Analysis of strong refractive effect within 11Li projectile structure

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Kassem O. Behairy, M. El-Azab Farid, Awad A. Ibraheem, Ola. Ramadan and M. Anwar. Analysis of strong refractive effect within 11Li projectile structure[J]. Chinese Physics C.
Kassem O. Behairy, M. El-Azab Farid, Awad A. Ibraheem, Ola. Ramadan and M. Anwar. Analysis of strong refractive effect within 11Li projectile structure[J]. Chinese Physics C. shu
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Analysis of strong refractive effect within 11Li projectile structure

  • 1. Physics Department, Aswan University, Aswan 81528, Egypt
  • 2. Physics Department, Assiut University, Assiut 71516, Egypt
  • 3. Physics Department, King Khalid University, Abha, Saudi Arabia
  • 4. Physics Department, Al-Azhar University, Assiut 71524, Egypt

Abstract: In the context of the double folding optical model, the strong refractive effect for elastic scattering of 11Li + 12C, 11Li +28Si systems at incident energy 29, 50 and 60 MeV/n has been studied. Real folded potentials are generated based on a variety of nucleon-nucleon interactions with the suggested density distribution for the halo structure of 11Li nuclei. The rearrangement term (RT) of the extended realistic density dependent CDM3Y6 effective interaction is considered. The imaginary potential was taken in traditional standard Woods-Saxon form. Satisfactory results for the calculated potentials are obtained, with a slight effect of the RT in CDM3Y6 potential. Successful reproduction with normalization factor closed to one for the observed angular distributions of the elastic scattering differential cross section has been achieved using the derived potentials. The obtained reaction cross-section is studied as a guide by extrapolating our calculations and previous results.

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    I.   INTRODUCTION
    • The study of halo nuclei which are far from the valley of β-stability and close to the nucleon drip-line has become one of the main topics in modern nuclear structure physics. This halo phenomenon for exotic nuclei has created much interest and many hundreds of papers since its discovery in the mid-1980s. The halo of nuclei is arising from the very weak binding of the last one or two valence (protons or neutrons) to the core containing all other nucleons which are tightly bound, this weak binding energy leads to the creation of neutron/proton tail. These halo nuclei are known by small separation energy as a result of the presence of a halo neutron/proton tail, extended nucleon density distributions or quite large root mean square (rms) radii and the emitted nucleons have narrow momentum distributions [1-3]. It has been proven that there are two sorts of halo nuclei: the neutron and proton halos, which is depending on the type of nucleon that arrangements the halo structure. Nuclei that have one proton halo include 8B and 26P, and that have a one neutron halo are 11Be and 19C. While nuclei that have a two proton halo are exhibited by 17Ne and 27S [4-6], and for a two neutron halo include 6He, 11Li, 17B, 19B, 22C, etc.[7-11]. These halo nuclei are consisting of three body systems (an inert core nucleus plus two valence neutrons or two valence protons).

      The exotic 11Li nuclei have attracted much theoretical and experimental interest as a typical case, where the first production of it was described by Postanzer et al. in 1966 [12] and some special properties such as its large matter radius were discovered by Tanihata et al. [1, 13]. 11Li nucleus has numerous interesting properties such as; i) It is consist of a core of three protons and six neutrons, and a halo of two loosely bound neutrons, where 11Li belongs to the three-body system (9Li+n+n) which is called as Borromean system [14, 15]. ii) Small separation energy (S2n) of the two-neutron halo is only S2n= 300 KeV [1] and in new research, it is found to be S2n= 378±5 KeV [15, 16], this small separation energy leads to an increase the matter radius of 11Li comparative to other Li isotopes and forming a low-density halo, where nuclear radii increase with the mass number A of a nucleus as A1/3 and it was found that the matter radius of 11Li nucleus largely differs from the A1/3 law whereas other Li isotopes follow this law exactly [16]. iii) Quite large rms compared to that of 9Li (2.32±0.02) fm [13] that was found to be (3.12±0.16) fm [1, 13] and latest measurements propose that it might be as large as (3.55±0.10) fm [2], where the size of 11Li nucleus is approximately equal to the size of 41Ca and as large as 208Pb [16, 17], this means that two halo neutron are mostly placed far from the 9Li core. iv) It Cannot be used as a target because it has a very short half-life time of 8.2 ms [18] for β- decay. All these properties made the nucleus of 11Li are an interesting halo nucleus that can be used as a projectile on several targets for studying the structure of it.

