Inelastic longitudinal electron scattering C42 form factors inTi nucleus

Inelastic longitudinal electron scattering form factors for secondexcited state C42 in 42Ti nucleus have been calculated using shellmodel theory. Fp shell model space with configuration (1f7/2 2p3/21f5/2 2p1/2) has been adopted in order to distribute the valenceparticles (protons and neutrons) outside an inert core 40Ca. Modernmodel space effective interactions like FPD6 and GXPF1 have beenused to generate model space vectors and harmonic oscillator wavefunction as a single particle wave function. Discarder space (coreorbits + higher orbits) has been included in (core polarization effect)as a first order correction in microscopic theory to measure theinterested multipole form factors via the model space.Gogny and Michigan sum of three-range Yukawapotential (M3Y-p2) have been utilized as a residual interaction tocouple the (particle-hole) pair across the model space active particlesand the excitation energy of the pair is (2ħω) and four options for theused effective and residual interactions were determined for thetransitions from (+0) to (+01,2,3), (+21,2,3) and (+41,2,3).


Introduction
The obtained information from the high energy of electron scattered by the nuclei depends on the magnitude of the de-Broglie wave length that is associated with the electron compared with the range of the nuclear forces. When the incident electron has energy 100 MeV and higher, 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 [1,2].
The scattering cross-section was derived for relativistic electron from spinless nucleus of charge Ze. The nuclear size can be taken into account by multiplying the Mott's crosssection by a factor called "nuclear form factor" which depends on the charge, current and magnetization in the target nucleus [3].
The change in nuclear root mean square (rms) radii for 46 Ti -50 Ti isotopes by electron scattering at 29 and 58 MeV had measured by Theissen et al. 1966 [4]. The differential cross-sections for inelastic electron scattering from 40,42,44 Ca, 46,48,50 Ti and 56 Fe isotopes have been studied by [5].
Inelastic electron scattering form factors for the 0 + →2 + transitions in some even Ti isotopes were studied in terms of the projected Hartree-Fock Bogoliubov wave function in the 2p-1f shell resulting from a slightly modified version of the Kuo-Brown effective interaction [6].
Han, in 2000 [7] explained that the full paired configuration mixing calculations in the even-even deformed nucleus 46 Ti shows that only small parts of the configuration components were important for the case of either ground states or excited states. Brandolini and Oliveira, in 2004 [8] discussed the experimental positive parity levels were discussed and compared with the theoretical and finding that the lowest 0 + , 2 + , and 4 + levels in 46 Ti are considered in addition to the ground state band. Dinca et al. 2005 [9] showed the even 52,54,56 Ti isotopes have been studied with intermediate-energy Coulomb excitation and absolute B(E; 0 + →2 + 1 ) transition rates had been obtained. These data confirm at neutron number N=32 in neutron-rich nuclei above the doubly magic nucleus 48 Ca. Large scale shell model calculations with the most recent effective interactions are unable to reproduce the magnitude of the measured strengths in the semi-magic Ti nuclei.
The level schemes and transition rates B(E2) of even-even 48,50,52,54,56 Ti isotopes have been studied by performing large scale shell model calculations with FPD6 and GXPF1 effective interactions [10]. Orbit 1f 7/2 had been adopted as a model space. The use of modern realistic M3Y effective nucleonnucleon interaction with two sets of fitting parameters (Ried fitting (M3Y-P1), and Paris fitting (M3Y-P0)) beside the use of MSDI had been done as a residual interactions within the calculation of core polarization effects in inelastic longitudinal electron scattering C6 form factor in 50 Ti within the framework of first order perturbation theory (microscopic theory) with 2ħω excitation energy [11].
Salman and Adeeb in 2013 [12] studied the form factors for the inelastic electron scattering to 2 + , 4 + states in 46,48,50 Ti in the framework of shell model. They used MSDI as effective interaction and M3Y as residual interaction. The calculation was performed in (0f 7/2 , 1p 3/2 , 0f 5/2 , 1p 1/2 ) model space as well as extended 6ħω model space. They were noticed that the core polarization effects were essential in obtaining a remarkable agreement between the calculated inelastic longitudinal F(q) , s and experimental data.

Theory
The single particle transition operator depends on the single nucleon, which is a proton or a neutron and it can be written as [13]: where:  n p T , is the single particle operator for proton/neutron. By rearranging equation, the transition operator can be written as: is the isoscalar part of the operator and  1  T T is the isovector part of the operator. The reduced singleparticle matrix of the isoscalar and isovector parts between two singleparticle states (α) and (β) are: Many particle matrix elements of the electron scattering operator T Λ are expressed as fallows [14]: The first and second terms in Eq.(4) are the model space and core polarization (CP) contributions respectively.
For a selected operator  JT T , the reduced matrix elements are written as the sum of the product of the one-body transition density matrix elements (OBDM) times the single-particle transition matrix elements [14]: is the multipolarity and the states are initial and final states of the nucleus. While α and β denote to the final and initial single-particle states, respectively (isospin is included).
The OBDM used in the present work are calculated by generating the wave functions of a given transition in the known nucleus using the shell model code OXBASH v. 2005 [15] which contains a complete library of shell model effective interactions. All the information about transitions of given multipolarities are contained in the OBDM which represents the combination of the model space wave functions.
The realistic M3Y effective nn interaction, which is used in electron scattering (V res =v 12 ) is expressed as a sum of the central potential part The three potentials are expressed as: [17]       , s  1 and s  2 are the nucleon spin operators, and S 12 is the tensor operator which is defined as [16,17]. The values of the best fit to the potential parameters ( shown in Table 1 [17].

