used inactual calculations. As long as heavy ions are concerned, it is necessaryto include explicitly the inelastic excitations of the involved nuclei usingthe coupled-channels approach [26–28]. A complete two-step Coupled Re-action Channel (CRC) approach was employed to study 8He(p, t)6He and6He(p, t)4He two-neutron transfer reactions in Refs. [17, 18].In Ref. [19], the experimental absolute cross sections of the one- and two-neutron transfer reactions induced by an 18O beam on a 12C target were re-produced without any scaling factor by means of Exact Finite Range (EFR)CRC calculations. Two approaches were used: the extreme cluster modeland the independent coordinate scheme. The description of these approachescan be found in Refs. [29] and [30], respectively. The same framework wasrecently applied to describe the absolute cross-section angular distributionsof the states populated by the 13C(18O,16O) reaction [21]. A new approachconsisting in a fully microscopic cluster calculation is introduced, performedby using the two-neutron spectroscopic amplitudes calculated in the shell-model framework. This new approach can be more extensively used withrespect to the extreme cluster model to evaluate the presence of clustercomponents in the wave functions involved. In this paper, the role of thedifferent partial waves in the microscopic calculation is discussed for someMicroscopic Cluster Model for the Description of (18O,16O) Two-neutron . . . 375transitions in 15C nucleus. In particular, the case of the state at 0.74 MeVis analysed, which displays some peculiar features. For this state, also two-step CRC calculations are performed to investigate the role of single-particlecomponents in the wave function.2. Experimental setup and resultsThe experiment was performed at the INFN-LNS laboratory using an18O6+ beam at 84 MeV incident energy on a thin 13C target. The samemeasurement was also done on 12C target to estimate the background com-ing from the 12C impurities in the 13C target. The MAGNEX spectrometerwas used to detect the ejectiles [6]. The particle identification and the datareduction technique are the same as described in detail in Refs. [12, 31]. Anexample of the obtained energy spectra for the 15C nucleus is shown in Fig. 1,in which the 14C background spectrum coming from the 12C impurities in the13C target is superimposed, after normalization. The strongest bound andresonant states appearing in the spectrum of the 15C nucleus are almost thesame that were strongly populated in the (t, p) reactions on 13C [32]. In par-ticular, below the one-neutron separation energy (Sn = 1.218MeV), only two15C bound states are identified, i.e. the ground (Jpi = 1/2+) and the 5/2+state at Ex = 0.74 MeV, that are characterized by a dominant single-particleconfiguration [33]. In the region between Sn and the two-neutron separa-tion energy (S2n = 9.39 MeV), narrow resonances at Ex = 3.103 (1/2−),0 2 4 6 8Ex (MeV)050100150200countsg.s.0.743.1034.224.665.866.847.3513C(18O,16O)15C θlab = 11.5°S n = 1.218 MeVFig. 1. (Colour on-line) Excitation energy spectrum of the 13C(18O,16O)15C reac-tion for 10◦ < θlab < 11◦. The hatched (red) area corresponds to the backgroundthat comes from 12C impurities in the target. The value of the one-neutron emissionthreshold is also indicated with the dashed line (Sn = 1.218 MeV).376 D. Carbone et al.4.22 (5/2−), 4.66 (3/2−), 6.84 (9/2−, 7/2−), 7.35 (9/2−, 7/2−) MeV arepopulated. All of these states are indicated to consist mainly of 2p–1h con-figurations (with respect to the 14Cgs vacuum state) [32]. Resonances with asingle-particle configuration of a 14Cgs+1n [34] are very weakly populated inthe present reaction. Examples of the obtained absolute cross-section angu-lar distributions for the ground state and the first excited state at 0.74 MeVare shown in Figs. 2 and 3.0 10 20 30 40 50 60θCM (deg)10-410-310-210-1100101102dσ/dΩ (mb/sr)extr. clustermicr. 1s+1p13C(18O,16O)15Cg.s.Fig. 2. (Colour on-line) Experimental cross-section angular distributions forthe ground state in 15C. Theoretical calculations: extreme cluster calculations(dashed/red line) and microscopic cluster model calculations including 1s + 1pwaves (solid/blue line).0 10 20 30 40 50 60θCM (deg)10-510-410-310-210-1100101102dσ/dΩ (mb/sr)micr. 1smicr. 1s+1pmicr. 1s+1p+1dmicr. 1s+1p+1d+2smicr. 1s+1p+1d+2s+1fmicr. 