Systematic analysis of elastic α scattering on A≈118-130 nuclei around the Coulomb barrier

Journal Pre-proofs Systematic analysis of Elastic α Scattering on A118 − 130 nuclei Around the Coulomb Barrier Zakaria M.M. Mahmoud, A. Hemmdan, Kassem O. Behairy PII: DOI: Reference:

S2211-3797(19)32579-3 https://doi.org/10.1016/j.rinp.2019.102892 RINP 102892

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Results in Physics

Received Date: Revised Date: Accepted Date:

26 August 2019 4 December 2019 18 December 2019

Please cite this article as: Mahmoud, Z.M.M., Hemmdan, A., Behairy, K.O., Systematic analysis of Elastic α Scattering on A118 − 130 nuclei Around the Coulomb Barrier, Results in Physics (2019), doi: https://doi.org/ 10.1016/j.rinp.2019.102892

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Systematic analysis of Elastic α Scattering on 𝐴 ≈ 118 ― 130 nuclei Around the Coulomb Barrier Zakaria M.M. Mahmouda,b, A. Hemmdanc, Kassem O. Behairyc aPhysics

Department, King Khalid University, Abha, Saudi Arabia of Physics, Faculty of Science, New – Valley University, Egypt cPhysics Department, Faculty of Science, Aswan University, Egypt

bDepartment

Highlights Global α-nucleus potential is constructed. The derived optical model potential is based on the folding model to describe the elastic scattering of α– particles from 118Sn, 124Sn and the isotope 120-130Te chain at energies of astrophysics interest. Two semi-microscopic JLM–R and DDM3Y1–FR potentials, in addition to microscopic JLM-RI potential are derived. In JLM calculations of the optical potential, the target density dependence is only considered. For the potential based on the DDM3Y1-FR effective interaction, both the α–particle and target density dependence were considered. In addition, the finite range exchange interaction was also considered. The calculated potentials have been fed into the auto–search code to compute the scattering observables. It is found that the obtained results are in good agreement with the measured data over both the energy and angular range. By looking deeply into the results, we have found that the DDM3Y1–FR is in better agreement with data than the JLM based potentials. It is observed that the obtained fitting parameters for the potentials show a very weak target mass dependence.

1

In addition, the average values of these parameters show clear energy dependence. Moreover, we have found that the calculated quantities like the volume integrals and total reaction cross sections are approximately target mass independent. The average values of these quantities have clear energy dependence. The results are compared with experimental and previous results for the same system and energy range we have found that our approach is quite successful and very promising.

Abstract Alpha elastic scattering angular distributions on target nuclei

118, 124Sn

and

120-130Te

at energies near the Coulomb barrier are analyzed.

We used three versions of the nuclear optical model potentials (OMP) to analyze twenty-nine data sets. JLM and the density and energydependent DDM3Y1 with finite range exchange term effective interactions were used to generate these potentials. Two real folded potentials based on JLM and DDM3Y1 effective interactions supplied with the conventional Wood–Saxon (WS) imaginary potential are used in the cross sections calculations. These potentials are denoted as JLM-R and DDM3Y1-FR, respectively. In addition, we used the full complex folded OMP, denoted as JLM-RI, to analyze the same data. These potentials successfully predict the cross section of the systems under study. We found that the JLM-R and DDM3Y1-FR could reproduce the measured cross sections over the full angular range very well compared to microscopic JLM-RI potential. Our calculated potentials have a clear energy dependence and in addition a weak target mass dependence. The total reaction cross section and volume integral are investigated in comparison to previous studies. The derived potentials are very promising for the construction of a new global α-nucleus potential and provide new parameters of α– induced reactions and scattering at the energy range 17∼27 MeV. 1. Introduction The basic input for the analysis of reactions involving α–particles in the entrance or exit channel is the α-nucleus potential. At low energies reactions (relevant to astrophysics interest), the knowledge of a parameterized α–nucleus optical potential is highly required for 2

statistical model calculations [1]. The global α–nucleus optical potential (GAOP) is important for understanding the production and the abundance of nuclei, heavier than iron, which is produced by neutron-capture in stars during He and C burning [1–6]. Therefore, for reactions of astrophysical interest, the parameters of the GAOP should be known at energies well below the Coulomb barrier. Several attempts were made to improve the knowledge of GAOP [7-10]. Avrigeanu et al. [8] used the double-folding formalism to obtain the real part of the optical potential. They used an energy-dependent phenomenological imaginary part and a dispersive correction to the microscopic real folded potential. The analysis of α-particle elastic scattering on A ∼ 100 nuclei at energies below 32 MeV was carried out. They found a regional parameter set of the OMP for low-energy (below 32 MeV) and nuclei in the mass range A ∼ 100. Avrigeanu et al. [10] extended the work of Ref. [8] for nuclei in mass region A ~ 50 – 120 and for energies between 13 and 50 MeV. They aimed to overcome the difficulties in the description of the data arising from the choice of the real and the surface imaginary potentials of Ref [8]. They Modified the former study to obtain an optical potential which describes equally well both the low-energy elastic scattering and α–induced reactions. Ashok Kumar et al. [9] attempted to determine a global α–nucleus optical model potential for A ∼ 12–209 and energies from the Coulomb barrier up to 140 MeV. They proposed for their study a hybrid microscopic-phenomenological approach. They found a good agreement with data at high energies for all the considered targets, while differ considerably in few cases at lower energies. Also, their obtained potential gives a good description of the compound nucleus reactions at which the α–particle exists in the entrance or exit channels. They concluded that the obtained global potential could be used for astrophysics reactions and alpha decay. Moreover, they claimed that this global potential could be fine-tuned to improve the quality of fits to specific data for different nuclei and energy ranges. Demetriou et al. [7] purposed a GAOP that works at energies 𝐸 < 40 MeV for the mass region 40 ≤ 𝐴 ≤ 208. This potential reproduced the existing elastic scattering data as well as the reactions (𝛼,𝛾), (𝛼, 𝑛), (𝛼, 𝑝) and (𝑛,𝛼). Their potential considered the energy dependence and the structure effects that characterize the α–nucleus interaction. The real part of their potential was calculated using a double-folding model based on the M3Yeffective nucleon–nucleon (NN) interaction. The imaginary potential was of Woods–Saxon (WS) form with a new parameterization. They used volume WS form (potential–I), volume plus surface WS (potential–II) and volume plus damped surface WS (potential–III). In addition, they used a dispersive relation to relate the real and imaginary parts and for further reduction of the ambiguities in optical model potential. From their analysis, it is found that the Potential–III was 3

