Analysis of Longitudinal Electroexcitation for Positive and Negative Parity States of 36,40 Ar Nuclei Using Different Model Spaces

The nuclear structure for the positive ( 1 , 2 , 3 2 + ) States and negative ( 1 3 − ) states of 36,40 Ar nuclei have been studied via electromagnetic transitions within the framework of shell model. The shell model analysis has been performed for the electromagnetic properties, in particular, the excitation energies, occupancies numbers, the transition strengths B( CL ) and the elastic and inelastic electron scattering longitudinal form factors. Different model spaces with different appropriate interactions have been considered for all selected states. The deduced results for the ( CL ) longitudinal form factors and other properties have been discussed and compared with the available experimental data. The inclusion of the effective charges are essential for obtaining a reasonable description for the data. The results of sdpf-model space with sdpfk and sdpfmu -interactions have a good improved for the ground state form factors for 36,40 Ar and for excitation properties of 1 2 + and 1 3 − state of 36 Ar nucleus.


Introduction
High-energy electron -nucleus scattering is precise tool for probing nuclear structure.The theoretical description of the electron scattering is challenging due to the significant influence of the nuclear electromagnetic properties.With the rapid development of the experiments great efforts have been devoted to the theoretical studies of the electron scattering of nuclei in the last decade [1][2][3][4][5].
The interest for exotic nuclei has increased enormously since the recent progress in nuclear accelerators.It is important and interesting to explore the properties of exotic nuclei both theory and experiment.Most of the nuclei are found to be deformed in their ground states [6,7,8].The present study sheds some light not only on the structured positive parity states ( Ar isotopes, but also on the available negative parity states which denoted by ( Ar and 40 Ar isotopes.The 36 Ar (N = Z = 18) nucleus has several interesting features therefore, it has attracted considerable experimental and theoretical investigations.The measurements of the g-factors, Lifetimes and the transition probabilities B (E2) of the 2 L + states in 36,38 ,40 Ar are performed by Speidel et al. [9], and explained by Large -scale shell model calculations.Recently, Riczu and Cseh.[10] reported gross features of the spectrum of 36 Ar nucleus.Their calculations are included the superderfomed and the hyperdeformed States In addition to the ground state region.They applied the multiconfigurational dynamical symmetry (Musy) reproduce these gross feature of the spectrum.The Musy is denoted to the connection between the deformation, clusterization and shell structure in terms of a dynamical symmetry [11].
In the present work, three different model spaces have been used for the positive -parity 2 L + states of 36 Ar.The first is sdpf-model space with sdpfk-interaction in which 36 Ar is considered as 16 O (N = Z = 8) inert core with active sd and fp-shell's.The second one is Hasp-model space with the hasn-interaction here four active protons and neutrons are covered the active shells 2s1/2, 1d3/2, 1f7/2 and 2p3/2 .The last one is the sd-shell model space with new USD -type interaction Viz the usdi-interaction [12].This interaction improved the predictions for separation energy in the entire sd-shell.Together with these interactions, we have been also used the sdpf -model space with the sdpfk and sdpfum-interactions for positive parity Ar and four active protons and neutrons for 40 Ar, distributed among active shells 1d3/2 and 1f7/2.The excitation energies, occupancies and B(C2) transition strengths are calculated .This work is extended to study the various CL Components longitudinal electron scattering form factors.The present calculations are performed using shell model OXBASH code [15].The shell model within the restricted model space is not enough to describe the electric properties.The core polarization effects are introduced through different effective charges.Bohr -Mottelson (B-M) effective charges [16], and standard (ST) proton and neutron effective charges ( ) [17] are adopted.

