Systematical study of optical potential strengths in reactions involving strongly,weakly bound and exotic nuclei on 120Sn

M. A. G. Alvarez, J. P. Fernandez-Garcıa, J. L. Leon-GarcıaDepartamento FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain

M. Rodrıguez-GallardoDepartamento FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain and

Instituto Carlos I de Fısica Teorica y Computacional, Universidad de Sevilla, Spain

L. R. Gasques, L. C. Chamon, V. A. B. Zagatto, A. Lepine-Szily, J. R. B. OliveiraUniversidade de Sao Paulo, Instituto de Fisica, Rua do Matao, 1371, 05508-090, Sao Paulo, SP, Brazil

V. ScarduelliInstituto de Fısica, Universidade Federal Fluminense, 24210-340 Niteroi, Rio de Janeiro, Brazil and

Instituto de Fısica da Universidade de Sao Paulo, 05508-090, Sao Paulo, SP, Brazil

B. V. CarlsonDepartamento de Fısica, Instituto Tecnologico de Aeronautica, Sao Jose dos Campos, SP, Brazil

J. CasalDipartimento di Fisica e Astronomia ”G. Galilei” and INFN-Sezione di Padova, Via Marzolo, 8, I-35131, Padova, Italy

A. AraziLaboratorio TANDAR, Comision Nacional de Energıa Atomica,

Av. Gral. Paz 1499, BKNA1650 San Martın, Argentina

D. A. Torres, F. RamirezDepartamento de Fısica, Universidad Nacional de Colombia, Bogota, Colombia

(Dated: October 22, 2021)

We present new experimental angular distributions for the elastic scattering of 6Li + 120Sn at threebombarding energies. We include these data in a wide systematic involving the elastic scatteringof 4,6He, 7Li, 9Be, 10B and 16,18O projectiles on the same target at energies around the respectiveCoulomb barriers. Considering this data set, we report on optical model analyses based on thedouble-folding Sao Paulo Potential. Within this approach, we study the sensitivity of the data fit todifferent models for the nuclear matter densities and to variations in the optical potential strengths.

PACS numbers: 25.70.Bc,24.10.Eq,25.70.Hi

I. INTRODUCTION

Nuclei present cluster structures [1]. Light, strongly orweakly bound, stable or exotic, nuclei such as 6He, 6,7Li,7,8,9Be, 12,13,14C, 16,18O, among others (isotopes and nu-clei), can be considered as results of n, 1,2,3H and 3,4Hecombinations. It has been evidenced by experimental ob-servations on break-up or transfer reactions (e.g. [2–5]).

The 4He possesses a significantly higher binding energyper nucleon than its light neighbors (see Table I), and afirst excited state with very high excitation energy (20.6MeV) that makes it a rather robust and inert nucleus.

Unlike 4He, 6He is an exotic nucleus that decays, bybeta minus emission, in 6Li, with a half-life of 806.7(15)ms [6]. It is a Borromean nucleus, i.e., the two sub-systems, 4He-n and n-n, are not bound. Reactions in-duced by 6He on different targets, at energies around theCoulomb barrier, exhibit a remarkable large cross sectionfor α particles production [3, 4]. It confirms a break-up

picture, which is associated to the weak binding of thehalo neutrons (S2n = 0.98 MeV - Table I) [6], that favoursthe dissociation of the 6He projectile.

7Li is one of the heaviest nuclides formed with verysmall yields during the primordial Big-Bang nucleosyn-thesis. Stable nuclei heavier than 7Li were formed muchlater through light nuclei reacting during stellar evolu-tion or explosions. Despite small amounts of 6Li and 7Libeing produced in stars, they are expected to be burnedvery fast. Additional small amounts of both, 6Li and 7Li,may be generated from cosmic ray spallation on heavieratoms in the interstellar medium, from solar wind andfrom early solar system 7Be and 10Be radioactive decays[7].

Both 6Li and 7Li have an anomalous low nuclear bind-ing energy per nucleon compared to their stable neigh-bors (see Table I). In fact, these lithium isotopes havelower binding energy per nucleon than any other stablenuclide with Z > 3. As a consequence, even being light,

arX

iv:1

912.

0913

0v1

[nu

cl-t

h] 1

9 D

ec 2

019

2

TABLE I: Binding energy per nucleon, proton and neutronseparation energies, possible mode of break-up and corre-sponding Q value for some nuclei. All energies are providedin MeV.

nucleus BE/A S1p S1n cluster Q

4He 7.07 19.81 20.586He 4.88 22.59 1.71 α+ n+ n −0.986Li 5.33 4.43 5.66 α+ d −1.477Li 5.61 9.97 7.25 α+ t −2.479Be 6.46 16.89 1.66 α+ α+ n −1.5710B 6.47 6.59 8.44 6Li + α −4.4611B 6.93 11.23 11.54 7Li + α −8.66

6,7Li are less common in the solar system than 25 of thefirst 32 chemical elements [8]. The 6Li and 7Li nuclei arestable weakly bound isotopes for which strong break-upeffects are expected in collisions with other nuclei. Theseisotopes can be considered as α + d and α + t clusters,with small Q values (see Table I).

Luong et al. [2] showed that break-up of 6Li into itsα+d constituents dominates in reactions with heavy tar-gets. However, break-up triggered by nucleon transfer ishighly probable. As an example, in the case of a 6Libeam focusing on a 120Sn target these processes couldbe:6Li + 120Sn → 121Sn + 4He + p;6Li + 120Sn → 121Sb + 4He + n.These strong break-up mechanisms triggered by nucleontransfer help in explaining the large number of α parti-cles observed in different 6Li reactions [2, 9]. In TableII, we present Q values of possible break-up processestriggered by transfer for systems involving some weaklybound projectiles on a 120Sn target.

