¨¸Ó³ ¢ —Ÿ. 2014. ’. 11, º 3(187). ‘. 363Ä380 ”ˆ‡ˆŠ ‹…Œ…’›• —‘’ˆ– ˆ ’Œƒ Ÿ„. ’…ˆŸ OPTICAL MODEL ANALYSIS OF p + 6He SCATTERING OVER A WIDE RANGE OF ENERGY Zakaria M. M. Mahmoud a , Awad A. Ibraheem b, c, 1 , M. El-Azab Farid d a Sciences Department, New-Valley Faculty of Education, Assiut University, Egypt b Physics Department, Al-Azhar University, Assiut, Egypt c Physics Department, King Khalid University, Abha, Saudi Arabia d Physics Department, Assiut University, Assiut, Egypt Optical model analysis of proton elastic scattering from 6 He has been carried out for eight sets of elastic scattering data at energies of 24.5, 25.0, 36.2, 38.3, 40.9, 41.6, 71.0 and 82.3 MeV/nucleon, respectively. The vector analyzing power and differential cross section for the elastic scattering of 6 He nucleus from polarized protons at 71 MeV have been analyzed in the framework of the optical model potential. The data are, ˇrst, analyzed in terms of phenomenological potentials using the WoodsÄSaxon form for the real and imaginary parts supplemented by a spin-orbit potential of Thomas form. The analysis has been then performed using microscopic single folded complex potentials. ·μ¢μ¤¨É¸Ö ´ ²¨§ Ê¶·Ê£μ£μ · ¸¸¥Ö´¨Ö ¶·μÉμ´μ¢ ´ 6 He ¤²Ö ¢μ¸Ó³¨ Ô´¥·£¨° · ¸¸¥Ö´¨Ö, 24,5, 25,0, 36,2, 38,3, 40,9, 41,6, 71,0 ¨ 82,3 ŒÔ‚/´Ê±²μ´ ¸μμÉ¢¥É¸É¢¥´´μ, ¢ · ³± Ì μ¶É¨Î¥¸±μ° ³μ¤¥²¨. ‚¥±Éμ·´ Ö ´ ²¨§¨·ÊÕÐ Ö ¸¶μ¸μ¡´μ¸ÉÓ ¨ ¤¨ËË¥·¥´Í¨ ²Ó´μ¥ ¸¥Î¥´¨¥ Ê¶·Ê£μ£μ · ¸¸¥Ö´¨Ö Ö¤¥· 6 He ¸ ¶μ²Ö·¨§μ¢ ´´Ò³¨ ¶·μÉμ´ ³¨ ¸ Ô´¥·£¨¥° 71 ŒÔ‚ ´ ²¨§¨·Ê¥É¸Ö ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ μ¶É¨Î¥¸±μ£μ ¶μÉ¥´Í¨ ² . ±¸¶¥·¨³¥´É ²Ó´Ò¥ ¤ ´´Ò¥ · ¸¸³ É·¨¢ ÕÉ¸Ö ¢ É¥·³¨´ Ì Ë¥´μ³¥´μ²μ£¨Î¥¸±μ£μ ¶μÉ¥´Í¨ ² ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ Ëμ·³Ò ‚Ê¤¸ Ä‘ ±¸μ´ ¤²Ö ·¥ ²Ó´μ° Î ¸É¨ ¶μÉ¥´Í¨ ² ¨ Ëμ·³Ò ¸¶¨´μ·¡¨É ²Ó´μ£μ ¶μÉ¥´Í¨ ² Ëμ·³Ò ’μ³ ¸ ¤²Ö ³´¨³μ° Î ¸É¨. ´ ²¨§ ¶·μ¢μ¤¨É¸Ö ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ ³¨±·μ¸±μ¶¨Î¥¸±μ£μ μ¤´μ³¥·´μ£μ ±μ³¶²¥±¸´μ£μ ¶μÉ¥´Í¨ ² . PACS: 25.70.Bc; 24.10.Ht; 27.20.+n; 21.60.Gx INTRODUCTION Over the last decades, since the discovery of the ®halo¯ phenomenon in nuclear physics [1], the detailed study of unstable (halo) nuclei has been at the forefront of nuclear physics research. The halo structure refers to highly neutron-rich (n-rich) or proton-rich (p-rich) light nuclei that lie, respectively, near the neutron- or proton-drip line and hence are totally ®unstable¯ systems. A number of such nuclei have now become available, both as the primary and secondary beams with various low, intermediate and high energies, called the radioactive nuclear beams. With the advent of radioactive nuclear beams and the discovery that nuclear matter, under certain conditions, may present a halo structure, a renewed interest 1 E-mail: awad ah [email protected] 364 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. has surged in the investigation of sizes and radial shapes of nuclei. In the case of light n-rich nuclei, this new halo structure is composed of an extended low-density distribution of loosely bound valence neutrons (halo) surrounding a core consisting of the majority of the nucleons. With these new radioactive beams, a new degree of freedom, the isospin, is now currently investigated to improve that knowledge and ˇnd new phenomena and properties of the nuclear matter. The structures of these nuclei are found to be different from the earlier known structures of nuclei at or near the β-stability line, and are referred to as halo structures. The 6 He nucleus is the prototypical example of a Borromean two-neutron halo nucleus; that is, the nucleus consists of three subsystems (a tightly bound 4 He core and two valence neutrons) and none of its binary subsystems has a bound state. The two valence neutrons extend well beyond the 4 He core with a separation energy S2n = 0.975 MeV [2]. The observed sudden rise in the measured interaction cross section in these nuclei has been attributed to the corresponding large increase in the nuclear root-mean-square radius [1]. Due to the very small separation energy of the last or the valance nucleons of these nuclei, the correct description of their wave functions plays a crucial role in the theoretical description of the scattering and reaction processes [3]. Considerable experimental and theoretical efforts have been devoted to the understanding of the structure of halo nuclei [3Ä12]. Traditionally, proton scattering has been one of the best means by which the matter densities of the nucleus may