Investigation of density and form factor of some F isotopes using Hartree-Fock and shell model calculations

Structure of unstable 21,23,25,26F nuclei have been investigatedusing Hartree – Fock (HF) and shell model calculations. The groundstate proton, neutron and matter density distributions, root meansquare (rms) radii and neutron skin thickness of these isotopes arestudied. Shell model calculations are performed using SDBAinteraction. In HF method the selected effective nuclear interactions,namely the Skyrme parameterizations SLy4, Skeσ, SkBsk9 andSkxs25 are used. Also, the elastic electron scattering form factors ofthese isotopes are studied. The calculated form factors in HFcalculations show many diffraction minima in contrary to shellmodel, which predicts less diffraction minima. The long tailbehaviour in nuclear density is noticeable seen in HF more than shellmodel calculations. The deviation occurs between shell model andHF results are attributed to the sensitivity of charge form factors tothe change of the tail part of the charge density. Calculations donefor the rms radii in shell model showed excellent agreement withexperimental values, while HF results showed an overestimation inthe calculated rms radii for 21,23F and good agreement for 25,26F. Ingeneral, it is found that the shell model and HF results have the samebehaviour when the mass number (A) increase.


Introduction
Most of our knowledge of nuclear physics is obtained from the study of stable nuclei on and near the stability line [1]. The development of radioactive isotope (RI) beam techniques [2][3][4] has opened a new field for the study of unstable nuclei far from the stability line.
Nuclear charge density distributions are very important to understanding the internal structure of nuclei [5]. For many years, electron-nucleus scattering has proven to be an excellent tool for the study of nuclear charge size and charge distribution. In the near future, elastic electron scattering off exotic nuclei will be realized. Thus, it is interesting and necessary to study electron scattering off exotic nuclei theoretically to provide the future experiments with some useful instructions in advance [6]. Several theoretical and experimental groups have devoted their work on studying the exotic nuclei [7][8][9][10][11][12].
Chu yan-Yun et al. [13] studied the electron scattering of unstable 17 F, 18 Ne, and some neutron rich N=8 isotones nuclei using relativistic mean field theory and phase shift analysis. The ground state charge density distributions, form factors and rms radii of 12 C and 16 O nuclei were calculated in shell model by Radhi et al. [14] using core plus valence and ab-intio. Calculations are compared with the results of self-consistent mean field using selected Skyrme forces. Recently, Radhi et al. [15] studied inelastic electron scattering form factors, energy levels and transition probabilities for positive and negative low-lying states using shell model and HF calculations.
The aim of the present work is to study the nuclear density and elastic electron scattering form factors for 21,23,25,26 F nuclei using shell model and HF calculations. The nuclear shell model calculation is performed using sd-model space which consist the active shells 1d 5/2 , 2s 1/2 and 1d 3/2 above the inert 16 O nucleus core. USD-type Hamiltonians called SDBA [16] has been used to provided realistic sdshell wave functions for ground state. The radial wave functions for the single-particle matrix elements were calculated by using the harmonicoscillator potential (HO) and the OBDM elements are computed from the shell model code oxbash [17]. For HF method, the effective nucleonnucleon interaction Skyrme forces SLy4 [18], Skeσ [19], SkBsk9 [20] and Skxs25 [21] parameterizations are used.

Theoretical formulations
The expectation value of the HF Hamiltonian of the system is given by [22]: contains all parts of nucleon-nucleon forces. This forces consists of some two-body terms together with a three-body term [23]: ) (r R l n is the radial part of the HO wave function and one body density matrix element. The matter density distribution of Eq.(5) may also be expressed as The corresponding rms radii are given by where g represents the corresponding number of nucleons. The neutron skin thickness (t), can be defined as The corresponding elastic scattering (J=0) form factor (C0) is written in the following form nucleon form factor and center of mass correction, respectively, given by [24]: and where A in Eq. (9) represents the mass number of the nucleus under study.

Results and discussion
In order to explain the nuclear structure of unstable 21,23,25,26 F nuclei nuclear radii, nuclear density distributions and form factors are studied using shell model calculations with sd-model space which consist the active shells 1d 5/2 , 2s 1/2 and 1d 3/2 above the inert 16 O nucleus core. USDtype Hamiltonians called SDBA [16] has been used to provided realistic sdshell wave functions for ground state. Also, self-consistent mean field with selected Skyrem forces (SLy4, Skeσ, SkBsk9 and Skxs25) are used. The HO size parameters for 21 F, 23 F, 25 F and 26 F are taken to be (1.77, 1.71, 1.9 and 2) fm, respectively.
The calculated proton, neutron and matter rms radii with neutron skin thickness (t) are tabulated and compared with experiment data in Tables 1 to 4      The obtained values of the proton density for these isotopes at center region and the long tail (which is noticeably seen in the distribution of the density at r ˃ 4 fm) decreased with increasing number of neutron.   26 F denoted as shaded area [25] are displayed in Fig. 12. It is clear from this figure that the calculated density with HF and shell model calculations give good agreement with the experimental data indicated with its error bars by the shaded area.   [26]. These figures give the conclusion that the form factors is not dependent on detailed properties of the distributions of neutron density.
It is apparent from Fig. 13 that the HF calculations for all Skyrme almost coincide in range of q ˂ 1.5 fm -1 . The deviation occurs between shell model and HF results at q ˃1 fm -1 , since the form factors are sensitive to the change in the tail part of the charge density. As one can see that both of shell model and experimental data has one diffraction minimum. The location of the minimum of shell results has forward shift as compared with the minimum of HF results. The longitudinal C0 elastic electron scattering form factors of 23 F nucleus are shown in Fig. 14

Conclusion
In this study, structure of unstable 21,23,25,26 F isotopes have been investigated using shell model and HF calculations.
In shell model calculations, results of rms radii showed excellent agreement with experimental data, while HF results showed an overestimation in the calculated rms radii for 21,23 F and good agreement for 25,26 F. In general, it is found that the shell model and Hartree -Fock results all have the same behaviour when the mass number (A) increase. The calculated rms of proton, neutron, matter and neutron skin thickness with shell model and the Skyrme HF have approximately been increased with increasing number of neutron. It is clear from the result of the density distribution that the calculated density are quite consistent with all the Skrme forces and shell model in the central region. It is useful to remark that the obtained values of the proton density for these isotopes at this region decreased with increasing number of neutron, while the neutron density has increased. The long tail behaviour in neutron density is noticeable seen in HF more than shell model calculation. Thus, in form factors calculations the deviation occurs between shell model and HF results since the form factors are sensitive to the change of the tail part of the charge density. As one can see that each HF results has two diffraction minimum, while shell model results has only one