TABLE II: Q values of some break-up processes triggered (ornot) by transfer for weakly bound projectiles colliding with a120Sn target.

projectile reaction products Q(MeV)

6Li 121Sn + α+ p 2.4726Li 121Sb + α+ n 2.092

7Li 122Sn + α+ p 4.0367Li 122Sb + α+ n 1.247

9Be 121Sn + α+ α 4.5979Be 120Sn + 8Be + n 4.505

10B 121Sn + 2α+ p -1.98910B 121Sb + 2α+ n -2.368

Unlike 6Li, 7Li presents a first excited state with rel-atively low excitation energy (E∗ = 0.48 MeV). The 7Linucleus also has a small binding energy for the α+t break-up, which is, however, about 1 MeV higher than that for

6Li (see Table I). Even so, in reactions of 7Li, the break-up channel of the α + t cluster is relevant [2]. Notwith-standing, 8Be formation (with subsequent α + α decay)through a proton pick-up transfer process (Q = 6.658MeV) is more probable.

The 9Be nucleus presents a Borromean structure com-posed of two α particles and one weakly bound neutron[10]. It has a binding energy for the α + α + n break-up comparable to that for 6Li (see Table I). The 1n-separation energy of 9Be is quite small in comparisonwith those for the other nuclei of Table I. Thus, when col-liding with a target nucleus, 9Be tends (with high proba-bility) to transfer its weakly bound neutron, with α+α or8Be formation (the later followed by α+α decay). In [11],Arazi et al. demonstrated the importance of couplings tounbound states to obtain theoretical agreement with the9Be + 120Sn data set, at energies around the Coulombbarrier, corroborating break-up as an important process.

Similar to 7Li, 10B also presents a first excited statewith low excitation energy (E∗ = 0.72 MeV). However,compared to 6,7Li and 9Be (Table I), its most favorablebreak-up channel, 10B → 6Li + 4He, is energeticallyhigher and, therefore, less probable. In addition, con-sidering the different values of the 1n-separation energy(Table I), break-up triggered by nucleon transfer is notas favored for 10B as it is for 9Be. In [12], we demon-strated that couplings to the continuum states are notimportant to obtain a good agreement between theoret-ical calculations and experimental data for 10B + 120Sn,at energies around the Coulomb barrier, indicating thatbreak-up is not an important process in this case. Theabove mentioned features indicate a very different reac-tion dynamics for 9Be and 10B weakly bound projectilesreacting with 120Sn.

Studying reactions involving weakly bound stable nu-clei is a crucial step towards a better understanding oftheir abundances. The structural models of these nu-clei are fundamental to determine how they interact and,therefore, to shed light on such abundances. Weaklybound nuclei, in general, have fundamental structuralcharacteristics, such as the above mentioned low break-up thresholds and cluster structures. Break-up can leadto a complex problem of three or more bodies, and canoccur by direct excitation of the weakly bound projectileinto continuum states or by populating continuum statesof the target [3, 13–18].

Weakly bound stable nuclei can easily be produced andaccelerated, with high intensities, in conventional particleaccelerators. Within this context, complementary exper-imental campaigns are being developed in two laborato-ries: the 8 MV tandem accelerator of the Open Labora-tory of Nuclear Physics (LAFN, acronym in Portuguese)in the Institute of Physics of the University of Sao Paulo(Brazil), and the 20 MV tandem accelerator TANDAR(Buenos Aires, Argentina). The aim of the joint collabo-ration is to study the scattering involving stable, stronglyand weakly bound, nuclei on the same target (120Sn), atenergies around the respective Coulomb barriers. These

3

measurements allow systematic studies that involve thecomparison of behavior for the different projectiles.

Many data, obtained in our experiments, with 120Sn astarget, have already been published [11, 12, 19, 20]. Inthe present paper, we present new experimental angulardistributions for the elastic scattering of the 6Li + 120Snsystem, at three bombarding energies. We include thesedata in a wide systematic involving the elastic scatter-ing of 4,6He, 7Li, 9Be, 10B and 16,18O projectiles, on thesame target, at energies around the respective Coulombbarriers. We analyze the complete data set within the ap-proach of the optical model (OM), assuming the double-folding Sao Paulo Potential (SPP) [21] for the real partof the optical potential (OP) and two different modelsfor the imaginary part. With this, we study the behaviorof the OP as a function of the energy for the differentprojectiles.

In the next section, we present a summarized review ofthe experiments. It will be followed by the explanation ofthe theoretical approach and corresponding applicationto the experimental data. Then, we discuss and comparethe behaviors of the OPs that fit the data for differentprojectiles. Finally, we present our main conclusions.

II. THE EXPERIMENTS

The measurements for the 6,7Li, 10,11B + 120Sn sys-tems are part of the E-125 experimental campaign, de-veloped at the LAFN, and correspond to the followingenergies: 1) 6Li at ELAB = 19, 24 and 27 MeV, reportedfor the the first time in this paper; 2) 7Li at ELAB = 20,22, 24 and 26 MeV [19]; 3) 10B at ELAB = 31.35, 33.35,34.85 and 37.35 MeV [12, 20]. The experimental setup isbased on SATURN (Silicon Array based on Telescopes ofUSP for Reactions and Nuclear applications). SATURNis installed in the 30B experimental beam line of the lab-oratory, which contains a scattering chamber connectedto the accelerator. The SATURN detection system hasbeen mounted with 9 surface barrier detectors in angularintervals of 5o. With this, in 3 runs we cover an angularrange of 120o, from 40o to 160o. The targets contained120Sn and 197Au, the latter used for the purpose of nor-malization. Further details are found in [12], [19] and[20].

The experimental data for 9Be+120Sn were obtained atthe TANDAR laboratory, at ELAB = 26, 27, 28, 29.5, 31,42 and 50 MeV. An array of eight surface barrier detec-tors, with an angular separation of 5◦ between adjacentdetectors, was used to distinguish scattering products.All details about data acquisition and analysis are pre-sented in [11].