      In the last decades, there are a lot of researches has generated widespread interest and excitement for the interaction cross section of the 11Li nucleus as a projectile with several different targets [19-27]. The quasi-elastic scattering of 11Li on 12C was measured by Kolata et. al. [20] at 60 MeV/n, and on 28Si at 29 MeV/n by Lewitowicz et. al. [21]. They used the couple-channels (CC) method to reproduce the data, where the energy resolution did not allow for the separation of the true elastic scattering from inelastic scattering. The recent experiment for 11Li on 12C is measured by Peterson, et.al. [22] at 50 MeV/n with their success for clearly separate the elastic and inelastic reaction channels in the critical forward-angle region. All the previous experimental studies [20-22] of quasielastic scattering of the neutron-halo nucleus of 11Li with low-Z targets have reported and confirmed the strong refractive effect of this scattering process as first predicted by Satchler et. al. [23]. This refractive has very long-range absorption due to 11Li breakup, which increases the reaction cross section, has been needed to reproduce the experimental data. Mermaz [24] showed that it was necessary to use a surface potential peaked at very far outside of the nucleus in order to reproduce the forward interference minimum core. The necessity of adding this long tail that was introduced by Mermaz to the real part of the optical model potential to fit 11Li on 12C and 28Si has been verified by via S-matrix inversion techniques for Cooper and Mackintosh [25]. Several studies [19, 26, 27], treated the refractive long tail using the dynamic polarization potential (DPP), generated by the strong coupling of the breakup channels with the elastic one for 11Li scattering. Khalili [27], proposed a theoretical interpretation of the data using Faddeev three-body wave functions within a four-body Glauber model but the results are not sensitive to details of the 11Li wave function. The problem is that the required tail is of a much greater extent than predicted by simple double folding (DF) model calculations and DPP corresponding to these models are repulsive than attractive in nature.

      Recently, the São Paulo potential (SPP) [28-31] has been successful in describing the elastic scattering and peripheral reaction channels for a large number of heavy-ion systems in a very wide energy region, besides, to describe the total reaction. Also, it has been used for analyzing the elastic scattering of stable, weakly bound and exotic nuclei on a variety of targets [32]. In otherwise, khoa et. al. [33] have generated the rearrangement term (RT) of the nucleon optical potential (OP) to modify the density and energy dependence of the CDM3Yn interactions, which it has succeeded in reproducing the Airy oscillation.

      In this respect, we reanalyze the elastic scattering of 11Li within a two-body model (core+halo) on 28Si at 29 MeV/n and on 12C at 50 and 60 MeV/n in an attempt to get an interpretation for strong refractive effect within 11Li projectile structure. Our model is based upon semimicroscopic approach. The real part is calculated microscopically by folding several versions of M3Y effective NN interaction (DDM3Y and CDM3Y6) in addition to the SPP. Three different densities for 11Li nucleus in conjunction with one density for 12C and 28Si targets are used. The imaginary part is calculated in the phenomenological Wood-Saxon (WS) form. Moreover, the effect of RT in cross section calculations is investigated. The present paper is planned as follows: theoretical formalism is presented in Sec. 2 while Sec. 3 is showing the results of analysis and discussion. Finally, concluding remarks are summarized in Sec. 4.

    II.   THEORETICAL FORMALISM
    • The optical nucleus-nucleus potential used in the present work is given by:

      $ U\left(R\right)={U}_{C}\left(R\right)+{N}_{R}{V}_{DF}\left(R\right)+i{W}_{v}\left(R\right), $

      (1)

      where, $ {U}_{C}\left(R\right) $ is the Coulomb potential $ {V}_{DF}\left(R\right) $ is the DF real potential and Wv(R) is the imaginary potential in WS form. The real DF potential is calculated as,

      $ {{\rm{V}}_{{\rm{DF}}}}\left( R \right) =iint {\rho _{\rm{P}}}\left( {\overrightarrow {{r_1}} } \right){\rho _{\rm{T}}}\left( {\overrightarrow {{r_2}} } \right){v_{NN}}\left( S \right){\rm{d}}\overrightarrow {{r_1}} {\rm{d}}\overrightarrow {{r_2}} $

      (2)

      here $ {\rho }_{P}\left({r}_{1}\right) $, $ {\rho }_{T}\left({r}_{2}\right) $ are the nuclear matter density of the projectile and the target, respectively while $ {v}_{NN}\left(S\right) $ is the effective NN interaction between two nucleons, $ S=\overrightarrow{R}-\overrightarrow{{r}_{1}}+\overrightarrow{{r}_{2}}. $

      $\begin{aligned}[b] {U_C}\left( R \right) =& \frac{{{Z_1}{Z_2}{e^2}}}{{2{R_C}}}\left( {3 - \frac{{{R^2}}}{{{R_C}}}} \right)\;\;\;\;{\rm{for}}\;\;\;\;R \leqslant {R_C},\\{\rm{or}}&\;\;\frac{{{Z_1}{Z_2}{e^2}}}{{{R_C}}}\;\;\;\;{\rm{for}}\;\;\;\;R \geqslant {R_C} \end{aligned}$

      (3)

      and the phenomenological imaginary WS potential is defined as,

      $ W\left(R\right)=\frac{{W}_{0}}{1+exp\left[\frac{R-{R}_{i}}{{a}_{i}}\right]} $

      (4)

      W0, Ri and ai are the depth, the radius and the diffuseness of the potential, respectively. $ {R}_{i}={r}_{i}\left({A}_{p}^{\frac{1}{3}}+{A}_{T}^{\frac{1}{3}}\right), $ where, i refers to V, W and AP, AT are the projectile and target mass number

    • A.   Nuclear matter density distributions

    • For the projectile nucleus, three different densities distributions are used, all these densities consider the structure of 11Li nucleus is consist of 9Li as the core and two halo neutrons. The first one is the cluster-orbital shell model approximation (COSMA) which can be written as [34],

      $\begin{aligned}[b] \rho \left(r\right)=&{N}_{CX}\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-{r}^{2}/{a}^{2}\right)}{{\pi }^{\frac{3}{2}}{a}^{3}}+{N}_{VX}\frac{2\mathrm{e}\mathrm{x}\mathrm{p}\left(-{r}^{2}/{a}^{2}\right)}{3{\pi }^{\frac{3}{2}}{b}^{5}}\\&\times\left[A{r}^{2}+\frac{B}{{b}^{2}}{\left({r}^{2}-\frac{3}{2}{b}^{2}\right)}^{2}\right], \;\; X=Z,N \end{aligned}$