be studied. Therefore, in order to investigate the structure of 6 He, several elastic scattering and interaction cross sections measurements have been performed for the p + 6 He reaction at energies of 721 [13], 717 [14], 297 [15], 151 [16, 17], 71 [8, 12, 18], 41.6 [8, 19], 40.9 [8, 20], 38.3 [21], 36 [23] and 24.5 [23, 24] MeV/nucleon. These data have been analyzed either in the framework of the Glauber diffraction theory [25, 26] or using the standard optical model through the single folding (SF) approach based upon the energy- and density-dependent JLM [21, 22, 24], the SBM [27] or the DDM3Y [28] effective nucleonÄnucleon (N N ) interactions. However, due to the low intensities of the available exotic beams, it is only recently that the inelastic scattering and transfer reactions on light particles could be undertaken to probe deeply the structure of these nuclei, i.e., to acquire further insight into the radial density distribution pertinent to these exotic nuclei [29]. The angular distributions of p + 6 He inelastic scattering to the ˇrst 2+ excited state at 1.87 MeV have been measured and analyzed using the SF optical potential at 24.5 and 40.9 MeV/nucleon [24, 20, 30]. In spite of this fair amount of earlier work performed to examine the sensitivity of the elastic scattering data to the physical structure of the exotic helium nucleus, there is not full agreement in the literature to the strength of sensitivity of the elastic protonÄnucleus differential cross section at intermediate energies to the structure calculation of the target nucleus 6 He. On the other hand, spin observables in scattering experiments have been rich sources for understanding nuclear structure and reactions. Recently, the analyzing power of an unstable beam of 6 He on a polarized proton target at an energy of 71 MeV/nucleon was measured for the ˇrst time [31]. It was found that at this energy the polarization changes sign from positive to negative at around 50◦ , which is in contradiction with some theoretical predictions [27, 32]. From the optical model analysis [31], it was implied that the p − 6 He spin-orbit potential might extend to a larger radius compared with the p − 6 Li case. In a recent theoretical study [33] of the same reaction, the elastic differential cross section and analyzing power observables at 297 MeV/nucleon were calculated using the impulse approximation to the single scattering term of the multiple scattering expansion of the optical potential. They Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 365 found that the polarization observable for p + 6 He changes sign from positive to negative at around 30◦ , and that the analyzing power for the p + 4,6 He reactions are very similar. They [33], also, claim that an extended neutron distribution cannot be responsible for the spin-orbit radius. Microscopic models now exist that can predict results of both elastic and inelastic scattering reactions. When good detailed speciˇcation of the nucleon structure of the nucleus is used, those predictions usually agree very well with observations both in shape and in magnitude. Thus, it is evident that a priori information on the halo structure of a nucleus is of vital importance for the theoretical treatment of these weakly bound nuclei [34]. In a very recent study, Uesaka et al. [35] and Sakaguchi et al. [12] presented an accurate measurement of the vector analyzing power for the p + 6 He elastic scattering at 71 MeV/nucleon. There was used, for the ˇrst time, a newly developed polarized proton solid target operated in a low magnetic ˇeld and at high temperature. The angular distribution of the elastic scattering differential cross section was also measured at angular range (42Ä87◦ ) larger than that (20Ä49◦ ) measured in [18]. In order to obtain theoretical reproduction of the observed data, they [12, 35] employed several (phenomenological, semimicroscopic and fully microscopic) optical potential representations. It was concluded that the spin-orbit potential for 6 He is characterized by a shallow and long-ranged shape compared with the global systematic of stable nuclei. This may resemble the diffuse density of the n-rich 6 He nucleus. However, the obtained match to the data, in particular the analyzing power at large angles, was not perfect. This may indicate limitation of the structure model and/or contribution of unaccounted reaction mechanisms that inuence the larger momentum transfer results [12]. The main aim of the