Charge density distributions and electron scattering form factors of 25 Mg , 27 Al and 29 Si nuclei

An effective two-body density operator for point nucleon systemfolded with the tenser force correlations( TC's), is produced and usedto derive an explicit form for ground state two-body charge densitydistributions (2BCDD's) applicable for 25Mg, 27Al and 29Si nuclei. It isfound that the inclusion of the two-body TC's has the feature ofincreasing the central part of the 2BCDD's significantly and reducingthe tail part of them slightly, i.e. it tends to increase the probability oftransferring the protons from the surface of the nucleus towards itscenteral region and consequently makes the nucleus to be more rigidthan the case when there is no TC's and also leads to decrease the1/ 2r 2 of the nucleus. It is also found that the effect of the TC's and theeffect of increasing the values of  on the 2BCDD's, elasticelectron scattering form factors and r2 1/ 2 are in the same directionfor all considered nuclei.


Introduction
Electron scattering is an excellent tool for studying the nuclear structure because of many reasons. Since the interaction between the electron and the target nucleus is relatively weak and its known where the electron interacts electromagnetically with the local charge, current and magnetization densities of nucleus. Besides, the measurements can be obtained without greatly disturbing the structure of the target [1,2]. The effect of the tensor correlation on the alpha-alpha interaction in 8 Be using an alpha cluster model. They had used the wave function of the alpha particle calculated by the projected Hartree-Fock method, which could treat the effect of the tensor correlation Sugimoto et al. [3]. Radhi et al. [4] studied the elastic longitudinal electron scattering form factors for 9 Be in the frame work of 1pshell model, which considered as the core of 4 He with five nucleons distributed out of the core. Their results are in good agreement with the experimental data for both models considered. The reduced transition probabilities B(C2) calculated for the two kinds of model space and for the effective charges which are used in this work. Hamoudi et al. [5] studied the Nucleon Momentum Distributions (NMD) and elastic electron scattering form factors of the ground state for 1pshell nuclei with Z=N (such as 6 Li, 10 B, 12 C and 14 N nuclei) in the frame work of the Coherent Density Fluctuation Model (CDFM) and expressed in terms of the weight function f |x| 2 . Hamoudi et al. [6] studied the nucleon momentum distributions (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 inclusion of 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 [7,8,9]. The aim of the present work is to derive an expression for the ground state 2BCDD, based on the use of the two -body wave functions of the harmonic oscillator in order to employ it for studying of the effects of the TC's and oscillator parameter on the root mean square charge radii r 2 2 1 , 2BCDD, elastic electron scattering form factors for 25 Mg, 27 Al and 29 Si nuclei.

Theory
The one body density operator of Eq. (1) could be transformed into a two-body density form by the following transformation [10] In fact, a further useful transformation can be made which is that of the coordinates of the two -particles, i r  and j r  , to be in terms of that relative ij r  and center -of -mass ij  R coordinates [11] ,i.e.
subtracting and adding (3-a) and (3-b) we obtain Eq. (4) may be written as where the following identities [12] have been used For closed shell nuclei with N=Z, the two -body 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. [13] will be adopted, i.e.
TC's presented in the Eq.
where the sum  , in Eq.(10), 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 zeroth order of the spherical Bessel function and q is the momentum transfer from the incident electron to the target nucleus. Eq. (12) may be expressed as 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 [14]. where A is the nuclear mass number. Introducing these corrections into Eq.(13), we obtain In the limit of q  0, the target will be considered as a point particle, and from Eq. (16), 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 languages.

Results and discussion
The dependence of the ground state 2BCDD's (in fm -3 ) on r (in fm) for 25 Mg, 27 Al and 29 Si nuclei are displayed in Figs.1 (a, b and c) respectively, where all parameter required to the calculation presented in Table 1. In Figs.1 (a, b and c) the calculated 2BCDD's without TC's (the dashed curves with 0   ) and with TC's (the solid curves with 0   ) are compared with experimental results (the dotted symbols) [15]. From these figures, the dashed curves deviate slightly from the solid curve especially at small r. Introducing the effect of TC's in the calculations tends to remove these deviations from the region of small r as seen in the solid curves. It is evident from these figures that the calculated 2BCDD's represented by the solid curves are in excellent accordance with those of experimental data hence they coincide with each other throughout the whole range of r.  Fig.1(a, b and c) In Fig. 2a we explore the calculated results for the form factors of 25 Mg nucleus. It is evident from this figure that the calculated results obtained in both of the dashed and solid curves are in coincidence with each other for the region of momentum transfer q  2.4fm -1 and they are in agreement behavior with those of experimental data [16] at this region of q. The first diffraction minimum which is known from the experimental data is well reproduced solid curves. It is noticed that the first diffraction minimum at the region of q =1.4fm -1 . It is clear from the Fig. 2a, the location of second diffraction minimum was shifted to the region of q =3.2fm -1 . It is seen from this figure that there is a disagreement between the experimental and calculated form factors of this nucleus at the region of momentum transfer q>2.5fm -1 where it seems there is a second diffraction minimum in the experimental data which cannot be reproduced in the correct place by both of the dashed and solid curves. It is obvious if we look at this figure that the effect of TC's begins at the region of momentum transfer q>1.4 fm -1 where the solid curve deviates from the dashed curve at this region of q. The form factor of 27 Al nucleus is displayed in Fig. 2b. As we can see from this figure that the available data of 27 Al nucleus are restricted for a small region of momentum transfer q 1.4fm -1 , it is noticed that the first diffraction minimum at the region of q =1.6fm -1 . It is clear from the Fig. 2b, the location of second diffraction minimum was shifted to the region of q =2.6fm -1 and the location of third diffraction minimum was shifted to the region of q =3.3fm -1 . It is evident from this figure that the calculated results obtained in both of the dashed and solid curves are in excellent agreement with those of experimental data [17]. It is also noted that the effect of considering the TC's becomes more effective at higher momentum transfer of q>2.5fm -1 and even it becomes progressively larger with increasing q. Besides, this effect enhances the calculated F(q)'s at q>2.5fm -1 as seen in the solid curve of this figure which overestimates the dashed curve at this region of q. The form factor of 29 Si nucleus is displayed in Fig. 2c. As we can see from this figure that the available data of 29 Si nucleus are restricted for a small region of momentum transfer q 1.3fm -1 , it is noticed that the first diffraction minimum at the region of q =1.4fm -1 It is clear from the Fig.2c, the location of second diffraction minimum was shifted to the region of q =2.4fm -1 , the location of third diffraction minimum was shifted to the region of q =3.2fm -1 . It is evident from this figure that the calculated results obtained in both of the dashed and solid curves are in excellent agreement with those of experimental data [17]. It is also noted that the effect of considering the TC's becomes more effective at higher momentum transfer of q>1.6fm -1 and even it becomes progressively larger with increasing q.

Conclusions
The two-body TC's exhibits a mass dependence due to the strength parameter  (A) and the two-body TC's have the feature of increasing the central part of the 2BCDD's significantly and reducing the tail part of them slightly, i.e. it tends to increase the probability of transferring the protons from the surface of the nucleus towards its central region (the central region of the nucleus towards its surface) and consequently makes the nucleus to be more rigid than the case when there is no TC's and also leads to decrease the 2 / 1 2 r of the nucleus.