Core Polarization Effects on the Inelastic Longitudinal C2 Form Factors of Open Sd-Shell Nuclei

Inelastic longitudinal 2 C form factors for + +


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, 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  [4,5,6], where the shape of the transition densities for these excitations is given by Tassie model [7], or by using a microscopic theory [8,9] 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, we have derived an expression for the ground state twobody charge density distributions (2BCDD) of light nuclei, based on the use of the two -body wave functions of the harmonic oscillator and the two-body correlation functions, which take account of the effect of the strong short range repulation and the strong tensor force in the nucleon-nucleon forces.Our aim is 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 2BCDD.This study is devoted on and Si 30 nuclei.

Theory
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 [5,6]: The model space matrix element has the form [10]: is the transition charge density of model space given by [4]: The core-polarization matrix element is given by [4]: where is assumed to have the form of Tassie shape and given by [7].
where N is a proportionality constant.It is determined by adjusting the reduced transition probability B(CJ) and can be given as: , , ( It is clear that eq. ( 9) contains two types of correlations: 1.The two body short range correlations (SRC) presented in the first term of eq. ( 9) and denoted by where the two partical wave function is given by [12] ( ) ( ) where the notation The longitudinal form factor is related to the charge density distribution through the matrix elements of multipole operators where Z is the proton number in the nucleus and ) (q F cm is the centre of mass correction, which removes the spurious state arising from the motion of the center of mass when shell model wave function is used, and given by [11]: The experimental data for the second 1 2 + state is mostly determined by the core polarization form factor, where the sd-shell contribution is small.The data are slightly over estimated when core polarization effect is included.The core polarization effects play the major role for this transition.The C2 form factors of 30 Si nucleus are shown in Fig. 4   q (fm -1 ) q (fm -1 ) PDF created with pdfFactory Pro trial version www.pdffactory.com state two-body charge density distribution derived as follow; we have produced an effective two-body charge density operator by folding the two-body charge density operator with the twobody correlation functions ij f ~ as[10]:

R
are relative and center of mass coordinates and the form of ij f ~ is given by [11]: 10) where c r (in fm) is the radius of a suitable hard core and 2 25 − = fm µ [11] is a correlation parameter.2. The two-body tensor correlations (TC) presented in the second term of eq.(9) are induced by the strong tensor component in the nucleon-nucleon force and they are of longer range.Here 2 and called a transformation bracket.For the purpose of extending the calculation to open shell nuclei we replaced the factors i j ˆ and j j ˆ in eq.(14) as PDF created with pdfFactory Pro trial version www.pdffactory.comto (zero or 1) for closed shell nuclei while for open shell nuclei they are larger than zero or less than one (i.e.0<

+ 2 +
where A is the nuclear mass number and b is the harmonic oscillator size parameter.The function ) (q F fs is the finite size correction, considered as a free nucleon form factor and assumed to be the same for protons and neutrons, and it takes the form[We first discuss the effects of two-body SRC and TC on the ground state 2BCDD in the open sdshell nuclei 22 Ne,26 Mg and 30 Si.The parameters required in the calculations of 2BCDD's such as the occupation probabilities η 's of the states, the values of ) r (in fm) for 22 Ne, 26 Mg and 30 Si nuclei are displayed in Fig. 1.The dashed and solid distributions are the those of experimental data [14], denoted by dotted symbols.It is clear that the dashed distributions deviate from the experimental data especially at small r.Introducing the effects of SRC and TC tends to remove these deviations from the region of small PDF created with pdfFactory Pro trial version www.pdffactory.com in the solid duistributions.It is evdient from these figures that the calculated 2BCDD's represented by the solid curves are in excellent agreement with those of experimental data hence they coincide with each other throughout the whole range of r(fm).Considering the effect of higher state occupation probabilities and the effects of SRC and TC are generally, essential in getting good agreement between the calculated result and experimental data.Core polarization effects on the inelastic longitudinal C2 form factors for some open sd shell nuclei, are discussed.The core polarization effects on the form factors are based on the Tassie model [7] together with the calculated 2BCDD.We adopt the universial sd (USD) interaction of Wildenthal [4] to generate the sd model space matrix elements, using the shell model code OXBASH [15].The inelastic longitudinal C2 form factors for the transitions to the 1 states are displayed in Figs. 2 to 4. The dash-dotted curves represent the contribution of the model space where the configuration mixing is taken into account, the dashed curves represent the core polarization contribution where the collective modes are considered 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 are represented by solid circles.The C2 form factor of the lowest 1 2 + states (1.275 MeV and 4.457 MeV) in 22 Ne nucleus are shown in Fig.2 and compared with the experimental data of Ref.[16].The sd-shell model space calculation fails to describe the data in qdependent form factor.The corepolarization effects give a strong modification to the form factors, where the core polarization effects enhance the form factors at the first maximum and bring the calculated values state is well described by the sd-shell model, and the core polarization effect enhances the form factor.The C2 form factors of the lowest 1 2 + states (1.809 MeV and 2.938 MeV) in 26 Mg nucleus are shown in Fig. 3.The model space calculations underestimate the data at all region of q .The core-polarization effects give a strong modification to the form factors, where the core polarization effects enhance the form factors and bring the calculated values very close to the experimental data.

Figure ( 1 ) 2 Figure ( 2 ) The Coulomb 2 C form factors for the transitions to the 1 2 + ( 1
Figure (1) Dependence of the 2BCDD on r(fm) for 22 Ne, 26 Mg and 30 Si Nuclei respectively.The dotted symboles are the experimental data of Ref [14].

Table ( 1) Parameters to the ground state 2BCDD's for some open shell nuclei.
for the PDF created with pdfFactory Pro trial version www.pdffactory.com PDF created with pdfFactory Pro trial version www.pdffactory.com