the most physically justified potential. From the studies [7–10], one can notice that the combined study of alpha elastic scattering and alphanucleus reaction data is the ‘doorway’ to develop a reliable and accurate global alpha-nucleus optical potential at low energies below the Coulomb barrier which are relevant for astrophysical applications. Recently, several α–nucleus elastic scattering and α–induced reactions for even-even nuclei were measured in a wide angular range [11– 23]. These measurements aimed to examine the behavior of α–nucleus potential near the region of astrophysical interest. Fueloep et al. [12] measured the elastic scattering cross section of 92Mo(α, α)92Mo in a wide angular range at the center of mass energies 13, 16, and 19 MeV. They extracted from the data an optical model potential that could be extrapolated down to astrophysical relevant energies around and below the Coulomb barrier. Their extrapolated real folded potential fits well the known systematic behavior of α–nucleus potentials while the imaginary part is not unique. Kiss et al. [14] measured, with high accuracy, the elastic scattering α–106Cd at energies 15.5, 17 and 19 MeV. Galaviz et al. [13] measured, with high precision, the elastic scattering

112,124Sn

(α,α)112,124Sn at energies above and below the Coulomb barrier. They

performed their analysis in the framework of the Optical Model. They found that the volume and surface (dominance at energies close to the Coulomb barrier) WS imaginary potentials were needed for a better description of the experimental data. The obtained potentials of Galaviz's [13] were used to reproduce elastic scattering data at higher energies for both nuclei. Mohr et al. [11] accurately measured the elastic scattering angular distribution of 144Sm (α, α)144Sm scattering at the energy 20 MeV. Kiss et al. [15] measured with high-precision the elastic data of 89Y(α, α)89Y scattering at 15.51 and 18.63 MeV in the center of mass system. They formulated a folding model to generate semi-microscopic potentials. A definite optical potential from the analysis of their experimental data have been extracted with the systematics of the previous α–nucleus optical potentials (see Ref. therein). These systematics were used to extrapolate the optical potential with very limited uncertainties to the astrophysical relevant energy 9.5 MeV in the center of mass. They found that all global parameterizations failed to reproduce the scattering pattern ratio at large angles. Palumbo et al. [16] performed a series of accurate measurement s at the Notre Dame FN Tandem accelerator of α– elastic scattering from 106Cd, 118Sn, and 120,124,126,128,130Te at energies both below and above the Coulomb barrier. These measurements were for a better understanding of the systematic of the α–nucleus optical potential and to test the reliability of the Hauser–Feshbach (HF) formalism at 4

energies of astrophysical interest. They derived a new parameterization of the α–nucleus optical potential from their elastic scattering analysis. The obtained potential employed to calculate the α– induced reactions on the considered nuclei using the HF approach. They concluded from their studies that, further experiments with an extended Z and mass numbers to determine how the optical potential evolves on the nuclear chart. Mohr et.al [24] analyzed the elastic scattering angular distribution of α–89Y,

92Mo, 106,110,116Cd, 112,124Sn

and

144Sm

at energies around

Coulomb barrier. They used the folding model to obtain the real part of their optical model potential, while the imaginary part was considered in surface WS form. Their aim was to construct a comprehensive α–nucleus optical potential with very few adjustable parameters. Their potential reasonably predicted the angular distributions and the total reaction cross sections for the investigated α–nucleus systems. From these previous studies [11–16], we noticed that considerable efforts were devoted to enhance our knowledge of α–nucleus potential but still these systematic studies limited in isotope and energy ranges. Therefore, a reliable α–nucleus global potential is still an open question. Undoubtedly, the reliability of GAOP is to provide theoretical support and experimental instruction for α–nucleus reactions in a wide range of target and incident energy when the nuclear reaction experimental data is not available. A set of recent precisely measured experimental data of α–induced reactions are also given in refs. [17-23]. One conclusion can be drawn from these articles that a modification of low energy α–nucleus optical potential is necessary. Therefore, the main goal of the present study is to provide a parameterization using folding potentials to investigate the elastic scattering of α–particles from isotopic heavy targets

118,124Sn

and

120,124,126,128,130Te

near the Coulomb barrier in the framework of the optical model. The

analysis of the scattering data is performed using the complex JLM effective nucleon-nucleon interaction developed by Jeukenne, Lejuene, and Mahaux [25]. In addition, we used DDM3Y1-FR, which is based on the G-matrix element of the soft-core interaction [26]. The real part of the α–nucleus potential is obtained within the framework of the double folding model. The presence of the high precision elastic scattering data [13,16] at low energies around the Coulomb barrier encouraged us to reanalyze these data. The importance of these precise data comes from the possibility of obtaining less ambiguous α–nucleus potential in this energy regime. 5

2. Theoretical formalism In the present work, we analyzed the elastic angular distributions in the framework of the optical model (OM). Using this model, the elastic scattering cross section can be calculated from the solution of the Schrödinger equation with a complex nucleus–nucleus potential. Usually, the potential is a function of the separation distance between the centers of projectile and target nuclei (𝑅 ) as: (1)

𝑈(𝑅) = 𝑉𝐶(𝑅) + 𝑁𝑟[𝑅𝑒 𝑉(𝑅)] + 𝑖𝑊(𝑅)

here 𝑉𝐶(𝑅) is the Coulomb potential which is obtained by assuming a uniform charge distribution with a radius 𝑅𝐶. 𝑁𝑟 is the renormalization factor of the folded real part and 𝑊(𝑅) is the imaginary term of the nuclear OMP. In this work, we used the folding model to generate the nuclear OMP using two different effective NN interactions, namely, the JLM and DDM3Y1-FR effective NN interactions. Three OMP versions are extracted based on these effective NN interactions. Two versions are semimicroscopic potentials where the real parts are based on the folding model and the imaginary parts are of surface WS with the following form, 𝑑

[

𝑅 ― 𝑅𝑖

𝑊(𝑅) = 𝑊D𝑑𝑅 1 + exp (

𝑎𝑖

)

]

―1

(2)

.

These semi-microscopic potentials are denoted as JLM-R and DDM3Y1-FR. The third one is the complex folded potential JLM-RI. In this case, the imaginary potential 𝑊(𝑅) is calculated using the folding procedure with JLM interaction multiplied by an imaginary renormalization factor 𝑁𝑖. The folding model serves as a good tool for predicting the bare OMP. In the folding model, the nuclear matter densities and the proper effective NN interaction are the most important ingredients. These two ingredients make the OMP more predictable, especially the real part, for any projectile and target combination over a wide range of energy.