1s+1p+1d+2s+1f+2p13C(18O,16O)15C0.74Fig. 3. (Colour-online) Experimental cross-section angular distributions for thestate at 0.74 MeV in 15C. Microscopic cluster model calculations including: 1s(solid/violet line); 1s+1p waves (dashed/blue line); 1s+1p+1d (dash-dotted/redline); 1s+ 1p+ 1d+ 2s (dash-double-dotted/magenta line); 1s+ 1p+ 1d+ 2s+ 1f(double-dashed-dotted/orange line); 1s+1p+1d+2s+1f+2p (dotted/black line).Microscopic Cluster Model for the Description of (18O,16O) Two-neutron . . . 3773. Theoretical analysisExact finite-range coupled reaction channel (CRC) calculations were per-formed to describe the cross section, using the FRESCO code [35], with thesame ingredients as in Ref. [19]. Firstly, a direct, simultaneous transfer ofthe two particles was considered. The two-particle wave functions for theone-step two-neutron transfer mechanism were obtained considering two dif-ferent schemes: (i) the extreme cluster model, in which the relative motionbetween the two transferred neutrons is frozen and separated from the core,and the two neutrons are coupled antiparallel to an intrinsic angular mo-mentum S = 0, with spectroscopic amplitudes for both target and projectileset to 1.0 and (ii) the microscopic cluster model, which was introduced inRef. [21]. The 12C is assumed as a closed core and the model space includesthe 1p1/2, 1d5/2, 2s1/2 orbits. The effective phenomenological ZBM interac-tion [36] was used. When a cluster model is used, the parameters relevantfor the definition of the wave function are the principal quantum numberN and the orbital angular momentum L relative to the core. These pa-rameters are obtained from the conservation of the total number of quantain the transformation of the wave function of two independent neutronsin orbits (ni, li) (i = 1, 2) into a cluster with internal state (n, l) [31]:2(n1− 1)+ l1+2(n2− 1)+ l2 = 2(N − 1)+L+2(n− 1)+ l. In the extremecluster model hypothesis, we consider n = 1 and l = 0 so that the clusteris in a 1s internal state. The spectroscopic amplitudes for both target andprojectile were set to 1.0. The resulting differential cross section using theextreme cluster hypothesis for the ground state is shown in Fig. 2. Usingthis extreme approximation, the calculations give in this case a cross sectionlarger than the experimental one. The main reason for this overestimationmay lie in the approximation that the two neutrons are coupled to the totalspin S = 0 with 100% of probability.A natural way to go beyond the assumptions of the extreme clustermodel is to introduce both parallel and antiparallel couplings for the twoneutrons. Realistic spectroscopic amplitudes are required for all possiblecombinations of single-particle configurations in this enlarged space that canbe derived from shell-model calculations. To achieve this goal, we made useof transformations from individual (j–j coupling) to relative and centre-of-mass coordinates (LS coupling) for the harmonic-oscillator wave functionsof the two-particle system [21], using the Moshinsky brackets [37]. Thisapproach is what we call microscopic cluster calculations. We performedmicroscopic cluster calculations considering that the cluster relative motionstate is represented by n = 1 and l = 0 quantum numbers, i.e. the clusteris in the 1s intrinsic state, and n = 1 and l = 1 (1p intrinsic state). The378 D. Carbone et al.results are shown in Fig. 2 for the ground state. We see that the resultsof (1s + 1p) microscopic cluster calculations are in rather good agreementwith the experimental angular distributions, and this is true also for thetransitions at 3.103, 4.22 and 4.66 MeV.3.1. Analysis of the 0.74 MeV stateThe behaviour of the state at 0.74 MeV is very different from the others.Passing from the microscopic calculation with only the 1s wave to the (1s+1p) waves, the obtained cross sections decrease. This is probably due toa destructive interference between the 1s and the 1p waves. We tried toadd more waves in this case, in particular the 1d (n = 1 and l = 2), 2s(n = 2 and l = 0), 1f (n = 1 and l = 3) and 2p (n = 2 and l = 1). Theresults are shown in Fig. 3. The inclusion of the 1d wave decreased evenmore the calculated cross section, then with adding step by step the otherwaves the results start to increase, up to a saturation for the (1s+1p+1d+2s + 1f) and (1s + 1p + 1d + 2s + 1f + 2p) cases. The only wave whichcontributes a lot to the calculated cross section is the 1f . This correspondsto a configuration of |13Cgs(1/2−)〉 coupled with two neutrons with l = 3,which gives the |15C0.74(5/2+)〉 state. The discrepancy still found betweenthe calculations and the data is probably due to the dominant single-particleconfiguration of this state, built as |15C0.74(5/2+)〉 =|14Cgs(0+)〉⊗(1d5/2)νwith spectroscopic factor close to 1 [33]. For this reason, the cluster modeladopted for these calculations is not the suitable approach for such kind ofstates, which do not present strong cluster components in the wave function.0 10 20 30 40 50 60θCM (deg)10-310-210-1100101dσ/dΩ (mb/sr)micr. 1s+1p+1d+2s+1f+2ptwo-step CRCcoherent sum13C(18O,16O)15C0.74Fig. 4. (Colour on-line) Experimental cross section angular distributions for thestate at 0.74 MeV in 15C. Microscopic cluster model calculations including 1s +1p + 1d + 2s + 1f + 2p (dotted/black line), two-step CRC calculations (solid/redline) and coherent sum between them (dashed/blue line).Microscopic Cluster Model for the Description of (18O,16O) Two-neutron . . . 