      (5)

      where, Z and N refers to the atomic number (protons) and neutrons, respectively, and NCX and NVX are fixed numbers, where NCX refers to the number of protons and neutrons in the core nucleus while NVX refers to the number of the neutrons in the halo term. since, A= 0.81, B = 0.19, $ {N}_{CZ} $= 3, $ {N}_{CN} $ = 6, $ {N}_{VZ} $= 0, $ {N}_{VN} $= 2, $ a $ = 1.89 fm and $ b $ = 3.68 fm [34].

      The second density distribution is taken in the semi-phenomenological density (SPD) form, where the total matter density distribution can be taken as,

      $ \rho \left(r\right)={\rho }_{n}\left(r\right)+{\rho }_{p}\left(r\right) $

      (6)

      and,

      $ {\rho }_{n}\left(r\right)={\rho }_{core}\left(r\right)+{\rho }_{tail}\left(r\right) $

      (7)

      Since, $ {\rho }_{core}\left(r\right) $ is the density of the neutrons in the core nucleus while, $ {\rho }_{tail}\left(r\right) $ represents the 2n-halo neutrons density distribution. The core part of both neutrons and protons density distributions can be written in the following expression [35, 36],

      $\begin{aligned}[b] {\rho }_{i}\left(r\right)=&\frac{{\rho }_{i}^{0}}{1+{\left[\left(\frac{{1+\left(\dfrac{r}{R}\right)}^{2}}{2}\right)\right]}^{{\alpha }_{i}}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(\dfrac{\left(r-R\right)}{{a}_{i}}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(\dfrac{-\left(r+R\right)}{{a}_{i}}\right)\right]},\\ i=&p,n \end{aligned}$

      (8)

      where p stands for the protons and n for neutrons. The central densities $ {\rho }_{p}^{0} $ and $ {\rho }_{n}^{0} $ are determined from the normalization conditions:

      $ 4\pi \int {\rho }_{n}\left(r\right){r}^{2}dr=N , $

      (9)

      $ 4\pi \int {\rho }_{p}\left(r\right){r}^{2}dr=Z , $

      (10)

      where N (Z) is the total number of neutrons (protons) in the nucleus and the other parameters ($ {\alpha }_{i},{a}_{i}) $ can be determined in detail through Refs. [35, 36]. The tail/halo part can be written as [35],

      $ {\rho }_{tail}\left(r\right)={N}_{0}\left(\frac{{r}^{2}}{{\left({r}^{2}+{R}^{2}\right)}^{2}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-r}{{a}_{t}}\right), $

      (11)

      where, N0 is determined from the normalization,

      $ 4\pi \int {\rho }_{tail}\left(r\right){r}^{2}dr=2 $

      (12)

      here, 2 refers to the number of halo neutrons. Finally, the third density distribution form is that deduced using the simpler non-relativistic Hartree-Fock (HF) model [37].

      For 12C and 28Si targets nucleus, only the two-parameter Fermi model (FM) is used [38]

      $ {\rho }_{T}^{12C\left(28Si\right)}=\frac{0.207\left(0.175\right)}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[\left(\dfrac{r-2.1545\left(3.15\right)}{0.425\left(0.475\right)}\right)\right]} $

      (13)

      where, the rms radius which extracted from this density is 2.298 and 3.012 fm for 12C and 28Si, respectively.

    • B.   The NN effective interactions

    • Based upon the M3Y interactions which is designed to reproduce the G-matrix elements for Paris and Raid [39, 40] effective NN interactions, two versions of the M3Y effective NN interaction are used. These versions are depending upon two different forms of the density and energy-dependent (DDM3Y and CDM3Y6). In addition to the SPP, the CDM3Y6 interaction that was recently modified by the RT [33] is also used. The first method is the DDM3Y effective interaction, defined as [38],

      $ \upsilon \left(E, \rho ,S\right)=g\left(E, S\right)F\left(E, \rho \right), $

      (14)

      where,

      $ F\left(E, \rho \right)=C\left(E\right)\left[1+\alpha \left(E\right)\mathrm{e}\mathrm{x}\mathrm{p}(-\beta \left(E\right)\rho )\right], $

      (15)

      $ C, \alpha $ and $ \beta $ are the energy dependent parameters while, $ \rho ={(\rho }_{P}+{\rho }_{T}) $. The factor $ g\left(E, S\right) $ is represents the original zero rang M3Y-Reid NN interaction [38] that can be written in the form,

      $ g\left(E, S\right)={\upsilon }_{D}\left(R\right)+{\rm{\hat J}}\left(E\right)\delta \left(S\right), $

      (16)

      since, E is the laboratory energy per projectile nucleons, $ \overrightarrow{s} $ is the inter-nucleon separation, $ \delta \left(s\right) $ is the delta function, $ {\rm{\hat J}}\left(E\right) $ is the effects of the knock-on exchange between interacting nucleons and $ {\upsilon }_{D}\left(r\right) $ that represents the direct parts. The $ {\rm{\hat J}}\left(E\right) $ is taken as [38]