In addition to our data, other experimental elastic scat-tering cross sections, for systems involving 120Sn as tar-get, were obtained from [22–29].

III. THE THEORETICAL APPROACH

Data of heavy-ion nuclear reactions have been suc-cessfully described in many works assuming double-folding theoretical models for the nuclear potential [30–39]. Among these models, the SPP [21] associates thenuclear interaction to a dependence on the local veloc-ity. The model includes a systematic of nuclear densitiesobtained for stable strongly bound nuclei and, in this con-text, it does not contain any free parameter. The SPP isrelated to the double-folding potential through:

VSPP(R) = VFold(R) e−4v2/c2 , (1)

where c is the speed of light and v(R) is the local rela-tive velocity between projectile and target. At energiesaround the Coulomb barrier (as in the present analysis)the velocity is much smaller than the speed of light andwe have: VSPP(R) ≈ VFold(R). The folding potential isrepresented as:

VFold(R) =

∫ ∫ρ1(~r1)ρ2(~r2)V0 δ(~R− ~r1 + ~r2) d~r1 d~r2.

(2)Here, ρ1 and ρ2 are the projectile and target matter dis-tributions, and V0 δ(~r) is the zero-range effective inter-action (with V0 = −456 MeV fm3). This V0 value wasobtained in [21], through a very wide systematic involv-ing phenomenological potentials extracted from elasticscattering data analyses for many systems. For a partic-ular nucleus, the respective nucleon distribution is foldedwith the matter density of one nucleon to obtain the cor-responding matter density of the nucleus (see [21]).

An important point that stands out against obtain-ing a systematical description of the elastic scatteringprocess with an OP (within the OM) is the difficulty indescribing the imaginary part of the interaction from fun-damental grounds. A fully microscopic description basedon the Feshbach theory is specially difficult at energieswhere collective as well as single particle excitations areimportant in the scattering process [40–42]. To face thisproblem within a simple model, an extension of the SPPto the OP imaginary part was proposed in [43], consid-ering the following OP:

UOP (R) = VSPP(R) + iNI VSPP(R). (3)

Elastic scattering data for many systems, at high en-ergies, have been described using NI ≈ 0.78 [43]. Atenergies around the Coulomb barrier, the SPP has alsobeen valuable in coupled channel calculations for sys-tems involving strongly (see e.g. [44]) and weakly bound(e.g. [12, 19, 20]) projectiles. Furthermore, the SPP hasaccounted for data of systems with exotic nuclei (e.g.[45, 46]). Besides being successful in elastic scatteringdata analyses, the SPP has also provided good descrip-tions of data for the fusion process of many systems (e.g.[47–51]).

4

In the present work, we propose the SPP theoreti-cal approach in the context of the OM to systemati-cally study the elastic scattering data for the 4,6He, 6,7Li,9Be, 10B, 16,18O + 120Sn systems, at energies around theCoulomb barrier. We assume equation (4) to describethe OP:

UOP (R) = NR VSPP(R) + iNI VSPP(R), (4)

where NR and NI represent multiplicative factors thatdetermine the strengths of the OP (real and imaginaryparts) and simulate, in a simple form, the effects ofthe polarization potential. The polarization arises fromnonelastic couplings. According to Feshbach’s theory[38, 52], it is energy dependent and complex. The imag-inary part comes from transitions to open non-elasticchannels that absorb flux from the elastic channel. Thereal part arises from virtual transitions to intermediatestates (inelastic excitations, nucleon transfer, among oth-ers). As already commented, standard average valuesobtained in [43] are NR = 1 and NI = 0.78.

For the purpose of comparison and with the aim of ac-counting only for the internal absorption (fusion) frombarrier penetration, without taking into account the ef-fect of the couplings, we also perform OM calculationsbased on equation (5):

UOP (R) = VSPP(R) + iW (R), (5)

where W (R) has a Woods-Saxon (WS) shape,

W (R) = W0/ [1 + exp (R−R0)/a] , (6)

withW0 = −100 MeV, R0 = r0

(A

1/31 +A

1/32

), r0 = 1.06

fm and a = 0.25 fm. Due to the small diffuseness value,such an internal imaginary potential just simulates thefusion process (without couplings) and does not take intoaccount the absorption by the peripheral channels.

Before proceeding with the OM analyses, we first ex-amine the effects of the densities on the nuclear interac-tion. As already commented, the SPP involves a system-atics of densities that makes the interaction a parameter-free model. However, one can question if the use ofthis systematics for weakly bound nuclei is appropriate.Thus, we have calculated nuclear densities through the-oretical Hartree-Bogoliubov (HB) calculations [53], as-suming two different interactions: the NL3 and DDME1models [54, 55]. Figure 1 shows a comparison of dif-ferent approaches for the matter densities of light weaklybound nuclei: the two-parameter Fermi systematic of theSPP and the theoretical HB. In the cases of 6Li and 10B(where N = Z), we also present in Fig. 1 the exper-imental charge density (obtained from electron scatter-ing) multiplied by 2. Except for 6He, all these densitiesare very similar, and therefore the use of the systematicsfor densities of the SPP is justified. We have also veri-fied that very similar values of cross sections are obtainedfrom OM calculations using these different models for thedensities. In the 6He case, the theoretical HB density is

rather different from that of the systematics at the sur-face region. Thus, we have taken an “experimental” den-sity for this nucleus, obtained from data analyses of pro-ton scattering at high energies [56]. The dashed-dottedorange line in Fig. 1(d) represents this “experimental”matter density (obtained from folding the nucleon distri-bution with the matter density of the nucleon, accordingto [21]). The “experimental” density is quite similar tothat from the systematics of the SPP (blue line). Thus,we consider that, even in the 6He case, the use of theSPP systematics for densities is justified.

