at different energies

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1 Pramana J. Phys. (2018) 90:66 Indian Academy of Sciences 16 C-elastic scattering examined using several models at different energies M N EL-HAMMAMY 1, and A ATTIA 2 1 Department of Physics, Faculty of Science, Damanhur University, Damanhour, Egypt 2 Department of Physics, Faculty of Science, Alexandria University, Alexandria, Egypt Corresponding author. marwa1374@yahoo.com MS received 30 March 2017; revised 28 October 2017; accepted 29 November 2017; published online 19 April 2018 Abstract. In the present paper, the first results concerning the theoretical analysis of the 16 C + preactionby investigating two elastic scattering angular distributions measured at high energy compared to low energy for this system are reported. Several models for the real part of the nuclear potential are tested within the optical model formalism. The imaginary potential has a Woods Saxon shape with three free parameters. Two types of density distribution and three different cluster structures for 16 C are assumed in the analysis. The results are compared with each other as well as with the experimental data to give evidence of the importance of these studied items. Keywords. PACS Nos Elastic scattering; single folding; cluster structure Dz; I; Ht; Cm; Cc 1. Introduction The neutron-rich carbon isotopes are crucial for studying neutron halos (skins) or cluster structures. The extended density distribution, called halo is one of the characteristic features of the nuclei close to the dripline. These nuclei have small separation energy of the last nucleon(s). As for the abundance of neutrons over protons at the nuclear surface; it is indicated by skin. Be that as it may, there is no correct meaning of the expressions skin and halo. For instance, 6 He and 8 He are regularly alluded to as neutron-skin as well as neutron-halo nuclei. The impact of halo or skin comes into view as new properties of the nuclei [1,2]. Information of halos in carbon isotopes is, to a specific degree, equivocal. For instance, 17 C has moderately small separation energies, and so extended density distribution is proposed [3]. Then again, it is appeared in [4] that there is no halo structure in 17 C. Similarly, 16 Cisa three-body system, but it is not a Borromean system because the two-body subsystems ( 14 C + n) are bound as 15 C. In addition, it has larger two neutron (S 2n ) separation energy equal to MeV [5,6], because it is closer to the stability line than other 2n-halo nuclei. The 2n-halo structure in 16 C was recommended [7] for clarifying a huge upgrade in the reaction crosssection. Even so, the pairing effect of the addition of two neutrons to 14 C seems to play an important role [8 10]. Other studies on the exotic structure of carbon isotopes have been done in [11 15]. As of late, theoretical calculations recommended that 16 C might be available in the two structures; 10 Be + 4 He +2n and 10 Be + 6 He [16]. In the first structure, the additional neutrons can form covalent bonds between core centres with the subsequent increase of the nuclear stability eventually leading to the conceivable formation of the purported nuclear molecules [17]. With respect to the second structure, a sign of conceivable new state is reported from binary ( 10 Be + 6 He) cluster decomposition of 16 C[16]. Furthermore, we propose a third structure ( 4 He+ 6 He+ 6 He) for examining various internal cluster structures of 16 C. This brief presentation gives a reasonable view on the significance of our study on 16 C isotope. For building up the examination on 16 C, calculations were performed by taking either different density distributions or different cluster structures as contributions in single-folding (SF) formalism. First, two types of nuclear density distributions were utilised, the Gaussian (G) and Gaussian oscillator (GO). G accepts the non-halo structure, while GO accepts core plus two neutrons halo structure [1,18]. Secondly, 16 C was inspected through 10 Be+ 6 He as S1, 10 Be+ 4 He + 2n as S2 and

2 66 Page 2 of 8 Pramana J. Phys. (2018) 90:66 4 He + 6 He + 6 He as S3 cluster structures. These contributions to effective nucleon nucleon (NN) interaction in a zero-range approximation are utilised to drive the two types of real folding potential for proton and 16 C. The imaginary potential was taken as Woods Saxon (WS) type. Both potentials inside the SF formalism in the light of OM had been used to analyse the angular distributions of the elastic scattering of proton with 16 Casaprojectile of incidence energy 50 and 300 MeV, in comparison with phenomenological optical model potential (OMP) parameters in which the real and imaginary potentials fitted as WS shape. The experimental data are provided in [19,20]. Our principal aim is not to fit the experimental data, but to study the effect on elastic scattering observables of different descriptions of 16 C structure by assuming simple models and checking the strength of these applied models with a unit real normalisation factor (N R ). This work has five sections. A brief presentation is given in 1, followed by the single-folding formalism in 2. The analysis procedure is given in 3. The results and