      ${\rm{\hat J}}\left( E \right) = - 276\left[ {1 - 0.005\left( {E/A} \right)} \right]\;{\rm{MeV}}.{\rm{f}}{{\rm{m}}^3},$

      (17)

      with the direct part in the M3Y-Paris as:

      ${\upsilon _D}\left( R \right) = \left[ {11062\frac{{{\rm{exp}}( - 4R)}}{{4R}} - 2538\frac{{{\rm{exp}}( - 2.5R)}}{{2.5R}}} \right]\;{\rm{MeV,}}$

      (18)

      The next NN effective interaction used is the other density dependent version (CDM3Y6) for the direct and exchange terms, there for the full CDM3Y6 interaction form is defined as [41],

      $ {\upsilon }_{D\left(Ex\right)}\left(\rho ,R\right)=g\left(E\right)F\left(\rho \right){\upsilon }_{D\left(Ex\right)}\left(R\right), $

      (19)

      since the direct part $ {\upsilon }_{D}\left(r\right) $ as Eq. (18) and the knock-on exchange parts in the infinite-range exchange are taken as:

      $\begin{aligned}[b] {\upsilon }_{Ex}\left(R\right)=&\left[-1524\frac{\mathrm{e}\mathrm{x}\mathrm{p}(-4R)}{4R}-518.8\frac{\mathrm{e}\mathrm{x}\mathrm{p}(-2.5R)}{2.5R}\right.\\&\left.-7.847\frac{\mathrm{e}\mathrm{x}\mathrm{p}(-0.7072R)}{0.7072R}\right]\;{\rm{MeV}}\end{aligned} $

      (20)

      with the function $ F\left(\rho \right) $ is written as [41, 42],

      $ F\left(\rho \right)=0.2658\left[1+3.8033\mathrm{e}\mathrm{x}\mathrm{p}\left(-1.41\rho \right)-4.0 \rho \right], $

      (21)

      while $ g\left(E\right) $ is the energy dependent factor given as [41],

      $g\left( E \right) = \left[ {1 - 0.003\left( {E/A} \right)} \right],$

      (22)

      For reproducing the saturation properties of symmetric nuclear matter (NM) in the standard HF calculation and to have a reliable density dependent interaction for use at different energies (the high-momentum part of the HF single-nucleon potential), the modified CDM3Y6 interaction (CDM3Y6-RT) with the RT contribution has been carried out. The density dependence of ΔF$( \rho ) $ obtained from the exact expression of the RT given as [33]:

      $ \varDelta F\left(\rho \right)=1.5\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.833 \rho \right)-1\right], $

      (23)

      In the other hand, we use the SPP, where the radial and energy dependence is written in the following [43],

      $ V\left(R,E\right)={V}_{F}\left(R\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-4{\beta }^{2}\right), $

      (24)

      where β= v/c, v is the local relative velocity between the two nuclei and VF (R) is a folding potential such as the expression in Eq. (2) where $ {\upsilon }_{nn}(S=\overrightarrow{R}-\overrightarrow{{r}_{1}}-\overrightarrow{{r}_{2}}) $ is a physical nucleon-nucleon interaction given by; $ {\upsilon }_{nn}\left(S\right)= {V}_{0}\delta \left(S\right) $ using $ {V}_{0}=-456\; MeV\;{fm}^{3} $ and the usage of the delta function is corresponding to the zero range approach. Systemizations of the nuclear densities were achieved in Ref. [44], to provide a good explanation of matter and charge distribution. In this work, for the target 12C, we assume that $ {a}_{m}=0.53 $ fm and $ {a}_{c}=0.56 $ fm for the matter and charge diffuseness, respectively. But the matter and charge distribution radius is given by $ {R}_{M}=1.3{A}^{1/3}-0.84 $ and $ {R}_{C}=1.76{Z}^{1/3}-0.96 $, respectively. The matter density of the projectile nucleus is used as tabulated in numerical form.

    III.   PROCEDURE
    • The analysis of 11Li+12C, 28Si elastic scattering is carried out by using the OP’s generated from Eq. (1). The DF real part of the OP’s are evaluated using several versions of the M3Y density- and energy dependent effective interactions as DDM3Y (14), CDM3Y6 (19) and the CDM3Yn-RT (23), beside the SPP. The imaginary potential is calculated by Eq. (4) as a phenomenological WS form with three parameters. Then the obtained potentials are fed into the HIOPTM-94 [45] computer code to calculate the elastic scattering differential cross sections. For obtaining best fits, searches on the potential parameters are done using the HIOPTM-94 code to achieve minimum $ {\chi }^{2} $ value, defined as [46, 47]:

      ${\chi ^2} = \frac{1}{N}\sum\nolimits_{K = 1}^N {{{\left[ {\frac{{{\sigma _{th}}\left( {{\theta _K}} \right) - {\sigma _{ex}}\left( {{\theta _K}} \right)}}{{\Delta {\sigma _{ex}}\left( {{\theta _k}} \right)}}} \right]}^2}} $

      (25)

      $ {\sigma }_{th}\left({\sigma }_{ex}\right) $ is the theoretical (experimental) cross section at an angle $ {\theta }_{K} $ in the c.m. system, $ {\Delta \sigma }_{ex} $ is the experimental error, and N the number of the data points. For the experimental errors of all measured data, an average value of 10% is used. For semi-microscopic analysis, searches are carried out on four parameters (the real renormalization factor (NR) in conjunction with the three WS imaginary potential parameters W0, Ri and ai). On the other side, for the phenomenological WS approach, searches are performed on six free parameters (three WS real potential parameters Vv, Rv and av plus three of imaginary potential part W0, Ri and ai).