present work is to calculate differential cross sections of elastic 6 He + p scattering at different energies studying the possibility to describe the existing experimental data with as minimal number of ˇtting parameters as possible. First, a phenomenological optical potential of square WoodsÄSaxon (WS) potential supplied with a Thomas form for the spin-orbit potential was used to describe the experimental data. Second, the single folding (SF) procedure is used to construct the real part of the optical potential (OP). For the construction of the folded potential, two main ingredients are required: (a) an effective nucleonÄnucleon (N N ) interaction in-medium, allowing for the mean ˇeld as well as Pauli blocking effects; and (b) a credible model of structure for the nucleus that is nucleon-based. For the effective interaction the density- and isospin-dependent M3Y effective interaction is used. For this kind of isospin-dependent effective interaction, the real folded potential receives contributions from both isoscalar and isovector components. Usually, in the usage of the complex optical model potential, for analyses of the differential cross sections, their imaginary part and the spin-orbit terms are determined in a phenomenological way. 1. THEORETICAL FORMALISM Usually, the real part of the nucleonÄnucleus optical potential is assumed to be a result of an SF of the effective N N potential with the nuclear density, i.e., this is a particular case of the double folding (DF) [36] in which a δ(r1 ) function has to be used for the density of the incoming particle ρ1 (r1 ). The beauty of the folding model lies in the fact that it directly links the density proˇle of the nucleus with the elastic scattering cross sections. Formally, the SF 366 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. potential is given as ρ2 (r2 ) υD(EX) (|s|, ρ, E) d3 r2 , V (R) = (1) where s = R − r2 . Exotic nuclei usually have nonzero isospin and it is necessary to make explicit the isospin degrees of freedom. For that reason the present calculations have been performed using υD(EX) (|s|, ρ, E), inside the integral of Eq. (1) for the SF procedure, as the DDM3Y effective [37] interaction given by D(EX) υD(EX) (|s|, ρ, E) = υIS D(EX) D(EX) (|s|, ρ, E) + υIV (|s|, ρ, E), (2) D(EX) (|s|, ρ, E) and υIV (|s|, ρ, E) are the isoscalar and isovector components of where υIS the effective nucleonÄnucleon interaction. A realistic separable energy and density-dependent DDM3Y of the following form has been used: D(EX) υIS D(EX) (|s|, ρ, E) = g(E)FIS D(EX) (|s|, ρ, E) υIV = D(EX) (ρ2 )υ00 (|s|), (3) D(EX) D(EX) g(E)FIV (ρ2 )υ01 (|s|). (4) The explicit radial strengths of the isoscalar (IS) and isovector (IV) components of the M3Y interaction based on the G-matrix of the Reid N N potential are given in the following form [38]: D (|s|) = 7999 υ00 e−2.5|s| e−4|s| − 2134 , 4|s| 2.5|s| EX υ00 (|s|) = −J00 (E) δ(|s|) = −276(1 − 0.005E/AP ), D (|s|) = −4886 υ01 −4|s| (5) (6) −2.5|s| e e + 1176 , 4|s| 2.5|s| EX υ01 (|s|) = J01 (E) δ(|s|) = 228(1 − 0.005E/AP ). (7) (8) Equations (6) and (8) mean that the knock on exchange potential is treated approximately by adding a zero-range pseudopotential [38]. This zero-range approximation has been used with some success in the DF model calculations of the heavy ions (HI) optical potential at low energies [36] where the data are sensitive only to the OP at the surface (near the strong absorption radius), it has been shown to be inadequate [39] in the case of rainbow scattering where the data are sensitive to the real OP over a wider radial domain. The g(E) in Eqs. (3) and (4) represents energy-dependent factor (scaling factor) which takes into account the empirical energy dependence of the nucleonÄnucleus optical potential. This scale factor for Reid effective N N interaction takes the form [37] g(E) = 1 − 0.0025E/AP , (9) D(EX) where E is the incident particle energy, while AP is the projectile mass number. The FIS is the realistic density-dependent factor which is included to reproduce the saturation properties D(EX) is to reproduce the empirical symmetry of symmetric nuclear matter, while the factor FIV energy and so to construct a realistic equation of state for asymmetric nuclear matter. The functional forms of these density-dependent factors are D(EX) FIS,IV (ρ2 ) = CIS,IV (1 − γρ), (10) Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 367 CIS = 1.2253, CIV = 0.7597, γ = 1.5124 fm3 . Through this density