Theoretical Formalism
The differential cross section in the plane -wave Born approximation ( PWBA ) for the scattering of electron with initial total energy Ei and final total energy Ef from a nucleus of mass M and charge Ze through an angle (Ө) is given by [18,19]; The Mott cross section Mott  for relativistic electron scattering from a massive point charge is given by; Where α = e 2 / ħc is the fine-structure constant.The target-recoil factor ƞ is given by; F of a given multipolarity (L), as a function of momentum transfer (q) between initial (Ji) and final (Jf ) nuclear states can be written in terms of Manynucleon matrix elements reduced in both angular momentum (J) and isospin (T) as follows [20]; ) The normalization (4π/Z 2 ) ensures that the longitudinal elastic scattering form factor equates to unity at zero momentum transfer Tz=(Z-N)/2. () is the longitudinal multipole operator, L is determined from the parity selection rules, the bracket ... ...
denotes the 3j-symbols.The center of mass correction is 22 /4 ..

q b
A cm F e = [21] is the correction for the lack of translation invariance in the shell model and the nucleon finite size (Ff.s.) form factor is [22]  L operator can be written as the sum of the product of the one-body density matrix (OBDM) times the reduced single-particle matrix elements [23]; , ( ) ( , , , , , ) ( , ) For elastic longitudinal electron scattering, the sum includes the core level, for inelastic scattering, the sum extends across all pairs of single-particle states , jj  (isospin is included).
The reduced single particle matrix elements used in the present work are those of Brown et al. [24].The (OBDM) in spin-isospin formalism are obtained in the form [25]; In the second quantization formalism, the OBDM takes the form, The form factor is related to the reduced transition probability at the photon point as [24]; where

Results and Discussion
The nuclear structure properties for positivenegative parity states of 36 Ar, and 40  , transitions (with L = 1,2,3).The single particle harmonic oscillator (HO) Potential with size parameter (b) is used to generate the single -particle wave functions.The oscillator length (b) is obtained from [25]; All calculations are carried out using the OXBASH code [15] for different model spaces with an appropriate interactions.The SD, HASP, D3F7 and SDPF model spaces are used with the usdi, hasn, Wo, sdpfk and sdpfmu interactions, respectively.The calculations with only model space wave functions are inadequate for reproducing the electron scattering data [26].Therefore, the contributions of the higher configurations outside the model space are essential to improve the predicted results.The core polarization effects are included through two different proton and neutron effective charges.Bohr -Mottelson (B-M) [16] Also, the standard (ST) proton and neutron effective charges ( ) [17].
Discussion of the present results will be divided into two parts; The first one pertains to elastic longitudinal electron scattering and the second one is related to inelastic longitudinal electron scattering.Both parts include, the longitudinal electron scattering form factors, excitation energies, transition strengths B(CL).

Elastic Longitudinal Electron Scattering Form Factors (J π =0)
The Co longitudinal form factors for 36 Ar nucleus are calculated using model spaces with three different interactions and shown in Figure (1-a), usdi-interaction (green curve), hasninteraction (purple curve) and sdpfmu-interaction (blue curve).For 40 Ar nucleus, the Co form factors are calculated using, full SDPF model space with the sdpfmu-interaction (blue curve) and sdpfk-interaction (black curve) as shown in Figure (1-b).The experimental data (red full circle) [26], are covering the range of momentum transfer (0.539 ≤ q ≤ 0.96) fm -1 .The Co form factor is clearly dominated at low q-region, with further diffraction structures at q >1.0 fm -1 .The diffraction minima are located at the same region of q for both interactions in 40 Ar, with some deviation in the second minimum of hasn-interaction for both 36 Ar and 40 Ar nuclei an overall agreement is obtained for all interactions through all the experimental q-regions.

Figure (1-a):
Elastic longitudinal form factor for 0 + state in 36 Ar compared with experimental data of Ref. [26].