discussions are given in 4, and conclusions are given in Single-folding formalism To examine the elastic scattering for the reaction of 16 C + p, the program code DFPOT [21] is utilised. V SF (r) is the standard type of SF potential presenting the integration of effective NN interaction over the groundstate density distribution of the 16 C nuclei ρ16 C (r 1 ) [22]. Different investigated density distributions will be discussed later. V FR (r) is assessed from the expression V SF (r) = ρ16 C(r 1 ) V NN (s)dr 1. (1) V NN (s) is the effective NN interaction potential (s = r r 1 ) which consists of two parts; the direct part V D (s) and the exchange part V EX (s) in the form V NN (s) = V D (s) + V EX (s). (2) The direct part has the usual form of two Yukawa terms V D (s) = 7999 e 4.0s 2134e 2.5s 4.0s 2.5s, (3) while the exchange part has only one delta term in zerorange approximation V EX (s) = V 0 (E)δ(s). (4) The exchange part represents the simple exchange between the incident nucleon and the nucleon of the target. We intend to examine the important role of the exchange part by applying two different renditions of V NN based on two types of delta terms: The first [23]is V 0 (E) = 276 and the second is [21] ( E A ) MeV, (5) V 0 (E) = 456 MeV. (6) E is the energy in the laboratory system and A is the mass number of the projectile. We called the folded potential related to the first and second forms as SFP- 1 and SFP-2, respectively. SFP-2 is associated with an exponential term and V LE (r, E) is named as the local equivalent potential (LEP). It is proposed in [24] asa correction for eq. (1) due to the non-locality exchange potential impact, which increases as energy increases. Here, we utilised the straightforward generalised local equivalent potential form V LE (r, E) = V FR (r) exp{ γ [E V c V LE (r, E)]}, (7) where γ = β and β = m 0b0 2 A proj. 2 h 2. This equation is solved for V LE (r, E) by an iterative method. The parameter b 0 is the nucleon nucleus nonlocality parameter and m 0 is the nucleon mass. 2.1 Density distributions In this work, we have used two types of densities of 16 C nucleus to calculate the real part used in explaining the proton scattering cross-section of 16 C[25 27]. The first one is Gaussian density (G) of the following structure: ρ(r) = Nρ 0 exp ( r 2 ) b 2, ( ) b 2 = 2R2 m 1 3/2 3, ρ 0 = πb 2, where N is the number of nucleons. (8) The second one is the Gaussian oscillator density (GO). It consists of two parts: ρ c (r) = N c ρ 0c exp ( r 2 ) (9) ρ v (r) = N v 2 3 ρ 0v ( r 2 b 2 v b 2 c ) exp b 2 c = 2R2 c 3, b2 v = 2R2 v 5, ( r 2 b 2 v ),

3 Pramana J. Phys. (2018) 90:66 Page 3 of 8 66 Table 1. Root mean square (r.m.s.) radii of the core and the valence nucleons that are used for densities of the different nuclei considered. Nucleus Configuration Density type R c (fm) R v (fm) R m (fm) Ref. 16 C 16 C G [5] 14 C +2n GO [5] 10 Be 10 Be G [29] 6 He 6 He G [30] 4 He + 2n GO [31] 6 He 4 He G [26] ρ 0 j = ( 1 πb 2 j ) 3/2, j = c, v. (10) From eqs (8) (10), it is clear that the G has one free parameter; R m, the root mean square (r.m.s.) radius. It describes the nucleus as a whole and does not differentiate between the core and the valence nucleons unlike the GO density. Therefore, the GO density is the simplest one that is used to describe the halo nuclei with two free parameters; core density ρ c (r) and valence density ρ v (r). In this context, we assume that the 16 C density has (core 14 C + 2n) densities. R c and R v denote the r.m.s. radii of the core and the valence nucleons, respectively. R m is given by [ Nc Rc 2 R m = + N v Rv 2 ] 1/2, (11) A where A is the mass number of the nucleus. N c and N v are the number of the core and valence nucleons, respectively. The total matter distribution density is written as follows [28]: ρ m (r) = ρ c (r) + ρ v (r). (12) From interaction cross-section experiments and via Glauber model calculations, matter r.m.s. radius is indicated for 16 C[5]asR m = 2.70 fm. 2.2 Cluster structures The conflict presented in the core plus nucleons assumption and observation of nuclear molecules for 16 C reflected the necessity of applying different cluster structures. At this point, we examined the most adequate cluster structure of 16 C nucleus. Three structures can be taken into account with different configurations of 16 C nucleus. It is assumed that the 16 C nucleus density distribution of each structure is the sum of the densities of the elemental nuclei [29 31]. These structures are identified as S1 for 10 Be + 6 He, S2 for 10 Be + 4 He +2n and S3 for 4 He + 6 He + 6 He. Assume that either S2 or S3 depends on the subdivision of one of the two elemental nuclei in the first general configuration S1. So, 16 C density is the sum of the densities of the 10 Be and 6 He nuclei in S1. It is written as ρ16 C(r) = ρ10 Be(r) + ρ6 He(r). 10 Be and 6 He densities used as a part of our investigation are the same as in eq. (8). In S2, the 16 C density is given as ρ16 C(r) = ρ10 Be(r) + ρ4 He(r) + 2ρn(r). 