Matter density distribution and longitudinal form factors for the ground and excited states of 17 Ne exotic nucleus

12 Matter density distribution and longitudinal form factors for the ground and excited states of Ne exotic nucleus R. A. Radhi, G. N. Flaiyh, E. M. Raheem Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq E-mail: esn_esn64@yahoo.com Abstract


Introduction
The progress of the new generation of experimental facilities on radioactive ion beams opens the opportunity to investigate unknown regions of exotic nuclei, far from the beta stability line, where these nuclei having a large asymmetry in the proton-toneutron ratio. With access to exotic nuclei at the very limits of nuclear stability, the physics of the neutron and proton driplines has become the focus of interest. The driplines are the limits of the nuclear landscape, where additional protons or neutrons can no longer be kept in the nucleus and they literally drip out [1].
The field of halo nuclei has generated much excitement and many hundreds of papers since its discovery in the mid-1980's. While early β-and γ-decay studies of many of these nuclei yielded information about their lifetimes and certain features of their structure, credit for their discovery should go mostly to Tanihata [2,3] for the work of his group at Lawrence Berkeley Laboratory's Bevalac in 1985 on the measurement of the very large interaction cross sections of certain neutron-rich isotopes of helium and lithium, along with Hansen and Jonson for their pioneering paper two years later in which the term 'halo' was first applied to these nuclei [4]. Halo nuclei are very weakly-bound exotic states of nuclear matter in which the outer one or two valence nucleons (usually neutrons) are spatially decoupled from a relatively tightly bound core such that they spend more than half their time beyond the range of the binding nuclear potential. In this sense, the halo is a threshold phenomenon in which the 'halo' nucleons quantum tunnel out to large distances, giving rise to extended wave function tails and hence large overall matter radii. The halo nucleons tend to be in low relative orbital angular momentum states so as not to be confined by the centrifugal barrier.
In the proton-rich or neutron-rich nuclei, a few exotic features have been observed. These includes a large extension of mass density distribution, referred to as halo or skin structure [2], a narrow momentum distribution [5] and a large concentration of the dipole strength distribution at low energies [6][7][8]. There are two main classes of halo state; the two-body halos with one nucleon surrounding the core, like the oneneutron halos 11 Be and 19 C and the oneproton halo 8 B and the Borromean threebody halos with two valence nucleons around the core like 6 He, 11 Li and 14 Be. The so called Borromean structure has also been discussed extensively [9,10]. The Borromean is defined as a three-body bound system in which any two-body subsystem does not bound. The pairing interaction between the valence neutrons plays an essential role in stabilizing these nuclei [9].
The extraction of the nucleon density distribution and nuclear radius from experimental total reaction cross section of nucleus-nucleus collisions has been carried out almost exclusively by using the Glauber model in the optical-limit approximation. The first series of measurements of interaction cross-sections using radioactive beams was performed by Tanihata and coworkers in 1985 [2,3]. The σ I were measured with transmission-type experiments. Their classical results for He and Li isotopes were one of the main experimental hints of the existence of halo states in nuclei. The measured interaction cross sections were used to extract root mean square radii (rms) using Glaubermodel analysis. The standard shell model fails to describe many of the essential features of halo nuclei (although it has proved to be of importance in providing spectroscopic information on a number of exotic nuclei) and many theorists acknowledge that there is a real need to go beyond the conventional shell model [12].
The two-frequency shell-model approach (TFSM) was employed successfully on halo nuclei [13,14], for both valence energy and rms radii. Within this model, one uses harmonic-oscillator (HO) wave functions with two oscillator size parameters, b core and b halo for the core and halo orbits, respectively. This technique will enable one to work freely on each part by changing b core(halo) till one can get a fit with some experimental results.
