Study of charge density distributions , elastic charge form factors and root-mean square radii for 4 He , 12 C and 16 O nuclei using Woods-Saxon and harmonic-oscillator potentials

The nuclear charge density distributions, form factors andcorresponding proton, charge, neutron, and matter root mean squareradii for stable 4He, 12C, and 16O nuclei have been calculated usingsingle-particle radial wave functions of Woods-Saxon potential andharmonic-oscillator potential for comparison. The calculations for theground charge density distributions using the Woods-Saxon potentialshow good agreement with experimental data for 4He nucleus whilethe results for 12C and 16O nuclei are better in harmonic-oscillatorpotential. The calculated elastic charge form factors in Woods-Saxonpotential are better than the results of harmonic-oscillator potential.Finally, the calculated root mean square radii usingWoods-Saxonpotentials how overestimation in comparison with experimental dataon contrary to the results of harmonic-oscillator potential.


Introduction
The radial distributions and sizes of nuclear matter and charges are basic properties of nuclei. They are important to test the validity of the nuclear single-particle wave functions used especially in density folding models [1]. The harmonic-oscillator (HO) potential is not accurate to describe the nuclear central confining potential because the potential continues to give a contribution even for much larger (distance from the center of nucleus) and does not become zero, besides the radial wave functions obtained from HO have a Gaussian fall-off behavior at large r which does not reproduce the correct exponential tail. In this field, Elton and Swift [2] firstly reproduced singleparticle radial wave functions in a parameterized single-particle local Woods-Saxon (WS) potential and adjusted the parameters so as to fit the shape of the wave functions to elastic electron scattering data and the eigenenergies to the proton separation energies in the 1 and 2 1 shell nuclei. Gibson et al. [3] studied the ground state of the 4 He nucleus using the single-particle phenomenological model.
Wave functions were regenerated from a WS potential whose parameters are chosen to regenerate the correct neutron separation energy. The proton separation energy and electron scattering form factors were then calculated. Gamba et al. [4] calculated the parameters of a WS potential well for ten p-shell nuclei by fitting the electron scattering form factors and single-particle separation energies. Brown et al. [5] described a new method for calculating nuclear charge and matter distributions which is complementary to the Hartree-Fock method taking into account shell model configuration mixing but it is only semi-self-consistent because the potential was allowed to vary linearly with the density. The method was applied to the core nuclei 16 O and 40 Ca. Lojewski et al. [6] used realistic singleparticle WS potential to evaluate the mean-square charge radii for eveneven nuclei. Lojewski and Dudek [7] evaluated the proton and neutron separation energies and mean square charge radii within the WS plus BCS model for even-even nuclei with 40 256. In [8] some properties of the solutions to the Dirac equations with WS potential were studied, the results obtained for spherical nuclei were compared to those of the relativistic mean field theory. In [9] the single-particle energies and wave functions of an axially two-center WS potential were computed. The spinorbit interaction was included in the Hamiltonian. In [10] the WS potential has been considered to compute the eigenvalues by using Numerov method for a Sturm-Liouville problem. In [11] the Schrödinger equation has been solved by using the Pekeris approximation, for the nuclear deformed WS potential within the framework of the asymptotic iteration method. The energy levels have been worked out and the corresponding normalized eigen functions have been obtained in terms of hypergeometric function.
The aim of the present work is to calculate ground state matter, proton, charge densities, and neutron root mean square ( ) radii, charge density distributions (CDD), elastic charge form factors for stable 4 He, 12 C, and 16 O nuclei using the radial wave functions of WS and HO potentials.

Theoretical formulations
The Schrödinger equation for the single-particle radial wave function can be written as [5]: where 1 / is the reduced mass of the core 1 and single nucleon, is the nucleon mass, is the atomic mass, is the single nucleon separation energy, is the radial eigenfunction of WS potential, , , and are the principal, orbital angular, and total quantum numbers.
For the local potential , the WS shape is used in the compact form shown below [2,4]: . . (2) where (3) represents the central part of , is the strength or depth of central potential, the is the diffuseness and 1 / is the radius parameter. 1 is the radius parameter of spin-orbit and and are the angular momentum and the spin operators respectively.
Finally, in Eq. (2) indicates the Coulomb potential generated by a homogeneous charged sphere and can be written as [12]: for protons and 0 for neutrons, with 1.44 . . Therefore, Eq. (2) can be written as: The point density distributions of neutrons, protons, and matter can be written respectively as [13]: where , represents the number of neutrons, protons, or nucleons in the nlj-subshell. It is worth mentioning that the summation in Eq. (7) spans all occupied orbits.
In order to compare the calculated point proton density distributions with the experimental densities, the finite proton size is required to be included.
The charge density distribution (CDD) is obtained by folding the proton density into the distribution of the point proton density in Eq. (7)as follows [14]: where in Eq. (10) denotes (number of neutrons), (atomic number which is the same for proton and charge) and , respectively. In the first Born approximation the elastic neutron, proton, charge and matter form factors are Fourier transforms of their corresponding density distributions [14]: where takes the same definition in Eq. (10).

