Proton Momentum Distributions and Elastic Electron Scattering from 70 Zn , 72 Ge and 74 Se nuclei

In the framework of correlation method so-called coherent density fluctuation model (CDFM) the proton momentum distributions (PMD) of the ground state for some even mass nuclei of fpshell like Zn,Ge and Se nuclei are examined. Proton momentum distributions are expressed in terms of the fluctuation function(|f(x)|) and determined from theory and experiment. The main characteristic feature of the PMD obtained by CDFM is the existence of high-momentum components, for momenta k ≥ 2 fm. For completeness, also elastic electron scattering form factors F(q) are evaluated within the same framework.


Introduction
The nuclear properties such as nuclear binding energies and nuclear density distributions ρ(r) have been gain more attention in theoretical studies for the last few decades.These quantities have been investigated successfully using the zero range Hartree-Fock method since they are not sentient to the correlations of the short range nucleon-nucleon interactions.On the other hand, the high energy interactions between particles and a target nucleus more over some low energy phenomena, giant resonance for example, have been becomes the scope of the experimental works in the resent years.These works make some additional quantities able to be done such as the nucleon momentum distributions n(k) [1,2] of nuclei which expand the range of the nuclear ground state theory.The nucleon momentum distribution is connected to some process, which are concerning by the experimental works, such as nuclear photo effect and absorption of the mesons by nuclei.Since the short range of the nucleon-nucleon force dose not included in the Hartree-Fock method, it is becomes impossible to investigate density and momentum distributions at the same time using this method where n(k) is sensitive to the short range and tensor correlations.For this reason we need a nuclear correlation methods to make a assured by exclusive and inclusive electron scattering measured data on nuclei.In general, for the nuclear interactions calculations of the cross-section it is important to know the momentum distribution of the interacting nucleons.The coherent density fluctuation model (CDFM) based on the local density distribution as a variable of the theory has been suggested in [1,2] to investigate the nuclear structure and nuclear reactions using the essential results of the infinite nuclear matter theory.
Hamoudi et al. [8,9,10] have studied the NMD and elastic electron scattering form factors for p-shell [8], sd-shell [9] and fp-shell [10] nuclei using the framework of CDFM.They [8,9,10] derived an analytical form for the NDD based on the use of the single particle harmonic oscillator wave functions and the occupation number of the states.The derived NDD's, which are applicable throughout the whole p-shell [8], sd-shell [9] and fp-shell [10] nuclei, have been used in the CDFM.The calculated NMD and elastic form factors of all considered nuclei have been in very good agreement with experimental data.
In this research, we follow the work of Hamoudi et al. [8,9,10] and utilize the CDFM with weight functions originated in terms of theoretical charge density distribution (CDD) of some fp-shell nuclei such as 70 Zn, 72 Ge and 74 Se nuclei.It is found that the theoretical weight function |()| 2 based on the derived CDD is capable to give information about the proton momentum distributions (PMD) and elastic charge form factors as do those of the experimental data [11].

Theory
The charge density distribution CDD of one body operator can be written as [12]: Using this supposition in Eq. ( 1), the ground state CDD of 70 Zn, 72 Ge and 74 Se nuclei is obtained as: Where Z is the atomic number, b is the harmonic oscillator size parameter.
The corresponding the mean square radius (MSR) is The central CDD,   ( = 0) is obtained from Eq. ( 2) as Then  1 is obtained from Eq. ( 4) as The PMD, (), of the considered nuclei is studied using two distinct methods.In the first, it is determined by the shell model using the single particle harmonic oscillator wave functions in momentum representation and is given by [13]: Which is the probability of finding a particle with momentum k at position r .
Using the basic relationships of the   () and () with the WDF, And One can obtain the corresponding expressions for   () and () using the WDF from Eq. (7), that is, And In eq. ( 10),   () has the following form [1,2]: In the case of monotonically-decreasing density distributions (/ < 0) one can obtain from (10) Where  = (9/8) 1/3 with a normalization condition: The form factor ) (q F of the target nucleus is also expressed in the CDFM as [6]: is the form factor of uniform charge density distribution given by [6]: Inclusion the corrections of the nucleon finite size ) (q F fs and the center of mass corrections ) (q F cm in the calculations requires multiplying the form factor of equation ( 16) by these corrections.Here, ) (q F fs is considered as free nucleon form factor which is assumed to be the same for protons and neutrons.This correction takes the form [14]: The correction Fcm (q) removes the spurious state arising from the motion of the center of mass when shell model wave function is used and given by [14]:  as Moreover, introducing the derived CDD of Eq. (2) into Eq.( 13

Results and Discussion
In the present work the CDFM has been used to investigate () and F(q) for some nuclei namely; 70 Zn, 72 Ge and 74 Se.In order to calculate the proton momentum distributions n(k), obtained from Eq. ( 14), we need to investigate the CDD for both experiment, such as, 2PF and 3PF [11] and theoretical consideration using Eq. ( 2) which includes some parameters needed

Vol: 13
No:2 , April 2017 DOI: http://dx.doi.org/10.24237/djps.1302.182BP-ISSN: 2222-8373 E-ISSN: 2518-9255 for calculations.These parameters have been calculated and presented in Table (1) together with other parameters employed for the selected nuclei used in the present work.The parameter α1 is determined by introducing the harmonic oscillator size parameter b, which gives the experimental root mean square (rms) radii and the experimental central density   (0) into Eq.(5), while the parameter  2 is assumed as a free parameter to be adjusted to obtain agreement with the experimental CDD.It is important to remark that when α1=α2=0, Eq. (2) is reduced to simple shell model prediction.The (2 −  1 ), ( − 20 −  2 ) and ( 1 +  2 ) proton occupations numbers for 2s, 1f and 2p orbitals, respectively, have been also calculated and tabulated in

Figure ( 1 )
Figure(1) demonstrates the CDD's for 70 Zn, 72 Ge and 74 Se nuclei which have been calculated using Eq.(2) and denoted in this figure as dashed and solid curves with  1 =  2 = 0 and  1 ≠  2 ≠ 0, respectively.For comparison, the experimental data[11,15] have been also presented in Figure(1) and denoted by the filled circle along with calculated CDD curves.This