Study of matter density distributions , elastic charge form factors and size radii for halo 11 Be , 19 C and 11 Li nuclei

In this work, the calculation of matter density distributions, elastic charge form factors and size radii for halo 11Be, 19C and 11Li nuclei are calculated. Each nuclide under study are divided into two parts; one for core part and the second for halo part. The core part are studied using harmonic-oscillator radial wave functions, while the halo part are studied using the radial wave functions of Woods-Saxon potential. A very good agreement are obtained with experimental data for matter density distributions and available size radii. Besides, the quadrupole moment for 11Li are generated.


Introduction
The development of the radioactive ion beams has allowed studies of nuclei far from stability. This technical headway led to the discovery of halo nuclei on the neutron-rich side of the valley of stability [1,2]. These weakly bound nuclei have a strongly clusterized structure [3][4][5][6]. In a simple model, they are seen as a core that contains most of the nucleons, to which one or two neutrons are loosely bound. Due to this poor binding, the valence neutrons tunnel far outside the classically allowed region and form a sort of halo around the core [7].
The charge (nucleon) density distribution can be determined by electron (hadron) scattering experiments. Among hadron scattering, proton elastic scattering at intermediate energies besides, nucleus-nucleus scattering using Glauber models are good tools to probe the nucleon density [8].
Al-Khalili and Tostevin [9] re-examine the matter radii of diffuse halo nuclei as deduced from reaction cross section measurements at high energies. They took 11 Li, 11 Be and 8 B as examples and showed that data require significantly larger matter radii than previously reported. Al-Khalili, and Thompson [10], studied root mean square (rms) matter radii of halo nuclei. They reviewed a basic measure in constructing, constraining, and assessing theoretical models of halo structures. They took loosely bound two-and three-body systems 11 Be, 6 He, 11 Li, and 14 Be. Esbensen et al. [11] investigated the consistency of the measured charge radius and dipole response of 11 Li within a three-body model. They showed how these observables are related to the mean square distance between the 9 Li core and the center of mass of the two valence neutrons. Alkhazov and Sarantsev [12] cleared up the sensitivity of the nucleus-nucleus scattering to the nuclear matter distributions of exotic halo nuclei, They have calculated differential cross sections for elastic scattering of the 6 He and 11 Li nuclei on several nuclear targets at the energy of 0.8 GeV/nucleon with different assumed nuclear density distributions in 6 He and 11 Li. Kanungo et al. [13] derived proton radii of [12][13][14][15][16][17][18][19] C densities from first accurate charge changing cross section measurements at 900A MeV with a carbon target. A thick neutron surface evolves from ~0.5 fm in 15 C to ~1 fm in 19 C. The halo radius in 19 C is found to be 6.4 0.7 fm as large as 11 Li. Their Ab initio calculations based on chiral nucleon-nucleon and threenucleon forces reproduced well the radii. Karataglidis et al. [14] Diverse means are used to investigate 17,19 C, their estimates have been made using a shell model for the systems. Information from those shell model studies were then used in evaluating cross sections of the scattering of 70 A MeV 17,19 C ions from hydrogen.

Theoretical formulations
The ground density distributions of point neutron and proton for halo nuclei can be written as [15]: where the density distributions of core ( ( )) are calculated using the radial wave functions ( ( )) of harmonic-oscillator (HO) potential [16] as follows: In Eq. (2), represents the number of neutrons/protons in the shell, where and represent the principal and orbital quantum numbers, respectively. and , represent the HO size parameters and isospin quantum number ( for proton and for neutron). It is worth mentioning that the summation in Eq. (2) spans all occupied orbits in the core for any nuclear sample under study. The matter density distribution for whole core nucleon can be written as a sum the densities of core neutrons and protons ( ( ) ( ) ( )). ( ) in Eq. (1), represents the density distribution for halo part and it is calculated using Woods-Saxon (WS) radial wave functions and can be written as: (3) and (4), and represent the occupation and fractional occupation numbers of neutrons, protons, or nucleons, respectively in the sub-shell , where represents the total angular quantum number. Besides, represents probability of the existence of the halo nucleons in higher sub-shells. Again, the matter density distribution for whole halo nucleons can be written as a sum the densities of halo neutrons and protons ( ( ) ( ) ( )). ( ) is the solution to the radial part of Schrödinger equation [17]: is the reduced mass of the core ( ) and single nucleon, is the mass of nucleon, is the atomic mass, is the separation energy of single halo nucleon.
The local potential ( ) in Eq. (5) is taken to be WS as shown in the compact form below [18,19]: represents the central part of ( ), is the strength or depth of central potential, is the diffuseness and where ( ) with and .
Eq. (8) represents the spin-orbit part of ( ), is the pion mass, is the strength or depth of spin-orbit potential, is the diffuseness of spin-orbit part, ( ) is the radius parameter of spin-orbit and ̂ and ̂ are the angular momentum and the spin operators respectively.
Finally, in Eq. (6) ( ) indicates the Coulomb potential generated by a homogeneous charged sphere and can be written as [20]: For protons and ( ) for neutrons, with . Therefore, Eq. (6) can be written as: The charge density distribution ( ) (CDD) is obtained by folding the proton density into the distribution of the point proton density in Eq. (1) as follows [18]: where ( ⃗) is taken to have a Gaussian form, as follows [18]: where . Such value of reproduces the experimental charge radius of the proton, 〈 〉 ( ) . The ground state density distributions of point neutron and proton for stable nuclei can be written as: The radii of neutron, proton, charge and matter can be directly calculated from their density distributions [18]: In Eq. (14), denotes to (number of neutrons), (atomic number which is the same for proton and charge) and ( ), respectively.
The longitudinal electron scattering form factors in the first Born approximation can be written as [21,22]: represents the momentum transfer from electron to nucleus during scattering. Eq. (15) can be simplified to the following [23]: where ( ) and ( ) are spherical Bessel function and charge transition density distribution, respectively. The total longitudinal form factors are given by: The Coulomb form factor in Eq. (16) can be reduced to: From Eq. (19), the multiparticle reduced matrix element of electric multipole operator can be written as: The quadrupole moment is related to the multiparticle reduced matrix element of electric multipole operator by the relation [20]: Therefore, the quadrupole moment can be reduced to the following final formula: The transition density distribution in Eq. (16) are coming from contribution of core-polarization (CP) and model-space (MS) as follows [24]: In the present work, ( ) is calculated using Tassie [25][26][27] and Bohr-Mottelson [28] models, respectively as: where, in the above two equations are found so as to reproduce the experimental quadrupole moments.
( ) can be written as [29]: where is the proton or neutron one body density matrix element and and stand for the single-particle states.
The incoherent sum of the longitudinal form factor for ground C0 and C2 parts can be written as: The ( ) CDD coming from contributions of one protron in the model-space is obtained by folding the single protron charge density into the distribution of the point proton density ( ( )) in Eq. (25) with the same procedure followed in Eq. (11).
The ( ) CDD coming from contributions of neutrons in the modelspace is obtained by folding the single neutron charge density into the distribution of the point neutron density ( ( )) in Eq. (25) as follows [30]: where ( ⃗) is taken to have a Gaussian form, as follows [30]: where the parameter and from Chandra and Sauer are given in Table 1. For 11 Li, the shell model calculation is done using the computer for windows [31] with the interaction [32] where the core is taken with be ( ⁄ ) while the modelspace spans all subshells ( ⁄ ⁄ ⁄ ⁄ ⁄ ) .

