Density Distributions and Elastic Electron Scattering Form Factors of Proton-rich 8B, 17F, 17Ne, 23Al and 27P Nuclei

In this work, the nuclear density distributions, size radii and elastic electron scattering form factors are calculated for proton-rich 8B, 17F, 17Ne, 23Al and 27P nuclei using the radial wave functions of Woods-Saxon potential. The parameters of such potential for nuclei under study are generated so as to reproduce the experimentally available size radii and binding energies of the last nucleons on the Fermi surface.


Introduction
The chosen radial wave functions are played a crucial role in understanding the nuclear structure in the aspect of nuclear sizes and radial distributions of nucleons in any nuclear system [1]. The experimental matter density distributions of exotic nuclei are mainly characterized by long tail behavior at large r [2]. Electron scattering and muonic atoms ( ) are pa owerful tool to probe the charge distributions and radii of stable nulei [3]. Unfortunately, due to very short lifetime of halo nuclei there is no ability to make tthe arget of them; therefore, such nuclei are studied in inverse kinematics [4]. The optical isotope shift provides important information about the very tiny quantities of radioactive nuclei [5]. Experimental matter rms radii are available through nucleus-nucleus scattering in Glauber model [6,7], besides the proton rms radii can be obtained from charge-changing cross section [8]. The naive radial harmonic-oscillator (HO) wave functions are unable to predict the ISSN: 0067-2904

Noori and Ridha
Iraqi Journal of Science, 2019, Vol. 60, No.6, pp: 1286-1296 78>= long-tail characteristic behavior at large for density distributions of halo nuclei [9][10][11][12]. Modifications are adopted to improve such shortcoming such as using two HO size parameters for core and halo parts to regenerate the asymptotic behavior and size radii but with adequate results [13][14][15]. The transformed HO wave functions in local-scale transformation opnened new approach to reform the performance of the radial wave functions of halo nuclei in [16][17][18].The single-particle wave functions of Woods-Saxon (WS) potential are used with very good agreements with experimental data for both stable and exotic nuclei [19][20][21][22].
In present work, we undertook the calculation of rms radii, density distributions and elastic electron scattering form factors for proton-rich 8 B, 17 F, 17 Ne, 23 Al and 27 P nuclei utilizing the realistic radial wave functions of WS potential.

Theoretical formulations
The nuclear transition proton ( )/neutron ( ) density distributions can be written as [13] where and denote the single-particle state ( ). represents the weight of transition obtained from shell-model calculations. In Eq. (1), the radial wave functions of WS potential ( ) are taken from the solution to radial part of the Schrodinger equation using Woods-Saxon potential given below [23] ( ) In the above equation, is the reduced mass of the core and single nucleon, is the mass of single nucleon, is the atomic mass and represents the binding energy of single nucleon in the state . The potential can be written as (3) The three terms in the Eq. (3), stand for the central, spin-orbit and Coulomb parts, respectively, given by [23]  The matrix element of 〈 ̂ ̂〉 written in the second term of Eq. (4) is given by The last term in Eq. (4), indicates the Coulomb potential generated by a homogeneous charged sphere. Besides, the radial wave functions of HO potential is used in Eq. (1) in order to compare with the results of WS. The ground point proton/neutron density distribution can be written from Eq. (1) as The total charge density distribution can be calculated from Eq. (6) by folding the point density distributions of protons and neutrons into the charge density of each one separately as follows ⁄ ⁄ (7) in the above equation, the first and second terms can be written as and ⃗ [25] in Eqs. (8) and (9) takes, respectively the following forms and The parameters in Eqs. (10) and (11) are taken from Ref. [24] and [25], respectively. The rms radii of neutrons, protons, charge and matter are calculated from where stands for (number of neutrons), (atomic number), charge and (mass number), respectively. Finally, in the first Born approximation, the longitudinal electron scattering form factors can be written as [26,27] ∑ The Coulomb multipole operator in Eq. (14) is given by ∫ ̂ ⃗ (15) The many-body matrix element in Eq. (14) can be written in terms of single matrix element as follows [27] 〈 ‖ ‖ 〉 ∑ ⁄ 〈 ‖ ‖ 〉 (16) The single Coulomb multipole opertor is given by (17) The Coulomb form factor in Eq. (14) can be simplified to For component in the above equation can be written as The transition density distribution is calculated from the contribution of core-polarization (CP) and model-space (MS) [27]: (20) In the present work, is calculated using Tassie [27] and Bohr-Mottelson [28] models, respectively.
(21) and (22) where, in the above two equations are found so as to reproduce the experimental multipole moments and is calculated from Eq. (7).
is related to the point proton/neutron transition density distributions and calculated from Eq.(1). The incoherent sum of multipole coulomb form factors can be written as