10-2

10-1

0 1 2 3 410

-3

10-2

10-1

1 2 3 4 5

9Be

ρ (f

m-3)

10B

(a)

(c)ρ

(fm

-3)

r (fm)

NL3

DDME1

Systematics

Charge x 2

(b)

6Li

(d)

6He

r (fm)

From Ref. [49]

FIG. 1: (Color online) Matter densities for 10B, 9Be, 6Liand 6He, calculated through different models (see text fordetails).

IV. STANDARD OPTICAL MODELCALCULATIONS

Before providing the results of the elastic scatteringdata fits, we present a comparison of the experimentalangular distributions with OM cross sections obtainedassuming the standard models for the OP. By stan-dard models we mean Equation (4) with NR = 1 andNI = 0.78, and Equation (5) (internal imaginary poten-tial). From now on, we refer to these standard modelsas Strong Surface Absorption (SSA) and Only InternalAbsorption (OIA), respectively. In order to illustrate theregion of energy of the data, for each angular distributionwe provide the value of the reduced energy, defined as:

ERed = Ec.m. − VB , (7)

where Ec.m. represents the center of mass energy and VBis the s-wave barrier height, obtained for the respectivesystem with the SPP. In Table III we present the barrierheights, radii and curvatures (~w) [47], for the systemsstudied in the present work.

5

TABLE III: Values of the s-wave barrier parameters obtainedwith the SPP for systems composed by projectiles focusingon 120Sn.

projectile VB(MeV) RB(fm) ~w(MeV)

4He 14.22 9.48 4.926He 12.78 10.52 3.356Li 19.76 10.16 4.207Li 19.45 10.34 3.869Be 25.78 10.40 3.9310B 32.38 10.34 4.1716O 50.79 10.56 4.1418O 50.05 10.74 3.86

Figure 2 presents four experimental angular distribu-tions for the strongly bound 4He projectile [22]. The en-ergies of the angular distributions vary from 5.1 to 19.1MeV above the barrier (5.1 ≤ ERed ≤ 19.1 MeV). Toavoid overlapping results, the cross sections for two angu-lar distributions have been displaced by a constant factorof 0.5. The solid blue and dashed green lines representthe theoretical results obtained with SSA and OIA, re-spectively. Both standard models provide rather similarresults, but the SSA accounts for the data with slightlybetter accuracy.

0 20 40 60 80-1.0

-0.5

0.0

0.5

1.0

0 20 40 60-1.0

-0.5

0.0

0.5

1.0

σ El./σ

Ruth

.

θc.m.

(degree)

ERed

= 5.1 MeV

ERed

= 12.0 MeV

(a)(b)

ERed

= 15.6 MeV

ERed

= 19.1 MeV

θc.m.

(degree)

4He +

120Sn

FIG. 2: (Color online) Experimental angular distribu-tions for the elastic scattering of the 4He + 120Sn system[22, 23]. To avoid overlapping results, the cross sections forERed = 12.0 and 19.1 MeV have been displaced by a con-stant factor of 0.5. The solid and dashed lines representtheoretical OM cross sections obtained with the SSA andOIA models, respectively.

Figure 3 presents experimental and theoretical (SSAand OIA) angular distributions for the strongly bound16O and 18O projectiles. In the 16O case, all energiesare below the corresponding barrier height. At the low-est energy (ERed = −4.0 MeV) the data are compatiblewith internal absorption (OIA), while for higher energiesthey approach to the results of strong surface absorption(SSA). In the case of 18O, the energy is slightly above the

barrier and the SSA reproduces well the data set, exceptat the rainbow region (θc.m. ≈ 90o).

0 60 120 1800.0

0.2

0.4

0.6

0.8

1.0

0 60 120 1800.0

0.2

0.4

0.6

0.8

1.0

1.2

16O

18O

σ El./σ

Ruth

.

θc.m.

(degree)

- 4.0 MeV

- 3.1MeV

- 2.3 MeV

(a)(b)

ERed

= 2.1 MeV

θc.m.

(degree)

FIG. 3: (Color online) Experimental angular distributionsfor the 16,18O + 120Sn systems [24, 25]. To avoid overlappingresults, the cross sections for some distributions have beendisplaced by a constant factor. The solid and dashed linesrepresent theoretical OM cross sections obtained with theSSA (solid) and OIA(dashed) models, respectively

Figure 4 presents data and theoretical predictions(SSA and OIA) for the elastic scattering of the exotic6He on 120Sn [26, 27]. Again the SSA provides a gooddescription of the data, with some deviation for the low-est ERed = 3.8 and 4.4 MeV, due to transfer/break-upchannels [27].

0 30 60 90-0.8

-0.4

0.0

0.4

0.8

1.2

0 30 60 90-0.8

-0.4

0.0

0.4

0.8

1.2

6Heσ E

l./σ

Ruth

.

θc.m.

(degree)

3.8 MeV

4.4 MeV

6.1 MeV

(a) (b)

ERed

= 6.7 MeV

ERed

= 8.4 MeV

θc.m.

(degree)

FIG. 4: (Color online) The same as Fig. 3, for the 6He +120Sn system. Data were extracted from [26, 27].

In Fig. 5, with the present new data, we show theoret-ical predictions for 6Li + 120Sn, at energies around thebarrier, in linear (a) and logarithmic (b) scales. The SSAcross sections (solid blue lines) are in good agreementwith the data, including at ERed = −1.66 MeV, which in-dicates strong surface absorption even in the sub-barrierenergy region. For comparison, in Fig. 6 we present

6

an excitation function for the elastic scattering of 6Li +120Sn from earlier measurements [28]. The data corre-spond to an angular range of 160o ≤ θLab ≤ 170o. Thesolid line represents the SSA cross sections at the averageangle θc.m. = 165.7o. There is a reasonable agreement be-tween experimental and theoretical results, but the slopeof the data is somewhat different from that of the OMcalculations.