In the present work, the two proton halo structure of 17 Ne is studied with the assumption that the two valence protons forming the halo. Shell-model configuration mixing is carried out by using a model space for the 15 O core nucleus different from that of the two halo protons where the spatial space of the valence protons is much larger than the core. The elastic electron scattering form factor, matter density distribution of the ground state (

Theory
The longitudinal (Coulomb) one-body operator for a nucleus with multipolarity J and momentum transfer q is given by [15]: the spherical harmonics and ) ( r   is the density operator, which is given by: The longitudinal (Coulomb) one-body operator becomes: where e(k) is the electric charge for the k-th nucleon. Since e(k) = 0 for neutron, there should appear no direct contribution from neutrons; however, this point requires further attention: The addition of a valence neutron will induce polarization of the core into configurations outside the adopted model space. Such core polarization effect is included through perturbation theory which gives effective charges for the proton and neutron. Eq. (3) can be written as: which can be written as: are the isoscalar and isovector charges, respectively. The bare proton and neutron charges are denoted by p e and n e , respectively. The reduced matrix element in both spin-isospin spaces of the longitudinal operator  T is expressed as the sum of the product of the elements of the one-body density matrix (OBDM) ) , (      i f X times the single-particle matrix elements, and is given by [16]: where  and  label single-particle states The role of the core and the truncated space can be taken into consideration through a microscopic theory, which combines shell model wave functions and configurations with higher energy as first order perturbation to describe EJ excitations: these are called core polarization effects. The reduced matrix elements of the electron scattering operator  Ô is expressed as a sum of the model space (MS) contribution and the core polarization (CP) contribution, as follows: which can be written as: According to the first-order perturbation theory, the single particle core-polarization term is given by [17]: where the operator Q is the projection operator onto the space outside the model space. The single particle core-polarization terms given in Eq. (11) are written as [17]:  exchanged with an overall minus sign (12) where the index 1  runs over particle states and 2  over hole states and  is the singleparticle energy, and is calculated according to [17]: Higher energy configurations are taken into consideration through 1p-1h n   excitations. For the residual two-body interaction V res , the M3Y interaction of Bertsch et al. [18] is adopted. The form of the potential is defined in Eqs. (1)-(3) in Ref. [18]. The parameters of 'Elliot' are used which are given in Table 1 of the mentioned reference. A transformation between LS and jj is used to get the relation between the twobody shell model matrix elements and the relative and center of mass coordinates, using the harmonic oscillator radial wave functions with Talmi-Moshinsky transformation [19,20].
Using Wigner-Eckart theorem, the single particle matrix elements reduced in both spin and isospin, are written in terms of the single-particle matrix elements reduced in spin only: with: where t z = 1/2 for a proton and -1/2 for a neutron. The single particle matrix element of the electric transition operator reduced in spin space is: where  n is the single-particle radial wave function. The reduced single-particle matrix element of the longitudinal operator becomes: where T e is the isoscalar (T = 0) and isovector (T = 1) charges.
Electron scattering form factor involving angular momentum J and momentum transfer q, between initial and final nuclear shell model states of spin J i,f and isospin T i,f are [21]: is the center of mass correction which is given by  [22].
The reduced electromagnetic transition probability B(CJ↑) can be obtained from the longitudinal form factor evaluated at The relation between the B(CJ) values for the emission ↓ and absorption ↑ process is [17]: (20) where i and f are the initial and final states, respectively.
For electromagnetic transition, the B(EJ) value can be calculated directly in terms of the electric multipole transition operator [17]: replacing the operator T in all the above equations by the operator O, Eq.(17) for the reduced single particle matrix element becomes: The reduced electromagnetic transition probability B(EJ) is defined as [17]: which can be written as: Then the isoscalar and isovector effective charges are given by: The proton and neutron effective charges can be obtained as follows: The longitudinal form factor, ) (q F J can be written as:  (18) and (27), the nucleon transition density can be found to be [16]: the corresponding mean square radius is given in terms of the nucleon density as [24]: (29) or it is given in terms of the occupation number as : is the average occupation number in each orbit.