dependence the DDM3Y is denoted as BDM3Y1. From Eq. (1) to (10) the direct part of the Re-OP (VD ) has the following form of the IS and IV contributions, correspondingly: D D VIS (R) = g(E) ρ2 (r2 )F (ρ2 )υ00 (|s|) d3 r2 , (11) D D (R) = g(E) δρ2 (r2 )F (ρ2 )υ01 (|s|) d3 r2 , (12) VIV ρ2 (r2 ) = ρ2,p (r2,p ) + ρ2,n (r2,n ), δρ2 (r2 ) = ρ2,p (r2,p ) − ρ2,n (r2,n ). (13) (14) Here ρ2,p (r2,p ) and ρ2,n (r2,n ) are the proton and neutron densities in the target nucleus. We consider a density for 6 He, which is described with a realistic wave function obtained by the variational Monte Carlo (VMC) wave function used in [14, 40]. This density is composed of ten Gaussian terms as 10 (Pk + Nk ) exp (−Ak · r2 ). (15) ρ(r) = k=1 The parameters Pk , Nk and Ak are listed in Table 1. The corresponding rms radii from the VMC wave function density are 2.56, 1.96 and 2.81 fm for nucleon, proton and neutron distributions, respectively. Table 1. Parameters of the VMC density in Eq. (15) k Pk Nk 1 Ä4.777580124879105· 10−3 Ä7.022185461489483· 10−3 4.0 2 0.929250185852335 · 10−2 2.859012640320818 · 10−2 2.56 0.579676566914048 · 10−2 1.6384 3 Ä0.166091230435732· 10 −2 Ak −2 4 0.230728830390548 Ä5.484897586013483· 10 5 Ä0.177513962911145 9.683921866054336 · 10−2 6 0.17863413483804 Ä4.977104280767115· 10 7 Ä4.037779402389877· 10−2 0.101382894392589 −2 8 2.248885252174397 · 10 9 Ä6.644139893014976· 10−3 10 1.058642564729591 · 10 −3 −2 0.67108864 0.4294967296 0.274877906944 Ä2.450757473603717· 10 −2 1.667264722270956 · 10−2 7.720490110559399 · 10 1.048576 −4 0.17592186044416 0.112589990684262 0.205759403792794· 10−2 Using Eqs. (1)Ä(14), one can obtain the following forms of the direct part of the IS Re-OP expressed by integrals in the coordinate and momentum space, correspondingly: D D VIS (R) = CIS g(E) [ρ2 (r2 ) − γ ρ2 (r2 )] υ00 (|s|) d3 r2 , (16) D (R) VIS CIS g(E) = 2π 2 ∞ D [ρ2 (q) − γ ρ2 (q)] υ00 (q)j0 (qr) q 2 dq, 0 (17) 368 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. where ρ˜2 (r2 ) is given as ρ˜2 (r2 ) = ρ22 (r2 ). (18) Similarly, exchanging ρ2 (r2 ) by δρ2 (r2 ) (Eq. (14)), one can obtain the IV part direct part of Re-OP as D D (R) = CIV g(E) [δρ2 (r2 ) − γδ ρ˜2 (r2 )] υ01 (|s|) d3 r2 , VIV D (R) = VIV CIV g(E) 2π 2 D VIV of the (19) ∞ D [δρ2 (r2 ) − γδ ρ˜2 (r2 )] υ01 (q) j0 (qr) q 2 dq, (20) 0 where δ ρ˜2 (r2 ) is given as δ ρ˜2 (r2 ) = [δρ2 (r2 )]2 . The Fourier transforms of ρ2 (r2 ), δρ2 (r2 ), ρ˜2 (r2 ), δ ρ˜2 (r2 ), from the following relation: and D υ01 (|s|) are given ∞ f (q) = (21) D υ00 (|s|) e iqr 3 f (r) j0 (qr) r2 dr. f (r) d r = 4π (22) 0 The j0 (qr) is the spherical Bessel of order zero. The exchange part of the Re-OP Eqs. (6) and (8) makes it easy to compute in coordinate space because of the presence of delta functions. 2. RESULTS AND DISCUSSION We perform our calculations on the cross section and analyzing power (at 71 MeV) of p + 6 He elastic scattering using phenomenological optical model potentials and the single folded potential. A search for the phenomenological nuclear potential parameters as well as for the normalization parameter for the single folded potential is carried out using the optical model code HERMES [41]. Best ˇts are obtained by minimizing χ2 , where 2 N 1 σcal (θi ) − σexp (θi ) 2 , (23) χ = N i=1 Δσexp (θi ) where σcal (θi ) and σexp (θi ) are the calculated and experimental cross sections, respectively, at angle, θi Δσexp (θi ) is the experimental error and N is the number of data points. An average value of 10% is used for the experimental errors of all considered data. 2.1. Phenomenological Analysis. Optical-model analysis of proton elastic scattering from 6 He has been carried out for 8 sets of scattering data at energies of 24.5, 25.0, 36.2, 38.3, 40.9, 41.6, 71.0, 82.3 MeV [40], respectively. These data have, in general, been analyzed in terms of an optical model in which the interaction is represented as the scattering of a point particle (proton) by a potential of the standard form, d M F (R)+ dR i 2 dFso (R) Lσp + Uc (R), (24) + [Vso + iWso ] R dR Uop (R) = −V 0 FrN (R) − iWiv FiM (R) + 4iaW is Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 369 where the ˇrst term is volume real part of the optical potential U (R) = −V 0 FrN (R), the second part is the imaginary potential and usually represented by two terms, volume Wv (R) = d M Wiv FiM (R) and surface Ws (R) = 4aWis F (R) ones. The third part is the spin-orbit dR i potential (real and imaginary) and in general, it takes