40 80 120 160-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

40 80 120 160

10-3

10-2

10-1

100

6Li +

120Sn

σ El./σ

Ruth

.

θc.m.

(degree)

- 1.66 MeV

3.10 MeV

5.95 MeV

(a) (b)

θc.m.

(degree)

FIG. 5: (Color online) Experimental and theoretical SSA(solid lines) and OIA (dashed lines) elastic scattering an-gular distributions for 6Li + 120Sn. Note the change fromlinear (a) to logarithmic (b) scale. To avoid superposition,the cross sections for two distributions are displaced (a) ordivided (b) by constant factors.

-8 -6 -4 -2 0 2 410

-2

10-1

100

6Li +

120Sn

σ El./σ

Ruth

.

ERed

(MeV)

θc.m.

= 165.7o

FIG. 6: (Color online) Experimental excitation functionfor 6Li + 120Sn [28] (see text for details). The solid linerepresents the SSA cross sections at θc.m. = 165.7o.

Figures 7 and 8 present results for 7Li + 120Sn [19].In this case, the SSA provides even better agreement be-tween data and theory than for 6Li.

Figures 9 and 10 present results for 9Be + 120Sn.Again, the SSA provides cross sections in reasonableagreement with the data.

40 80 120 160-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

40 80 120 16010

-4

10-3

10-2

10-1

100

7Li

σ El./σ

Ruth

.

θc.m.

(degree)

- 0.6 MeV

1.3 MeV

3.2 MeV(a)

(b)

θc.m.

(degree)

FIG. 7: (Color online) The same as Fig.5, for 7Li + 120Sn.Data were extracted from [19].

40 80 120 160-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

40 80 120 160

10-4

10-3

10-2

10-1

100

(a)

7Li

σ El./σ

Ruth

.

θc.m.

(degree)

5.1 MeV

7.0 MeV

8.9 MeV

(b)

θc.m.

(degree)

FIG. 8: (Color online) The same as Fig.5, for 7Li + 120Sn.Data were extracted from [19, 29].

Finally, Fig. 11 presents results for 10B + 120Sn. TheSSA does not work as well as in other cases of weaklybound nuclei. However, the reduced energy region in thecase of 10B is low and the results for this nucleus aresimilar to those shown for 16O in Fig. 3(a).

V. COMPARISON OF THE BEHAVIOR OF THEOPTICAL POTENTIAL FOR DIFFERENT

PROJECTILES

As commented in the previous section, the SSA pro-vides an overall reasonable description of the completedata set studied here. Even so, small deviations betweendata and theoretical predictions are observed. In thissection, we assume Equation (4) with two adjustable pa-rameters, NR and NI , in order to fit the data more ac-curately, and compare the behavior of the correspondingOP parameter values obtained for different projectiles.

7

0 40 80 120 160-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 40 80 120 160

10-4

10-3

10-2

10-1

100

9Be

σ El./σ

Ruth

.

θc.m.

(degree)

(a)

(b)

θc.m.

(degree)

- 1.6 MeV

- 0.7 MeV

0.3 MeV

1.7 MeV

FIG. 9: (Color online) The same as Fig.5, for 9Be + 120Sn.Data were extracted from [11].

0 40 80 120-0.8

-0.4

0.0

0.4

0.8

1.2

0 40 80 120 160

10-4

10-3

10-2

10-1

100

(a)

3.1 MeV

13.3 MeV

20.7 MeV

9Beσ E

l./σ

Ruth

.

θc.m.

(degree)

(b)

θc.m.

(degree)

FIG. 10: (Color online) The same as Fig.5, for 9Be + 120Sn.Data were extracted from [11].

A. The uncertainties of the NR and NI values

In this subsection, we discuss some ambiguity inher-ent to the extraction of the NR and NI best fit valuesand their respective uncertainties. For this purpose, wehave performed several calculations in order to verify thesensitivity of the data fit on variations of the NR andNI parameter values. Just as an example, we illustratehere the results obtained with the data set for 18O atERed = 2.1 MeV. The corresponding best data fit is ob-tained with NR = 0.739 and NI = 0.877, with reducedchi-square of χ2 = 5.80.

In Fig. 12 (a), we present the values of χ2 as a func-tion of NI for several (fixed) values of NR. For each NR,there is an optimum NI value that provides the smallestχ2. Here, we can observe the strong correlation betweenthe NR and NI parameters. This correlation can be evenbetter observed in Fig. 12 (b), which presents the opti-mum NI value as a function of NR. Clearly, for largerNR values we have smaller values of optimum NI . In Fig.

0 40 80 120 160-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 40 80 120 160

10-4

10-3

10-2

10-1

100

10B

σ El./σ

Ruth

.

θc.m.

(degree)

(a)

(b)

θc.m.

(degree)

- 3.4 MeV

- 1.6 MeV

- 0.2 MeV

2.1 MeV

FIG. 11: (Color online) The same as Fig.5, for 10B + 120Sn.Data were extracted from [12, 20].

12 (d), we show the χ2 (obtained with the optimum NI)as a function of NR. In Fig. 12 (c), we show three curves(in the NR - NI plane) that correspond to different levelsof χ2 (and also the point that provides the best χ2 = 5.80for this data set).

Within the context of the theory of errors, the un-certainty of an adjustable parameter can be approxi-mately estimated considering variations of the reducedchi-square by about 1/N around the minimum χ2 value(which should be close to 1), where N is the number ofdata. The experimental angular distribution, adoptedas an example, contains 18 data points, and therefore1/N ≈ 0.06. Since the best χ2 = 5.80, one should con-sider the range χ2 ≤ 5.86 for the determination of theerror bars of the parameters. Nevertheless, the OM isonly a simple (in fact simplified) theoretical model to de-scribe the experimental phenomenon, and one can notexpect the theory of errors to work perfectly in this case.For instance, the best χ2 = 5.80 is very far from theexpected value χ2 ≈ 1 of the theory.