As halo nuclei are consist of a compact core plus a number of outer nucleons loosely bound and spatially extended far from the core, it is suitable to separate the density distribution of Eq. (28) into two parts. The first is connected to the core nucleons while the second is connected to the halo nucleons, so the matter density of the whole halo nucleus becomes:

Results and discussion
The lightest bound isotope of neon is 17 Ne and its two outer protons separation energy is S 2p = 950 keV. The Borromean character of 17 Ne combined with its low two-proton separation energy makes it an obvious candidate to be a two-proton halo nucleus [25,26] The size parameters of the core and the outer two halo protons are chosen to reproduce the rms matter radius which is consisting with the measured value. b core is fixed at 1.633 fm, this value gives the rms matter radius of 15 O core nucleus equal to 2.44 fm, which is consisting with the measured value 2.44  0.04 fm [11]. The size parameter for the outer two protons halo b halo is chosen to be 2.368 fm to reproduce the rms matter radius of 17 Ne nucleus 2.75 fm that is consisting with the measured value 2.75  0.07 fm [11]. The proton and neutron rms of the 17 Ne nucleus are calculated as rms p =2.954 fm and rms n =2.430 fm and the difference between these values is 0.524 fm means that the 17 Ne nucleus has a halo structure.
The ground state matter density distributions ) (r m  in (fm -3 ) for 17 Ne nucleus are calculated and plotted in Fig.(1) as a function of nuclear radius r in (fm). The plus symbols are the calculated matter density of 15 O core plus two protons with the assumption that the outer two protons move in the sd model space. The dash-dotted, dashed and solid curves are the calculated matter density of 15 O core plus two protons when the outer two protons move in the pure 2s 1/2 , pure 1d 3/2 and pure 1d 5/2 orbits, respectively. The filled circles are the experimental matter density deduced from the Glauber model using the fitting procedure with (HO+HO) density function [30]. The poor agreement in Fig.1(a) between all calculated ground state matter density distributions with (b core =1.633 fm and b halo =2.368 fm), and the fitted data motivate us to recalculate the size parameters and the corresponding rms values of the core and the outer two halo protons, where b core is fixed at 1.606 fm and b halo is fixed at 2.640 fm. These values gives rms matter radius of 15 O core nucleus equal to 2.40 fm and rms of 17 Ne nucleus equal to 2.82 fm which are consisting with the measured values 2.44  0.04 fm and 2.75  0.07 fm, respectively. The proton and neutron rms are calculated as rms p =3.086 fm and rms n =2.390 fm and rms p -rms n =0.696 fm. Fig.1(b) shows the calculated ground state matter density distributions for all considered configurations of 17 Ne with (b core =1.606 fm and b halo =2.640 fm), and the fitted data. It is evident that the solid curve describes the fitted data in most of the q values more than the other curves, and we can say that the dominant configuration of the 17 Ne nucleus is ( 15 O core plus two protons halo in pure 1d 5/2 orbit) with occupation numbers (1s 1/2 ) 4 , (1p 3/2 ) 7.9144 , (1p 1/2 ) 3.0856 and (1d 5/2 ) 2 . The calculations showed a long tail in the density distribution, which is interpreted as evidence for the two valence protons halo.   =1.606 fm, b halo =2.640 fm), the calculated ground state matter density distribution of 17 Ne with 15 O core-two protons in 1d 5/2 (TFSM; dashed curve, b core =1.633 fm, b halo =2.368 fm), the calculated ground state matter density distribution with only one size parameter for all orbits (one size parameter ; dotted curve, b=1.827 fm) and the fitted data (filled circles). The poor agreement between dotted curve and fitted data especially at the long tail region of the fitted data is due to carry out the calculations by taking the whole model space of 17 Ne nucleons as a one part (one harmonic oscillator size parameter) where it must dividing into two parts one for core nucleons and the other for extra two halo protons to get a coincidence with the fitted data as in the dashed and solid curves where the solid curve is markedly interpreting the long tail behavior because of using a suitable values of the size parameters. The longitudinal form factors C0 for elastic electron scattering from 17 Ne nucleus are calculated for three configurations which are 15 O core plus two protons in (sd-shell, 1d 3/2 orbit and 1d 5/2 orbit). Appropriate oscillator size parameters which used are either (b core =1.633 fm and b halo =2.368 fm) or (b core =1.606 fm and b halo =2.640 fm) to get a good agreement between the calculations (solid curves) and the experimental data of 20 Ne nucleus (open circles) as shown in Figs. (3)(4)(5).