Results and discussion
The nuclear shell model is used to calculate CDDs, form factors and corresponding proton, charge, neutron, and matter radii for 4 He, 12 Table 2.  The results of the calculated charge, matter, proton, and neutron radii for 4 He, 12 C, and 16 O nuclei are presented in Table 3. For 4 He nucleus, the results in WS potential for the charge and matter radii showed overestimation in comparison with experimental data on contrary to the results of HO potential which can reproduce such experimental data. Regarding the calculated proton and neutron radii in both potentials, there is appreciable variation between the results of both potentials. Unfortunately, there are no available experimental data to compare with. For 12 C nucleus, the calculations in both WS and HO potentials for the calculated charge and matter radii showed very good agreement with experimental data. For the calculated proton radii, the results of both potentials are almost equal on contrary to the results of the calculated neutron radii which showed large deviation for both potentials. In 16 O nucleus, the calculated charge radius in WS and HO potentials are both in excellent agreement with experimental data, while the results for the calculated matter radii showed slight overestimation in WS potential in comparison with experimental data on contrary to the results of HO potential which agree with the experimental data. The calculated proton radii in WS and HO potential are also almost the same. Appreciable deviation is observed for the calculated neutron radii in both potentials. The calculated charge density distributions are depicted in Fig. 1 for  4 He (a), 12 C (b), and 16 O (c) nuclei in WS (solid curve) and HO (dashed curve) potentials. For 4 He nucleus, it is clear from Fig. 1 (a) that the result from WS is better than the result from HO potential which showed a large deviation from experimental data at central region. In Fig. 1 (b), the calculated CDDs for 12 C nucleus in both WS and HO potentials are depicted. It is clear that the results of WS and HO potentials are almost the same in central region with slight deviation upwards of the WS potential from experimental data. Finally, the results of the calculated CDDs in WS and HO potentials are shown in Fig.1(c). It is obvious that the result of HO potential is better than WS potential in central region on contrary to result of HO potential which showed an appreciable underestimation in the central region with behavior going well with experimental data in central region. The calculated charge form factors are illustrated in Fig.2 for 4 He (a), 12 C (b), and 16 O (c) nuclei in WS (solid curve) and HO (dashed curve) potentials. For 4 He nucleus ( Fig. 2(a)), it is clear that the result of WS is better than the result of HO potential which completely failed to reproduce the first diffraction minimum in comparison with experimental data. For 12 C nucleus ( Fig.2 (a)), the result of WS potential predicts the existence of second diffraction minimum. The result for HO potential is slightly better than the result of WS potential for all regions. Finally, in Fig. 2(c), the charge form factor for 16 O nucleus is illustrated. The results in HO potential failed to reproduce the second diffraction minimum while the result of WS potential is very good at low and medium regions. At high region, the result for WS potential slightly overestimates the position of second diffraction minimum by roughly 0.1 , and underestimates the calculated charge form factors downwards at second diffraction minimum and beyond.

Conclusions
The nuclear charge density distributions (CDD), form factors, and corresponding proton, charge, neutron, and matter radii besides single nucleon binding energies for stable 4 He, 12 C, and 16 O are calculated in both Woods-Saxon (WS) and harmonicoscillator (HO) potentials. The results showed an overestimation in the calculated charge, matter, proton, and neutron radii in WS potential for 4 He nucleus in comparison with available experimental data on contrary to the results of HO potential which easily reproduce the available experimental data. For 12 C nucleus, the charge, matter, and proton radii are almost well generated in both WS and HO potential but with appreciable deviation for neutron radii for both potentials. For 16 O nucleus, the results of the calculated charge and proton radii are roughly the same. The deviation appreciably noticed in matter and neutron radii for both potentials where the result for HO potential is better than the result for WS potential in comparison both with available experimental data. In general, there is an overestimation in the calculated radii in WS potential. For the calculated CDDs, the results for WS potential in 4 He nucleus are much better than results for HO potential. For 16 O nucleus, the behaviors in both potentials are the same but in HO potential is better. For 12 C nucleus, the results in HO potential are much better than results of WS potential. Regarding the calculated charge form factors, for 4 He nucleus, the results in WS potential is much better in HO potential which completely failed to predict the existence the first diffraction minimum. For 12 C nucleus, the results for both potentials are the same at all regions with the difference that there is a second diffraction minimum predicted by WS potential. Finally, for 16 O nucleus, the results for WS potential are much better in comparison with experimental data than the results for HO potential which completely failed to reproduce the second diffraction minimum.  [18,19] for 4 He and [20] for both 12 C and 16 O nuclei.