Charge density distributions and electron scattering form factors of 19 F , 27 Al and 25 Mg nuclei

An effective two-body density operator for point nucleon system folded with two-body correlation functions, which take account of the effect of the strong short range repulsion and the strong tensor force in the nucleon-nucleon forces, is produced and used to derive an explicit form for ground state two-body charge density distributions (2BCDD's) and elastic electron scattering form factors F (q) for 19F, 27Al and 25Mg nuclei. It is found that the inclusion of the two-body short range correlations (SRC) has the feature of reducing the central part of the 2BCDD's significantly and increasing the tail part of them slightly, i.e. it tends to increase the probability of transferring the protons from the central region of the nucleus towards its surface and to increase the root mean square charge radius ˂ r 2˃ 1/2 of the nucleus and then makes the nucleus to be less rigid than the case when there is no (SRC). It is also found that the effects of two body tensor correlations (TCs) on 2BCDD's and ˂ r 2˃ 1/2 are in opposite direction to those of (SRC).


Introduction
The charge density distributions (CDD) is the most important quantities in the nuclear structure which was well studied experimentally over a wide range of nuclei. This interest in CDD is related to the basic bulk nuclear characteristics such as the shape and size of nuclei, their binding energies, and other quantities which are connected with the CDD. Besides, the density distribution is an important object for experimental and theoretical investigations since it plays the role of a fundamental variable in nuclear theory [1]. The inclusion of shortrange and tensor correlation effects is rather a complicated problem especially for the microscopic theory of nuclear structure. Several methods were proposed to treat complex tensor forces and to describe their effects on the nuclear ground state [2,3].
A simple phenomenological method for introducing dynamical short range and tensor correlations has been introduced by Dellagiacoma et al. [4]. In that method, twobody correlation operator is introduced to act on the wave function of a pair of particles. A similar correlation operator was proposed earlier by Da Proveidencia and Shakin [5] as well as Malecki and Picchi [6] for describing the shortrange correlation effects.
The effect of the short range correlations due to the repulsive part of two-body interaction on the charge form factor of several p-shell nuclei has been analyzed in detail [7] with an independent particle model (IPM) generated in the harmonic oscillator (HO) well [8,9]. In Ref. [7], it was shown that the high-momentum parts (q>3 fm -1 ) of the form factors calculated with and without correlations behave in completely different ways, which indicates that electron scattering at high momentum transfer could give useful information on the short-range correlations. Hamoudi et al. [10] had studied an effective two-body density operator for point nucleon system folded with twobody correlation functions, which take account of the effect of the strong short range repulsion and the strong tensor force in the nucleon-nucleon forces, is produced and used to study the ground state two-body charge density distributions and elastic electron scattering form factors F(q) for 2s -1d shell nuclei with Z =N (such as 20 Ne, 24 Mg, 28 Si and 32 S nuclei). Hamoudi et al. [11] studied the effects of short range correlation and occupation probabilities of single particle orbits for various closed and open shell nuclei with N=Z. Hamoudi et al. [12] studied the NMD for the ground state and elastic electron scattering form factors in the framework of the coherent fluctuation model and expressed in terms of the weight function (fluctuation function). The aim of the present work is to study the effects of short range correlations and tensor correlations on the ground state two body charge density distributions, root mean square charge radii and elastic electron scattering form factors for 19 F, 25 Mg and 27 Al nuclei.