Since 17 Ne halo nucleus is unstable (short-lived), the calculated longitudinal elastic form factors C0 are compared with the experimental one for 20 Ne (stable isotope) which is taken from Refs. [31,32]. In spite of the nucleons in 20 Ne nucleus are more than these in 17 Ne nucleus, the significant difference between the C0 form factors of the 17 Ne halo nucleus and that of stable 20 Ne nucleus is mainly attributed to the harmonic oscillator size parameters of both nuclei, which is in 17 Ne nucleus bigger than that in 20 Ne nucleus. According to this bases, we can concludes that the differences between the calculated C0 form factors of 17 Ne and experimental data of 20 Ne nucleus are attributed to the last two protons in the two nuclei. It is so apparent from the Figs. (3)(4)(5) that there is a reasonable interpretation of the experimental results by the calculations.  (dashed and solid curves, respectively). The filled circles are the experimental data extracted from Glauber model [30].

Excited States
The C2 coulomb form factors of the inelastic electron scattering from 17 Ne nucleus are calculated for three configurations: 15 O core plus two protons in (sd-shell, pure 1d 3/2 orbit and pure 1d 5/2 orbit). The calculations are executed for the first two excited states of each configuration which are: state In this transition, the nucleus is excited from the ground to state ( 2 ). The experimental reduced transition probability B(C2  ) value of this transition is equal to 18 25 66   e 2 fm 4 [33].
We first consider the two halo protons of 17 Ne nucleus are distributed over sd-shell orbits. The excitation energy of this state is of 2.104 MeV. The harmonic oscillator size parameters that used in calculations are b core =1.633 fm and b halo =2.368 fm, where these values are calculated in terms of the experimental rms when 17 Ne nucleus in its ground state. In Fig.(6), the theoretical calculations of the C2 Coulomb form factors are presented. Four theoretical curves are shown: Dotted curve displays the calculations in terms of the bare nucleon charges (e p =1.0 e , e n =0.0 e), dash-dotted and dashed curves are the calculations in terms of effective charges that chosen to account for the core polarization effects (e p =1.033 e, e n =0.033 e) and (e p =1.3 e, e n =0.3 e), respectively; while the solid curve shows the result with standard nucleon charges (e p =1.3 e, e n =0.5 e). It is apparent from Fig. 6, that all calculated C2 form factors are approximately coinciding with each other within the momentum transfer range q =0 -2.3 fm -1 . The calculated C2 with bare charges (dotted curve) have only one diffraction minimum located at q =1.36 fm -1 , whereas those calculated with standard and effective charges have two diffraction minima. The theoretical and experimental results of the reduced transition probability B(C2  ) are given in Table 1. It is so clear from the table that the calculations with all used nucleon charges overestimates the experimental value (excessively large).
The second configuration of the two halo protons suppose that these protons are in a pure 1d 3/2 . The excitation energy of these protons is 2.260 MeV. The harmonic oscillator size parameters that used in the calculations are b core =1.606 fm and b halo =2.640 fm, where these values lead to a significant improvement in the calculated ground state nucleon density distributions.
The longitudinal C2 electron scattering form factors are shown in Fig. 7, where three values of the nucleon charges are used and each value gives a specific curve. Dotted curve displays the calculations in terms of the bare nucleon charges (e p =1.0 e, e n =0.0e), dashed curve reflect the calculations by using the core polarization effective charges (e p =1.111 e, e n =0.265 e) and solid curve refers to using standard nucleon charges (e p =1.3 e , e n =0.5 e). All curves are deviate from each other beyond q  1.25 fm -1 , and there is one diffraction minimum associated with dotted and solid curves while dashed curve gives two diffraction minima. The agreement between the calculated and the experimental transition strengths B(C2  ) is remarkably good especially with using bare and effective nucleon charges, as that shown in Table 1. The third configuration assumes that the two halo protons are in a pure 1d 5/2 orbit. The size parameters are taken to be b core =1.633 fm and b halo =2.368 fm. The calculations for the C2 transition from the ground state (  2  3  2 ) state at excitation energy of 0.955 MeV are shown in Fig. 8. Three theoretical curves are shown: Dotted, dashed and solid curves correspond to the calculations with using bare, effective and standard nucleon charges respectively. Two diffraction minima are exhibited by dashed curve, besides there is one diffraction minimum given by dotted and solid curves. As shown from Table 1, there is a remarkable agreement between experimental value of the transition strength B(C2  ) and those of calculated values especially at using bare and effective nucleon charges.   [33].