the Thomas form. The functional form −j R − Rk j j of the radial form factors Fk (R) are usually of WS form, Fk (R) = 1 + exp , ak 1/3 Rk = rk A (k = r for real, i for imaginary, so for spin-orbit potentials, respectively), raised to power j. The L is the relative angular momentum between the proton and 6 He nucleus and σp is the Pauli spin operator of the proton. The last part, Uc (R), is the Coulomb potential due to a uniform distribution of appropriate size (radius Rc = r0c A1/3 ) and total charge, ⎧ 2 2e ⎪ ⎪ ⎪ , R > Rc , ⎪ ⎨ R Uc (R) = (25) 2 ⎪ R 2e2 ⎪ ⎪ , R Rc , ⎪ ⎩ 2Rc 3 − Rc r0c is ˇxed at 1.3 fm. We aim in the present analysis to get or extract OP for p + 6 He elastic scattering over the considered energy range. For this purpose the usual WS and square WS potentials for the real and imaginary potentials supplied with spin-orbit potential of Thomas form are used. The OP of the usual WS is denoted as Set-1. In this set the shape parameters of both the real and imaginary parts are different and ˇxed with energy. The OP denoted as Set-2 is of square WS form. The shape parameters of this set are also different for both real and imaginary potentials and ˇxed with energy as Set-1. The OP denoted as Set-3 is of square WS form but the shape parameters of both the real and imaginary potentials are the same. The shape parameters of spin-orbit potential for these sets are chosen according to the best ˇtting of the analyzing power data at 71 MeV incident energy. The shape parameters of spin-orbit potential of Set-1 and Set-2 are rso = 1.248 fm and aso = 0.910 fm. For Set-3 these parameters take the values rso = 1.301 fm and aso = 1.032 fm. In Fig. 1, the calculations of the observables made with the OPs of Sets-1, 2, and 3 are shown together with the experimental data. The results of these calculations are collected in Table 1. Calculations with Sets-1, 2 and 3 are shown in Fig. 1 by solid, dashed and dash-dotted lines, respectively. The calculations with all the potential sets reproduce both dσ/dΩ and Ay at 71 MeV incident energy over the whole angular region except for the most backward data point of Ay . In these calculations, real spin-orbit potentials are used in Sets-1 and 2 where real and imaginary spinorbit potentials are used in Set-3. From Table 1 it is shown that addition of surface imaginary potential for the three sets is needed to reproduce the data. Also, it is shown that the real, imaginary and spin-orbit potentials have no clear energy dependence. This may be attributed to the effect of the breakup of the 6 He nucleus or to the enhancement of the coupling to the continuum which leads to a greater inuence on the nuclear OP of p + 6 He system [21]. The calculations based on Sets-1 and 2 result in appreciable similar dσ/dΩ and Ay data as shown in Fig. 1. The calculations based on Set-3 gave dσ/dΩ and Ay different from that of Sets-1 and 2. The calculations of Sets-1 and 2 are near to reproduce the data more than that of Set-3. From all of these calculations it is shown that the phenomenological optical model analyses suggest that the Ay data can be reproduced with a shallow and long-ranged spin-orbit 370 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. Fig. 1. The angular distribution of elastic p + 6 He scattering differential cross section, σ, with respect to Rutherford (Coulomb) cross section, σR , at 24.5, 25.0, 36.2, 38.3, 40.9, 41.6, 71.0, 82.3 MeV deduced using phenomenological WS potentials in comparison with measured data Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 371 potential, as is clear from Table 1. The phenomenological analysis indicates that the spinorbit potential between a proton and 6 He is characterized by long-ranged radial dependence. Intuitively, these characteristics can be understood from the diffused density distribution of 6 He. Also, from the table it is clear that the range of the spin-orbit potential for p + 6 He scattering is larger than that of the real and imaginary potentials. 