6

2.1. OMP based upon JLM effective interaction: The JLM effective NN interaction is a nuclear matter based effective NN interaction. This interaction is a realistic complex effective NN interaction that is constructed on the base of nuclear density and energy [25]. The way in which this potential is constructed takes care of higher order effects like nucleon exchange and anti-symmetrization. Hence the Hartree term alone can reproduce the entire nucleon–nucleus and nucleus–nucleus optical potential without explicitly treating the exchange contribution. Using the extracted matter densities [40] with the JLM effective NN interaction, we get first the nucleon–nucleus potential (n-N) as: 𝑉𝑛 ― 𝑁(𝑅,𝐸𝑛) = ∫𝜌𝑇(𝑟) 𝜐𝑛𝑛(|𝑠|,𝜌,𝐸𝑛) 𝑑3𝑟, s = R ― r,

(3)

In general, the JLM effective NN interaction is factorized as: 𝜐𝑛𝑛(|𝑠|,𝜌,𝐸𝑛) =

gr(|𝑠|) 𝜌𝑇

[𝑉0(𝜌𝑇,𝐸𝑛) + 𝛼𝜏𝑉1(𝜌𝑇,𝐸𝑛)] gi(|𝑠|)

+𝑖

𝜌𝑇

[𝑊0(𝜌𝑇,𝐸𝑛) + 𝛼𝜏𝑊1(𝜌𝑇,𝐸𝑛)],

(4)

Here 𝜌𝑇 is the matter density of the target nuclei. The radial factors, gr(|𝑠|) is of the following Gaussian form [27: 3

2 gk(|𝑠|) = (1 𝑡𝑘 𝜋) 𝑒𝑥𝑝( ― |𝑠| 𝑡2𝑘),

(5)

𝑘 = 𝑟 𝑜𝑟 𝑖

Then we obtain the α-N OP as: Vα ― N(R,En) = ∑j = p,n∫ραj(rα)VjN(|s|,En)d3rα,

|s| = |R ― rα| (6)

7

where the 𝜌𝛼𝑗(𝑟𝛼) is the matter density of the alpha particle point proton density (𝑗 = 𝑝) or point neutron density (𝑗 = 𝑛), respectively. For the range parameters, we adopted the values 𝑡𝑟 = 1.25 𝑓𝑚, 𝑡𝑖 = 1.35 𝑓𝑚 [28-30]. In previous studies, the commonly accepted values for 𝑡𝑟 and 𝑡𝑖 are found to locate between 1 and 1.4 𝑓𝑚 [31–35]. 𝛼 = (𝜌𝑛 ― 𝜌𝑝) (𝜌𝑛 + 𝜌𝑝) is the asymmetry term, τ = +1( ― 1) for proton (neutron) projectile, respectively. 𝐸𝑛 is the incident nucleon energy (for incident proton 𝐸𝑛 should be replaced by 𝐸𝑛 ― 𝑉𝐶𝑜𝑢𝑙), 𝑉0, 𝑉1, 𝑊0 and 𝑊1are the potential components. It is worth noting that the iso–vector component of the effective interaction has no contribution since the α – particle is 𝑁 = 𝑍 nuclei. 2.2. OP based upon DDM3Y1 effective interaction: For the sake of comparison and confirmation, the real part of the α–nucleus OMP is obtained using the density–dependent M3Y-Reid [36] effective NN interaction (DDM3Y). The M3Y and its different density dependent versions are the most widely and successfully used effective NN interactions in the last few decades in understanding nuclear scattering and reactions [2]. In this work, we used the DDM3Y1-FR version of M3Y effective NN interaction. The functional form [2] of this version is expressed as follows: (7)

𝜈𝐷(𝐸𝑋)(𝜌,𝑠,𝐸) = 𝑔(𝐸)𝐹(𝜌)𝜐𝐷(𝐸𝑋)(𝑠),

where 𝜈𝐷(𝑠) is the direct part, 𝜈𝐸𝑋(𝑠) is the knock-on exchange parts, 𝑔(𝐸) and 𝐹(𝜌) are the energy–and the density–dependent factors, respectively. The explicit functional forms of the DDM3Y1-FR [37-39] are parameterized as: 𝑒 ―4𝑠

𝜐𝐷(𝑠) = 7999.0

4𝑠

𝑒 ―2.5𝑠

(8)

―2134.25 2.5𝑠 , 𝑒 ―4𝑠

𝜐𝐸𝑋(𝑠) = 4631.38

4𝑠

𝑒 ―2.5𝑠

𝑒 ―0.7072𝑠

(9)

―1787.13 2.5𝑠 ―7.8474 0.7072𝑠 ,

𝑔(𝐸) = [1 ― 0.002(𝐸/𝐴.)],

(10)

8

(11)

𝐹(𝜌) = 0.2843[1 + 3.6391𝑒𝑥𝑝 ( ―2.9605 𝜌)].

For the overlapping density 𝜌 in Eq. (11), we used for simplicity the arithmetic average approximation of the individual densities as 𝜌 =

(𝜌𝛼(𝑟𝛼) + 𝜌𝑇(𝑟𝑇)) 2. The folding model with M3Y or DDM3Y effective NN interaction predicted only the real part of the nucleus-nucleus potential. In such a case, a phenomenological imaginary potential must be added to the real part to describe elastic scattering. The direct part is usually obtained by the folding procedure as: VD(E,R) = ∫ρα(rα)ρT(rT)νD(ρ,E,|𝑠|)drαd𝑟𝑇 ,𝑠 = 𝑅 + 𝑟𝑇 ― 𝑟𝛼

(12)

Here 𝜈𝐷(𝜌,𝐸,𝑠) is the direct part of the effective NN interaction and 𝜌𝛼(𝑟𝛼) and 𝜌𝑇(𝑟𝑇) , are the densities of the α–particle and target nuclei, respectively. Exact non-local exchange part is too complicated in real calculations and an equivalent local potential could be obtained by representing the relative motion wave function of nucleons as a plane wave. In such a case the exchange potential is obtained by the following fully anti-symmetrized matrix element of the exchange part 𝜈𝐸𝑋(𝑠) of the effective NN interaction as: 𝑉𝐸𝑋(𝑅) = ∑𝑖 ∈ 𝛼, 𝑗 ∈ 𝑇⟨𝑖𝑗│𝜈𝐸𝑋(|𝑠|)│𝑗𝑖⟩

(13)

where |𝑖⟩ and |𝑗⟩ refer to the single-particle wave functions of the α–particle and target nuclei, respectively. Using explicitly the one-body density matrix 𝜌(𝑟,𝑟), one can write (12) as: 𝑉𝐸𝑋(𝐸,𝑅) = ∫𝜌𝛼(𝑟𝛼, 𝑟𝛼 + 𝑠)𝜌𝑇(𝑟𝑇,𝑟𝑇 ― 𝑠)𝜈𝐸𝑋(𝜌,𝐸,𝑠)𝑒𝑥𝑝(𝑖𝑘(𝑅).𝑠 𝑀)𝑑𝑟𝛼𝑑𝑟𝑇, (14)

