Study of Matter Density Distributions, Elastic Electron Scattering Form factors and Root Mean Square Radii of 9 C, 12 N, 23 Al, 11 Be and 15 C Exotic Nuclei

: The ground state densities of neutron-rich ( 11 Be, 15 C) and proton-rich ( 9 C, 12 N, 23 Al) exotic nuclei are investigated using a two-body nucleon density distribution (2BNDD) with two frequency shells model (TFSM). The structure of the valence one-neutron of 11 Be is in pure (1p 1/2 ) and of 15 C in pure (1d 5/2 ) configuration, while the structure of valence one-proton configuration is in 9 C, 12 N are to be in a pure (1p 1/2 ) and 23 Al in a pure (2s 1/2 ) . For our studied nuclei, an efficient (2BNDD) operator for point nucleon system folded with two-body correlation operator's functions is used to investigate nuclear matter density distributions, elastic electron scattering form factors, and root-mean square (rms) radii. The effect of the strong tensor force (TC) in nucleon-nucleon forces is taken into account in the correlation. The wave functions of a single particle harmonic oscillator are used with two different oscillator size parameters, β c and β v , the former for core (inner) orbits and the latter for valence (halo) orbits. The measured matter density distributions of these nuclei clearly show the long tail results. The plane wave born approximation (PWBA) is used to investigate the elastic electron scattering form factors for these exotic nuclei.


Introduction:
Since the structure and decay modes of several nuclei far from stability are still unknown, exotic nuclei research is one of the most exciting research areas of modern nuclear physics [1]. It has become a challenging topic in nuclear physics since the discovery of neutron halo phenomena in 11 Li [2]. Weak binding energies describe the proton-and neutron-rich regimes in the nuclei chart, resulting in "exotic" features known as halos. [3]. Exotic nuclei are distinguished by the fact that they are far from the stability valley and have an abnormal N/Z ratio, implying that they have more protons or neutrons than stable nuclei; thus, these nuclei are referred to as proton-rich nuclei or neutron-rich nuclei. Matter density distribution ρm(r) of a nucleus is essential to describe the nuclear structure. This fascination with ρm(r) stems from fundamental bulk nuclear properties such as nuclei shape and size, binding energies, and other quantities linked to ρm(r). Experimental matter density distributions of exotic nuclei are mainly characterized by long tail behavior at large r [4]. The inclusion of strong tensor force is one of the most fundamental nuclear forces, but its firstorder effect on the shell structure has been clarified only recently in studies on exotic nuclei. The tensor force can change the spin-orbit splitting depending on the occupation of specific orbits [5] The two-frequency shell model (TFSM) and the binary cluster model (BCM) are used to investigate the ground state densities of unstable proton-rich 9 C, 12 N, and 23 Al exotic nuclei [6].A simple phenomenological method for introducing dynamical short range and tensor correlations was presented by Dellagiacoma et al. [7]. Two versions of the density distribution of the one-proton halo 17 F nucleus have been taken into account in order to derive the double folding potentials. The measured angular distributions of elastic scattering differential cross section and the corresponding reaction cross sections have been successfully reproduced at different energies using the derived potentials [8]. Sultan [9] has used the binary cluster model BCM to investigate neutron, proton, and matter densities in the ground state of the exotic 14 B and 17 C nuclei.

Theory:
The one body density operator can be transformed into a two-body density form by the following transformation [10]: Rahi and Flaiyh Subtracting and adding Eq. (4a) and Eq. (4b) the following relations can be obtain: Introducing Eq. (4c) and (4d) into Eq. (2) yields; Finally, an effective two-body nucleon density operator (to be used with uncorrelated wave functions) can be produced by folding the operator of Eq.(4) with the two-body correlation functions ij f as : In this work , a simple model form of the two-body tenser correlation operators used by fiase et al. [12] was adopted, i.e The two-body tensor correlations (TC) presented in Eq.(6) are induced by the strong tensor component in the nucleon-nucleon force and they are of longer range. Here 2  is a projection operator onto the 1 3 S and 1 3 D states only. However, Eq. (6) can be rewritten as : where the sum  , in Eq. (7), is overall reaction channels, ij S is the usual tensor operator, formed by the scalar product of a second-rank operator in intrinsic spin space and coordinate space and is defined by is the strength of tensor correlations and it is non zero only in the As the halo nuclei is oversized and easily broken system consisting of a compact core plus a number of outer nucleons loosely bound and specially extended far from the core, it is suitable to separate the ground state density distribution of Eq. (5) into two parts, one is connected with the core nucleons and the other with the halo nucleons, so matter density distribution for the whole halo nucleus becomes [13]: The normalization condition of the above ground state densities is given by: Here g (r) represents one of the following densities: matter, charge, core, or halo densities. The rms radii of corresponding above densities are given by: Elastic electron scattering form factor from spin zero nuclei ( 0  J ), can be determined by the groundstate charge density distributions (CDD). In the Plane Wave Born Approximation (PWBA), the incident and scattered electron waves are considered as plane waves and the CDD is real and with spherical symmetry , therefore the form factor is simply the Fourier transform of the CDD . Thus [14,15] where ) (q F fs is the finite nucleon size and ) (q F cm the center of mass corrections. ) (q F fs is considered as free nucleon form factor and assumed to be the same for protons and neutrons. This correction takes the form [15]: The correction ) (q F cm removes the spurious state arising from the motion of the center of mass when shell model wave function is used and is given by [14]:.
Where A is the nuclear mass number.

