Study of positive parity states form factors for 17 O nucleus with Skyrme-Hartree-Fock method.

Elastic and inelastic electron scattering form factors for the 17 O nucleus are calculated for the positive parity low-lying states in the momentum transfer interval 0.05-3 fm − 1 in the framework of shell model and Skyrme-Hartree-Fock calculations. Also, protons, neutrons, mass and charges root mean square, r.m.s., radii and the charge spatial density distribution are calculated. The ZBME model space out of 12 C nuclei core with rewile interaction have been used. For all selected ground and excited states, Skyrme interactions with SkXcsb and SLy4 parameter, harmonic and Wood-Saxon potentials are adopted in Hartree-Fock theory to generate the mean potential and hence calculate the elements of single-particle matrix and form factors. Also, we have used Tassie model to account for core-polarization effect. Finally, to check the reality


Introduction:
Electron scattering (ES) is considered by a strong tool for investigation of the nuclei properties and structures, and it offers a major source of experimental data to check the reality of theoretical calculations and models for nuclear ground and excited states [1], [2]. Additionally, it offers the most accurate data on nuclear size, charge distribution, and electromagnetic current distributions inside the nucleus [3].
The first-born approximation in which one photon exchanges with the nucleus, is considered to be a successful approximation for tiny α (Zα ≪ 1), is the framework in which we perform our computations [4], where Z denotes the atomic number. Also we divided the 17 O nucleus shells into the 12 C core and the model space (valence) shells, the core nucleons have been put to be non-interacting and only taken the interactions of valence nucleons, we take care this problem by introducing nucleons effective charges. Applying the effective charges plays the role of polarizing the protons of core by the valence neutrons and protons and mostly called the core-polarization effect (CPE) [4]. Because the nuclear system is complex, for the simplicity of calculation we divide in to three parts, the core part which is make from some or all closed shells, the model space which consists of some states out of the closed core, and the extra space which are the remnant states beyond the model space. We put the core nucleons to be inert, that is non-interacting, while indeed they have grate contribution in interaction. So we have to correct this approximation. We do this through a model called Coulomb valance Tassie model (CVTM) or simply Tassie model (TM) through NushellX@MSU code which treats this through nucleon effective charges that enables uniform mass distribution and charge density. This process called core-polarization effect in which the core nucleons due to interaction with the incident electron jump to excited states and the core will be polarized [5].
The first experiment in electron excitation of nuclei was done by Collins and Waldman in 1940, while the first theoretical treatment was pointed out by Mamasachlisov in 1943 [6]. The nucleus 17 O have about 112 excited states under 24 MeV excitation level [7]. This isotope of oxygen thus offers a great possibility to understand the abundant level structure of the light and stable nucleus [8], [9]. Many theoretical and experimental research have been done to explore the static and dynamic properties of this fascinating system of nucleons.
In 1975, the first work was done by Kim et al. [9] to study the inelastic ES for 17 O at momentum transfers 0.6 up to 1.1 fm −1 .
The transverse form factor (FF) for the 17 O nucleus ground state has been determined through elastic ES in the effective momentum transfer interval 0.55-2.8 fm −1 by Hynes et al. [10]. Hicks [11] theoretically found the root mean square (r.m.s.) radii with M1 and M3 multipoles FFs for the 1d 5/2 17 O neutron state. He also drew the information on the distinction the M3 and M5 contributions quantitatively.
The radial algebraic wave function parameters which are correspondent to the 17 O nucleus valence neutron state of 1d 5/2 have been calculated by Coon and Jaqua [12] via fitting to the magnetic ES data at high momentum transfer. Radhi et al. [13] investigated the transverse FFs of elastic magnetic ES for the odd-A stable nuclei including 17 O at 5/2 + state in the sd configuration, taking into consideration CPEs. Elastic scattering for 17 O has been studied by Han et al. [14]. E. Strano et al. [15] observed the ground state scattering for 17 O on 58Ni nuclei at some different energies. Alzubadi et al. [7] studied the transition probabilities, energy levels and FFs for both elastic and inelastic ES for states close to ground state of positive and negative parities in the momentum transfer 0.5 upto 1.5 fm −1 . Inelastic ES for nucleus 17O has been studied by taking in to account the higher states outside 16 O core by Abbas et al. [16]. El-Hammamy et al. [17] analyzed the 17  The aim of the current work is to calculate protons, neutrons, mass and charges r.m.s. radii, the charge spatial density distribution, and ESFFs for positive low-lying parity states for the momentum transfer values from 0.05 to 3 fm −1 with 0.05 fm −1 resolution. The ZBME model space has been used for the core 12 C and the interaction type used is rewile Hamiltonian which is used for generate the positive parity states actual wave functions. As a correction for core nucleons which have been put as inert, non-interacting, we use CPE through Tassie model.