Figure (1-b):
Elastic longitudinal form factor for 0 + state in 40 Ar compared with experimental data of Ref. [26].The excitation energies of these states are calculated with the usdi, hasn, sdpfk-interactions for 36 Ar nucleus, also with sdpfk and sdpfmu-interactions for 40 Ar nucleus, and displayed in Figure (3).The calculated energies are listed in Table (1) together with the measured values [26,28], and other results for comparison.For 36 Ar, the results of 1 2 + state with usdi, hasn, sdpfkinteractions (1.818MeV, 1.82MeV and 1.766MeV respectively), provide a good agreement with both measure value (1.97±0.05MeV)and that of Ref. [3] as well as for other 2 2 + and 3 2 + levels.For 40 Ar, the results with sdpfk-interaction for 1 2 2 , 2 + + , and 3 2 + levels (1.546MeV, 3.06MeV and 3.887MeV respectively), reproduced the measured values (1.46±0.05MeV,2.524±0.01MeVand 3.207 ± 0.013MeV respectively) and slightly better than that of sdpfu-interaction.The result of sdpfmu-interaction (1.028MeV) for the first level is underestimated compared with the measured value (1.46±0.05MeV)and with that of Ref. [3] (1.28MeV).The excitation energies for the negative parity state ( 13 − ) of both 36 Ar and 40 Ar nuclei are calculated with Wo, hasn and sdpfmu-interactions.The results of Wo-interaction are the best for both 36 Ar and 40 Ar (4.797MeV and 4.018MeV respectively) in comparison with the measured values (4.178±0.01MeVand 3.680±0.012MeV),but becomes significantly worse for sdpfmuinteractions for both 36 Ar and 40 Ar (8.0MeV and 7.5MeV respectively).The results of sdpfmuinteraction over-predict the data by about a factor of two.

The Inelastic C2 Form Factor for Positive Parity states (J
The inelastic longitudinal form factors are calculated using different model spaces with different interactions, as mentioned previously.The results of each interaction with B-M effective charges, are denoted by solid curve, and the dashed curve for the results with the standard effective charges ( with both sdpfk and hasn-interactions (black and purple curves respectively) explain the C2 data [26] very well over all regions of momentum transfer.The C2 form factors for the usdiinteraction with both effective charges (green curves) are over predict the q-data.The predicted transition strength with usdi-interaction (B(C2) = 323.2e 2 fm 4 ) is very close to both measured value [26] (B (C2) = 301( − + ) e 2 fm 4 ) and to that of Ref [3].The results of sdpfk and hasn interactions (259 e 2 fm 4 and 274.6 e 2 fm 4 respectively) are also in a good agreement with measured value within the experimental error as shown in Table (2).
The longitudinal C2 form factors for 2 2 + and 3 2 + states for all three interactions with both B-M and ST effective charges are shown in Figure (4-b) and Figure (4-c), respectively.There is no available experimental data for these states.The results of usdi and hasn-interactions with B-M for 2 2 + state are close to each other in the range of 0.1<q<1.8fm -1 , with two diffraction minima of hasn-interaction, while the first diffraction minima are coincidentally located at q ≈1.1 fm -1 .The same result and behavior can be noticed for 3 2 + state for sdpfk and hasninteractions with B-M effective charges.It is obvious that the result of sdpfk-interaction shows different behavior with a clear discrepancy compared from that of both usdi and hasninteractions.No diffraction minima appeared with sdpfk interaction for 2 2 + state and appeared at q ≈1.3 fm -1 for 3 standard values are compared together with that of bare nucleon charges as well as with the available experimental data [26].The predicted C2 form factor for 1 2 + state is compared with available experimental data [26] and depicted in Figure (5-a).In spection of these curves reveals that the results of sdpfk and sdpfmuinteractions with bare nucleon charges (dashed-dot curves) exhibit qualitative similarity to the shape of the experimental data and underestimate in its magnitude.The core polarization effects with standard effective charges (dashed curves) for both interactions give a good description of the experimental data and slightly overestimate the data with B-M effective charges at q ≈ 1.0 fm -1 .The predicted B(C2) for 1 2 + state with sdpfk-interaction (299.7 e 2 fm 4 ) and with sdpfmuinteraction (223.9 e 2 fm 4 ) have well improved the description of the measured B(C2) of Ref. [29] (332±17e 2 fm 4 ) within the experimental error.The sdpfmu-result is close to that of Ref [3] (245 e 2 fm 4 ) as shown in Table (2).The longitudinal C2 fare factors for 2 2 + state for both sdpfk and sdpfmu-interactions are compared with available experimental data [26] and displayed in Figure (5-b).It is clear that the results of ST effective charges (dashed curves) are close to that of bare nucleon charges (dashed-dot curves) for both interactions, and overestimate state the experimental dada at q > 0.6 fm -1 .The predicted C2 form factors with B-M effective charges for both interactions are in a reasonable agreement with the experimental data.The measured transition strength B(C2) (45+24 e 2 fm 4 ) [26], is well reproduced with both sdpfk and sdpfmuinteractions (30.2 and 31.7 e 2 fm 4 respectively).
The inelastic C2 form factors for 3 2 + state with sdpfk and sdpfmu-interaction are shown in Figure (5-c).The sdpfmu-interaction with bare nucleon charges (dashed-dot curves) gives a remarkable agreement with the available experimental data [26], and slightly overestimate the data with B-M effective charges.The C2 form factors for sdpfk-interaction with bare and effective charges are broadly consistent with the major trends of the experimental data.This behavior is clearly appeared in the predicted B(C2) strengths as shown in Table (2).   Ar and 40 Ar respectively.For 36 Ar, the predicted C3 form factors with bare and effective charges for all interactions are displayed in Figure (6-a).the light of the best agreement with the experimental data is due to the extended SDPF and HASP model spaces (blue and purple Curves).The results of sdpfmu and hasn interactions reproduce more precisely the experimental data over all q-regions.The longitudinal C3 form factor of Wo-interaction has the same behavior in the shape for that of other interactions and underestimated in magnitude.The predicted B(C3) strengths for sdpfmu and hasn-interactions (0.76E+4 and 0.614E+4 e 2 fm 6 respectively) are underestimated by about a factor of 2. While that of Wo-interaction is far away from the measured value of B(C3) [26] (1.62±0.2)E+4e 2 fm 6 ).For 40 Ar, the Longitudinal C3 form factors for 1 3 − state at 3.68MeV with Wo, hasn and sdpfmu interactions are compared  The hasn interaction yield an adequate description of the experimental data and better than that of other interactions which they a slight deviation below the experimental dada.The predicted B(C3) strength (0.3E+4 e 2 fm 6 ) is underestimated by about a factor of 3 for hasninteraction and more than this for other interaction as listed in Table (2).