(

𝑠

) ( (|

𝑠

|) )

(15)

𝜌𝑖(𝑟, 𝑟 ± 𝑠) = 𝜌𝑖 𝑟 ± 2 𝑗1 𝑘𝑖𝑓 𝑟 ± 2 𝑠 , 9

sin 𝑥 ― 𝑥cos 𝑥

𝑗1(𝑥) = 3

𝑥3

(16)

,

The Fermi momentum 𝑘𝑓, is considered in the following approximation 𝑘𝑖𝑓(𝑥) =

[

3𝜋2 2

]

𝜌𝑖(𝑥)

23

+

5𝐶𝑆[∇𝜌𝑖(𝑥)]2 3𝜌2𝑖 (𝑥)

5 ∇2𝜌𝑖(𝑥)

+ 36

𝜌𝑖(𝑥)

(17)

,

𝐶𝑆 is the Weizsaecker strength which represents the surface contribution to the Thomas–Fermi kinetic energy. 𝑘(𝑅) is the relative-motion momentum and is given by: 𝑘(𝑅) =

2𝑚𝑀 ℏ2

[𝐸𝐶𝑀 ― 𝑉(𝑅) ― 𝑉𝐶(𝑅)],

(18)

𝑀 = 4𝐴𝑇/(4 + 𝐴𝑇) and 𝐸𝐶𝑀 are the reduced mass and the relative energy in the center of mass system and 𝑚 is the nucleon mass. Thus, the real folded potential for α – nucleus becomes 𝑉(𝑅) = 𝑉𝐷(𝑅) + 𝑉𝐸𝑋(𝑅).

(19)

The imaginary potential with this real folded potential is of the surface WS form as Eq. (2). As stated in previous studies [24] (and references therein) the surface imaginary potential is dominate at the near Coulomb barrier energies. 2.3. The matter density distribution: In the present work, the matter density distribution of the α–particle is taken in the simple Gaussian form [40] as: 𝜌𝛼(𝑟) = 0.4229𝑒𝑥𝑝( ―0.7024𝑟2),

(20)

and the matter density distribution for targets nuclei are taken in the form of Fermi–density distribution as [41]: 10

[

)]

𝑟 ― 𝐶𝑛,𝑝

(

ρ𝑛,𝑝(r) = 𝜌𝑛,𝑝(0) 1 + 𝑒𝑥𝑝

0.55

―1

(21)

where C𝑖 represents the central radius of the distribution and it is defined as:

[ ( ) ],

𝐶𝑖 = 𝑅𝑖 1 ―

𝑏 2

(𝑖 = n, p)

𝑅𝑖

(22)

𝑅𝑖 is the effective sharp radius parameter which is given as, 1

𝑅𝑖 = 1.28𝐴1/3 ―0.76 + 0.8𝐴

―3

(23)

𝑓𝑚

while, 𝜌𝑛,𝑝(0) is given by 3𝑁1

𝜌𝑛(0) = 4𝜋 𝐴 𝑟3 , 0𝑛

with 𝑟𝑜𝑝 = 1.128 𝑓𝑚 ,

3𝑍1

(24)

𝜌𝑝(0) = 4𝜋𝐴𝑟3

0𝑝

𝑟𝑜𝑛 = 1.1375 + 1.875 × 10 ―4A 𝑓𝑚

3. Results and discussion The folding calculations are carried out to obtain the α-particle OMP's from the targets

120,124,126,128,130Te, 118Sn

at energies 17, 19, 22, 24.5

and 27 [16] MeV and 124Sn at 19.51 [13] MeV as explained in Sec. 2. The twenty-nine sets of data are analyzed in the framework of our derived JLM–R, JLM–RI and DDM3Y1–FR OMPs. The computer code HIOPTIM-94 [43] is used to calculate the angular distributions of the elastic scattering differential cross sections for the twenty-nine data sets that are considered in the present work. A search is carried out on optical potential parameters to get the best fitting to experimental data by minimizing χ2, using the equation:

11

1

N

σcal(𝜃i) ― σexp(θi) 2

(

χ2 = N∑i = 1

Δσexp(𝜃i)

).

(25)

σcal(𝜃𝑖) and σexp(𝜃𝑖) are the calculated and experimental cross-sections at angle𝜃𝑖, respectively, Δσexp(𝜃𝑖) is the corresponding experimental error and 𝑁 is the number of data points. The search is performed on four free parameters, depths W0, radius (Ri) and diffuseness (ai) parameters of the imaginary surface in addition to real renormalization factor Nr, for the real JLM–R and DDM3Y1–FR potentials. For the complex JLM–RI potential only two free parameters are searched on, the real and imaginary renormalization factors (Nr and Ni). The obtained best-fit parameters are listed in Tables-(17) while the predicted angular distributions of elastic scattering differential cross section are shown in Figs.1–7 in comparison with the corresponding experimental data. The relations between the best-fit parameters in Tables-(1-7) with the target mass and energy for the twentynine sets are investigated. We found that the searched parameters show a very weak target mass dependence. So, we present the target mass mean values of these parameters (target mass average values) in Table-8. These mean values are plotted as a function of energy in Figs. 9-11. The bars (the standard deviation) in Fig. 9 –11 represent the target mass dependence at each energy. 3.1 Elastic scattering cross sections Many systematic α–OMPs were proposed in the past few decades to generate α–nucleus global potential. Four global α – nucleus potentials of McFadden and Satchler [44], Avrigeanu et al. [8], Frohlich and Rauscher [45–47], and Demetriou et al. [7] were widely used to analyze the α–nucleus reactions. Palumbo et al. [16] tested these potential models in studying and analyzing the considered data sets in their work. Demetriou et al. [7] model is the only model that provides a consistent agreement with experimental data, but it under-estimates the cross sections oscillations at large angles, while significant discrepancies for the other models were found [16]. The calculated elastic scattering angular distributions for the considered systems using our derived potentials are displayed in comparison with the corresponding experimental data [16] in Fig. 1 for α-

118Sn

and Figs.2–6 for α-120-124-126-128-130Te at the energies 17,19,22,24.5 and 27, 12

respectively. In addition, the calculated elastic scattering angular distributions in comparison with the corresponding experimental data [13] for the system α-