Theory and methodology:
The ESFF decomposed in to longitudinal component, F L (q), and a transverse one, F T (q). They depend only on momentum transfer q and energy loss ω, but not on scattering angle θ . They can be divided into different multipoles contributions [18]: where F C J , F E J , and F M J are longitudinal, transverse electric, and magnetic transition FFs with multipolarity J respectively. The permitted values for J in eq.(1) are limited by is the ground state total angular momentum and J f is those for excited states of 17 O nucleus. Two other electromagnetic selection rules restrict the values of J further, which are the parity conservation rules. They are (-1) J for electric (CJ,EJ) interactions and (-1) J+1 for magnetic (MJ) interactions [8]. The longitudinal and transverse FFs could be represented with respect to double bar matrix elements of the electromagnetic transition operators of ES [8].
Here η = C, E, M, β = 1 for η = C and β = (J + 1)/J for η = E, M. And J i and J f are respective initial and final nuclear states. T η J,t z (q) denotes each of Coulomb, transverse electric and magnetic multiple operators of ES respectively. F 2 cm (q) represents the translational-invariance lack, center of mass, correction in the shell model and F f s (q) is the FF for finite size of nucleons (fs). The double bar ES operator T J (⃗ r, t z ) matrix elements for model space is the total of matrix elements of the one-body density (OBDM) X J J f J i (t z ,j i ,j f ) times the elements of single particle matrix. It is determined by where j i and j f specify different single-particle states in the momentum space of shell model and t z = 1/2 and -1/2 for a proton and a neutron respectively. On the other hand we use Skyrme force as a nucleon-nucleon effective interaction. This force consists of two parts, a two-body part and a zero-range three-body part [19], [20].
where v (2) i j is the two-body part and v i jk is the three-body part. The three-body part could be reduced to a two-body density dependent part. The Skyrme potential can be decomposed into its constituent terms The term of t 3 coefficient is the three-body part and other terms are two-body part with δ 12 = δ r 1 -r 2 and The operators k and k ′ are the two nucleons relative wave vectors operates on the wave functions to the direction of right and left respectively and P σ is the spin-exchange operator which is [21].
Putting two-body Skyrme interaction into Hartree-Fock equations gives a one body mean field potential. The neutron, proton, charge and mass densities can be determined by the relation [22].
where R β r denotes radial part, k represents neutron, proton, charge and mass, w β is the probability of occupying the nucleon state β and j β stand for total angular momentum of a nucleon in state β . The density distribution is a direct explore of the nucleus size, therefore the neutron, proton, charge and mass distributions r.m.s. radii can be calculated from the relation [23].
The polarization of core (CP) charge density in CVTM model depends on the ground state charge density of the nucleus. The ground state charge density is expressed in terms of the two-body charge density for all occupied shells including the core. The transition density of the core polarization of the Tassie shape is given by the collective modes of its nuclei [5].
where C is a constant of proportionality and ρ 0 is the charge density distribution for the ground state two-body. Now the Coulomb form factor for this model becomes:

