Gogny interaction and nuclear charge distribution in 48Ca Nucleus

Charge multipole Coulomb scattering form factors in 48 Ca nucleus have been reproduced utilizing the theory of nuclear shell. The efficient two-body nuclear potential fpbm is considered to construct the-spin orbit term LS vectors with Harmonic Oscillator HO potential as a wave function of single particle in Fp shell. Discarded spaces ( core + higher configuration) are taken into account through the Core polarization effect by model space with accurate two-body potential of Gogny to interact the LS operating particles with the discarded space pair ( particle-hole) with energy of excitation equal to 2ћω. Gogny interaction has been selected as it had succeeded in nuclear shell theory. The computed results were compared with the available experimental data.


Introduction
Electron scattering probabilities, elastic and inelastic have been studied in wide varietal models and approaches which are succeeded or failed to reproduce the experimental results.Relativistic mean field theory is one of the most important 1 .Diffraction probabilities of electrons have been studied for some low mass number isotopes, like .They had been performed in terms of the mixing configuration shell model.The off model space orbits were included in a so called core-polarization effects microscopically considering particle-hole excitations from the core orbits to the upper orbits via fp shell with 2ħω excitations.The core polarization matrix components were calculated using the theoretical Michigan sum of three range Yukawa (M3Y) interaction 2 .An account of significant neutron-proton cooperation was detected and single-particle momentum distributions for each isospins in 48 Ca computed from these spectral functions approve that a higher portion of high-momentum protons than neutrons are caused by neutron excess 3 .The associated neutron transfer with the role of diverse Skyrme forces effect was studied in the reactions view 40 Ca, 48 Ca+ 96 Zr.The synthesis cross-section has been computed using Wong formula and extended Wong model 4 Theoretical approaches to the quadrupole response achieve disparate findings, whereas experimental data unambiguously show the states' multipolarity and one-phonon nature.Furthermore, cross section calculations are insensitive enough to detangle between quadrupole states that are thought to be attributable to a distinct excitation mode in one of the theoretical approaches 5 .The short range of two-body potential (SRCs) was considered as most active term for the tail with great momentum of the momentum allocation of the nucleon, n(k) 6 .The two-body Skyrme type Sly5 interaction is used for the core polarization matrix elements.FPBM model space-effective interactions of Richter are adopted to generate the model space wave functions, the oscillator potential was utilized to construct the LS shell single particle basis vectors, where an analytical solution is possible.Comparisons were made with the available experimental readings for different multipolarities 7 .
An algorithm of (M3Y) written in FORTRAN 90 had been modified to include realistic Gogny potential in the original code to calculates the twobody matrix elements which are 1 st order effects of core polarization.
The goal of this work is to use a realistic (NN) Gogny potential 8 as an effective residual interaction to read the effects of cast-off orbits microscopically, with a best and careful selection of LS effective two body potential which construct the LS vectors (shell model wave functions) and highly energy wave function with the same transition operations.Oscillator single particle vectors will be selected as a single basis vectors.We shall clarify the corepolarization process on the form factors of elastic and inelastic electron scattering for the low lying states of 2p1f-LS shell in 48 Ca isotope.Gogny interactions of H. Nakada (2002) 8 with sets P2 of Ried fitting of parameters are to be applied as a discarded space effective potential for the core polarization calculations.

Electron scattering
The diffraction probabilities of electrons from a target nucleus of atomic number Z and mass number A in the first-order Born approximation (PWBA) are provided by 9,10 )  is the cross-section of Mott for point spinless nucleus high energy scattered electron from a, is given by: Where  =  2 ℏ ⁄ = (1/137) is the fine structure constant which is representative the interaction order, ө is the diffraction angle and E i is the incident electron energy 2 The nucleus recoil factor is written as 10 : The momentum transfer components   are provided by, (ћ=c=1 for calculation reasons) 2 :

Core polarization effects
Including the cast-off space microscopically as a 1 st order particle-hole state perturbation (p-h), via model space to compute these effects as a residual interaction, Gogny -type hamiltonian 10 .