124Sn

is shown in Fig.7. From these figures and the 𝜒2 𝑁 values, we found that both JLM–R and DDM3Y1–FR potentials are

consistent to describe the experimental data very well overall the energy and angular ranges. Both potentials reproduce successfully the positions and magnitude of the cross-section’s oscillations at large angles. The prediction of the complex microscopic JLM–RI potential to describe the data is limited to low energy data. As shown in the figures, the JLM–RI potential significant underestimates the predicted cross sections and oscillations at backward scattering angles for energies (E ≥ 22 MeV). This means that the microscopic imaginary potential, based on JLM complex effective NN interaction, failed to predict the radial shape of the imaginary part of α–nucleus optical potential over a wider radial range. This may be turned to the threshold anomaly which is observed at low energy and occurred due to an effective closure of open reaction channels [48]. The success of our derived potentials in reproducing the experimental data under investigation is also consistent with the previous studies [12,16,49]. This result indicates the success of our approach to predict the radial shape and strength of the real potential. In contrary, the less success of the JLM–RI potential implies that the complex JLM effective interaction could not predict the correct radial shape and potential strength of the imaginary part. This could be attributed to the choice of the imaginary range parameter of the JLM radial form factor. This parameter could be left as a free search parameter and see how it affects the calculated cross sections. To see whether the calculated potentials depend on the chosen density distributions for targets, we calculated the α–118Sn DDM3Y1-FR real folded potential using experimental matter densities for 118Sn nucleus based on electron scattering [42]. In Fig.8, we compare the different density distributions for

118Sn

nucleus (2pF-1, 2pF-2 and 3pG) of Ref [42] with the present (2pF-density)-panel (a) and the corresponding real

folded potentials -panel (b). The root mean square (RMS) for these densities is 4.679, 4.676, 4.634 [42] and 4.683 fm, respectively. As shown in this figure, the present 2pF, 2pF-1, 2pF-2 densities are almost identical and different from the 3pG density for distance above 5 fm. The corresponding calculated folded potentials are denoted as DDM3Y-2pF, DDM3Y-2pF-1, DDM3Y-2pF-2, and DDM3Y-3pG respectively. From the behavior of these densities, it is noticed that DDM3Y-2pF, DDM3Y-2pF-1, DDM3Y-2pF-2 folded real potentials are identical and different 13

from the DDM3Y-3pG potential at the tail. As shown in panel (b), the DDM3Y-3pG is less attractive at distance above 9 fm. In panel (c), we present the corresponding differential cross sections for the four calculated real potentials. These potentials reproduce the differential cross sections in equal footing. This means that the cross sections are sensitive to the calculated potentials for a distance less than 9 fm, where the potentials are very close to each other. 3.2 Energy and mass dependence From Tables- (1-7), we have found that the best-fit parameters show an energy and weak target mass dependence. For example, for a given energy the searched parameters are slightly fluctuating around some average value for all nuclei. The mean values for these parameters are listed in Table-8 and shown in Figs. (9-11) as a function of energy. These average values change with energy and show clear energy dependence. It is obvious from these tables that, the renormalization factors 𝑁𝑟 for all derived potentials have nearly the same behavior as they slightly decrease with mass number. Accurately, the approximate average for 𝑁𝑟 varies from 0.8 to 0.795 for JLM–R and from 0.98 to 0.95 for DDM3Y1–FR potentials while, approximately constant for microscopic potential JLM–RI with nearly average value 0.78. On other hand, the imaginary renormalization factor 𝑁𝑖 for the imaginary part slightly increases with energy and target mass. The energy dependence of the average WS imaginary potential parameters is shown in Fig. 9. As shown, the surface depth (W0) and radius (Ri) linearly increase with increasing energy while the diffuseness (ai) parameters linearly decrease with energy. The energy dependence of these average parameters’ values may be formulated by a linear relation as given by Eq. (26). The parameters of these linear relations are listed in Table-9. 𝐴 = 𝐹 + 𝐾 𝐸𝐿𝑎𝑏, (A = 𝑊0, 𝑅𝑖, 𝑎𝑖, 𝐽𝑟, 𝐽𝑖)

(26)

These linear relations are presented to show the general trend of the energy dependence of the searched parameters and the calculated quantities. It is worth mentioned that the energy range is not wide enough to make a conclusion about the actual energy dependence. As shown in Fig. 9, the imaginary depth parameter seems to be saturated at some value as energy increases. This reflects the dispersive nature of the optical potential 14

and the energy dependence should be considered in more detail for an extended energy range with an appropriate energy step. For the shape parameters, we found that the radius parameter has a weak increasing energy dependence and it is approximately constant as energy increases. The diffuseness parameter has weak decreasing energy dependence and saturated at some value as energy increases. Concerning the volume integrals, it is noticed that the real volume integrals slightly and linearly decrease with increasing energy for our derived potentials, while the imaginary ones linearly increase with energy. Both the real and imaginary volume integrals show slight deviations with target mass as is shown in the Table- (1-7). It should be mentioned that the present results are consistent with the results of previous studies [12, 16, 45] for both real and imaginary potentials. The energy dependence of the target mass average real and imaginary volume integrals is shown in Fig. 10. In this figure, we found that the imaginary volume integral increases with energy up to 22 MeV and saturated above this energy for JLMR and DDM3Y1-FR. This energy behavior reflects the dispersive nature of the optical potential. This means that the energy dependence of the real volume integrals should be considered in more detail with an extended energy range. We have presented this energy dependence for the real volume integral to guide the eyes about the general weak decreasing energy dependence in this very limited energy range. As we see from the figures (Figs. 9-10), the linear fittings have the same behavior of all the potentials for the same calculated quantity. It is interesting to investigate the obtained values of the absorption (total reaction) cross sections 𝜎𝑅 from the analysis of the considered systems. Reaction cross sections data are important to verify the analysis of elastic scattering reaction and to find a unique optical potential which can eliminate ambiguities in optical potentials. From Tables-(1-7), we find that 𝜎𝑅 for a given energy slightly change from nucleus to nucleus. So, the mass average 𝜎𝑅 over the target mass number at given energies is listed in Table-8. The standard deviation of the average 𝜎𝑅 is hardly seen in the figure. We present the energy dependence of our mass averaged 𝜎𝑅 in Fig. 11. At energies below the Coulomb, the reaction cross section is governed by the tunneling probability through the Coulomb barrier. So 𝜎𝑅 should have an exponential growing with increasing energy from below to above the Coulomb barrier. We compare our calculated mass averaged 𝜎𝑅 with that calculated using McFadden and Satchler [44] global α – nucleus potential as shown in Fig.11. It is obvious that the present calculated 𝜎𝑅 are in consistence with that calculated using McFadden and Satchler [44] global α–nucleus potential as well as with the previous studies [16]. 15