Taking into account this point, in many works, the es-timate of uncertainties of the OM adjustable parametersis performed considering a different level of reduced chi-square, for instance, an increase of 10% or 20% relative toits minimum value. Nevertheless, oftentimes the correla-tion between the parameters (as that for NR with NI) isnot considered when determining uncertainties. In thiscase, the uncertainties can be largely underestimated.

Just to illustrate this point, let us suppose that wechoose the level χ2 = 7 (about 20% above the bestχ2 = 5.80) to determine the uncertainties. This levelis represented by the dashed line in Fig. 12 (a). Thesolid red curve in this figure corresponds to the variationof χ2 as a function of NI for the fixed (and also the bestfit value) NR = 0.739. If one neglects the NR - NI cor-relation, the uncertainty of the NI parameter is foundaccording to the intersections of the solid red curve withthe χ2 = 7 level (dashed line). The blue arrows in fig-ure 12 (a) show the corresponding region of uncertainty:

8

0.86 ≤ NI ≤ 0.90 (relative uncertainty of about 4.5%).However, an inspection of the curve corresponding to theχ2 = 7 level in Fig. 12 (c) shows that, when consideringthe NR - NI correlation, a better estimate for the un-certainty of NI is 0.76 ≤ NI ≤ 1.01, therefore a muchlarger range of about 28% for the relative uncertainty.The same could be said about the NR uncertainty. Thedashed line in Fig. 12 (d) also represents the level χ2 = 7.The correspondingNR region is 0.61 ≤ NR ≤ 0.84 (about32% of relative uncertainty in NR). This region alreadycontains the effect of the correlation (since the χ2 ver-sus NR curve of Fig. 12 (b) was obtained consideringthe variation of the optimum NI value with NR). Theblue arrows that connect Figs. 12 (b) and (d) illustratethe effect of the correlation on the uncertainties of theNR and NI parameter values. In our example, the con-sideration or not of the correlation affects the parameteruncertainty values by a factor about 6.

0.6 0.7 0.8 0.9 1.0 1.1

0

4

8

12

16

20

0.6 0.7 0.8 0.90.7

0.8

0.9

1.0

0.8

1.0

1.2

0.6 0.7 0.8 0.9 1.0

4

8

121.01

0.86 0.90

NR = 0.8

1.0

NR = 0.6

0.739

NR = 0.9

NI

χ2

(a)

0.76(c)

NI

NR

χ2 = 7

χ2 = 6.5

χ2 = 6

χ2 =

0.76

1.01

(b)

NI

0.840.61

(d)

χ2

NR

5.8

FIG. 12: (Color online) The figure shows results obtainedby fitting an experimental angular distribution for 18O atERed = 2.1 MeV. (a): χ2 versus NI for some NR values.(b): Optimum NI value as a function of NR. (c): Curvesin the plane NR - NI that correspond to some levels of χ2.(d): χ2 obtained with the optimum NI as a function of NR.See text for details.

Other important question can be raised here. Whatwould be a good χ2 level to estimate uncertainties? Inour example, we chose 20% above the best (minimum)χ2. The best χ2 is obtained with NR = 0.739 andNI = 0.877, while the borders (χ2 = 7) correspond to twopossible pairs: NR = 0.61 and NI = 1.01 or NR = 0.84and NI = 0.76. In Fig. 13 we present, in linear (a) andlogarithmic (b) scales, the experimental angular distri-bution for 18O at ERed = 2.1 MeV, and three theoreticalcurves. Two of them, the solid black and dashed red lines,correspond to the best χ2 = 5.80 and to one case whereχ2 = 7. These two lines are almost indistinguishable,indicating that this increase of 20% in χ2 is probably toosmall to represent actual significance. The other curve(dotted blue lines in the figure) represents the result of

a fit, in which NR = 1 was fixed and only NI was con-sidered as adjustable parameter. The corresponding op-timum NI = 0.616 was found, with χ2 = 15.54. Despitethe difference of a factor of about three between the re-spective χ2 values, both OPs (of the best χ2 = 5.80 andthat with χ2 = 15.54) provide a quite reasonable datafits (see the black and blue lines in Fig. 13). The largedifference between the respective χ2 is mostly related tothe fit in the backward angular region (in particular forthe datum at the last angle θc.m. ≈ 150o). On the otherhand, the fit with χ2 = 15.54 (dotted blue lines in the fig-ure) clearly provides a slightly better data description inthe rainbow region (θc.m. ≈ 90o). Thus, one might ask:taking into account the physical behavior, does the fitwith χ2 = 5.80 actually describe the experimental datain a better way than that with χ2 = 15.54?

0 40 80 120 1600.0

0.2

0.4

0.6

0.8

1.0

1.2

0 40 80 120 160

10-1

100

18O

σ El./σ

Ruth

.

θc.m.

(degree)

χ2 = 5.80

χ2 = 7

χ2 = 15.5

NR = 0.739

NI = 0.877

NR = 0.61

NI = 1.01

NR = 1

NI = 0.616

(a) (b)

ERed

= 2.1 MeV

θc.m.

(degree)

FIG. 13: (Color online) Experimental angular distributionfor 18O at ERed = 2.1 MeV, in linear (a) and logarithmic(b) scales. The solid black lines represent the best data fit(χ2 = 5.80) obtained within the OM. The dashed red lineswere obtained with NR = 0.61 and NI = 1.01 that provideχ2 = 7. The dotted blue lines correspond to the result ofthe fit in which NR = 1 was fixed and only the NI value wasadjusted (χ2 = 15.54).