2.2. Single Folding Analysis. Usually the real part of the optical model potential is obtained using the folding model. In this calculation, the SF procedure (Eq. (1)) is used to obtain the real part of the OP. The imaginary part is treated phenomenologically either by using the WS forms or by normalizing the SF potential by an imaginary normalization factor Ni . The spinorbit part is also treated phenomenologically by using the usual Thomas form. The elastic scattering data of p + 6 He system have been analyzed using this real folded potential over the energy range considered above. The calculations based on the real SF potential supplied with imaginary potential of WS form are denoted as SFWS1 in both Fig. 2 and Table 2. The calculations based on real folded and imaginary potential of square WS are denoted as SFWS2 and SFWS3. In Table 2 it is shown that the shape parameters of the spin-orbit potentials of SFWS1 and SFWS3 are the same and different from that of SFWS2. The shape parameters of spin-orbit potentials used with SFWS1, 3 are rso = 1.248 fm, aso = 0.910 fm with root2 1/2 = 3.612 fm. The shape parameters of spin-orbit potentials mean-square (rms) radius rso 2 1/2 used with SFWS2 are rso = 1.118 fm, aso = 1.134 fm with rms radius rso = 3.990 fm. From the ˇgures it is seen that all the potential sets (SFWS1, 2 and 3) reproduce the elastic scattering data equally well except at 38.3, 40.9 and 82.3 MeV. For the energy 38.3 MeV the calculations based on the three sets are very similar up to an angle of around 57◦ and then deviate. The set SFWS1 gives the best result since it is within the experimental errors over the whole angular range considered. For the energy 40.9 MeV, the results of SFWS1 and SFWS3 are very similar and reproduce the data very well over the whole angular range. The difference between the results of SFWS1, 3 and those of SFWS2 starts at an angle of around 32◦ . For the energy 82.3 MeV, SFWS2, 3 potentials give very similar results and start to deviate from that of SFWS1 at angle of around 22◦ . For the other energies all the three sets give very similar results and reproduce the data nicely over the considered angular range. The experimental data of the analyzing power Ay are presented besides those of differential cross sections at the energy 71.0 MeV. This is considered as a good test for the considered potential. From Fig. 2 it is seen that the SFWS1 and SFWS3 results are near to reproduce the experimental data over the whole angular range, except for most backward data points of Ay at 71 MeV, more than those of SFWS2. As another alternative for the imaginary potential, the folded potential is used and normalized by an imaginary normalization factor Ni . The spin-orbit potential is treated as usual by using the Thomas form. The calculations based on this imaginary folded potential are denoted as SFRI in Fig. 3. The best ˇtting parameters of these calculations are collected in Table 3. It is found that a spin-orbit potential of shape parameters, rso = 1.362 fm, aso = 0.786 fm 2 1/2 = 3.479 fm, gives the best results. As observed in Fig. 3, the SFRI and of rms radius rso gives satisfactory results of dσ/dΩ over the considered energy range. For Ay , the results of SFRI are within the experimental errors except for the last two angles. Finally, for consistency, the spin-orbit term is taken extracted from the SF potential as Vso (R) = (Nrso + iNiso ) 2 dV (R) , R dR (26) 372 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. Fig. 2. Same as Fig. 1, but using the derived SF real potentials Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 373 Table 2. Phenomenological optical potential ˇtting parameters obtained using Eq. (23) for p + 6 He elastic scattering using WS central real and imaginary potentials and spin-orbit term of Thomas form. Dx and Dxx correspond to the depths of the potential (V0 , Wiv , Vso and iWso ) and Wis , respectively. Same deˇnition for radius and diffuseness parameters (Rx and ax ) in fm, real and imaginary volume integrals (Jx , x = 0, I and So) in MeV · fm3 , total reaction cross section (σR ) 1/2 in mb and rms radii rx2 in fm E, MeV Set Set-1 24.5 Set-2 Set-3 Set-1 25.0 Set-2 Set-3 Set-1 36.2 Set-2 Set-3 Set-1 38.3 Set-2 Set-3 Potential Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Dx , MeV 37.303 14.374 2.154 39.212 16.422 3.977 45.64 11.507 2.459 1.163 39.153 18.957 0.279 43.577 16.652 0.775 45.64 11.715 2.459 1.163 39.615 3.988 0.133 40.366 17.499 2.500 40.322 11.559 2.459 1.163 38.826 32.206 0.197 41.632 23.251 0.544 40.322 11.559 2.459 1.163 Dxx , MeV Å 5.317 Å Å 0.584 Å Å 2.971 rx , fm 1.115 0.850 1.248 1.455 1.298 1.248 ax , fm 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 Å 1.301 1.032 Å 4.635 Å Å 1.024 Å Å 2.918 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 Å 1.301 1.032 Å 10.988 Å Å 1.078 Å Å 2.463 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 Å 1.301 1.032 Å 0.799 Å Å Ä1.105 Å Å 2.463 