Results and Discussion
Experimental data must be available to conclude that the accepted selected theoretic approach reproduces the experimental reads with an appropriate deviation or variances.The experimental readings must be sufficient modern in regard to the environments that are concerning the computation and contrast.Calculations should be sound and gently plotted to reveal the measurements perfectly.The LS effectual potential FPBM was used to generate the (1f7/2 1f5/2 2p3/2 2p1/2 )  48  shell model wave functions.
The discarded orbits were comprised to calculate the configurating contributions from model-space outside in the operation under interest.
FP LS shell is considered in this work for  48  (shell model space having  40  as an inert core) to all higher orbits with 2ℏ excitation for normal transitions.
The One Body Density Matrix elements (OBDM) for all transitions considered will be referred to subsequently as each form factors are considered in turn.
The Cp effects are calculated with the Gogny N-N potential 10 as a residual interaction.The twobody matrix elements in a specific single-particle basis were deduced directly from the scattering phase shift using this technique.The HO potential is used to determine the radial component of the single-particle wave functions, with the size parameter (b) fitted to obtain the nuclei's root mean square radius (r.m.s).

The 48 Ca Nucleus
The nucleus  48  is the dual behavior nucleus because it is an inert core and active nucleus with a neutron excess.It proved to be a good nuclear probing nucleus and thus provides an excellent examining base for nuclear models.Really, the isotope  48  is more inert than  40 ,  48  and  56  because of the filled neutron orbit 1f 7/2 so that it is a useful isotope in fp LS shell nuclei.We use the single particle basis vector of the harmonic oscillator (HO) and parameter of oscillation ( b= 1.988 fm).

The charge Form Factor for 0 + 4 State
Charge form factors (C0) is plotted compared with the experimental readings in Fig. 1, the calculated scattering probabilities were resulted by using Gogny two-body potential, readings are in an excellent agreement with the experimental data at the range of (q≤1).While for the higher (q) values the results deviate from the experimental data because of the limited actions of harmonic wave vectors.The C2 Longitudinal Form Factor For 2 + 4 state First 2 + state at Ex=3.83MeV.Inelastic longitudinal form factors (C2) have calculated using Gogny as shown in Fig. 2 the calculated form factors using Gogny have underestimated the experimental data at all regions of (q) except at q=1, 1.8 and 2.1 fm -1 which have a good agreement.The theoretical result shows three peaks while the experimental data have two peaks which shifted than it.The behaviors of results as two peaks as the experimental data but the first was underestimating the data at all (q) region which means its need more correction.

Baghdad Science Journal
Qualitatively it has a good agreement in behavior especially at q=1.4 and q=1.5 fm -1 as shown in Fig. 2 Open The quadruple C2 charge form factor for 48 Ca 1f2p shell model space is illustrated in Fig. 3 using of Gogny , as a residual interaction, there is a good agreement in results with the experimental data which shifted in all region of (q), the theoretical calculations show three peaks which overestimate the data at all (q) but gave a good agreement at q=1, 1.6 and near q=2.7 fm -1 .This type of residual interaction cannot explain the results very well (Fig. 3).The fourth 4 + state for 48 Ca at Ex=7.79MeV As shown in Fig. 7 which represented the total C4 form factors as a function of residual interactions shows disagreement with the experimental data at all (q) regions while the results underestimate the data at all (q) regions.The experimental data are taken from ref.

Conclusions:
Harmonic oscillator wave function is still valid especially for q<2.2 fm -1 as it is known and the approach depends on the first order Born approximation then there are more modifications need to beused to make the reproduced values closer to measured values.Gogny has fair action in a comparison with M3Y versions, but it might need modifications especially for fitting parameters and the use of zeroth range meson interaction is active.Density dependence realistic versions are more useful than the independent one and the pion exchange potentials gives better as a results of charge exchange process.

Figure 1 .
Figure 1.Coulomb C0 scattering probabilities for the first 0 + 4 vector in 48 Ca isotope.The experimental readings are red from ref. 2

Figure 2 . 2 Fourth 2 +
Figure 2. Coulomb C2 scattering probabilities for the first 2 + 4 vector in 48 Ca isotope.The experimental readings are red from ref. 2

Figure 7 .
Figure 7. Coulomb C4 scattering probabilities for the fourth 4 + 4 vector in 48 Ca isotope.The experimental readings are red from ref. 2

Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: November 2022 E-ISSN: 2411-7986
|  ⟩ and |  ⟩ are labeled as the many body LS vectors,    are the initial and final single particle states 2 .
are the elements of the off fp LS shell matrix.