Thus, uncertainties of adjustable parameter values ob-tained from OM data fits should be considered just asrough estimates. If the strong correlation between NR

and NI is taken into account (and it should be), the un-certainties of these parameters become quite large. Inaddition, as commented in the previous paragraph, it ispossible to obtain a quite reasonable description of theexperimental angular distribution (18O at ERed = 2.1MeV) assuming very different NR values. The reason forthis behavior is also related to the correlation betweenthe NR and NI parameters. As illustrated in Fig. 13,the (best fit) pair NR = 0.739 and NI = 0.877 producesOM cross sections similar to those obtained with (fixed)NR = 1 and (adjusted) NI = 0.616 (despite the large dif-ference of a factor of 3 in the corresponding χ2 values).

This behavior observed for the angular distribution of18O at ERed = 2.1 MeV is also present in many other

9

cases (projectiles and energies). The correlation betweenNR andNI implies a wide ambiguity in the determinationof these parameter values, when simultaneously adjustedwithin the context of the OM data fits. In principle, theeffect of the polarization due to inelastic channels wouldaffect both: the real and imaginary parts of the OP. Evenso, in order to avoid this question of correlation and con-sequent ambiguity, from now on we assume NR = 1 inthe OM calculations, and adjust only the NI parametervalue in the data fits.

B. The sub-barrier region

When comparing data for different systems, it is im-portant to take into account the region of energy con-sidered. Thus, in this section we compare NI values ob-tained for different projectiles in approximately the sameregion of reduced energy.

As illustrated in Figs. 3 (a) and 11, at energies belowthe barrier, both systems, with 16O and 10B, present datawith behavior in between the theoretical results of OIA(internal absorption = weak surface absorption) and SSA(strong surface absorption). In Fig. 14, we present theresults obtained through OM data fits, for 16O (a) and10B (b). The figure also shows the NI values obtained foreach angular distribution. The 16O case presents the ex-pected behavior of strongly bound nuclei: vanishing sur-face absorption at 4 MeV below the barrier (NI = 0.09),and increasing NI = 0.25 and 0.50 values when approach-ing the barrier. On the other hand, the NI values for 10Bare quite similar (about 0.35) for the three energies belowthe barrier, indicating that the surface absorption doesnot decrease significantly even at sub-barrier energies.

0 40 80 120 160

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

40 80 120 160

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-2.3 MeV

-3.1 MeV

NI = 0.09

NI = 0.50

ERed

= - 4.0 MeV

σ El./σ

Ruth

.

θc.m.

(degree)

NI = 0.25

16O

(a) (b)10

B

-0.2 MeV

-1.6 MeV

-3.4 MeV

NI = 0.35

NI = 0.35

NI = 0.32

θc.m.

(degree)

FIG. 14: (Color online) Experimental angular distributionsfor 16O and 10B and corresponding theoretical OM crosssections obtained from data fits. The reduced energies andNI values are indicated in the figure.

In Fig. 15 (a), we present results for the weakly bound6Li, 9Be and 10B projectiles, at energies about 1.6 MeVbelow the barrier. In pannel (b), we have 7Li (instead

6Li) and again 9Be and 10B, at energies about 0.5 MeVbelow the barrier. Even in this low energy region, thethree projectiles present non vanishing NI values, thelargest being those for 9Be (NI ≈ 1.3), followed by thosefor the lithium isotopes (about 0.7), and the small one(0.35) being that for 10B. This behavior indicates moreabsorption at sub-barrier energies, probably due to thebreak-up process, for 6,7Li and 9Be [2, 11].

0 40 80 120 160-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 40 80 120 160-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

10B

9Be

6Li

NI = 0.35

NI = 1.30

-1.5 MeV

-1.6 MeV

σ El./σ

Ruth

.

θc.m.

(degree)

ERed

= -1.7 MeV

NI = 0.75

(a) (b)

10B

9Be

7Li

NI = 0.35

NI = 1.25

NI = 0.60

-0.2 MeV

-0.7 MeV

-0.6 MeV

θc.m.

(degree)

FIG. 15: (Color online) The same as Fig. 14, for othersystems and energies.

C. The above-barrier region

Now we analyze angular distributions at energies abovethe barrier. Again we present comparison of data onlyin similar reduced energy regions. For a good appreci-ation of the results, the figures contain both linear andlogarithmic scales.

Fig. 16 presents angular distributions for 7Li and 9Beat ERed ≈ 1.5 MeV. The NI values of about 0.6 for 7Liand 1.2 for 9Be are quite similar to those obtained atsub-barrier energies (see Fig. 15).

Figure 17 presents results for 6He, 6Li, 7Li and 9Be atabout 3.5 MeV above the barrier. The best fit NI = 4 ob-tained for 6He is a very large value. However, we pointout that, due to the large error bars of the cross sec-tion data, the sensitivity of the χ2 to the NI parametervalue is very weak for this angular distribution, and muchsmaller NI values also provide a good data fit. The NI

values obtained for the weakly bound 6Li, 7Li and 9Benuclei are large, again indicating strong surface absorp-tion in these cases.

Figure 18 presents results for 6He, 6Li and 7Li (twoenergies) at ERed ≈ 6 MeV. The 6He and 6Li OM fitsresult in NI values larger than 1. The two energies for 7Liprovide, consistently, similar values around NI ≈ 0.89.

Finally, Fig. 19 presents results for the strongly bound4He and the weakly bound 9Be nuclei, at very high ener-gies ERed ≈ 20 MeV. A striking difference of about one

10

0 40 80 120 160

10-2

10-1

100

0 40 80 120 160

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

ERed

= 1.7 MeV9Be

7Li

σ El./σ

Ruth

.

θc.m.