Results and discussions
In the present work, the MDDs, proton, charge, neutron, and matter radii, besides elastic electron scattering charge form factor, are computed using the radial wave functions of HO potential for core part for all nuclei under study, the halo part are computed using the radial wave function of WS potential. Regarding the core part, two HO size parameters are used, one for protons ( ) and the second for neutrons ( ) in order to regenerate the available experimental radii. The depth of central part of WS potential ( ) in this work is chosen so as to reproduce the experimental single-nucleon separation energies for 11 Be, 19 C and 11 Li nuclei, while other parameters in Eq. (8) are fixed to be , , .
For halo 11 Be, 19 C and 11 Li nuclei, the chosen size parameters of HO potential for protons and neutrons in the core part and the parameters of WS potential are presented in Table 2. The configuration mixing included in our calculations as mentioned in Eq. (4) using probability amplitudes ( ). In this work the configuration mixing are chosen between the sub-shells, , and for 11 Be and 19 C, while the configuration mixing for 11 Li are chosen to be between and , as shown in Table 2.
The computed proton, charge, neutron, and matter radii are presented in Table 3, for the nuclei under study.
The computed results of the charge and matter radii are well predicted for the nuclei under study.
The calculated MDDs are shown in Fig.1 and compared with experimental data. The solid and dashed curves represent the calculated MDDs for nuclei under study using HO+WS and HO+HO, respectively. It is obvious that the inclusion of configuration mixing to insure the contribution from higher subshells leads to freely get match between the calculated matter density distribution and experimental at tail part. Very good agreements are obtained with experimental MDDs in all figures shown in Fig. 1.   11 Be, 19 C and 11 Li.
The calculated charge form factors are illustrated in Fig. 2(a) for 11 Be, 2(b) for 19 C and 2(c) for 11 Li, respectively, and compared with experimental data of the corresponding stable nuclei. In Fig. 2(a) and Fig. 2(b), the solid and dashed curves represent the calculated charge form factor in HO+WS and HO+HO, respectively. It is clear from aforementioned both figures that the results of HO+WS shift the results upwards to approach experimental data, on contrary to the results of HO+HO which underestimate the calculated charge form factors. In Fig. 2(c), the solid and dashed curves represent the calculated charge form factor using Tassie and Bohr-Mottelson models, respectively where the incoherent sum of the and components are applied on 11 Li according to the Eq. (26). All results in Fig. 2 provide predictions for the future experiments on the electronradioactive beam colliders where the effect of the neutron and proton halo or skin on the charge density distributions and charge form factors is planned to be studied, where the normalization constant in Eq. (23) and (24) are computed using the CDD of HO+WS so as to reproduce the experimental quadrupole moments, [38]. 11 Be, 19 C and 11 Li nuclei compared with stable 9 Be, 12 C and 7 Li nuclei.

Conclusions
The MDDs of one-neutron halo in 11 Be and 19 C, and two-neutron halo in 11 Li have been calculated in HO for core parts and WS potential for halo parts. The elastic Coulomb electron scattering form factors for aforementioned nuclei have been studied in shell model. Besides, the quadrupole moment for are investigated in this work. A very good agreement with experimental data for the calculated charge and matter radii and MDDs are obtained for the three nuclei under study. The results of the calculated elastic charge form factors for the nuclei are controversial till future experiments on the electronradioactive beam colliders are settled.