Results and discussion:
In the present study, we first found the one-body density matrices elements (OBDM) for the nucleons from Hartree-Fock equations putting Skyrme two-body interaction for SkXcsb and SLy4 parameterizations and using SM code NUSHELLX-@MSU [24]. Hence we calculated the Coulomb and transverse FFs for states of low-energy and positive parity of 17 O. Furthermore, the Tassie model has been utilized to account for CPE, which is calculating it from effective charges of nucleons [5]. The effective charges e π = 1.5e for protons and e ν = 0.5e for neutrons have been adopted, it is tested very good for the sd shell [7]. The graphs were drawn using the Grapher program. Also, the finite size of nucleons has been considered in the computing the FFs. Besides the Skyrme interaction, the calculations for harmonic (HO) and Wood-Saxon (WS) single-particle potentials have been performed. The oscillator length constant b = 1.747 fm specified for HO potential. All calculations have been performed for ZBME model space, 1p 1/2 , 1d 5/2 , 2s 1/2 , 1d 3/2 orbits, and rewile interaction which is a special interaction of NushellX@MSU program for this model space . Then we compared the results with experimental the data [8], [25]. We divided the results and discussion into the radius, the charge densities distribution and the FFs sections.

The Radius:
The results for proton, neutron, mass and charge r.m.s. radii for 17 O for four different potentials mentioned above are presented in Table 1 parallel to the experimental for charge radii [25]. As shown in the table, the calculated charge radii values for all potentials agree with the experimental, but the best is for HO potential. The values are 2.8366, 2.8967, 2.9274 and 2.8686 fm for HO, SkXcsb, SLy4 and WS interactions, respectively, while the measured value is 2.6932 fm.

The Charge Densities Distribution:
The variation of charge density with distance from the nucleus center is shown in Figure 1.

The Form Factors:
The calculations were performed only for low-energy positive parity states for the Coulomb and magnetic FFs.

Form Factors of Ground state, J
The 17 O ground state spin and parity of is 5/2 + . From the rules of the summation of total angular momentum, |J i − J f | ≤ J ≤ J i + J f , and parity conservation, (1) J for Coulomb and (1) J+1 for magnetic, the allowed multipole values of Coulomb (or longitudinal) and transverse magnetic FFs are C0, C2, C4, and M1, M3, M5 respectively. The scattering here is elastic since the nucleus remains in the ground state without excitement. This is because no energy transfers from the bombarding electron to the nucleus. In order to show the ESFFs dependence on potentials, the total ongitudinal and transverse magnetic FFs are found utilizing the complete ZBME model space for SLy4, SkXcsb parametrizations of Skyrme, HO, and WS potentials. Also, to know the quantitative contribution of the multipoles in producing total FF, we plot multipole projections parallelism to the totals. The multipoles are drown only for SkXcsb parametrization, which they are C2, C4, M1, M3, and M5 multipoles. Notice that C0 is not plotted here, this is because the measured Coulomb values exist for a total of only C2 and C4 contributions. The totals are given by eq.(1) The graphs are shown in Figure 2.
For the longitudinal FF, Figure 2(a), which is performed at ground state, 0.0 MeV, there is a good agreement qualitatively and quantitatively by the calculated results for all potentials to the experimentals along all measured momentum transfer, q, values, although a small overvalue will be shown. In addition, during all q values except around 1.9 f m −1 the C2 multipole has the major contribution to the formation of the total longitudinal FF, and around 1.9 f m −1 the contribution dominated by C4 multipole. Also this agreement coming back to applying Tassie model for core polarization which generally has effect on the longitudinal form factors. For the transverse one, Figure 2(b), a very good consistency between all the potentials used with the experiment during all momentum transfer values will be shown, but the best one is SkXcsb parameterization. At low momentum transfer region, M1 multipole controls produce the total FF. At moderate region, each of M1, M3, and M5 contribution participates in making the total, and at high momentum transfer, just M5 dominates. For the J = 1/2 state from the rules mentioned above, only the multipoles C2 for Coulomb and M3 for magnetic FFs are allowed. These are plotted in Figure 3. The scattering due to this state and others come after, is inelastic, since this leads to taking the nucleus from the ground to exited sates. Inelastic longitudinal C2 multipole FFs for the first 1/2 + state of 17 O at 0.870 MeV are demonstrated in Figure 3(a). In comparison with the measured taken data [8], one can see that the results for all potentials perfectly agree with the experimental data quantitatively and qualitatively at the low and intermediate region of q with a small underestimate. However, at high q region, they spread and were rather in agreement with the experimental one, and the HO was best of them. The inelastic magnetic M3 multipole is shown in Figure 3(b). Below 1.5 f m 1 momentum transfer, all potentials match with the experimental result except the HO one, which slightly shifts above and the best one is SkXcsb parameterization, from 1.5 up to 3 f m 1 all potentials qualitatively are in good consistency with the measured values, but quantitatively they are a little below the measured values with the HO potential being their nearest to the experimental values. This discrepancy is due to the non-counting transverse electric contribution E2, which reproduces the total transverse FF together with M3 multipole.

