The study of nuclear structure for some nuclei

An analytical form of the ground state charge density distributionsfor the low mass fp shell nuclei ( 40  A  56 ) is derived from asimple method based on the use of the single particle wave functionsof the harmonic oscillator potential and the occupation numbers ofthe states, which are determined from the comparison between theoryand experiment.For investigating the inelastic longitudinal electron scattering formfactors, an expression for the transition charge density is studiedwhere the deformation in nuclear collective modes is taken intoconsideration besides the shell model space transition density. Thecore polarization transition density is evaluated by adopting theshape of Tassie model together with the derived form of the groundstate charge density distribution. In this work, we devote ourinvestigation on 0 3 2 3 1 1   transition of Ti 50 , 0 1 2 1 1 1   transitionof Cr 50 and 0 2 2 2 1 1   of Cr 52 nuclei. It is found that the corepolarization effects, which represent the collective modes, areessential for reproducing a remarkable agreement between thecalculated inelastic longitudinal C2 form factors and those ofexperimental data.

) is derived from a simple method based on the use of the single particle wave functions of the harmonic oscillator potential and the occupation numbers of the states, which are determined from the comparison between theory and experiment. For investigating the inelastic longitudinal electron scattering form factors, an expression for the transition charge density is studied where the deformation in nuclear collective modes is taken into consideration besides the shell model space transition density. The core polarization transition density is evaluated by adopting the shape of Tassie model together with the derived form of the ground state charge density distribution. In this work, we devote our investigation on nuclei. It is found that the core polarization effects, which represent the collective modes, are essential for reproducing a remarkable agreement between the calculated inelastic longitudinal 2 C form factors and those of experimental data.

Introduction
Charge density distributions, transition densities and form factors are considered as fundamental characteristics of the nucleus. These quantities are usually determined experimentally from the scattering of high energy electrons by the nucleus. The information extracted from such experiments is more accurate with higher momentum transfer to the nucleus. Various theoretical methods [1][2][3] are used for calculations of the charge density distributions (CDD), among them the Hartree-Fock method with the Skyrme effective interaction the theory of finite Fermi systems and the single particle potential method. Calculations of form factors [4] using the model space wave function alone is inadequate for reproducing the data of electron scattering. Therefore effects out of the model space, which is called core polarization effects, are necessary to be included in the calculations. These effects can be considered as a polarization of core protons by the valence protons and neutrons. Core polarization effects can be treated either by connecting the ground state to the J-multipole   n giant resonances [4], where the shape of the transition densities for these excitations is given by Tassie model [5] or by using a microscopic theory [6-10] which permits one particle-one hole (1p-1h) excitations of the core and also of the model space to describe these longitudinal excitations. Comparisons between theoretical and observed longitudinal electron scattering form factors have long been used as stringent test of models of nuclear structure.
In this study, derive an analytical form for the ground state charge density distributions of low mass fp shell nuclei ( 56 40   A ) using the single particle wave functions of the harmonic oscillator and the occupation numbers of the states, which is determined from the comparison between theory and experiment.
This study is aimed to investigate the inelastic longitudinal electron scattering form factors, where the deformation in nuclear collective modes (which represent the core polarization effects) is taken into consideration besides the shell model space transition density. Core polarization transition density is evaluated by adopting the shape of Tassie model together with the derived form of the ground state charge density distribution. Our investigation is devoted on nuclei. It is found that the core polarization effects are essential for reproducing a remarkable agreement between the calculated inelastic longitudinal 2 C form factors and those of experimental data.

Theory
The interaction of the electron with charge distribution of the nucleus gives rise to the longitudinal or Coulomb scattering. The longitudinal form factor is related to the CDD through the matrix elements of Coulomb multipole operators ) ( q T L J and is given by [4] ) where J and Z are the multipolarity and the atomic number of the nucleus, respectively. ) (q F cm is the center of mass correction, which remove the spurious state arising from the motion of the center of mass when shell model wave function is used and is given by [4]   where A is the nuclear mass number and b is the harmonic oscillator size parameter. ) (q F fs Is the free nucleon form factor and assumed to be the same for protons and neutrons and it takes the form [4]. The longitudinal operator is defined by [11]  

 
is the spherical harmonic wave function and ) , ( z t r  is the charge density operator. The reduced matrix elements in spin and isospin space of the longitudinal operator between the final and initial many particles states of the system including the configuration mixing are given in terms of the one body density matrix (OBDM) elements times the single particle matrix elements of the longitudinal operator [4]   where the OBDM are calculated in terms of the isospin -reduced matrix elements.
The model space matrix elements is not adequat to describe the absolute strength of the observed gamma-ray transition probabilities, because of the polarization in nature of the core protons by the model space protons and neutrons. The many particle reduced matrix elements of the longitudinal operator, consists of two parts one is for the model space and the other is for core polarization matrix element [6], i.e.

 
where the model space matrix element has the form [12]   is the transition charge density of model space and is given by [4]   is the core-polarization transition density which depends on the model used for core polarization. To take the core-polarization effects into consideration, the model space transition density is added to the corepolarization transition density that describes the collective modes of nuclei. The total transition density where cor J Z   is assumed to have the form of Tassie shape and given is by [5] dr (11) where N the proportionality constant. In Eq. (11) ). Using this assumption together with the single particle wave functions of the harmonic oscillator potential, the ground state charge density distribution for the low mass fp shell nuclei is obtained as The parameter where the value of Using Eq. (13) in (14), we obtain: where the harmonic oscillator size parameter b is obtained by introducing the experimental MSR of considered nuclei into Eq. (15).

Results, discussion and conclusions
The 2 C longitudinal form factors of the low mass fp shell nuclei are calculated for the transition and Cr 52 , 50 nuclei. The deformation in nuclear collective modes (which represent the core polarization effects) is taken into consideration by using the shape of Tassie model [5], that depends on the ground state . CDD The ground state CDD for the low mass fp shell nuclei ( ), which is used in Tassie model, is derived and formulated in terms of the single particle wave functions and the occupation numbers of the states as is given by Eq. (12). These occupation numbers are determined from the comparison between the calculated and experimental . CDD The single particle wave functions are those of the harmonic oscillator potential whose oscillator size parameters b are chosen in such a way that to reproduce the experimental root mean square ( rms ) charge radii . nuclei are given in Table 2.
Inelastic Fig. 1. The dashed curves represent the contribution of the model space where the configuration mixing is taken into account and the solid curves represent the total contribution, which is obtained by taking the model space together with the core polarization effects. The experimental data of Ref. [14] are represented by solid circles. It is very clear from Fig. 1 that the core-polarization effects give a strong modification to the form factors, where the core polarization effects enhance the C2 form factors at the first and second maximum and bring the calculated values very close to the experimental data. Besides, the locations of the diffraction minimum, in all considered nuclei, are located in the correct places when the core polarization effects are included in the calculations. However, the available experimental data [14] presented in Fig. 1 is very well reproduced by the present calculations throughout all momentum transfer values. It is concluded that the core polarization effects, which represent the collective modes, are essential in obtaining a remarkable agreement between the calculated 2 C longitudinal form factors of the low mass fp shell nuclei and those of experimental data.