    IV.   RESULTS AND DISCUSSION

      A.   Potentials and densities

    • The radial shape of the considered different types of the density distributions of 11Li are shown in Fig. 1, in linear and logarithmic scale. The corresponding calculated rms of protons, neutrons and the matter for all used densities distributions of the 11Li nucleus are listed in Table 1 as compared with previous works and experimental data. The present results confirm the previous calculations that predict a long tail halo for the 11Li nucleus than experimentally data as it is obvious, from Table 1. Also, it's noticed, our calculations agree well with the previous rms values, however, there is a difference slight compared to the experimental data. Moreover, other previous studies have extracted these quantities [24, 48-51] which inter and agree with present calculations, but it is noticed that the important study of Mermaz [24] extracted a value ($ {\langle {R}_{m}\rangle }^{\frac{1}{2}}=3.74 $) which is greater than experimental and other theoretical calculations.

      Figure 1.  The densities of 11Li as logarithmic and linear scales.

      DensityRpRnRrms
      Calc.Ref.Calc.Ref.Calc.Ref.
      COSMA2.322.32 [35]3.623.3183.21 [51]
      SPD2.162.17 [36]3.673.67 [36] 3.21 [13]3.3253.32 [36]
      3.12 [1, 13]
      HF2.243.082.8722.846 [48]
      Exp.2.88$ \pm 0.11 $[13]3.21$ \pm 0.17 $ [13]3.12$ \pm 0.16 $ [13]

      Table 1.  The calculated rms for proton (Rp) and neutron (Rn) radii, and the mass rms radii (Rrms) in Fermis of the three NM densities for 11Li nucleus.

      These results of the considered density distributions are illustrated well in Fig. 1. It can be seen that COSMA density has a larger value than of the SPD and HF at small distances (r ≤ 1 fm), while the latter density has the smallest value. At large distances (r ≥ 10 fm), we can see that all the used densities have a long tail in due to the structure of 11Li, as a core with two valence neutrons. The SPD density has the longest tail, so it gives a successful description for the largest radius of 11Li. On the other hand, investigation of the halo densities for the 11Li nucleus and the considered NN effective interactions effects are through the calculation of the DF potential of Eq. (2). The real part of the OP’s are calculated with the DDM3Y, CDM3Y6 and CDM3Y6-RT effective NN interactions in addition to the SPP potential folded with the considered three densities COSMA, SPD and HF at energies 50 and 60 MeV/n for the 11Li scattered by 12C and at 29 MeV/n for 11Li+ 28Si system. The physical observation related to the resulted potentials for 12C at the energy 60 MeV/n are listed in Table 2 and shown in Fig. 2, as one case.

      MeV/nDensity-J (MeV fm3)Rrms (fm)
      CDM3Y6CDM3Y6-RTSPPDDM3YCDM3Y6CDM3Y6-RTSPPDDM3Y
      11Li + 12C
      50COSMA301.12271.42271.0252.744.6924.884.404.789
      SPD300.84271.28275.5251.994.6914.864.454.772
      HF302.57274.86271.4254.374.2724.414.034.358
      60COSMA281.71256.50254.5235.634.6944.894.394.799
      SPD281.50262.70257.1234.994.6934.874.334.772
      HF283.31259.91253.2237.434.2784.423.904.360
      11Li + 28Si
      29COSMA358.79311.71314.7297.645.1285.314.795.20
      SPD358.07311.30309.4296.435.1275.294.655.18
      HF359.04314.73314.1297.494.7374.874.454.81

      Table 2.  The calculated volume integrals and rms radii of the folded potentials for 11Li+12C, 28Si systems at 50 and 60 MeV/n for 12C and at 29 MeV/n for 28Si 29 MeV/n respectively, using the four NN effective interactions with the three densities without NR.

      Figure 2.  The real DF for 11Li+12C at 60 MeV/n using the three densities with the DDM3Y, CDM3Y6, CDM3Y6-RT effective NN interactions and the SPP with NR=1.0 in the linear scale.

      The folded potentials in the present work are calculated as a sum of two parts for projectile (11Li) density, halo (neutrons) density plus core density is folded with target density and NN effective interaction. The effect of different densities of 11Li structure and the considered effective interaction on our potential model is representing the focused study point for us.

      Generally, as we have seen from Fig. 2 and Table 2, the folded potentials using the considered densities and effective interaction have approximately the same depths and behavior for the density-dependent effective NN interactions with a slight difference. From Table 2, we notice that the long potential rms is to be achieved with COSMA and SPD densities, CDM3Y6-RT, and DDM3Y effective interactions while, the smallest potential tail is to be achieved with HF density, SPP and CDM3Y6 effective interactions. Also, the energy dependence for the real potential volume integral of the 12C target is observed, where it decreases with increasing energy. Fig. 2, is confirming these results where the SPP has the largest depth compared to that using the DDM3Y and CDM3Y6 while the CDM3Y6-RT is the smallest between all; this arises in the interior region at r (r $ < $ 2 fm). The folded potentials of the other (50 MeV/n) energy and those for 28Si target at 29 MeV/n have the same behavior as it at 60 MeV/n, but with different depths, where the depths of the potentials increase as decreasing energy. The obtained potential agree well with the systematic suggested potential models, by the break up potential for Yabana [52], DP potential for Khalili [27] and the folded potentials by Khoa [26]. However, the extracted real folded potentials have a shorter tail in facing the successful Mermaz potential [24].