Excited State J π = 3/2 + :
In J = 3/2 state, we have C2 and C4 Coulomb multipoles. They belongs to 5.084 MeV state. The total FFs, CT = C2 + C4, are calculated for all potentials, but the different multipoles are only demonstrated for SkXcsb parametrization. The plots are illustrated in Figure 4 parallel to the experimental values. The predicted values rather have a different shape to the experimental values and overestimate them below 1.7 f m −1 , and underestimate above 2 f m −1 . But in the region 1.7 up to 2 f m −1 the predicted curves pass through the experiment ones. According to Alzubadi [7], this discrepancy is because the single-particle radial wave functions in the self-consistent mean field code are very good for the states of the holes but not for those of particles. Therefore excited states consists mainly by those particle states that may deliver uncertain FFs. Furthermore, in the results, there is a minimum, a bottom, near 1.9 f m −1 which is not exist in observed values. Along all q values region except around 1.9 f m −1 the C2 multipole has a dominant contribution in reproducing the total FFs, and around 1.9 f m −1 the total results made by C4 contribution.

Excited State
In the 7/2 + excited state, the longitudinal C2 FF at 7.576 MeV is depicted in Figure 5. Obviously, the theoretical FF formed by the predicted ZBME model space has the same shape of the experimental curve with a small underestimate below 1.5 f m −1 . The experimental data do not exist above 2 f m −1 to comparison. The theoretical curves coincide together below this q value, but after that, they scatter. For all states, the FF curves relatively have a bell shape that is initially increase with q and then drop along it, this is according to eq. (2).

Excited State
Finally, the Coulomb contribution C2 FFs for 9/2 + excited state at 8.470 MeV are plotted and shown in Figure 6. The calculated results quantitatively and qualitatively are a reasonable agreement to the experimental values, although there is a slightly overestimate below the momentum transfer 1.6 f m −1 .

Conclusion:
In the present work, we studied the structure of 17 O nucleus. From the results, we conclude that employing SHF calculations with SM are be able to describe the nuclear structure and properties, for instance, elastic and inelastic electron scattering form factors for low-lying positive parity states. Also Skyrme, HO, and WS potentials are very successful to represent effective nucleon-nucleon interactions, and give a reasonable form factor and charge radii to the experimental values, with the Skyrme one is the best. On the other hand, it is clear from the results that the HF method and equations can produce an appropriate mean field to the nucleons from Skyrme effective nucleon-nucleon interaction in the frame work of shell model for studying the nuclear structure. In general, The spatial behavior of the densities are be able to give by the three interactions. However, the constant characteristic near the nucleus center can be reproduced by the Skyrme