The first configuration of the two halo protons of 17 Ne nucleus suppose that these protons are distributed over sd-shell orbits and excited with excitation energy of 2.133 MeV. The size parameters of the harmonic oscillator that used in the calculations are b core =1.633 fm and b halo =2.368 fm. Four values of the nucleon charges are employed and the C2 Coulomb form factors that results from these values are plotted in Fig. 9 and denoted by dotted curve which is calculated by means of the bare nucleon charges (e p =1.0 e, e n =0.0 e), dash-dotted and dashed curves that are calculated in terms of the core polarization effective charges (e p =1.033 e , e n =0.033 e) and (e p =1.3 e , e n =0.3 e), respectively, and solid curve which reflects the use of the standard nucleon charges (e p =1.3 e , e n =0.5 e). The dotted curve gives only one diffraction minimum located at (q =1.34 fm -1 ), while the others gives two diffraction minima. There is a good conformation between all curves up to q  2.4 fm -1 , and the deviation between them is appear beyond this value. The calculated transition strength B(C2  ) values are incompatible with the experimental value for any used nucleon charges as that shown in Table 2.
In the case of the two protons are in a pure 1d 3/2 orbit (second configuration), the calculated C2 coulomb form factors for the transition from the ground state ) with excitation energy of 0.487 MeV are presented in Fig. 10. The size parameters are taken to be b core =1.606 fm and b halo =2.640 fm. Dotted curve, dashed curve and solid curves are deduced in terms of the nucleon bare charges (e p =1.0 e , e n =0.0 e), core polarization effective charges (e p =1.102 e , e n =0.255 e) and standard nucleon charges (e p =1.3 e, e n =0.5 e), respectively. Dashed and solid curves gives two noticeable diffraction minima and dotted curve gives only one diffraction minimum. The experimental value of the reduced transition probability B(C2  ) is well described by the calculated values with bare and effective nucleon charges, as that shown in Table 2.  According to the third configuration, the two halo protons of 17 Ne nucleus are assumed to be in a pure 1d 5/2 orbit. The calculated C2 coulomb form factors are presented in Fig. 11 with excitation energy 0.789 MeV and size parameters b core =1.633 fm and b halo =2.368 fm. The Dotted, dashed and solid curves represents the calculated C2 form factors with bare nucleon charges, core polarization effective nucleon charges (e p =1.102 e , e n =0.255 e) and standard nucleon charges, respectively. There is one diffraction minimum given by dotted curve and two diffraction minima given by dashed and solid curves. The theoretical and experimental values of the reduced transition probability B(C2  ) are given in Table 2, where the inclusion of the core polarization effective nucleon charges enhances the calculated transition strengths to get an excellent agreement with the experimental value.   (  and with the assistance of Tables 1 and 2, we can say that the sd-shell configuration of the two halo protons of 17 Ne nucleus fails to describe the data. In addition to the good agreement between the calculated and experimental transition strength values B(C2  ) when the two halo protons considered to be in a pure 1d 3/2 and a pure 1d 5/2 , we can also say that the dominant configuration of the 17 Ne halo nucleus is 15 O core plus two protons in 1d orbit.

Conclusions
The inclusion of the two frequency shell model approach and the effective nucleon charges in the calculations that related to matter density distribution, longitudenal form factors C0 and C2, and the electric transition strengths B(C2) for 17 Ne exotic nucleus lead to a markedly interpretation of the experimental results. Two different size parameters of the single particle wave functions of harmonic oscillator potential introduced in calculations and the result showed a longdensity tail of the matter density distribution which is coincident with the fitted data and interpreted as evidence for the two valence protons halo in addition to the noticeable difference that is found between the calculated proton and neutron rms matter radii which also indicates that the two valence protons forming the halo. It is found that the dominant configuration of the 17 Ne halo nucleus is 15 O core plus two protons in 1d orbit.