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 1.301 1.032 Å Jx , rx2 1/2 , 3 MeV · fm fm 467.2 3.003 245.7 3.084 21.11 3.612 408.7 2.907 205.4 3.621 38.98 3.612 475.7 2.907 243.7 3.145 25.38 3.964 12.00 490.4 3.003 267.3 3.052 2.738 3.612 454.2 2.907 230.3 3.657 7.600 3.612 475.7 3.907 243.6 3.140 25.38 3.964 12.00 496.2 3.003 295.6 3.198 1.307 3.612 420.7 2.907 242.6 3.657 25.50 3.612 420.3 2.907 223.1 3.123 25.38 3.964 12.00 486.3 3.003 284.9 2.941 1.926 3.612 433.9 2.907 190.0 3.418 5.332 3.612 420.3 2.907 223.1 3.123 25.38 3.964 12.00 σR , mb 414.5 401.5 421.3 428.6 434.6 417.7 404.0 383.6 330.0 366.0 309.8 330.0 374 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. The end of Table 2 E, MeV Set Set-1 40.9 Set-2 Set-3 Set-1 41.6 Set-2 Set-3 Set-1 71.0 Set-2 Set-3 Set-1 82.3 Set-2 Set-3 Potential Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Real Imag. Re-SO Real Imag. Re-SO Real Imag. Re-SO Im-SO Dx , MeV 39.494 Ä4.650 1.334 44.193 0.766 0.093 45.922 7.007 2.459 1.163 35.434 Ä3.265 5.491 43.538 40.050 6.428 41.781 11.559 2.459 1.163 24.991 23.243 2.643 28.808 31.918 3.054 29.626 11.583 2.459 Ä0.613 23.666 41.157 2.500 30.217 53.275 0.241 28.081 8.773 2.459 1.163 Dxx , MeV Å 10.505 Å Å 3.318 Å Å 2.875 rx , fm 1.115 0.850 1.248 1.455 1.298 1.248 ax , fm 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 Å 1.301 1.032 Å 22.329 Å Å Ä0.006 3.019 Å 3.875 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 Å 1.301 1.032 Å Ä1.068 Å Å Ä4.239 Å Å Ä0.737 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 1.301 1.032 1.115 0.850 1.248 1.455 1.298 1.248 0.690 0.719 0.910 1.053 1.500 0.910 1.455 1.053 1.301 1.032 Å Ä0.469 Å Å Ä4.987 Å Å 2.875 Å Jx , rx2 1/2 , 3 MeV · fm fm 494.7 3.003 212.8 3.284 13.08 3.612 460.6 2.907 181.2 3.958 0.914 3.612 478.7 2.907 192.8 3.195 25.38 3.964 12.00 443.8 3.003 506.9 3.246 53.82 3.612 453.8 2.907 426.2 3.556 63.00 3.612 435.5 2.907 281.8 3.174 25.38 3.964 12.00 313.0 3.003 166.3 2.868 25.90 3.612 300.3 2.907 118.9 2.602 29.93 3.612 308.8 2.907 90.04 2.735 25.38 3.964 Ä6.322 296.4 3.003 328.5 2.907 24.50 3.612 315.0 2.907 307.3 3.158 2.360 3.612 292.7 2.907 211.2 3.171 25.38 3.964 12.00 σR , mb 322.7 308.8 300.2 496.9 494.2 370.8 194.7 148.8 120.3 286.5 293.4 230.8 where the form Fso (R) is replaced by the folded potential given by Eq. (1) and the real and imaginary depths Vso and Wso are replaced with the normalization factors Nrso and Niso , respectively. By using this procedure, the total optical potential takes the form Uop (R) = −(Nr + Ni ) V (R) + (Nrso + iNiso ) 2 dV (R) + Uc (R). R dR (27) Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 375 Fig. 3. Same as Fig. 1, but for the derived complex potentials 376 Mahmoud Zakaria M. M., Ibraheem Awad A., El-Azab Farid M. Table 3. Optical model best ˇt parameters for p + 6 He elastic scattering using central real folded and WS imaginary potentials with spin-orbit term of Thomas form E, PotenMeV tial SFWS1 24.5 SFWS2 SFWS3 SFWS1 25.0 SFWS2 SFWS3 SFWS1 36.2 SFWS2 SFWS3 SFWS1 38.3 SFWS2 SFWS3 SFWS1 40.9 SFWS2 SFWS2 SFWS1 41.6 SFWS2 SFWS3 SFWS1 71.0 SFWS2 SFWS3 SFWS1 82.3 SFWS2 SFWS3 Nr 1.088 1.034 1.069 1.163 1.178 1.175 1.190 1.136 1.127 1.193 1.044 1.000 1.215 1.218 1.123 1.109 1.187 1.085 0.780 0.841 0.783 0.949 0.883 0.854 Wi , MeV 14.193 15.716 22.319 23.406 17.492 11.367 4.242 16.656 7.921 33.152 14.572 0.172 Ä5.644 15.000 Ä8.893 5.223 17.861 28.178 13.241 39.927 24.332 Ä24.58 68.706 68.027 WD , MeV 5.426 2.038 0.708 3.153 2.230 3.583 10.582 2.555 4.365 0.122 1.294 4.709 10.784 1.648 6.070 17.988 5.670 3.667 Ä1.587 Ä5.018 Ä3.282 20.631 Ä6.876 Ä6.844 Wsr , MeV 4.044 4.512 1.533 0.141 0.108 0.205 2.518 0.924 2.266 0.745 2.500 0.861 1.160 2.500 2.127 6.028 1.670 7.175 4.294 3.578 4.276 0.318 Ä0.279 0.947 Wsi , Jr , Ji , Jsr , MeV MeV · fm3 MeV · fm3 MeV · fm3 1.059 466.5 259.9 39.64 1.608 443.4 240.8 41.66 0.827 458.3 243.9 15.02 0.002 497.7 282.6 1.386 Ä0.001 504.0 266.3 0.995 0.452 502.7 268.8 2.007 1.017 490.6 303.6 24.68 0.157 468.4 273.0 8.530 1.067 464.5 271.1 22.21 0.968 488.1 290.7 7.304 2.335 427.5 230.4 23.08 1.366 409.4 212.9 8.435 1.642 492.7 222.9 11.37 0.009 494.2 216.5 23.08 0.356 455.7 187.8 20.85 1.004 448.8 498.8 59.08 Ä0.118 480.4 424.2 15.42 1.739 439.0 432.4 70.32 0.291 283.4 74.88 42.09 Ä0.034 305.6 154.4 33.04 0.527 284.3 84.01 41.91 0.885 329.8 306.8 3.116 0.198 306.6 344.5 Ä2.575 0.231 296.6 339.5 9.282 Jsi , MeV · fm3 10.38 14.85 8.110 0.0152 Ä0.0013 4.432 9.971 1.453 10.45 9.483 21.56 13.39 16.09 0.085 