Study of the nuclear structure of halo nuclei 23 O and 24 F using the two-body model

The nuclear structure included the matter, proton and neutron densities of the ground state, the nuclear root-mean-square (rms) radii and elastic form factors of one neutron 23O and 24F halo nuclei have been studied by the two body model of  within the harmonic oscillator (HO) and Woods-Saxon (WS) radial wave functions. The calculated results show that the two body model within the HO and WS radial wave functions succeed in reproducing neutron halo in these exotic nuclei. Moreover, the Glauber model at high energy has been used to calculated the rms radii and reaction cross section of these nuclei.


Introduction
One of the most exciting discoveries in the recent experimental progress using radioactive nuclear beams is the neutron halo in some light neutron-rich nuclei [1,2]. On the other hand, the search for proton halo in light proton-rich nuclei has attracted much attention in recent years. The theory based on the relativistic mean filed (RMF) approximation predicts the existence of proton halo or proton skin structure in some proton-rich nuclei [3,4] and experiments have subsequently observed some of the theoretical predictions [5,6].
The observed neutron halo nuclei, such as 11 Li, 11 Be, 14 Be and 17 B, are in their ground state and are located near the drip line. In principle, the neutron halo appears in very loosely bound nuclear system with a low density which results in substantially large spatial extension of the wave function for the neutron (proton) halo in these nuclei [7].
It is well known that Glauber model analysis on reaction cross section is an important and feasible tool to deduce 30 the nuclear size, which directly connects the reaction cross section to the nuclear density distribution. The Glauber theory in the optical limit (OL) approach has been successfully applied to describe unstable nucleusnucleus scattering [8].
A. Alsajjad and Abdullah [9] used Woods-Saxon (WS) and Harmonic Oscillator (HO) potentials wave functions within the two-body model of [ + ( )] to study the ground state features such as the neutron, proton and matter densities, the associated root mean square (rms) radii and the elastic form factors of exotic 17 F and 19 C nuclei. According to the calculated results, it was found that this model provides a good description on the nuclear structure of the above exotic nuclei. The reaction cross sections for these nuclei have been studied using the Glauber model with an optical limit approximation (OLA) at high energy region. Abdullah [10] studied the ground state densities of halo 6 He, 11 Li, 12 Be and 14 Be nuclei using a three-body model of (core + n + n). The long tail property was shown in the calculated neutron and matter densities of these nuclei. The calculated results of matter densities were in good accordance with the experimental results. Abdullah [11] used the two body model of (Core+n) within the radial wave functions of the cosh potential to investigate the ground state features such as the proton, neutron and matter densities, the rms nuclear proton, neutron, charge and mass radii of unstable neutron-rich 14 B, 15 C, 19 C and 22 N nuclei. The calculated results showed that the two body model with the radial wave functions of the cosh potential succeeds in reproducing neutron halo in these nuclei.
Abdullah [12] used the radial wave functions of the Bear-Hodgson potential to study the ground state features such as the proton, neutron and matter densities and the associated rms radii of two neutrons halo 6 He, 11 Li, 14 Be and 17 B nuclei. These halo nuclei were treated as a three-body system composed of core and outer two-neutron (Core + n + n). The radial wave functions of the Bear-Hodgson potential were used to describe the core and halo density distributions. The interaction of core-neutron takes the Bear-Hodgson potential form. The outer two neutrons of 6 He and 11 Li interact by the realistic interaction REWIL whereas those of 14 Be and 17 B interact by the realistic interaction of HASP. The obtained results showed that this model succeeds in reproducing the neutron halo in these nuclei. Elastic charge form factors for these halo nuclei were analyzed via the plane wave Born approximation.
In this work, the WS and HO potentials wave functions will be used within the two-body model of ( + ) to study the nuclear structure including the matter, proton and neutron densities of the ground state, the nuclear root-mean-square (rms) radii and elastic form factors of one neutron 23 O and 24 F halo nuclei. Moreover, the Glauber model with an OLA will be used to calculate the rms radii and the of these nuclei.

Theory
The distributions of matter density for unstable exotic nuclei are given as [11]: where ( ) and ( ) are the core and halo densities, respectively and can be written as [12]: where ( ) ℓ refer to proton or neutron number in the sub-shell ℓ and ℓ ( ) is the radial wave function. The distributions of matter density in Eq.(1) can be written in terms of proton [ ( )] and neutron [ ( )] densities as [10]: where ( ) and ( ) can be written as [10]: In the present work, the calculated results are analyzed using two density parameterizations which are WS and HO.
In the WS parameterization, the core and halo densities are parameterized with WS radial wave function which are taken from the solution to the radial part of the Schrodinger equation using WS potential [10]: ℓ is the single-particle energy, , ℓ and are the principle, orbital angular and total quantum numbers and is the reduced mass of the core and single nucleon = ( − 1/ ), A is the nuclear mass number and m is the nucleon mass. The ( ) is the potential of the core given as [13]: ( ) = 0 ( ) + ( ) . + ( )(7) where 0 ( ) is the central potential that takes the form of WS potential [10]: ( ) is the spin-orbit potential expressed as [10]: : is depth of spin-orbit potential, and ( ): is the Coulomb potential (for protons only) generated by a homogeneous charged sphere of radius [14]: In the HO parameterization, the core and halo densities are parameterized with harmonic oscillator radial wave function given by [15]: is the HO size parameter (which is the length parameter for the HO well), is the mass of the proton and is the angular frequency.
The rms radii of the neutron and proton distributions can be calculated by [16]: The elastic form factors in the plane wave Born approximation (PWBA) are written as [17]: where Z is the atomic number, 0 ( ) the Bessel function and q the momentum transfer to the target nucleus from the incident electron. The reaction cross section ( ) using the Glauber model within OLA is given by [18]: where ( ) is the transparency function at impact parameter . The ( ) in OLA is given in terms of the elastic − S matrix for the projectiletarget system as [19]: (17) ( ) is the overlap of the densities of ground state for target and projectile ( and , respectively).