6 He in S1 can be handled as the core 4 He and two valence neutrons. The 10 Be and 6 He densities are the same as in eq. (8)andeqs(9) and(10), respectively. Finally, we examine S3 for the 16 C nucleus, where 10 Be nucleus in S1 is supplanted by its constituents 4 He and 6 He. Here, the density of 16 C is written as ρ16 C(r) = ρ4 He(r) + ρ6 He(r) + ρ6 He(r). We have used the same density distributions for the 4 He and 6 He formulated by eq. (8). R c, R v and R m of the different nuclei that are used in this work in addition to their assumed configurations are listed in table Analysis procedure Elastic scattering of 16 C+p is studied at 50 and 300 MeV for the first time. The theoretical calculations of the present study is divided into three parts. First, we concentrated on the investigation of two density distributions of the 16 C nucleus; G and GO. Secondly, we investigated the 10 Be + 6 He, 10 Be + 4 He + 2n, 4 He + 6 He + 6 He systems that represent cluster structures S1, S2 and S3, respectively for the 16 C nucleus. The calculations of these two parts have been completed in the framework of SF formalism in the light of OM with two adaptations of effective NN interaction. The produced real SFP parameters of every part have been embedded into DWUCK4 program [32] what is more with phenomenological imaginary parameters to compute

4 66 Page 4 of 8 Pramana J. Phys. (2018) 90:66 Table 2. WS optical potentials obtained from the analysis of 16 C+ p scattering. The Coulomb radius r c is fixed as 1.3 fm. Reaction E (MeV/n) V (MeV) r v (fm) a v (fm) W (MeV) r w (fm) a w (fm) Ref. 16 C+ p [19] Present work Present work Table 3. The optical potential parameters for the 16 C nucleus different densities and structures in the analysis of 16 C+ p system. The real folded potential is calculated within SPF-1 at 50 MeV. Density distribution N R W (MeV) r w (fm) a w (fm) σ R (mb) G GO S S S Table 4. The optical potential parameters for the 16 C nucleus different densities and structures in the analysis of 16 C+p system. The real folded potential is calculated within SPF-1 at 300 MeV. Density distribution N R W (MeV) r w (fm) a w (fm) σ R (mb) G GO S S S the differential scattering cross sections. The imaginary potential has WS shape as follows: [ ( )] r 1 Rw W I = W 1 + exp, (12) a w where R w = r w A 1/3, with A as the mass number of 16 C. W, R w and a w are depth, radius and diffuseness parameters, respectively. The imaginary parameters have been fluctuated to fit the experimental data. As a result, the main movable parameters are imaginary parameters. By contrasting the theoretical estimations and experimental data of scattering cross-section angular distributions, we picked one set of parameters that effectively explained the experimental data at the two energies, to be as a beginning stage for the calculations of the third part. In this part, we calculated the real LEP also with WS imaginary potential, as well, to explain the behaviour of the scattering cross-section angular distributions. 4. Results and discussion The elastic scattering of 16 C + p system at 50 and 300 MeV has been investigated by using single-folding model (SFM) for two different densities and three structures of the 16 C nucleus. The density parameters of the different nuclei that are studied in this work and their assumed configurations are given in table 1. In table 2, we have given the parameters obtained from the examination of 16 C+ p with regard to phenomenological OMP deliberately of correlation with SFM. WS shape of the real and imaginary potential has been utilised. Besides, the resulting parameters for imaginary parts accompanied by the SFP-1 are given in tables 3 and 4 in addition to reaction cross-sections σ R and real normalisation N R = 1.0. The 16 C density distributions used in SFM have been introduced into a comparative form in figure 1. Itis demonstrated that the GO density distribution has a tail longer than G density and this mirrors the intriguing structure of substantial range. So, in order to illustrate

5 Pramana J. Phys. (2018) 90:66 Page 5 of 8 66 Figure 1. G and GO density distributions of 16 C. Figure 2. Elastic scattering angular distributions obtained from the different density distributions and different structures of 16 C nucleus in the SF analysis using SFP-1 (dotted line) and fitted WS optical potential (solid line) in comparison with the experimental data (dots) at E16 C = 50 MeV. these differences, we have contrasted the theoretical results with the experimental data of the elastic scattering angular distribution in figures 2 and 3a. The two densities mold a similar portrayal of the experimental data onto the same model of using SFP-1 regardless of the oscillatory behaviour at 50 MeV. Therefore, we utilised one curve just for representation in figure 2. The angular distributions are correctly reproduced except for 60 c.m. angles which are underestimated. This might be due to the few experimental data points and small scattering angles. Whereas at 300 MeV as in figure 3a, the investigated densities show the general conduct of the experimental data, but none of them is perfect. Furthermore, we have acquired the scattering angular distribution of a considerable number of structures in correlation between each other and experimental data. In figure 2 for 50 MeV, only one curve is presented for Figure 3. (a) Elastic scattering angular distributions obtained from the G (dashed line), GO (dotted