Matter Density Distributions, Root-mean Square Radii and Elastic Electron Scattering Form Factors of Some Exotic Nuclei (17B, 11Li, 8He)

The two-neutron halo-nuclei (17B, 11Li, 8He) was investigated using a two-body nucleon density distribution (2BNDD) with two frequency shell model (TFSM). The structure of valence two-neutron of 17B  nucleus in a pure (1d5/2) state and in a pure (1p1/2) state for  11L and 8He nuclei. For our tested nucleus, an efficient (2BNDD's) operator for point nucleon system folded with two-body correlation operator's functions was used to investigate nuclear matter density distributions, root-mean square (rms) radii, and elastic electron scattering form factors. In the nucleon-nucleon forces the correlation took account of the effect for the strong tensor force (TC's). The wave functions of single particle harmonic oscillator are used with two different oscillator size parameters βc and βv, where the former is for the core (inner) orbits and the latter is for the valence (halo) orbits. The measured matter density distributions of these nuclei clearly showed long tail results. To investigate elastic electron scattering form factors the plane wave born approximation (PWBA) with two body nucleon density distribution (2BNDD's) was use.


Introduction
Over the last two decades, the world of atomic nuclei has changed dramatically and continues to do so. This is partially because the nuclear structure physics frontier has moved from stable to exotic nuclei. Exotic nuclei are atomic nuclei with shorter life times and in most situations, an unbalanced ratio between the neutron number (N) and the proton number (Z), whereas stable nuclei are atomic nuclei with sufficiently long or infinite life time's [1]. Tanihata et.al with his experiment's to measure the radius of the 11 Li nucleus using RI (radioactive ion or rare isotope) beams of this nucleus in 1985 was the first step in such developments. According to this experiment, 11 Li has a very high interaction cross section [2]. Many new progresses have been made in recent years. The discovery of halo phenomena in exotic nuclei is one of the typical examples. In general, the halo nucleus has large neutron or proton excess where a few outside nucleons are very weakly bound. Such halo systems are well described by the few-body models, which assume that halo nuclei consist of a core and a few outside nucleons [3]. While the inclusion of the strong tensor force is one of the most basic nuclear forces, its firstorder impact on the shell structure has only recently been explained in exotic nuclei studies Depending on the occupation of specific orbits, the tensor force will alter the spin-orbit splitting [1]. T. Frederico et al. [4] used zero-range interactions to study the structure of three-body halo nuclei produced by two neutrons and a core. Only the swave scattering lengths and two-neutron separation energy will fully parameterize the halo wave function. Low-energy properties of the one-and two-neutron halo of light exotic nuclei, which are dominated by s-wave short-range two-body interactions were studied by T. Frederico, et al. [5]. In the study of Raghad Imad , showed that the twobody tensor correlations exhibit a mass dependence due to the strength parameter α (A), while the two-body short range correlations did not exhibit this dependency [6]. The inelastic longitudinal form factors C2 determined using this transition charge density with excitation of the levels for 54,52,50 Cr nuclei was shown by G.N. Flaiyh [7]. The central polarization transition density is calculated using the Tassie model's shape and the derived form of the ground state two-body charge density distributions (2BCDD's) in his research. Ahmed N. Abdullah [8] studied the nuclear matter distributions of neutron rich 6 He, 11 Li, 14 Be and 17 B halo nuclei by the Bear Hodgson potential. These halo nuclei are 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 are 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.
The aim of the work is to derive an expression for the ground state twobody nucleon density distributions (2BNDD's) of some exotic nuclei ( 17 B, 11 Li, 8 He), based on the use of the twobody wave functions of the harmonic oscillator in order to employ it for the study of the effects of the (TC's), and oscillator size parameter on the root mean square radii, 2BNDD's and elastic form factors.

