Charge Density Distributions and Elastic Electron Scattering from 58 Ni , 64 Zn , 70 Ge and 76 Se nuclei using the occupation numbers of the states

The charge density distribution (CDD) and elastic electron scattering form factors, F(q), for some 1f-2p shell nuclei, such as Ni, Zn, Ge and Se nuclei have been evaluated using the wave functions of the harmonic oscillator and occupation numbers of states. It found that considering the effect of higher shells through introducing additional parameters namely δ1 and δ2 lead to astonishing accordance between the calculated and the observed results of the CDD and elastic form factors F(q).


Introduction
The charge density distribution (CDD) and form factors (a factor depends on the charge, current and magnetization distributions in the target nucleus) are fundamental properties of the nucleus and can be obtained experimentally from electron-nucleus scattering.The electron-nucleus interaction is considered by the first Born approximation as an exchange of a virtual photon.In this case the initial and final particles are considered free and can be represented by plane waves [1].The electron scattering from the nucleus at high energy gives important information about the nuclear structure.Information obtained from the high energy electron scattering by the nuclei depends on the magnitude of the de Broglie wave length that is associated with the electron which is compared with the range of the nuclear forces.When the energy of the incident electron is in the range of 100 MeV and more, the de Broglie wave length will be in the range of the spatial extension of the target nucleus.Thus with this energy, the electron represents a best probe to study the nuclear structure [2,3].Al-Rahmani and Mahdi have calculated the charge density distribution and elastic form factors for some 2s-1d shell nuclei using the plan-wave Born approximation and illustrated that the inclusion additional parameters in the calculations improves the results and makes them in remarkable accordance with the measured results [4].
The aim of the present work is to extend the calculations of Al-Rahmani and Mahdi to higher shells (such as the 1f-2p shell nuclei) and to derive an analytical form for the charge density distribution and elastic form factors F(q).

Theory
The charge density distribution (CDD) can be determined in terms of the harmonic oscillator radial wave function ( ℓ ) as [4,5]: According to the simple shell model, the 1f-2p shell nuclei are assumed as an inert core of filled 1s, 1p, 1d, and 2s while the 1f orbit is occupied by ( − 20) protons, where Z is the atomic number of nuclei.Using this assumption the CDD of 1f-2p shell nuclei can be written as: Where b is the harmonic oscillator size parameter.
The normalization condition of the  ℎ given in eq (1) is where the parameters  1 and  2 are the occupation number of higher shells.Using this assumption, with the help of Eq. ( 1), the charge density distribution can be written as: and the corresponding MSR is The central CDD,  ℎ ( = 0) is obtained from Eq. ( 5) as then  1 is obtained from Eq. ( 7) as The elastic electron scattering form factors is determined by the ground state charge density distribution (CDD) [4,5], i.e.
where is the zeroth order spherical Bessel function, q is the momentum transfer from the incident electron to the target nucleus and the  ℎ () is the CDD of the ground state.
An analytical form for elastic electron scattering form factor, F(q), can be obtained by introducing the form of the CDD of eq. ( 5) into eq.( 9), and performing the integration, i.e., Inclusion the corrections of the center of mass   () = ( 2  2 /4) [6] and the finite nucleon size   () = (−0.43 2/4) [6] in the calculations needs multiplying the form factor of Eq. ( 11) by these corrections.

Results and Discussion
The calculated CDD's for some even-A of 1f-2p shell nuclei, such as 58 Ni, 64 Zn, 70 Ge and 76 Se nuclei have been obtained using the analytical form of eq. ( 5) and compared with the fitted to   Figure 1 illustrates the calculated CDD's for 58 Ni, 64 Zn, 70 Ge and 76 Se nuclei.The blue and red curves are the calculated results using eq.( 5) with  1 =  2 = 0 and  1 ≠  2 ≠ 0, respectively whereas the filled circle symbols correspond to the measured data [7,8].This figure illustrates that the blue curves are in disagreement with the measured data, especially for small r.Inclusion of the parameters  1 and  2 (i.e., considering the higher orbitals) in the calculation leads to astonishing accordance with the measured data as demonstrated by the red curves.As it is evident from figure 1 that the red curves of 64 Zn and 76 Se deviates slightly from the measured data of 2PF and 3PF, respectively, especially at the region (1.5 <  < 2.8) fm.In general, considering the effect higher shells in Eq.( 5) improves strongly the calculated CDD of 64 Zn and 76 Se nuclei, but these higher shells are not enough for resolving completely the problem of slight deviation.This deviation may be attributed to the necessity of considering other higher shell, such as 1g shell.However, this deviation doesn't affect generally the very well agreement with the experimental data throughout the whole range of r.
The elastic electron scattering form factors from considered spin-zero nucleus are calculated in terms of the ground state CDD using the plane wave Born-Approximation (PWBA), where the form factor is a Fourier transform of CDD as given in eq (9).The form factors of 58 Ni, These figures show that the experimental form factors of 58 Ni [9] and 64 Zn [10] nuclei are in a very good agreement with those of calculated result up to q ≈0.9 fm -1 , while for higher q the calculated results for these nuclei underestimate the experimental data.It is evident from

Summary & Conclusions
Analytical expressions for the ground state CDD and elastic electron scattering from factors for some 1f-2p shell nuclei, namely 58 Ni, 64 Zn, 70 Ge and 76 Se nuclei has been derived from a method that based on used the wave function of the harmonic oscillator and occupation numbers of states.From the calculated charge density distributions and form factors, it is concluded that: 1-Including the effect of higher shells through introducing the parameters (

Vol: 13
No:2 , April 2017 DOI: http://dx.doi.org/10.24237/djps.1302.165AP-ISSN: 2222-8373 E-ISSN: 2518-9255 function of the momentum transfer.In this figure the filled circle symbols and red curves are the experimental data and the calculated results, respectively.The form factors of 58 Ni and 64 Zn nuclei are presented in figures 2(a) and 2(b), respectively.

figure 2 (
figure 2(a) that the diffraction minima and maxima of 58 Ni nucleus are reproduced in the correct places.

Figures 2 (
Figures 2(c) and 2(d) illustrate the form factor of 70 Ge and 76 Se nuclei, respectively.These figures give an indication that the observed data [8] of 70 Ge and 76 Se nuclei are in very good agreement with the calculated results for all momentum transfer values.It is evident from these figures that the observed first and second diffraction minima are quite well described by the calculated results.

Table 1 -Parameters of CDD for consider nuclei together with
().