line) density distributions of 16 C nucleus in the SF analysis using SFP-1 and fitted WS optical potential (solid line) in comparison with the experimental data (dots) at E16 C = 300 MeV. (b) Elastic scattering angular distributions obtained from the different structures S1 (solid line), S2 (dashed line), S3 (dotted line) of 16 C nucleus in the SF analysis using SFP-1 in comparison with the experimental data (dots) at E16 C = 300 MeV. all structures as well as different densities by reason of the congruent of their theoretical curves. But, we have noticed that using either S2 or S3, a good description to experimental data is produced compared to S1 at 300 MeV in figure 3b. In general, we have observed that different structure configurations convey the theoretical curves nearer to the data in comparison with the investigated density distributions at 300 MeV. Nevertheless, some diversities are in the minimums ever since this region is more sensible to the wave function and to the internal structure. On the other hand, this impact is absent at 50 MeV. Slightly different reaction cross-section (σ R ) values and consistent imaginary potential parameters have been obtained by utilising two densities and three structures

6 66 Page 6 of 8 Pramana J. Phys. (2018) 90:66 Figure 4. Real potentials of G, GO densities and S1, S2, S3 structures used for 16 C nucleus in the SF analysis. The symbols are indicated in figure G GO S1 S2 S Figure 5. Variation of real potential with incident 16 C energy for different investigated structures. as listed in table 3 for 50 MeV. The conduct of the real SFPs has been considered and represented in figure 4. We have seen that the real potential is characterised by attractive values. We have found that the depths are close to each other. At 300 MeV, the circumstances are different; the real potentials have showed repulsive behaviour with various depths. The deepest one is seen in the G density and the nearest is found for S2 and S3 structures. This is credited to the values that have been reported in table 4. Moreover, the increase in the values of the imaginary parameter compared to that in table 3 is related to the reduction of the real folding potential values. As a result, the potential depths decrease with increase in energy and this is summarised by the estimations of σ R at high energy than that at low energy. For more clarification, the variation of real potential for incident energies for different structures and densities is summarised in figure 5. We have applied the results of the relation, ( V = e 2 Z/4πε 0 d ) in which, V is the Coulomb interaction potential energy between 16 C and the proton, e is the charge of an electron, Z is the charge number of 16 C, ε 0 is the permittivity of the space and d is the distance between the centres of 16 Cand proton. The analysis showed that the potential begins to shift from attraction to repulsion at different E and d for different assumptions of 16 C nucleus; at E 200 MeV and d = fm for G, at E 170 MeV and d = fm for S1 and at E 160 MeV and d = fm for GO, S2, S3. Regarding the above results from our point of view, the proton deals with incident 16 C nucleus as the body consists of 16 particles as a whole, at low energy. So, this leads to the slight variation in real potential with unit normalisation factor, which is reflected on constant imaginary parameters with different densities and structures. Similar description of elastic scattering cross-section is shown in figure 2 by using SFP-1 model. On the contrary, at high energy, 16 C and P get closer to each other with a high probability of internal reaction contributions to elastic scattering. For that reason, proton deals with 16 C as different bodies according to our assumptions; one body as G, two bodies ( 10 Be, 6 He) as S1, three bodies (core 14 C, n, n) as GO, ( 4 He, 6 He, 6 He) as S3 and four bodies ( 10 Be, 4 He,n,n)asS2.Thissheds light on the increase in W, a w values and decrease in R w values related to the variation of real potentials at 300 MeV in comparison to 50 MeV. Therefore, the assumptions GO, S2 and S3 represented a good description for the experimental data than others as has been appeared in figures 3a and3b. Contingent upon these outcomes, we have chosen S3 as a model instance of study in the third part in this work at both energies. The calculated LEPs in correlation with SFP-2 are shown in figure 6. We have observed that LEP has marginally extraordinary shape compared to SFP-2. Tables 3, 4 and 5 demonstrate that the adopted parameters for the imaginary potential are all different. There is a direct proportionality between a w and σ R at each energy. In figures 7 and 8, the improvement to the calculations has been accomplished, where the experimental data are portrayed effectively with LEP as opposed to SFP-1. The analysis has been done in this work without any real normalisation to the calculations on the purpose of testing the strength of the potentials under the influence of each investigated item. Unfortunately, there are no other work which can be compared with the values acquired according to our calculations for this reaction. Also, the absence of other experimental data has restricted our work at two energies.