Results and Discussion
The radial wave functions of WS potential are used to calculate the density distributions, rms radii and elastic electron scattering charge form factors for 8 B, 17 F, 17 Ne, 23 Al and 27 P nuclei. The parameters of WS potential ( , , , and ) displayed in Table-1 are chosen so as to reproduce the experimental single-nucleon binding energies (for the last proton and neutron on Fermi surface) and available experimental rms radii for nuclei under study. The depth of spin-orbit ( ) is fixed to be 9.0 MeV.
In Table-2, the calculated and available experimental radii for nulei under study are displyed. It is clear that the experimental data are well generated for the used WS parameters. In Tables-(3-7), the single binding energies of protons and neutrons predicted by the fixed parameters of WS potential are presented for 8 B, 17 F, 17 Ne, 23 Al and 27 P nuclei. It is obvious that the last single nucleon binding energies for both protons and neutrons are well reproduced.
In Figures-(1,2) the calculated matter density distributions for 8 B (a), 17 F (b), 17 Ne (c), 23 Al (d) and 27 P (e) are represented by solid and dashed curves for WS and HO, respectively are depicted and compared with experimental data represented by the shaded area. From calculations, it is clear that results of WS are in very good agreement with experimental data for all r especially the tail region on contrary to the results of HO which completely failed to reproduce the tail region due to the Gaussian fall-off behavior of radial wave function at asymptotic region. Knowing that the 8 B, 23 Al and 27 P are 1p-halo, 17 F is 1p-skin and 17 Ne is 2p-halo [2,4,7]. The proton and neutron density distributions for nuclei under study are drawn in Figure- 17 Ne (c), 23 Al (d) and 27 P (e). The long tail characteristic is well generated in the proton distribution denoting the existence of halo in 8 B, 17 Ne, 23 Al and 27 P nuclei (due to the low binding energies of the last protons) and skin in 17 F.
In Figure- 27 Al (d) and 31 P (e) nuclei, respectively. For stable nuclei mentioned previously, the CKII interaction [29] with 1p-model space (for 10 B) and SDBA interaction [30] with sd-shell model space ( 19 F, 20 Ne, 27 Al and 31 P nuclei) are used. The shell model calculations with the aforementioned two effective interactions are done using OXBASH code [31]. It is clear that the 8 B, 17 F, 17 Ne and 27 P nuclei have forward and fluctuated downward shifts while 23 Al nuclei has backward and fluctuated downward shifts.
In Figure-4, the calculated charge form factors of stable 10 B (a), 19 F (b), 20 Ne (c), 27 Al (d) and 31 P (e) nuclei are depicted and compared with those of experimental data. The radial wave functions of WS potential are used to calculate the charge form factors. The C0+C2 and C0+C2+C4 components are calculated for 10 B and 27 Al while C0 components are calculated for 19 F, 20 Ne and 31 P. The higher CJ components for 10 B and 27 Al are calculated from the contribution of core-polarization, using Tassie (T) [27] (solid curve) and Bohr-Mottelson (BM) [28] (dashed curve) models, and the model space, using Eq. (1). The CKII [29] and SDBA [30] interactions are used for 10 B and 19 F, 20 Ne, 27 Al and 31 P, respectively. From the overall figures, there are in general good agreements with experimental data.

Conclusions
The nuclear density distribution and elastic electron scattering form factors have been calculated using the radial wave functions of WS potential. The parameters of such potential are selected so as to reproduce firstly the binding energy of the single nucleon on the Fermi surface and secondly the available experimental rms radii for proton-rich nuclei under study. For the calculated matter densities, the results are in very good agreement with those of experimental data. It is found a noticeble long tail in 8 B, 17 Ne, 23 Al and 27 P indicating the presence of halo and this is attriubuted to the low binding energies of the last proton(s), besides the low orbital quantum numbers where the last proton(s) are found to be in 1p 1/2 for 8 B, 2s 1/2 for 17 Ne, 23 Al and 27 P while in 17 F in spite of very low binding energy of the last proton, it is found a skin behavior due to the existence of such proton in 1d 5/2 (the centrifugal term in effective potential is high pushing the last proton to the core and diminishing the tail). For the calculated charge form factors, the C0 component for 19 27 Al are calculated and compared with available experimental data. The results for exotic nuclei are left for the future projects on electron-radioactive ion beams which proposed to build to study the access of protons or neutrons on charge form factors.