Results and Discussion:
The nuclear ground state properties of one-neutron ( 11 Be, 15 C) and one-proton ( 9 C, 12 N, 23 Al) exotic nuclei have been calculated using 2BNDD including the effect of two-body tenser correlations (TC) using two frequency shell model (TFSM). The calculations were based on using different model spaces for the core and the extra halo nucleon. The single particle harmonic oscillator wave functions were employed with two different size parameters of β c and β v . The parameters β c and β v used in the TFSM of the present study together with the calculated and experimental rms radii of exotic nuclei ( 9 C, 12 N, 23 Al, 11 Be, 15 C) are shown in Table-1 . The nuclear properties which include nucleons matter density, elastic electron scattering form factor and rms radii were programmed by Fortran 90 power station.

Tabel 1-The
Parameters β c and β v used in the TFSM of the present study together with the calculated and experimental rms radii of ( 11 Be, 15 C, 9 C, 12  One neutron exotic nuclei: 1. 11 Be nucleus. 11 Be ( is formed by coupling the core 10 Be ( ). The value of oscillator size parameter β c for core ( 10 Be) is equal to 1.62fm, which gives rms nucleon radii equal to (2.28fm), while the oneneutron exotic nuclei 11 Be assumed to be in a pure (1p1/2) with occupation number equal to 0.25 and oscillator size parameter β v =1.95fm were used to give rms nucleon radii equal to (2.80fm). These results of rms nucleon radii were obtained via calculating the matter density distribution and there was a good agreement between the theoretical results and experimental data as shown in Table-1.The two body nucleon density distribution (2BNDD) (in fm -3 ) of the ground state was plotted against r (in fm) of the ground state, as shown in Figure 1. The black line of Figure (1-a) represents the normal contribution of core 10 Be, the valence (one-neutron exotic nuclei in the state of 1p 1/2 ) is defined by the blue line, which has a long tail, and the matter density distribution (core + valence) is represented by the red line, which also has a long tail and has a good agreement with the experimental data of Fukuda et al. [16] for 11 Be represented by the shaded space. Figure (1-b) shows a comparison of matter density distribution of 11 Be (represented by red line) and the matter density distribution of stable nuclei 9 Be (represented by blue line).  11 Be with that of stable nuclei 9 Be.
Elastic electron scattering form factors of 2BNDD are shown in Figure 2, the filled circle symbol represents the experimental data of Glickman et al. [17] for 9 Be. The red line curve represents form factors with oscillator size parameter β =2.78fm ( β assumed to be the average of β c and β v ). Through comparing the theoretical results of the elastic electron scattering form factors for 11 Be nucleus with experimental data of stable 9 Be nucleus, the difference in behavior of first diffraction minimum at -1 for 11 Be nucleus was noted. 11 Be with experimental data from Glickman et al. [17].

15 C Nucleus:
is formed by coupling the core 14 ). A value of oscillator size parameter for core 14 C is equal to β c =1.60fm, which gives rms nucleon radii equal to (2.34fm), while the one-neutron exotic nuclei 15 C which is in a pure (1d 5/2 ) with occupation number equal to 0.083 and oscillator size parameter β v =1.88fm used to give rms nucleon radii equal to 2.78fm. These results of rms nucleon radii were obtained via calculating matter density distribution . Good agreement between the theoretical results and experimental data were noted, as shown in Table-1. The two body nucleon density distribution (2BNDD) in fm -3 of the ground state was plotted versus r in (fm) of the ground state( Figure 3). In Figure (3-a), the black line represents the normal contribution of core 14 C, the valence (one-neutron exotic nuclei in the state of 1d 5/2 ) is defined by the blue line, which has a long tail and the matter density distribution (core + valence) is represented by the red line, which also has a long tail and a good agreement with the experimental data of Fang, et al. [18] for 15 C and represented by the shaded region . Figure  (3-b) shows a comparison of matter density distribution of 15 C (represented by red line) with matter density distribution of 12 C (represented by blue line).