Theory
The following transformation could be used to convert the one-body density operator to a two-body density type [9].
In reality, a useful transformation of the coordinates of the two particles, i r → and j r → , to be in terms of that centerof -mass ij → R and relative ij r → coordinates, can be rendered [10].
Finally, by folding the operator of Eq. (4) with the two-body correlation functions ij f , the effective two-body nucleon density operator (to be used with uncorrelated wave functions) is produced: In this paper, we will use a simple model type of the two-body tenser correlation operators from [11], where the sum  , in Eq. (7), is over all 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 one with the halo nucleons, so the matter density distribution for the whole halo nucleus becomes [12]: The normalization condition of the above ground state densities is given by: One of the following densities is represented by g (r) matter, charge, center, and halo densities the rms radii of the above densities are calculated as follows: The cross section for 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 spherical symmetric, therefore the form factor is simply the Fourier transform of the CDD. Thus [13,14].
where ) (q F fs 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 [14]. 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 given by [13] : where A is the nuclear mass number.

Results and discussion
The nuclear ground state properties of two-neutron ( 17 B, 11 Li, 8 He) exotic nuclei was calculated using (2BNDD'S) including the effect of two-body tenser correlations (TC's) using two frequency shell model (TFSM). Different model spaces for the core and the extra halo nucleon were used in the calculations. The wave functions of a single particle harmonic oscillator were used with two different size parameters of βc (size parameters of core) and βv (size parameters of valence). Table 1 shows the parameters of βc and βv used in TFSM of the present study together with the calculated and experimental rms radii of exotic nuclei ( 17 B, 11 Li, 8 He). It is clear from this table there was a good agreement between the calculated and experimental result of rms radii for these exotic nuclei.
). The value of oscillator size parameter for core 15 B is equal to βc= 1.7 fm, which gives rms radii equal to 2.46 fm, while the valence (two-neutron) assumed to be in a pure (1d2/5) and oscillator size parameter equal to βv=2.2fm used to give rms radii equal to (2.80fm). The experimental and calculated nucleon rms radii for this nucleus are displayed in (Table 1). It was noted that there is good agreement between the theoretically calculated values of rms radii with the experimental data. Two body nucleon density distribution (2BNDD'S) in fm -3 of ground state were plotted versus r (in fm) as shown in Fig.1(a), the black line represents the normal contribution of core 15 B, the valence (two-neutron in state 1d2/5) is represented by the blue line, which takes the shape of a long tail in this distribution. The matter density distribution (core + valence) is represented by a red line, which has a long tail as well and agrees well with 17 B's experimental data, which were taken from Vagen et al. [19] and represented by the shaded space. Fig. 1(b) shows a comparison between the matter density distribution of 17 B (represented by the red line) and the matter density distribution of stable nuclei 10 B (represented by the blue line). It is clear from this figure that the red and blue curves are diverse. As the valence two neutrons in 17 B is weakly bounded, the red curve has a longer tail than that of the blue curve. Fig. 1 (a and b) provide the conclusion that the halo phenomenon in 17 B is connected to the valence neutrons but not to the core nucleons.

Figure 1: (a) The comparison between Matter density distributions of 17 B with that of experimental data, (b) The comparison between matter density distribution of 17 B and 10 B.
The elastic form factor are shown in the Fig.2 the red curve represents form factor with Oscillator size parameter (β=1.95fm) (assumed to be the average of βc and βv), while the line with the black filled circles represents the experimental data for stable nuclei 10 B which were taken from Cichocki et al. [20]. When comparing the theoretical result with the experimental data of elastic electron scattering form factor, a different behavior between exotic nuclei ( 17 B) and stable nuclei ( 10 B) was noted, where the first diffraction minimum for 17 B located a ≈ 1.7 -1 .

11 Li nucleus
The second nucleus studied in this work was 11 Li. The nucleus 11 Li ) is formed by coupling the core 9 Li ( ) the oscillator size parameter of core is equal to βc=1.70 fm, Which gives rms radii equal to (2.34 fm), while the valence (two-neutron) is assumed to be in a pure (1p1/2) with occupation number 0.5 and oscillator size parameter equal to 2.18 fm which gives rms radii equal to 3.13 fm. The experimental and calculated nucleon rms radii for this nucleus are displayed in Table 1. A good agreement between the theoretically calculated and the experimental results was noted.