(degree)

ERed

= 1.3 MeV

(a) (b)

NI = 1.25

NI = 0.59

9Be

7Li

θc.m.

(degree)

FIG. 16: (Color online) Experimental angular distributionsfor 7Li and 9Be, in logarithmic (a) and linear (b) scales. Theblack lines represent the OM fits (performed by adjustingonly the NI parameter). The corresponding NI and reducedenergy values are indicated in the figure.

0 40 80 120 160

10-4

10-3

10-2

10-1

100

0 40 80 120 160-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

3.1 MeV

3.2 MeV

3.1 MeV

6He

6Li

NI = 4.00

9Be

7Li

σ El./σ

Ruth

.

θc.m.

(degree)

ERed

= 3.8 MeV

(a) (b)

NI = 1.80

NI = 0.80

NI = 1.77

θc.m.

(degree)

FIG. 17: (Color online) The same of Fig. 16, for otherprojectiles and energies.

order of magnitude is observed for the corresponding NI

values: 0.20 and 1.90.

In Fig. 20, we show the NI values as a function ofthe reduced energy for several projectiles. We have notincluded the results for 6He and 4He, at low energies, be-cause the χ2 for these distributions are not very sensitiveto the NI values, due to the large error bars of the ex-perimental cross sections. The solid lines in this figureare only guides for the eyes. The dashed line correspondsto the standard NI = 0.78 value. Considering only thebehavior of the weakly bound nuclei, the figure indicatesincreasing NI parameter values in the following order:10B, 7Li, 6Li and 9Be.

0 40 80 120

10-5

10-4

10-3

10-2

10-1

100

0 40 80 120 160-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

7Li

7.0 MeV

5.1 MeV

6.0 MeV

6He

6Li

NI = 1.10

7Liσ E

l./σ

Ruth

.

θc.m.

(degree)

ERed

= 6.1 MeV

(a) (b)

NI = 0.79

NI = 0.99

NI = 1.40

θc.m.

(degree)

FIG. 18: (Color online) The same of Fig. 16, for otherprojectiles and energies.

0 20 40 60 80

10-3

10-2

10-1

100

0 20 40 60 80

-0.8

-0.4

0.0

0.4

0.8

1.2

ERed

= 20.7 MeV

4He

9Be

NI = 0.20

σ El./σ

Ruth

.

θc.m.

(degree)

ERed

= 19.1 MeV

(a) (b)

NI = 1.90

θc.m.

(degree)

FIG. 19: (Color online) The same of Fig. 16, for otherprojectiles and energies.

VI. CONCLUSIONS

In this paper, we have presented new data for the elas-tic scattering of 6Li + 120Sn at ELAB = 19, 24 and 27MeV. The corresponding angular distributions were con-sidered together with other elastic scattering data of sev-eral projectiles on the same target nucleus. The completedata set was systematically analyzed within the contextof the OM. We have demonstrated that the SPP in thecontext of the standard SSA provides a quite reasonabledescription of the data for all systems, without the ne-cessity of any adjustable parameter. We have obtainedmore accurate agreement between data and theoreticalcross sections by considering adjustable OP strengths inorder to improve the data fits.

We have illustrated the strong correlation between thereal and imaginary adjustable strength factors (NR andNI) in an example with one angular distribution. If thiscorrelation is taken into account, the uncertainties of theNR and NI best fit values become very large. In addition,

11

-4 0 4 8 12 16 200.0

0.5

1.0

1.5

2.0

NI

ERed

(MeV)

4He

6Li

7Li

9Be

10

B

16

O

18

O

FIG. 20: (Color online) NI values obtained from OM dataanalyses as a function of the reduced energy for several pro-jectiles. The solid lines in the figure are only guides forthe eyes, while the dashed line corresponds to the standardNI = 0.78.

different pairs of these parameters, with correspondingχ2 values that differ by a factor as large as 3, providerather similar theoretical angular distributions that agreewell with the data. This behavior is also found for otherprojectiles and energies. In order to avoid this ambiguity,we have assumed the SPP for the real part of the OP,with fixed standard NR = 1, and adjusted only the NI

parameter value in the OM data fits.As observed in Figs. 14 to 19, the theoretical cross

sections obtained through OM fits with only one free pa-rameter (NI) are in quite good agreement with the datafor all systems and energies. We have studied the behav-ior of the best fit NI value in different energy regions,

and compared results obtained for the various projec-tiles. The weakly bound 6,7Li, 9Be and 10B projectilespresent significant NI values at sub-barrier energies, in-dicating strong surface absorption even in this low energyregion, a characteristic probably related to the break-upprocess. Still considering these nuclei, increasing NI pa-rameter values are observed in the following order: 10B,7Li, 6Li and 9Be. This order is related to the bindingenergy of these nuclei (presented as Q values in Table I).This suggests a clear correlation between the break-upprobability and the absorption of flux from the elasticchannel.

Acknowledgments

This work was supported by the Ministry of Science,Innovation and Universities of Spain, through the projectPGC2018-096994-B-C21. This work was also partiallysupported by the Spanish Ministry of Economy and Com-petitiveness, the European Regional Development Fund(FEDER), under Project No FIS2017-88410-P and bythe European Union’s Horizon 2020 research and innova-tion program, under Grant Agreement No 654002. Thiswork has also been partially supported by Fundacao deAmparo a Pesquisa do Estado de Sao Paulo (FAPESP)Proc. No 2018/09998-8 and 2017/05660-0, ConselhoNacional de Desenvolvimento Cientıfico e Tecnologico(CNPq) Proc. No 407096/2017-5 and 306433/2017-6,and, finally, it is a part of the project INCT-FNA Proc.No 464898/2014-5. A. Arazi acknowledges Grant No

PIP00786CO from CONICET. D. A. Torres. and F.Ramirez acknowledge support from Colciencias undercontract 110165842984.

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