Rahi and Flaiyh
Iraqi Journal of Science, 2022, Vol. 63, No. 3, pp:  The elastic electron scattering form factors of 2BCDD are shown in Figure 4, the red line curve represents the form factors with oscillator size parameter β=2.68fm (β assumed to be the average of β c and β v ) ,while the filled circle symbol represents the experimental data for 12 C taken from Crannell [19]. When the theoretical elastic electron scattering form factors for the 15 C nucleus were compared to experimental results for the stable nucleus 12 C, it was found that they behave similarly in the first diffraction minimum for 15 C at 1.1fm -1 and 12 C at 1.7fm -1 for 12 C. One proton exotic nuclei: 1. 9 C Nucleus 9 C ( ) is formed by coupling the core 8 B ( ) with the valence one-proton ( ). A value of oscillator size parameter for core 8 B (β c =1.75fm) gives the rms nucleon radii equal to 2.34fm , while the valence one-proton exotic nuclei 9 C is to be in a pure(1p 1/2 ) with occupation probabilities 0.25 and oscillator size parameter( β v ) equal to 2.00 fm to give rms nucleon radii equal to( 2.75 fm). These results of rms nucleon radii were obtained via calculating the matter density distribution. The theoretical results showed good agreement with the experimental data, as shown in Table 1. The two body nucleon density distribution (2BNDD)( in fm -3 ) was plotted versus r (fm) in the ground state( Figure 5). In Figure (5-a), the black line represents the normal contribution of core 8 B,the valance (one-proton exotic nuclei in the state of 1P 1/2 ) is represented by the blue line, which has a long tail and the matter density distribution of (core +valence) is represented by the red line, which also has a long tail and has a good agreement with the 9 C experimental data of 9 C of Hong et al. [20][21][22][23] which is represented by the filled circle symbol. Figure (5 b) shows the comparison between the matter density distribution of 9 C nucleus (represented by red line) and matter density distribution of stable 12 C nucleus (represented by the blue line). The elastic electron scattering form factors of 2BNDD are shown in Figure 6, the red line curve represents the form factors with oscillator size parameter β=1.87fm (β assumed to be the average of β c and β v ), while the filled circle symbol represents the experiment data for 12 C of Crannell [19]. Through comparing the theoretical results of the elastic scattering form factors for 9 C nucleus, it was noticed that they have the same behavior of the experimental results of stable nucleus 12 C, but the first diffraction minimum for 9 C was at -1 and for 12 C at -1 . The elastic electron scattering form factors 2BCDD are shown in Figure 8, where the red line curve represents form factors with oscillator size parameter β=1.61fm ( β assumed to be the average of β c and β v ), while the filled circle symbol represents the experimental data for 12 N of Dally et al. [25]. Through comparing the theoretical results of the elastic electron scattering form factors for 12 N nucleus, it was noted that they have the same behavior of the experimental results of stable nucleus 14 N, but the first diffraction minimum for 12 N was at -1 and for 14 N at -1 .

23 Al Nucleus
is formed by coupling of the core 22 Mg ( . A value of the oscillator size parameter of core 22 Mg is equal to (β c =1.78fm) and it gives rms radii equal to (2.75fm), while valence one-proton be in pure (2s 1/2 ) with occupation number equal to 0.25 and oscillator size parameter β v equal to 1.87fm, which gives rms radii equal to (2.92fm) . These results of rms nucleon radii were obtained via calculating the matter density distribution. There was a good agreement between the theoretical results and the experimental data. The two body nucleon density distribution (2BNDD) in fm -3 of the ground state was plotted versus r (fm) in the ground state( Figure 9). In Figure (9-a), the black line represents the normal contribution of core 22 Mg , the valence (one-proton exotic nuclei in state 2s 1/2 ) is represented by the blue line, which has a long tail, the matter density distribution (core + valence) is represented by the red line, which also has a long tail and has a good agreement with the experimental data of Fang et al. [26] which is represented by the shaded space. Figure (9-b) represents the comparison between the matter density distribution of 34 Al(the red line) and the matter density distribution of 27 Al(the blue line).
(a) (b) Figure 9-(a) Comparison between the matter density distribution for 23 Al with that of experimental data.(b): Comparison between matter density of 23 Al nuclei with that of 27 Al nuclei.
The elastic electron scattering form factors for 2BCDD are shown in Figure 10, where the red line curve represents the form factors with oscillator size parameter β=1.83fm (β assumed to be the average of β c and β v ) , filled circle symbol represents the experimental data for 23 Al of Li et al. [27][28][29][30]. Through the comparison of the theoretical results of the elastic scattering form factors for nucleus 23 Al with the experimental results of stable nucleus 27 Al, it was found that they behave similarly. However, the first diffraction minimum for 23 Al was at fm -1 and for 27 Al at fm -1 .

Conclusions
Because of neutron valence or proton valence, which are considered to be a distinctive characteristic of halo nuclei:;in this work, the measured matter density via the framework of two body nucleon density distribution (2BNDD) with effect of tensor force (TC) and two different oscillator size parameters β c and β v for our exotic nuclei displayed a long tail at (r > 6fm) behavior. The measured matter density and rms radii of ( 9 C, 12 N, 23 Al, 11 Be, 15 C) exotic nuclei agreed well with the experimental results. The elastic electron scattering form factors of one-proton exotic nuclei ( 9 C, 12 N , 23 Al) have a similar behavior through the comparison with experimental results of stable nuclei( 12 C, 14 N , 27 Al).