    • B.   Differential cross sections

    • Many attempts have been achieved to generate appropriate OP for analyzing the angular distribution cross section of 11Li projectile reactions. The result drawn from all previous theoretical observations [19, 23-27] studies that the nearside/farside interference minimum is missing or greatly attenuated in the elastic scattering of 11Li with low-Z targets at intermediates energies. The presence of a sharp minimum at 40 in the calculated angular distributions, was due to the contamination by 9Li ions resulted from projectile breakup [22]. The successful theoretical work that explained this discrepancy at forward angles was proposed by Mermaz [24] by using phenomenological potential with coupling for low excited inelastic channel and also, with many adjusted free parameters.

      Frist the same kined of analysis as previously used has been presented, WS potential. To improve the agreement with the considered experimental data we use the widely used phenomenological WS potential with six parameters. More parameters in a such famous model mean more flexibility in order to obtain the best fit that can be used later as a guide. The elastic scattering for 11Li nucleus from 12C at the two energies 50 and 60 MeV/n, and from 28Si at 29 MeV/n have been analyzed in the framework of the conventional optical model by using the standard WS form as in Eq. (4). Moreover, a semi-microscopic optical DF folding potentials have been generated to study the sensitivity of NN interaction as well as the form of the halo density distributions of 11Li on reaction differential cross sections. The obtained results are listed in Tables 3 to 5, and shown in Figs. 3 to 6.

      Energy(MeV/n)V(MeV)rv(fm)av(fm)W(MeV)ri(fm)ai(fm)-J (MeV fm3)$ {\chi }^{2} $$ {\sigma }_{R} $
      JRJI
      11Li + 12C
      50116.40.830.812.832.190.057286.0386.6813.02009
      60295.920.211.3520.720.921.20233.3386.125.21400
      11Li + 28Si
      29.0261.910.411.787.0292.290.362201.5733.052.961495

      Table 3.  Best fit OP parameters of the WS model

      Figure 3.  The best fit angular distribution of 11Li scattered elastically from 12C and 28Si that calculated with the WS potential model. The symbol represents the experimental data taken from Refs. [20-22].

      MeV/nDensityPotentialNR-J (MeV fm3)W (MeV)ri (fm)ai (fm)$ {\chi }^{2} $$ {\sigma }_{R} $
      JRJI
      29 COSMA CDM3Y6-RT 0.64 198.76 45.12 14.69 2.188 0.767 4.53 1460
      DDM3Y 0.82 242.49 52.12 15.65 2.235 0.709 4.53 1528
      CDM3Y6 0.69 247.20 50.45 15.40 2.227 0.728 4.60 1518
      SPP 0.899 282.99 46.04 14.79 2.198 0.772 5.00 1485
      SPD CDM3Y6-RT 0.62 193.58 45.23 15.73 2.174 0.795 5.31 1457
      DDM3Y 0.81 238.51 51.81 15.61 2.234 0.718 5.01 1527
      CDM3Y6 0.68 243.41 50.53 15.60 2.220 0.740 5.34 1516
      SPP 0.896 277.11 46.15 14.82 2.198 0.770 5.85 1483
      HF CDM3Y6-RT 0.58 181.83 45.95 16.60 2.128 0.849 6.37 1442
      DDM3Y 0.77 229.06 52.00 16.11 2.218 0.736 5.62 1517
      CDM3Y6 0.64 233.83 50.95 16.16 2.204 0.750 6.22 1504
      SPP 0.87 273.86 51.08 17.76 2.148 0.803 7.17 1481

      Table 5.  The best fit parameters have obtained of the elastic scattering data for 11Li+28Si system at energy 29 MeV/n using NN interaction (DDM3Y, CDM3Y6, CDM3Y6-RT and SPP) with the considered three densities (COSMA, SPD and HF).

      Figure 6.  Same as Fig. 4, but for the elastic 11Li from 28Si scattering at energy 29 MeV/n. The experimental data are taken from Ref. [21].

      The results of angular distributions using the WS optical model are compared with experimental data in Fig. 3 and their potential parameters are listed in Table 3. As shown in Fig.3 (panel ‘a’), the best fitting with experimental data has been achieved for 28Si target without an anomaly at near or far angles. The results are more successful than previous studies that used a combination of elastic and inelastic parts to get a reasonable result [21]. On the other side, a reasonable fit with data has been obtained for 12C target at the two energies 50 and 60 MeV/n, as shown in Fig.3 (panel ‘b’ and ‘C’), but with a sharp minimum between (2.5)0 and 40.

      The results displayed in Tables 4 and 5 with Figures 4 to 6 have been performed by the semi-microscopic potentials. Firstly, it's noticed that the NR coefficients are close to one, 1.0±0.01 for energy 50 and 60 MeV/n on the 12C target while 0.8±0.01 with the 28Si target, and provides a satisfactory description with experimental data. These results are more successful than these have been achieved in previous studies that used, CC model [21], quasielastic potential [26], Glauber of four body model [27], Breakup effect [51] and finally the JLM potential [53].