Theory
The one body density operator can be transformed into a two-body density form as [1].
where r ⃗ r i ⃗ ⃗⃗ : is the Dirac delta function, A is the nucleon number In fact, a further useful transformation can be made which is that of the coordinates of twoparticles, i r  and j r  , to be in terms of that relative ij r  and centerofmass ij  R coordinates [13]. i.e.
Subtracting and adding (3-a) and (3- Eq. (4) may be written as where the following identities [14] have been used or closed shell nuclei with N=Z, the twobody charge density operator can be deduced from Eq. (6) as Finally, an effective two-body charge density operator (to be used with uncorrelated wave functions) can be produced by folding the operator of Eq. (7) with the two-body correlation In the present work, a simple model form of the two-body full correlation operators of ref. [15] will be adopted, It is obvious that this equation includes two types of correlations: In fact, the SRC's are central functions of the separation between the pair of particles which reduce the twobody wave function at short distances, where the repulsive core forces the particles apart, and heal to unity at large distance where the interactions are extremely weak. A simple model form of two-body SRC's is given by [15]   where the sum  , in Eq. (11), 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 where the two particle wave function is given by [16] It is important to indicate that our effective two body charge density operator of Eq. (8) is constructed in terms of relative and center of mass coordinates, thus the space-spin part JM j j J j i ) ( of two-particle wave function constructed in jj-coupling scheme must be transformed in terms of relative and center of mass coordinates. The nuclear mean square charge radius < r 2 > 1/ 2 is defined by [16] ∫ 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 [17,18] (18) 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 [19]. where A is the nuclear mass number. Introducing these corrections into Eq. (17), we obtain In the limit of q 0, the target will be considered as a point particle, and from Eq. (20), the form factor of this target nucleus is equal to unity, i.e. We also wish to mention that we have written all computer programs needed in this study using Fortran-90 languages.

Results and discussion
The calculations for the ground state two body charge density distributions (2BCDD's) ch (r) , the root mean square charge radii< r 2 > 1/ 2 and elastic electron scattering form factors F (q)' s are carried out for 19 F, 25 Mg and 27 Al nuclei. A choice for the single value of the hard core radius r c =0.4 fm is adopted for all considered nuclei. The strengths of the tensor correlations α A) are determined by fitting the calculated < r 2 > 1/ 2 with those of experimental data. All parameters required in the calculations of ch (r), < r 2 > 1/ 2 and F (q)'s, such as the harmonic oscillator spacing parameter, h , the occupation probabilities, ƞ's, of the states and α (A), are presented in Table 1 including both effects of SRC (with r c = 0.4 ƒm) and TC with α A) ≠0), are in very good agreement with those of experimental data [20]. The results for the dependence of ch r) in ƒm -3 ) on r in ƒm) for 19 F, 25 Mg and 27 Al nuclei are displayed in Figs.1. In Fig.1, the dashed and solid distributions are the calculated ch (r) of 19 F, 25 Mg and 27 Al nuclei without effects (r c =0 and α A) =0) and with the effects of SRC and TC (r c = 0.4 ƒm and α A) ≠0) included, respectively. These distributions are compared with those of experimental data [20], denoted by dotted symbols. In 19  In 19 F nucleus, the first diffraction minimum and first maximum which are known from the experimental data [20] are very well reproduced by the dashed and solid curves. In general, the calculated F (q)'s are in very good accordance with the data up to momentum transfer q ≈ 2.4 ƒm -1 . For higher q, the calculated form factors are in disagreement with the data, where the second diffraction minimum observed in q=2.5 ƒm -1 .
In 25 Mg nucleus, It is noticed from the figure the dashed and solid curves are in reasonable agreement with those of experimental data [20] throughout the range of momentum transfer q ≤ 1.5 ƒm -1 . It is noted that the effect of correlations begin at the region of q ˃ 2.3 ƒm -1 , where the solid curve deviates from the dashed curve at this region of q.
In 27 Al nucleus, the dashed and solid curves are in good agreement with the experimental data up to momentum transfer of q =2.3 fm -1 , and it underestimates clearly these data at the region of q >2.3 fm -1 . It is so clear that the location of the third observed diffraction minimum is not reproduced in the correct place by the dashed and solid curves. It is noted that the effect of correlations begin at the region of q˃2.5 fm -1 .

Conclusions
This study shows that the two-body tensor correlations exhibit a mass dependence due to the Strength parameter α A) while the two-body short range correlations do not exhibit this dependency. Including the effect of SRC alone increases the probability of transferring the protons from the central region of ch (r) towards its tail (i.e., the nucleus becomes less rigid than the case when there is no SRC) and then increases the calculated < r 2 > 1/ 2 . It is found that the effect of TC on ch (r) and <r 2 > 1/ 2 is in opposite direction to that of SRC.