Conclusions
In the present work, different model spaces with different interactions have been used to investigate the nuclear structure of positive ( 1,2,3 2 + ) states and negative ( 1 3 − ) parity states of 36,40 Ar nuclei.The contribution nucleon occupancy is dominant from (1d5/2) orbit.The SDPF model space has a remarkable improvement in both shape and magnitude for elastic and inelastic longitudinal form factors for positive ( 1, 3 2 + ) and negative ( 1 3 − ) states, while is broadly consistent with the major trends of the experimental data for positive ( 22 + ) state.The inclusion of the effective charges are adequate to obtain a reasonable agreement between the predicted results and the experimental data.The discrepancies in excitation energies (Ex) and transition strengths B(C3) with sdpfmu-interaction are clearly appeared, with good agreement for ( 1,2,3 2 + ) positive parity states.Extending the model space to include higher energy configuration explains the C2 form factor data remarkably well.
to low energy of a few Mev.The electromagnetic properties of first excited 2 L + states are primarily considered of both 36 Science, 2023, Vol.64, No. 10, pp: 5 Dakhil and Abdulhasan ,

Figure 3 :
Figure 3: Low-lying energy spectra for 36,40 Ar.Theoretical shell model results with different interactions for change parity states are compared with experimental data of Refs.[26, 28].
Factors for Negative Parity state (J π = 1 3 − ) The Longitudinal C3 form factors for 1 3 − stare at 4.178MeV of 36 Ar and at 3.680MeV of 40 Ar are calculated using D3F7, HASP and SDPF-model spaces with Wo, hasn and sdpfmuinteractions.The comparison with available experimental data are shown in Figure (6-a) and Figure (6-b) for
formulated the following expression for effective charges,

Table 1 :
[26,excitation energies (MeV) for positive & negative-parity states in36Ar and40Ar for different model space, compared with experimental data of Refs.[26, 28]; 2 + state.For 40 Ar, the inelastic longitudinal C2 form factors for 2 L + states are calculated using full sdpf model space with sdpfk and sdpfmu-interactions.The results with effective charges (