Figure 2: Comparison between the calculated elastic form factors of 17 B and the experimental data of 10 B.
Two body nucleon density distribution (2BNDD'S) in fm -3 of ground state are plotted versus r (in fm) as shown in Fig.3(a), the black line represents the normal contribution of core 9 Li, The valence (two-neutron exotic nuclei in the state of 1p1/2) is represented by the blue line, which takes the shape of a long tail in this distribution, the matter density distribution (core + valence) is defined by a red line, which has a long tail as well and agrees well with 11 Li's experimental data, which were taken from Dobrovolsky [21] and is represented by the shaded space. Fig. 3(b) shows a comparison between the matter density distribution of 11 Li which is represented by the red line, and the matter density distribution of stable nuclei 7 Li represented by the blue line. It is clear from this figure that the red and blue curves are diverse. As the valence two neutrons in 11 Li are weakly bound, the red curve has a longer tail than that of the blue curve. Fig.  3(a and b) provide the conclusion that the halo phenomenon in 11 Li is connected to the valence neutrons but not to the core nucleons.

Figure 3: (a) The comparison between matter density distributions of 11 Li with that of experimental data, (b) The comparison between matter density distribution of 11 Li and 7 Li.
The elastic electron scattering form factor 2BCDD's are shown in Fig.4, the red curve represents form factor with oscillator size parameter (β=1.94 fm ) (assumed to be the average of βc and βv) ,while the curve with black filled circles represents the experimental data for stable nuclei 7 Li, which were taken from Suelzle et al. [22]. When comparing the theoretical results with experimental data, it was noted that they have the same behavior of first diffraction minimum, where the first diffraction minimum for 11 Li is located at ≈ 1.6 -1 and for 7 Li at ≈ 2.5 -1 .

8 He nucleus
The third nucleus are studied in this work 8 He. The nucleus 8 He ( 2 , 0 , is formed by coupling the core 6 He ( 1 , 0 , ).The oscillator size parameter of core is equal to βc=1.80 fm, which gives rms radii equal to (2.27 fm), while the valence (two-neutron) is assumed to be in a pure (1p1/2) with occupation number 0.5 and oscillator size parameter equal to 1.75fm which gives rms radii equal to 2.55 fm. The experimental and calculated nucleon rms radii for this nucleus are displayed in Table 1. It is noted that there is a good agreement between the theoretical calculations and the experimental results. Two body nucleon density distribution (2BNDD'S) in fm -3 of ground state are plotted versus r (in fm) as shown in Fig. 5(a). The black line represents the normal contribution of core 6 He, the blue line represents the valence (two-neutron exotic nuclei in state of 1p1/2) through this distribution, it takes the form of a long tail, the red line represents the matter density distribution (core +valence) and it takes the form of long tail too and has a good agreement with the experimental data of 8 He, which were taken from Antonov [23] and is represented by the shaded area. Fig. 5(b) shows the comparison between the matter density distribution of 8 He, represented by the red line, and the matter density distribution of stable nuclei 4 He, represented by the blue line. It is clear from this figure that the red and blue curves are diverse. As the valence two neutrons in 8 He are weakly bound, the red curve has a longer tail than that of the blue curve. Fig. 5(a and b) provide the conclusion that the halo phenomenon in 8 He is connected to the valence neutrons but not to the core nucleons. The elastic electron scattering form factor 2BCDD's are shown in Fig.6, the red curve represents form factor with oscillator size parameter (β=1.77fm ) (assumed to be the average of βc and βv), while the black filled circles curve represents the experimental data for stable nuclei 4 He which were taken from J. S. McCarthy et al. [24]. When comparing the theoretical results with experimental data for stable nuclei ( 4 He) it was noted that they have the same behavior of first diffraction minimum, Where the first diffraction minimum for 8 He is located at ≈ 2.6 -1 and for 4 He at ≈ 3.1 -1 .

Conclusions
Because of two-neutron valence, which are considered to be a distinctive characteristic of halo nuclei, the measured matter density via the framework of two body nucleon density distribution (2BNDD's) with effect of tensor force (TC's) and two different oscillator size parameters βc and βv for our exotic nuclei display a long tail at (r > 6fm) behavior in this work. The measured matter density and rms radii of ( 17 B, 11 Li,