379Two-step sequential CRC calculations, introducing the 17O + 14C inter-mediate partition, were also performed to improve the results for the tran-sition to this state. The coupling scheme is the same reported in Ref. [21].The obtained results are in better agreement with the data with respect tothe microscopic cluster model calculations, as visible in Fig. 4. This confirmsthe importance of single-particle components for this state. Following thetechnique described in Ref. [19], we performed also the coherent sum of thesequential CRC cross sections and the 1s + 1p + 1d + 1f + 2p microscopiccluster and the result is shown in Fig. 4. The interference between the directand sequential mechanisms provides a little improvement in the descriptionof the experimental data.4. ConclusionsWe reported the spectrum and cross-section angular distributions ob-tained for the two-neutron transfer reaction 13C(18O,16O)15C at 84 MeVincident energy. The experimental cross sections for the population of thestates in 15C are reasonably well-described by one-step CRC calculationswith no need for any “unhappiness” factor. The new microscopic clustermodel allows to describe rather well the experimental cross sections for thetransitions in which cluster components are important. For the 0.74 MeVstate, which contains strong single-particle components, it was necessary toadd the sequential two-step mechanism to describe the cross section.This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovationprogramme (grant agreement No. 714625).REFERENCES[1] S. Mordechai et al., Nucl. Phys. A 301, 463 (1978).[2] P. Guazzoni et al., Phys. Rev. C 83, 044614 (2011).[3] P. Guazzoni et al., Phys. Rev. C 69, 024619 (2004).[4] G. Potel et al., Phys. Rev. Lett. 107, 092501 (2011).[5] G. Potel et al., Phys. Rev. Lett. 105, 172502 (2010).[6] F. Cappuzzello et al., Eur. Phys. J. A 52, 167 (2016).[7] D. Carbone et al., Phys. Rev. C 90, 064621 (2014).[8] D. Nicolosi et al., Acta Phys. Pol. B 44, 657 (2013).[9] D. Carbone et al., Acta Phys. Pol. B 45, 431 (2014).[10] D. Carbone et al., J. Phys.: Conf. Ser. 312, 082016 (2011).380 D. Carbone et al.[11] F. Cappuzzello et al., Nat. Commun. 6, 6743 (2015).[12] D. Carbone, Eur. Phys. J. Plus 130, 143 (2015).[13] F. Cappuzzello et al., Phys. Lett. B 711, 347 (2012).[14] M. Cavallaro et al., Phys. Rev. C 93, 064323 (2016).[15] H. Laurent et al., Nucl. Instrum. Methods Phys. Res. A 326, 517 (1993).[16] M. Cavallaro et al., Nucl. Instrum. Methods Phys. Res. A 700, 65 (2013).[17] N. Keeley et al., Phys. Lett. B 646, 222 (2007).[18] K. Rusek, L. Giot, P. Roussel-Chomaz, Eur. Phys. J. A 32, 159 (2007).[19] M. Cavallaro et al., Phys. Rev. C 88, 054601 (2013).[20] M.J. Ermamatov et al., Phys. Rev. C 94, 024610 (2016).[21] D. Carbone et al., Phys. Rev. C 95, 034603 (2017).[22] M.J. Ermamatov et al., Phys. Rev. C 96, 044603 (2017).[23] B. Paes et al., Phys. Rev. C 96, 044612 (2017).[24] F. Cappuzzello et al., Eur. Phys. J. A 51, 145 (2015).[25] F. Cappuzzello et al., J. Phys.: Conf. Ser. 630, 012018 (2015).[26] P.D. Bond et al., Phys. Rev. C 16, 177 (1977).[27] M.C. Lemaire, K.S. Low, Phys. Rev. C 16, 183 (1977).[28] D. Pereira et al., Phys. Lett. B 710, 426 (2012).[29] G.R. Satchler, Direct Nuclear Reactions, Oxford University Press, 1983.[30] I.J. Thompson, Comput. Phys. Rep. 7, 167 (1988).[31] F. Cappuzzello et al., Nucl. Instrum. Methods Phys. Res. A 763, 314 (2014).[32] S. Truong, H.T. Fortune, Phys. Rev. C 28, 977 (1983).[33] G. Murillo, S. Sen, S.E. Darden, Nucl. Phys. A 579, 125 (1994).[34] J.D. Goss et al., Phys. Rev. C 8, 514 (1973).[35] J.I. Thompson, http://www.fresco.org.uk/[36] A.P. Zuker, B. Buck, J.B. McGrory, Phys. Rev. Lett. 21, 39 (1968).[37] M. Moshinsky, Nucl. Phys. 13, 104 (1959).

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Microscopic cluster model for the description of (18O,16O) two-neutron transfer reactions

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