4. Summary and Conclusions The present work is an extension of our previous studies [50-52] for constructing a global α-nucleus potential. In this framework, we derived the OMPs based on the folding model to describe the elastic scattering of α– particles from

118,124Sn

and the isotopic

120-130Te

chain at

energies of astrophysics interest. Two semi-microscopic JLM–R and DDM3Y1–FR potentials, in addition to microscopic JLM-RI potential, are derived. Our aim is to obtain a systematic α–nucleus optical potential with very few search parameters. For this reason, we used a folded optical model potential based on the complex JLM effective NN interaction with only two adjustable parameters (real and imaginary renormalization factors). In addition, we used a renormalized real part of the folding optical potential for JLM effective NN interaction with a surface WS imaginary part. For completeness, we used a renormalized real folded potential based on DDM3Y effective NN interaction with a surface WS imaginary part. For the last two procedures, we have only four free parameters, the renormalization factor and the three imaginary potential parameters. In this work, no scaling or width parameter is introduced in the real folded potentials. In JLM calculations of the folded optical potential, the target density dependence is only considered. This potential is computed in two steps first, we calculate the nucleon–target potential (nucleon means proton or neutron in α–particle). In the next step, we folded the resulted potential with proton and neutron densities of the α–particle. For the potential based on the DDM3Y1-FR effective interaction, both the α–particle and target density dependence are considered. In addition, the finite range exchange interaction is also considered. The calculated potentials are feed into the auto–search code to compute the scattering observables. It is found that the obtained results are in quite agreement with the measured data over both the energy and angular range. By looking deeply into the results, we found that the DDM3Y1–FR is in better agreement with data than the JLM based potentials. Since we look for regional OMP for α–nucleus systems, we have studied the dependence of our potentials on both target mass and energy. We observed that the obtained fitting parameters show a very weak target mass dependence. These parameters fluctuate around some average values for each nucleus with small standard deviations at any given energy. In addition, the average values of these parameters show 16

clear energy dependence. Moreover, we found that the calculated quantities like the volume integrals and total reaction cross sections are approximately target mass independent. The average values of these quantities have clear energy dependence. By comparing our results with previous results for the same system and energy range, we found that our approach is quite successful and very promising. The present results encourage us to extend the analysis over wider target mass range for extending our regional α–nucleus potentials.

Acknowledgment The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/124/40. References 1. W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952). 2.

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Fig. 1. α–118Sn calculated elastic scattering angular distribution at the energy range 17-27 MeV compared with experimental data (open circles) [16]. Solid, dashed, and dashed-dot dot lines represent the calculated cross sections using JLM–R, JLM–RI and DDM3Y1–FR potentials, respectively. Fig. 2. The same as Fsig.1 but for α–120Te system. The experimental data are taken from [16] Fig. 3. The same as Fsig.1. but for α–124Te system. The experimental data are taken from [16] Fig. 4. The same as Fsig.1 but for α–126Te system. The experimental data are taken from[16] Fig. 5. The same as Fsig.1. but for α–128Te system. The experimental data are taken from[16] Fig. 6. The same as Fsig.1. but for α–130Te system. The experimental data are taken from [16] Fig. 7. The same as Fsig.1 but for α–124Sn system at the energy 19.51 MeV. The experimental data are taken from [13] Fig. 8. Panel (a) presents 2pF, 2pF-1, 2pF-2 and 3pG densities of

118Sn

nucleus, Panel(b) shows the corresponding DDM3Y1-FR real folded

potentials for α– 118Sn system at 17 MeV. Panel (c) shows the corresponding elastic scattering angular distributions. Fig. 9. Energy dependence of the best-fit parameters 𝑊0 , 𝑅𝑖 and 𝑎𝑖 of the imaginary potential part. Left Panels for JLM–R potential while the right panel for DDM3Y1–FR potential. 20

Fig. 10. Energy dependence of the volume integrals for JLM–R, JLM–RI and DDM3Y1–FR potentials, respectively. Left Panels for real volume integrals while the right Panels for the imaginary volume integrals. Fig. 11. Energy dependence of the total reaction cross-sections 𝜎𝑅 for JLM–R, JLM–RI and DDM3Y1–FR potentials respectively in comparison to that of McFadden/Satchler potential [44].

Table-1: Best-fit parameters of the calculated potentials for the α–118Sn system at the energies 17–27 MeV. 𝐸 [MeV] 17.00

19.00

Pot.

𝑁𝑟

𝐽𝑟 [MeV.fm3]

JLM-R JLM-RI DDM3Y1-FR

0.800 0.800 0.97

372.47 372.47 358.10

JLM-R

0.800

371.17

JLM-RI

0.800

371.17

DDM3Y1-FR

0.95

345.5

JLM-R

0.811

374.47

JLM-RI

0.793

365.91

22.00

〈𝑟2𝑟 〉 [fm] 5.52 5.45 5.52 5.45

〈𝑟2𝑖 〉

𝜒2

𝑊𝐷 [MeV]

𝑅𝑖 [fm]

𝑎𝑖 [fm]

𝐽𝑖 [MeV.fm3]

[fm]

27.366 1.110 26.15

1.432

0.525

1.355

0.589

76.86 77.44 74.30

7.34 6.64 7.10

3.1 2.4 2.1

537.4 572.4 528.1

33.493

1.484

0.404

77.14

7.46

13.1

729.6

71.86

6.63

9.5

831.4

1.025

𝑁

𝜎𝑅 [mb]

30.01

1.301

0.64

86.03

6.90

6.6

803.5

34.303

1.472

0.415

79.93

7.41

20.4

1026

69.63

6.61

42.5

1130.

5.51 0.990 21

24.00

27.00

DDM3Y

1.02

350.15

JLM-R

0.799

365.65

5.45

36.29

1.232

0.754

111.73

7.20

18.8

1225

34.474

1.442

0.453

84.21

7.30

19.2

1224

69.94

6.60

36.3

1318.

5.51 JLM-RI

0.801

366.22

DDM3Y1-FR

0.90

364.90

JLM-R

0.82

375.14

0.990 5.45

38.36

1.402

0.446

87.31

7.11

13.6

1172

36.3

1.430

0.47

91.53

7.28

34.9

1392

75.56

6.58

77.5

1469

86.36

7.10

29.3

1326

𝜒2

𝜎𝑅 [mb]

5.51 JLM-RI

0.780

356.84

DDM3Y1-FR

0.89

361.8

1.074 5.45

36.54

1.393

0.468

𝑊𝐷 [MeV]

𝑅𝑖 [fm]

𝑎𝑖 [fm]

𝐽𝑖 [MeV.fm3]

30.058 0.938 31.253

1.370

0.577

1.253

0.668

84.88 64.35 88.71

7.15 6.64 6.75

1.7 1.7 1.4

466.7 474.2 465.7

33.590

1.472

0.410

76.4

7.10

7.2

654.4

61.01

6.66

5.5

742.2

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-2: The same as Table-1 but for the α–120Te system. 𝐸 [MeV]

Pot.