3.493 9.835 Ä1.093 17.05 2.855 Ä0.317 5.167 8.676 1.830 2.266 σR , mb 429.8 432.4 430.2 440.9 460.9 469.3 412.8 404.3 409.7 371.8 348.6 342.3 332.0 334.0 312.9 496.7 497.5 489.0 103.5 177.0 110.9 313.1 293.4 290.2 The parameters Nr , Ni , Nrso and Niso are treated as variational parameters to reproduce the experimental cross section. The results of this procedure are denoted as SFRISO and shown by dashed line. The best ˇtting parameters obtained are collected in Table 5. The rms radius of the obtained spin-orbit potential using Eq. (25) at the energy 71.0 MeV is 3.061 fm. Comparing this value with that obtained using spin-orbit potential of phenomenological Thomas form, we note that the spin-orbit potential of Eq. (25) has a shorter range. So, to reproduce the experimental data by using this procedure, a correction term should be added to the folded spin-orbit potential [12, 35]. Also, a correction term could be added at least to the imaginary folded potential to simulate this effect. It is commonly surmised that, because 6 He is weakly bound, breakup has a large effect on the elastic scattering channel and is responsible for the reduction of Nr below unity. This effect can be represented by a dynamic polarization potential (DPP) which has a strongly repulsive real part in the surface and an additional absorptive (imaginary) part [42]. If the contribution from the DPP were simulated by a surface correction using splines added to both the M3Y and DDM3Y real DF potentials, Khoa et al. [42] could obtain successful descriptions of the 6 Li + 12 C elastic scattering data without using a normalizing factor, i.e., NR = 1, all over the energy range E = 10−53 MeV/nucleon. Optical Model Analysis of p + 6 He Scattering over a Wide Range of Energy 377 Table 4. Optical model ˇtting parameters for p + 6 He elastic scattering using central folded real and imaginary potentials with spin-orbit term of Thomas form E, MeV 24.5 25.0 36.2 38.3 40.9 41.6 71.0 82.3 Nr Ni 1.089 1.098 1.068 1.163 1.151 1.364 0.859 0.978 0.610 0.649 0.698 0.672 0.564 1.433 0.161 0.573 Wsr , MeV 3.315 7.159 3.130 0.588 7.715 2.416 2.744 3.338 Wsi , MeV 0.131 3.802 0.401 0.771 0.457 1.724 Ä0.335 0.905 Jr , Ji , Jsr , Jsi , MeV · fm3 MeV · fm3 MeV · fm3 MeV · fm3 467.1 261.4 34.83 1.381 470.0 277.7 75.21 39.94 440.3 287.6 32.88 4.214 476.0 275.1 6.176 8.101 466.8 228.9 81.05 4.804 551.8 580.0 25.38 18.11 312.0 58.39 28.82 Ä3.517 339.8 199.0 30.82 8.357 σR , mb 418.1 418.6 377.1 363.9 323.4 499.1 84.77 218.7 Table 5. Optical model best ˇt parameters for p + 6 He elastic scattering using central folded real, imaginary and spin-orbit potentials E, MeV 24.5 25.0 36.2 38.3 40.9 41.6 71.0 82.3 Nr Ni Nrso Niso 0.981 1.178 1.180 1.259 1.046 1.039 0.871 0.973 0.661 0.970 0.792 1.383 0.762 1.731 0.170 0.664 0.382 0.220 0.009 0.271 0.541 0.542 0.077 0.023 0.053 0.329 Ä0.004 0.501 0.119 0.342 0.004 0.128 Jr , Ji , Jrso , Jiso , MeV · fm3 MeV · fm3 MeV · fm3 MeV · fm3 420.7 283.5 121.5 16.79 504.0 415.3 69.68 104.3 486.6 326.5 2.687 Ä1.135 515.5 566.1 79.29 146.6 424.4 308.9 155.7 34.20 420.4 700.5 155.4 97.92 316.4 61.76 17.87 1.003 337.8 230.7 4.803 27.11 σR , mb 431.6 449.4 405.5 407.2 379.0 499.6 90.20 231.8 The obtained total reaction cross sections, σR , listed in Tables 2Ä4 for both the phenomenological and microscopic calculations are compared with only one available experimental value measured at 36.2 MeV/nucleon, (410 ± 21) mb [22, 44]. In general, σR decreases almost linearly as energy increases. It is clear that the values of σR corresponding to the SF calculations are more consistent with the measured value than the corresponding value related to the WS one. Unfortunately, no other reported values of σR , at the energy range considered in this work, in previous studies are found to be compared with our results. From this calculation it is expected that the addition of surface imaginary potential to the volume folded one may give better results than those obtained. CONCLUSIONS The SF optical potentials are generated based on the deduced density and the BDM3Y1 effective N N interaction. Eight sets of p + 6 He elastic scattering data at energies of 24.5, 25.0, 36.2, 38.3, 40.9, 41.6, 71.0 and 82.3 MeV are analyzed using both the derived real potentials and the phenomenological WS potentials in the framework of the DWBA mechanism. Successful reproductions of the data are obtained using the generated potentials. 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