      (MeV/n)DensityPotentialNR-J (MeV fm3)W (MeV)ri (fm)ai (fm)$ {\chi }^{2} $$ {\sigma }_{R} $
      JRJI
      50 COSMA CDM3Y6-RT 1.18 320.27 104.23 4.529 1.889 1.122 11.17 2300
      DDM3Y 1.15 289.27 105.71 4.926 1.840 1.124 14.60 2273
      CDM3Y6 0.92 278.36 95.06 4.393 1.852 1.092 12.96 2146
      SPP 0.86 233.51 80.11 4.727 1.675 1.169 14.72 1872
      SPD CDM3Y6-RT 1.16 314.55 105.07 4.217 1.959 1.041 13.97 2324
      DDM3Y 1.13 283.91 110.30 4.724 1.905 1.086 17.36 2356
      CDM3Y6 0.91 274.07 99.950 4.219 1.925 1.031 15.68 2229
      SPP 0.93 257.02 100.27 4.818 1.799 1.225 16.45 2236
      HF CDM3Y6-RT 1.06 291.83 103.05 3.900 2.019 0.918 14.57 2290
      DDM3Y 1.06 270.07 111.37 4.382 1.977 1.012 17.85 2389
      CDM3Y6 0.87 261.79 101.15 3.946 1.994 0.946 16.20 2258
      SPP 0.85 229.75 94.16 4.150 1.867 1.188 18.21 2190
      60 COSMA CDM3Y6-RT 1.18 303.45 136.83 50.00 0.674 1.371 6.21 1737
      DDM3Y 1.20 283.39 124.32 49.57 0.650 1.342 6.03 1625
      CDM3Y6 0.94 264.15 116.92 44.16 0.725 1.214 5.82 1484
      SPP 0.85 215.98 92.40 39.03 0.750 1.032 6.61 1187
      SPD CDM3Y6-RT 1.12 293.94 140.90 60.06 0.569 1.459 7.19 1811
      DDM3Y 1.18 276.33 126.92 53.02 0.607 1.400 6.76 1678
      CDM3Y6 0.92 258.60 120.01 47.52 0.684 1.269 6.50 1536
      SPP 0.85 213.14 95.12 39.81 0.742 1.066 6.74 1228
      HF CDM3Y6-RT 1.05 272.94 137.63 50.00 0.676 1.373 7.73 1741
      DDM3Y 1.07 254.89 131.36 84.24 0.384 1.490 7.33 1720
      CDM3Y6 0.84 238.26 124.09 67.91 0.517 1.355 7.11 1572
      SPP 0.80 204.70 101.33 50.90 0.628 1.182 7.24 1313

      Table 4.  The best fit parameters have obtained of the elastic scattering data for 11Li+12C system at energies 50 and 60 MeV/n using NN interaction (DDM3Y, CDM3Y6, CDM3Y6-RT and SPP) with the considered three densities (COSMA, SPD and HF).

      Figure 4.  The best fit angular distribution of elastic 11Li from 12C scattering at 50 MeV/n calculated with the semi-microscopic real DF potentials; CDM3Y6-RT (dashed-dotted line), DDM3Y (sold line), CDM3Y6 (dashed line), and SPP (dotted line). The symbol is a representation of the experimental data from Ref. [22]. Upper panel for COSMA density, middle panel for SPD density and bottom panel for HF density.

      Generally, the results corresponding to Figs. 4 to 6 shows that the considered effective interactions and densities predicting the best agreement to experimental results. It is clear that the three angular distributions resulted from, DDM3Y, DDM3Y6 and DDM3Y6-RT, are very similar to each other but with a small shift in the forward angle between (2.5)0 and 40 for SPP is noticed.

      Fig. 4, shows the results in comparison with the measured experiment data for 12C target at energy 50 MeV/n where the agreement with the data is reasonable and more quality than those obtained from the quasielastic calculation of Peterson et. al. [22]. Likewise, Fig. 5 shows The goodness of the predictions of our calculations with the experimental data for 12C target at energy 60 MeV/n against previous attempts by many authors [26, 27, 52, 53]. Meanwhile, the same results for 28Si target at energy 29 MeV/n, which are more appropriate with experimental data than the previous studies of [21, 48, 53, 54] have been achieved as shown in Fig. 6. Despite the successes achieved in the prediction of data analysis for practical data, anomalies at small angles still exist. In this angular range, the optical model amplitude is insensitive to potential used in due to a breakdown of the optical model in the explanination of the small-angles cross section for halo nucleus, which could be attributed this to an extreme peripheral nature of these reactions at intermediate energies in due to surface transparent potential.

      Figure 5.  Same as Fig. 4, but for energy 60 MeV/n. The experimental data are taken from Ref. [20].

    • C.   Total reaction cross sections

    • In order to understand the formation of a halo around the core of radioactive nuclei which is very difficult, it is useful to study the reaction cross section (RCS) in addition to differential cross sections of the exotic nucleus 11Li from targets. Also, the RCS $ {\sigma }_{R} $ represents an important constraint on the OP calculation. The average total RCS $ {\sigma }_{R} $ for the present calculations are listed in Table 6. Systematic for predicting the behavior of the reaction mechanism is extrapolated at a range of low and high energies (30, 75 and 85 MeV/n). These results are listed together with the previous values [19, 21, 23, 26, 48, 52] in conjunction with the experimental values of Refs. [55-57] and illustrated in Fig.7. Unfortunately, the $ {\sigma }_{R} $ was not measured at 50 MeV/n.