𝑁𝑟

𝐽𝑟 [MeV.fm3]

17.00

JLM-R JLM-RI DDM3Y1-FR

0.800 0.800 0.96

372.39 372.39 345.60

JLM-R

0.800

371.09

JLM-RI

0.800

371.09

DDM3Y1-FR

0.960

347.09

JLM-R

0.800

369.15

19.00

22.00

〈𝑟2𝑟 〉 [fm] 5.54 5.47 5.54 5.47

0.881

〈𝑟2𝑖 〉 [fm]

𝑁

32.594

1.252

0.675

91.37

6.75

3.5

737.3

36.294

1.439

0.451

87.41

7.32

17.4

976.8

58.62

6.64

25.3

1053.

5.53 JLM-RI

0.796

367.42

0.843 22

24.00

27.00

DDM3Y1-FR

0.97

343.37

JLM-R

0.800

376.53

5.47

39.461

1.400

0.427

85.33

7.12

18.9

902.

37.660

1.428

0.454

89.93

7.28

15.9

1168

58.57

6.62

28.6

1246

5.53 JLM-RI

0.800

367.53

DDM3Y1-FR

0.965

341.90

JLM-R

0.792

362.14

00.842 5.47

39.481

1.368

0.466

89.25

7.01

15.3

1108.

36.201

1.407

0.486

90.15

7.21

27.3

1336

65.83

6.60

64.4

1406

6.95

30.2

1261

5.53 JLM-RI

0.777

355.47

DDM3Y1-FR

0.975

339.69

00.947 5.47

41.940

1.355

0.479

95.25

𝑊𝐷 [MeV]

𝑅𝑖

𝑎𝑖

𝐽𝑖

[fm]

[fm]

[MeV.fm3]

[fm]

30.000

1.385

0.548

81.10

7.25

1.8

478.8

53.97

6.73

1.9

484.3

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-3: The same as Table-1 but for the α–124Te system. 𝐸 [MeV]

17.00

29.00 22.00

𝐽𝑟

Pot.

𝑁𝑟

JLM-R

0.800

371.49

JLM-RI

0.750

348.28

DDM3Y1-FR

0.91

346.39

JLM-R

0.800

370.20

JLM-RI

0.801

370.55

DDM3Y1-FR

0.91

344.9

JLM-R

0.800

368.26

[MeV.fm3]

〈𝑟2𝑟 〉 [fm] 5.58 5.51 5.58

5.58

0.789

〈𝑟2𝑖 〉

𝜒2

𝜎𝑅 𝑁

[mb]

33.00

1.270

0.636

88.03

6.83

2.6

475.2

34.294

1.359

0.571

93.20

7.16

7.2

757.3

58.64

6.71

4.9

767.0

0.853 33.994

1.246

0.681

94.04

6.79

3.1

7715.

36.230

1.437

0.453

86.49

7.40

20.9

1007.

23

24.00

27.00

JLM-RI

0.781

359.49

DDM3Y1-FR

0.93

342.68

JLM-R

0.800

368.26

JLM-RI

0.787

362.46

DDM3Y1-FR

0.95

341.21

JLM-R

0.799

364.59

JLM-RI

0.770

351.53

DDM3Y1-FR

0.96

339.02

0.842 5.51 5.58 5.51 5.58 5.51

58.11

6.69

25.1

1077.

37.73

1.391

0.445

82.43

7.17

18.1

941.5

36.000

1.433

0.453

85.55

7.38

21.4

1201.

62.31

6.68

33.2

1281.

0.902 37.73

1.391

0.459

85.96

7.12

16.0

1136.

35.491

1.419

0.472

86.27

7.33

25.6

1363.

63.64

6.66

56.1

1432.

88.95

7.06

12.7

1298.

𝜒2

𝜎𝑅

0.923 38.90

1.360

0.482

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-4: The same as Table-1 but for the α–126Te at 17, 19 and 22 MeV. 𝐸 [MeV]

17.00

19.00

𝐽𝑟

Pot.

𝑁𝑟

JLM-R

0.800

371.06

JLM-RI

0.773

358.45

DDM3Y1-FR

0.89

346.06

JLM-R

0.800

369.77

JLM-RI

0.800

369.93

DDM3Y1-FR

0.92

344.57

[MeV.fm3]

〈𝑟2𝑟 〉 [fm] 5.60 5.53 5.60 5.53

〈𝑟2𝑖 〉

𝑊𝐷 [MeV]

𝑅𝑖

𝑎𝑖

𝐽𝑖

[fm]

[fm]

[MeV.fm3]

[fm]

30.300

1.374

0.559

81.76

7.25

1.4

494.5

53.72

6.75

1.5

500.5

0.788

𝑁

[mb]

29.300

1.282

0.639

79.75

6.93

1.2

490.6

34.250

1.344

0.577

91.66

7.13

6.4

770.8

53.72

6.74

4.8

777.2

86.24

6.83

2.7

781.5

0.785 31.25 24

1.245

0.683

JLM-R

0.808

371.37

JLM-RI

0.782

359.62

DDM3Y1-FR

0.96

342.36

36.200

1.435

0.453

85.71

7.42

22.4

1022.

57.18

6.72

24.5

1090.

89.72

6.90

12.5

1054

〈𝑟2𝑖 〉

5.60 22.00

0.832 5.53

34.20

1.282

0.618

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-5: The same as Table-1 but for the α–128Te system. 𝐸 [MeV]

17.00

Pot.