Results and discussion
The nuclear structure including the matter, proton and neutron densities of the ground state, the nuclear rms radii and elastic form factors of one neutron 23 O and 24 F halo nuclei were studied by the two body model of ( + ) within the HO and WS radial wave functions. In addition, the Glauber model at high energies has been used to calculate the matter rms radii and reaction cross section ( ) of these nuclei.
For the 23 O and 24 F halo nuclei, the calculations were performed adopting a 22 O and 23 F cores, respectively, plus single valence neutron for the structure of these nuclei. The core configurations, half-life time and separation energy ( ) [20,21] for the studied nuclei are listed Table 1.
The WS parameters for stable nuclei ( 18 O, 19 F) and the depth of WS potential (V 0 ) for core nucleons (protons and neutrons) as well as the depth of spin-orbit (V so ) of halo nuclei 23 O and 24 F were taken from Brown and Rae [23]. The V 0 for outer neutron and the parameters (r 0 , r so , a 0 and a so ) were adjusted to reproduce the experimental of the outer neutron and the experimental matter rms radii. The parameters of the WS and HO parametrizations used in the present analysis are listed in Table 2.
The calculated rms radii obtained by the two density parameterizations and the corresponding experimental data are summarized in Tables 3 and 4 along with the available calculated results obtained from the relativistic mean field (RMF) method [25]. From the results shown in these tables, it can be clearly seen that the rms radii obtained by the two density parameterizations are consistent with the experimental data more than those of RMF method.      Fig. 1 presents the calculated matter (dashed-red curve), core (green curve) and halo (dark-blue curve) densities as well as the experimental data of matter densities (shaded region) [26] for 23 O (upper part) and 24 F (bottom part). The left and right panels correspond to the calculated results obtained by WS and HO parameterizations, consecutively. It is noted that, the calculated results obtained by the two density parameterizations give a good description of the experimental data and show halo component in the matter distributions. Moreover, the dashedred curves in Figs.1 (a and c) better describs the experimental data than those in Figs. 1 (b and d).
The calculated matter (dashed-red curve), proton (green curve) and neutron (dark-blue curve) densities for 23 O and 24 F obtained with two density parameterizations are plotted in Fig. 2.
It is evident from this figure that there is neutron halo in 23 O and 24 F as their neutron density distributions have a long tail.
In Fig. 3 the density distribution of matter for unstable exotic 23 O and 24 F nuclei are compared with those for their stable 18 O and 19 F isotopes. In the figure, the dashed-red and dark-blue curves are the density distributions of matter for unstable and stable nuclei, consecutively. It is noted that the matter density distributions for each pair of isotopes are different. The weak binding of the outer neutron in exotic 23 O and 24 F nuclei leads to extended their matter density distributions.
It is clear from Figs.1-3 that the behavior of the matter density in WS and HO parameterizations are diverse because different wave functions that used in these two parameterizations.
The elastic form factors of exotic 23 O and 24 F nuclei (red curves) calculated by PWBA using the WS and HO proton densities are compared in Fig. 4 with those for their stable 18 O and 19 F isotopes (dark-blue curves) together with the experimental data (dark-blue symbol) of stable isotopes [27,28]. It is noted that the form factors for each isotopes pairs are quite different although they have the same proton number. The minima position of the red curve has shifted to the left as compared with that of the dark-blue curve. This change is attributed to the variation in the proton densities due to the presence of the extra neutrons.    The calculated at high energy for 23 O + 12 C and 24 F + 12 C systems using the Glauber model with OLA along with experimental data are listed in Table 5. From the results shown in Table 5, it can be clearly seen that a good description of the experimental is obtained by the calculated results for both halo nuclei. To calculate the matter rms radius of halo nuclei from the reaction cross sections ( ), the calculated (red line) obtained by the Glauber model within OLA versus the matter rms radii for the halo nuclei 23 O and 24 F on 12 C target at high energy are plotted in Figs.5 (a and b), consecutively. The horizontal black line shows the experimental (given in Table 5) with error bar plotted by the shaded area. The intersection point of the red line with horizontal black line represents the obtained matter rms radius (〈 2 〉 1/2 ) for the halo nuclei. From Fig.5(a) [Fig.5

Conclusions
From this study the following conclusions can be drawn: 1-It was found that the two body model of ( + ) within the HO and WS radial wave functions is remarkably capable of providing theoretical predictions on the structure of halo nuclei and be in a satisfactory description with those of experimental data. 2-The halo structure of one neutron 23 O and 24 F exotic nuclei was emphasized through exhibiting the long tail performance in their calculated neutron and matter density distributions, where this performance is considered as a distinctive feature of halo nuclei. 3-The results of the matter density distributions calculations when the halo neutron in 23 O and 24 F has mixed configuration of (1d 3/2 ) and (2s 1/2 ) with dominant (2s 1/2 ) were in best agreement with experimental data. 4-It was found that the major difference between the calculated form factor of unstable nuclei 23 O and 24 F and those of their stable isotope 18 O and 19 F can be attributed to the variation in the proton densities due to the presence of the extra neutrons. 5-The Glauber model at high energy gave a good description for both matter rms radii and of these nuclei.