Results and discussion
In the present work, 40 Ca is used as inert core and the model space is the fp shell. The Titanium nucleus 42 Ti is considered in the fp-shell region. The OBDM for all transitions considered are calculated using the effective interactions FPD6 and GXPF1. The core-polarization effects has been included in order to account for the contribution of configurations from outside of the model space in the transition as shown in Table 2. Using a realistic effective nucleon-nucleon (nn) interaction as a residual interaction to calculate the CP effects through a (perturbation) microscopic theory, with a selection of model space effective interaction which generates the shell model space wave functions and highly excited states. Harmonic oscillator wave function was adopted as a single particle wave function. We shall discuss the cp effects on the inelastic electron scattering form factors for the low lying states of 2p1f-shell nuclei. M3Y and Gogny interactions of Nakada [17] is to be adopted as a residual interaction for the core polarization matrix elements. In the present work we will concentrate attention for the transition 4 + 2 1 in 42 Ti isotope, where 40 Ca was used as a closed core and two protons freely distributed in fp shell model space. We have chooses (GXPF1 and FPD6) as a model space effective interaction to generate the model space wave functions with (Gogny and M3Y) as a residual interaction. The total form factors (MS+CP) for the transition C4 2 is calculated from the core polarization (CP) contribution and model space (MS), where the protons contribute to the charge form factors.
For Fig. 1, when using GXPF1 as an effective interaction at Ex=5.408MeV with Gogny as a residual interaction will see that the total form factors is larger than the two contributions in the first and second lobe respectively and they were in the same phase approximately also the diffraction minima (the point of intersection between the first and Second lobe) with respect to q (momentum transfer) values are CP (at 1.85fm -1 ) > total (at 1.75fm -1 ) > MS (at 1.7fm -1 ) and the values for the same contributions with respect to ‫|‬F(q) ‫|‬ 2 are total (at 1.5*10 -7 ) > CP (at 2.5*10 -8 ) > MS (at 2.2*10 -8 ). Here the core part is in positive with respect to model space.
For Fig. 2, when using FPD6 as an effective interaction at Ex=5.961MeV with M3Y as a residual interaction will see that the total form factors lies between the two contributions in the first and second lobe respectively and they were in the same phase for (total and MS) except (CP) also the diffraction minima with respect to q (momentum transfer) the values are CP (at 2.2fm -1 ) > total (at 1.7fm -1 ) > MS (at 1.65 fm -1 ) and for the same with respect to ‫|‬F(q)‫|‬ 2 values are CP(at 1*10 -7 ) > total (at 4*10 -8 ) > MS(at 3.9*10 -8 ). Here the core part is in negative with respect to model space.

Fig. 2: Inelastic longitudinal form factors for the transition to the 4 + 2 state in 42 Ti with and without core-polarization effects by using M3Y as a residual interaction with FPD6 as an effective interaction at Ex=5.961 MeV.
For Fig. 3, when using GXPF1 as an effective interaction at Ex=5.408 MeV with M3Y as a residual interaction will see that the total form factors is in negative (between CP and MS) in the two contributions at the first and second lobe respectively and they were in the same phase for the (total and MS) except CP also the diffraction minima with respect to q values are CP (at 2.3 fm -1 ) > total (at 1.75fm -1 ) > MS (at 1.7 fm -1 ) and for the same with respect to‫|‬F(q) ‫|‬ 2 values are total (at 3.5*10 -8 ) > MS (at 2.2*10 -8 ) > CP (at 1*10 -8 ). Here the core part is in negative with respect to model space.

Fig. 3: Inelastic longitudinal form factors for the transition to the 4 + 2 state in 42 Ti with and without core-polarization effects by using M3Y as a residual interaction with GXPF1 as an effective interaction at Ex=5.408 MeV.
For Fig. 4, when using FPD6 as an effective interaction at Ex=5.961 MeV with Gogny as a residual interaction will see that the total form factors is larger than the two contributions in the first and second lobe respectively and they are in phase. The diffraction minima with respect to q values are CP (at 1.8 fm -1 ) > total (at 1.7 fm -1 ) > MS (at 1.65 fm -1 ) and for the same with respect to‫|‬F(q) ‫|‬ 2 values are total(at 1.5*10 -7 ) > MS (at 4*10 -8 ) > CP (at 1.9*10 -8 ). Here the core part is in positive with respect to model space.

Conclusions
The diffraction minima is near with each of them for two transitions when we used the residual interaction Gogny with effective interaction FPD6 or GXPF1 but the deviation in diffraction minima in the other transitions appear when we used the residual interaction M3Y with the two effective interaction.
M3Y as a residual interaction makes the C.P part in negative contribution with M.S for two cases when using it with effective interactions FPD6 or GXPF1. Gogny as a residual interaction makes the C.P part in positive contribution with respect to M.S for another two cases when using it with effective interactions. FPD6 or GXPF1.