      TargetE/APresent cal.Previous
      Cal.Exp.
      12C302942 ±21710 [23]2947 [55]
      502238.58±10.68
      601552.67±16.791274±30 [26]
      1391±35 [19]
      1520 [52]
      1500±50 [56, 57]
      751449±31401 [26]1430±60 [56]
      87.01264 ±21264 [26]1260±40 [57]
      28Si291493.17±2.451402 [21]
      1970 [48]
      1520 [52]
      2947±386a [48]
      a Measured at 25.5 MeV/n.

      Table 6.  The calculated mean value and the previous calculations comparable to the experimental results of the reaction cross section $ {\sigma }_{R} $ (mb) for 11Li+12C, 28Si systems at 50 and 60 MeV/n for 12C and at 29 MeV/n for 28Si respectively.

      Figure 7.  Energy dependence of the reaction cross section in the optical limit of the DF model using a present model for 11L+12C. The full triangle is the mean obtained results with considered potentials and the full star for previous calculations, while the full circles are the experimental data taken from Refs. [48, 55-57].

      As seen in Table 6, the present calculated $ {\sigma }_{R} $ generally agrees or better than previous theoretical calculations, and on average, our results which obvious in Fig.7 are closer to the experimental cross sections measured or that estimated from the systematics at relevant bombarding energies for the 12C target. The calculated energy dependence of the total cross sections is well approximated by the expression:

      $ {\sigma }_{R}^{Cal.}\left(E\right)=1.3 \pi {\left({A}_{p}^{\frac{1}{3}}+{A}_{T}^{\frac{1}{3}}\right)}^{2}\left(\frac{1459.4}{E}-\frac{11621.4}{{E}^{2}} \right) $

      (i26)

      here, $ E $ Represents the value of energy per nucleon. It is obvious from Fig.7 together with the results listed in Table 6 that, the behavior of the generated results reflect the success of our potentials to reproduce the exotic nuclei reactions. For the 28Si target, while the present result of $ {\sigma }_{R} $ agree with previous theoretical calculations, it is less than experimental data. A reason for this discrepancy could be the effect of the coupling of the low-lying exited states of 28Si to the elastic channels which must be included or separated from the elastic events by those inelastic states. In addition, this value for $ {\mathrm{\sigma }}_{\mathrm{R}}=2947\pm 386\;\mathrm{m}\mathrm{b} $ corresponds to an energy of 25.5 MeV / n.

    V.   CONCLUSION
    • Explaining the discrepancy between experimental and calculated differential cross sections at forward angles for the interaction of 11Li with low-Z targets at intermediate energies can be divided into two stages. First, the studies that were couldn't predict this anomaly as a highly refractive nature for a quasi-elastic scattering process. Second, the success generated by Meermaz [24] and confirmed by the recent experimental study of Peterson et. al. [22] in which the elastic and inelastic reaction channels were cleanly separated in the forward angle. They verified that the 9Li contamination data is responsible for the displayed interference minima. The purpose of the present paper is as a new attempt to describe the elastic scattering of 11Li from 12C and 28Si through the DF.

      In the context of the DF optical model potential and as an extension of our studies [58-60], the elastic scattering of 11Li+12C and 11Li+28Si has been reanalyzed. The nucleus-nucleus interaction potential has been constructed microscopically from folding the density-dependent effective NN interaction DDM3Y, CDM3Y6 and SPP over the nuclear matter distributions of the interacting nuclei. The resulting potentials are used as the real part of the OP, while the imaginary component is treated phenomenologically through the WS form to calculate the angular distributions of the elastic scattering differential cross sections. Successful predictions were obtained by using our potentials generated for different energies within the measured angular ranges. This result indicates that using a unified energy dependent real potential is adequate to successfully fit the data through the considered energy range. Looking at the chi-square values mentioned in the above tables, we found that with different versions for the NM density of the lithium nucleus, the smallest values at energy 50 and 60 MeV were obtained using CDM3Y6-RT and CDM3Y6 respectively.

      Due to the repulsive contribution of the RT to the real folding potential, especially at a small internuclear distance, this contribution has been shown to be vital in the application of the folding model to the study of elastic 11Li+12C and 11Li+28Si scattering in order to obtain the realistic shape and strength of the real potentials. The predicting power of the folding model for the 11Li+12C and 11Li+28Si seems much improved. For this kind of folding potentials, a systematic folding model studies of the elastic and inelastic nucleus-nucleus scattering over a wide range of energies are encouraged.

      On the other hand, the total reaction cross sections for the considered reactions together with an extended energies are investigated. It is observed that the reaction cross section increases linearly with rising energy at the lower energies. It is also noted that both the observed data and those theoretically obtained from previous elastic scattering studies are in good agreement with the values generated by the current elastic scattering calculations.

      Finally, we can argue that the simple calculated semi-microscopic potential of the DF model is considered as an advanced step and quite successful compared with the previous complicated studies. The present results encourage us to modify the present potential approach in the next step for full microscopically to explain the discrepancy of the interaction of 11Li with low-Z targets at intermediate energies and to extend the analysis for other halo reactions.

Reference (60)

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