7 Pramana J. Phys. (2018) 90:66 Page 7 of 8 66 Table 5. The optical potential parameters for (S3) structures for the 16 C nucleus in the analysis of 16 C+ psystem. The real folded potential is calculated within local equivalent potential (LEP). E(MeV) N R W (MeV) r w (fm) a w (fm) σ R (mb) Conclusions Figure 6. A comparison of LEP for 16 C + psystemat50 MeV (dashed line), 300 MeV (solid line) and SFP-2 (dotted line). Figure 7. Elastic scattering angular distributions obtained from the S3 structure of 16 C nucleus in the SF analysis using SFP-1 (dashed line) and LEP (solid line) compared to the experimental data (dots) at E16 C = 50 MeV. The theoretical predictions for the elastic scattering of 16 C+ psystemate16 C = 50 and 300 MeV have been carried by SFP within the OM formalism. Several models were tested for the real part of the nuclear potential by utilising effective NN interaction as a part of two forms (SFP-1 and LEP). In order to construct the SFPs, two types of density distributions of 16 C nucleus have been incorporated as one choice, while three different cluster configurations of 16 C nucleus: 10 Be + 6 He (S1), 10 Be + 4 He + 2n (S2) and 4 He + 6 He + 6 He (S3) have been selected as the other choice. Alongside the after effects of SFP-1 model, we consider that any of the two cluster (S2, S3) configurations as well as density (GO) could portray 16 C in an equal foot at high energy. On the other hand, there is no noticeable effect at low energy. For enhancing our outcomes, LEP model has been utilised as a rectification of SFP-1 model and the experimental elastic scattering differential cross-sections have been recreated superbly for S3 for instance. Then, better prediction of the experimental data is obtained. Concluding generally, various internal structures of the neutron-rich nuclei may give vital contributions to the theoretical studies in addition to suitable folding potential model of the nucleus nucleon interactions particularly at high energies. References Figure 8. Elastic scattering angular distributions obtained from the S3 structure of 16 C nucleus in the SF analysis using SFP-1 (dashed line) and LEP (solid line) compared to the experimental data (dots) at E16 C = 300 MeV. [1] I Tanihata, J. Phys. G 22, 157 (1996) [2] K Riisager et al, Europhys. Lett. 49, 547 (2000) [3] T Bumann et al, Phys. Lett. B 439, 256 (1998) [4] C Wu et al, Nucl. Phys. A 739, 3 (2004) [5] A Ozawa et al, Nucl. Phys. A 691, 599 (2001) [6] G Audi, O Bersillon, J Blachot and A H Wapstra, Nucl. Phys. A 624, 1 (1997) [7] T Zheng et al, Nucl. Phys. A 709, 103 (2002) [8] H Wuosmaa et al, Phys. Rev. Lett. 105, (2010) [9] M Wiedeking et al, Phys. Rev. Lett. 100, (2008) [10] Z Elekes et al, Phys. Lett. B 586, 34 (2004) [11] P J Leask, J. Phys. G: Nucl. Part. Phys. B 27, 9 (2001) [12] N I Ashwood et al, Phys. Rev. C 70, (2004) [13] M Freer et al, Phys. Rev. C 84, (2011)

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