𝑁𝑟

JLM-R

0.800

370.63

JLM-RI

0.760

363.49

0.92

345.76

JLM-R

0.800

369.34

JLM-RI

0.789

364.05

DDM3Y1FR

19.00

𝐽𝑟 [MeV.fm3]

〈𝑟2𝑟 〉 [fm] 5.62 5.56 5.62

𝜎𝑅

𝑊𝐷 [MeV]

𝑅𝑖

𝑎𝑖

𝐽𝑖

[fm]

[fm]

[MeV.fm3]

[fm]

30.550

1.364

0.565

81.87

7.24

1.9

508.3

49.89

6.78

1.9

504.5

𝜒2 𝑁

[mb]

33.55

1.279

0.599

84.49

6.89

1.3

480.2

34.300

1.342

0.575

90.65

7.15

6.5

782.4

52.38

6.77

5.2

786.4

0.768 25

22.00

24.00

27.00

DDM3Y1-FR

0.940

344.28

JLM-R

0.800

367.41

JLM-RI

0.769

364.63

DDM3Y1-FR

0.920

342.06

JLM-R

0.800

365.81

JLM-RI

0.800

365.69

DDM3Y1-FR

0.950

340.60

JLM-R

0.773

352.00

JLM-RI

0.770

350.78

DDM3Y1-FR

0.970

338.41

5.56 5.62

34.30

1.242

0.658

89.88

6.80

2.7

776.2

36.376

1.435

0.453

85.70

7.46

25.0

1036.

59.41

6.75

28.9

1099.

0.840

5.56 5.62

38.30

1.397

0.437

82.34

7.26

15.6

967.1

35.500

1.439

0.443

84.42

7.47

21.1

1223.

60.11

6.73

34.8

1309.

0.877

5.56 5.62

38.95

1.386

0.451

85.18

7.22

13.3

1163.

35.60

1.419

0.475

85.8

7.41

29.3

1388.

67.06

6.71

75.7

1465.

88.07

7.15

12.9

1330

〈𝑟2𝑖 〉

𝜒2

0.979

5.56

38.89

1.361

0.480

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-6: The same as Table-1 but for the α–130Te system. 𝐸 [MeV]

17.00

19.00

𝐽𝑟

Pot.

𝑁𝑟

JLM-R

0.800

370.21

JLM-RI

0.766

354.44

DDM3Y1-FR

0.95

345.44

JLM-R

0.800

368.93

JLM-RI

0.795

366.67

DDM3Y1-FR

0.96

343.96

[MeV.fm3]

〈𝑟2𝑟 〉 [fm] 5.65 5.58 5.64 5.58

𝑊𝐷 [MeV]

𝑅𝑖

𝑎𝑖

𝐽𝑖

[fm]

[fm]

[MeV.fm3]

[fm]

29.822

1.376

0.552

78.94

7.32

1.8

515.5

50.39

6.80

1.6

520.1

0.745

𝜎𝑅 𝑁

[mb]

29.94

1.291

0.625

79.58

7.01

1.2

509.8

34.100

1.352

0.565

89.18

7.21

7.7

791.7

49.48

6.79

5.7

798.9

81.87

6.95

3.2

778.3

0.728 31.10 26

1.274

0.634

22.00

24.00

27.00

JLM-R

0.761

349.2

JLM-RI

0.774

355.03

DDM3Y1-FR

0.95

341.75

JLM-R

0.800

365.40

JLM-RI

0.798

364.53

DDM3Y1-FR

0.97

340.28

JLM-R

0.775

352.40

JLM-RI

0.773

351.48

DDM3Y1-FR

0.99

338.09

5.64 5.58 5.64 5.58 5.64 5.58

34.29

1.34

0.610

0.840

94.63

7.51

29.7

1126.

57.33

6.77

31.8

1115.

35.35

1.437

0.437

76.51

7.33

18.8

982.1

36.450

1.445

0.433

82.53

7.52

23.9

1232.

60.34

6.75

43.3

1323.

0.883 37.45

1.395

0.443

81.01

7.29

15.2

1174

35.42

1.435

0.43

78.64

7.45

28.8

1372.

65.59

6.73

68.7

1479.

86.05

7.20

20.4

1347.

0.961 37.92

𝑊0 = 𝑁𝑖 for JLM-RI potentials

27

1.368

0.481

Table-7: The same as Table-1 but for the α–124Sn system. 𝐸 [MeV ]

Pot.

𝑁𝑟

JLM-R 19.51

JLM-RI DDM3Y1 -FR

𝐽𝑟 [MeV.fm3 ]

0.77 2 0.72 5 1.00

〈𝑟2𝑟 〉 [fm]

351.00 5.58 334.84 344.77

5.51

𝑊𝐷 [MeV ]

𝑅𝑖 [fm]

𝑎𝑖 [fm]

𝐽𝑖 [MeV.fm3 ]

[fm]

36.53 7

1.48 0

0.39 1

79.62

7.55

0.956

--

--

70.41

6.71

38.67

1.40 1

0.44 9

87.11

7.22

〈𝑟2𝑖 〉

𝜒2 𝑁 2.35 3 12.3 4 1.1

𝜎𝑅 [mb] 805. 7 909. 6 794. 2

𝑊0 = 𝑁𝑖 for JLM-RI potentials

Table-8: The mean value for the best fitting parameters in the previous tables. Pot.

JLM-R

JLM-RI

DDM3Y1-FR

Energy

17

19

22

24

27

WD [MeV]

29.683

34.005

35.616

36.017

35.802

Ri [fm]

1.384

1.392

1.426

1.437

1.422

ai [fm]

0.554

0.517

0.473

0.447

0.467

Jr [MeV.fm3]

371.375

370.083

366.643

368.330

361.254

Ji [MeV.fm3]

80.902

86.372

86.645

85.328

86.478

σR [mb]

498.2

743.8

1027.3

1203.0

1364.8

Jr [MeV.fm3]

361.587

368.910

362.020

365.286

353.220

Ji [MeV.fm3]

58.293

57.848

60.047

62.254

67.536

σR [mb]

509.3

783.9

1094.0

1295.4

1450.2

WD [MeV]

30.532

31.941

36.889

38.39

38.838

Ri [fm]

1.288

1.302

1.357

1.388

1.367

ai [fm]

0.626

0.610

0.520

0.453

0.478

Jr [MeV.fm3]

347.892

351.373

343.728

345.778

343.402

Ji [MeV.fm3]

82.477

83.532

88.010

85.742

88.936

σR [mb]

491.6

752.7

1012.0

1150.6

1312.4

28

Table-9: The average of best fit values for the real renormalization factors 𝐴 = 𝐹 + 𝐾 𝐸𝐿𝑎𝑏 obtained for JLM- R, JLM–RI and DDM3Y1–FR potentials. 𝑊𝐷

𝑅𝑖

𝑎𝑖

𝐽𝑟

𝐽𝑖

F

20.6±5.52

1.273±0.051

0.73±0.022

379.20±6.4

60.0±6.671

Unite

[MeV]

[fm]

[fm]

[MeV.fm3]

[MeV.fm3]

K

0.667±0.27

0.0053±0.0012

0.0021±0.0074

-0.91±0.16

0.77±0.10

Unite

MeV

[fm/MeV]

[fm/MeV]

[fm3]

[fm3]

29