Investigation of the ground state features for some Sn-isotopes in the framework of Skyrme-Hartree-Fock method

The nuclear ground state properties of some Sn-isotopes have been investigated by the Skyrme-Hartree-Fock (SHF) method with the Skyrme parameters; SII, SIII, SV, SKXce, SLy4 and SKT. These nuclear properties include the charge, proton and matter densities and their root mean square (rms) radii, neutron skin thickness, nuclear binding energy per particle and charge form factor. The obtained results are compared with the available measured data and the relativistic mean field theory (RMFT) results.


Introduction
The Skyrme-Hartree-Fock (SHF) method is widespread used to study the features of nuclei.This method is successfully used for a wide range of nuclear properties such as neutron and proton density, root mean square (rms) charge radii, binding energy, etc.The Hartree-Fock description of nuclear characteristics gives good results not only for stable spherical and deformed nuclei, but also for neutron-deficient and neutron-rich nuclei [1][2][3].One of the main quantities used to describe the nuclear properties is the nuclear charge density distribution.
The charge densities directly related to the wave functions of protons, therefore they are can be given us much detailed information on the internal structure of nuclei.Charge density distributions for stable nuclei have been good studied with electron-nucleus scattering method which is known to be one of the powerful tools for investigating nuclear charge density distributions [4,5].Tel et al. [3] have been used the Hartree-Fock method to calculate the ground-state properties for the neutron-rich isotopes of Boron and compared the calculated results with the experimental data.Huseyin et al. [6] studied the neutron, proton and charge densities, neutron, proton, mass and charge rms radii and neutron skin thickness for the some neutron-rich isotopes by the Hartree-Fock method.Shen and Ren [7] have been study the ground-state properties of He, Li, and Be nuclei using Skyrme-Hartree-Fock model with force parameters MSKA.They have been successfully reproduced neutron halo in nuclei 6 He, 8 He, 11 Li, and 14 Be.In this research, we investigate the applicability of the SHF method with the Skyrme parameters; SII [8], SIII [2], SV [9], SKXce [10], SLy4 [11] and SKT [12] to study the static nuclear properties for 116,118,119,120,122,124 Sn stable nuclei and compared the obtained results with the results of relativistic mean field theory (RMFT) and the available experimental data.

Theory
The Skyrme force is an effective interaction with a two-body and three body parts [2]: The Skyrme forces with the three-body term replaced by a density-dependent two-body term are unified in a single form [12] and used for the central potential as an extended Skyrme force [13]: where Where  ̂ the tensor force.It is interesting to note that the Skyrme interaction is particularly restrictive with respect to the spin-orbit terms  ̂. ,  ̂ is the spin exchange operator and  ̂ is the spin operator and  0 ,  1 ,  2 ,  3 ,  4 ,  0 ,  1 ,  2 ,  3 , and  are Skyrme force parameters, 12 = ( 1 −  2 ) and the  ̂ and  ̂′ operators are the relative wave vectors of two nucleons acts to the right and to the left (i.e. the complex conjugate wave functions, with coordinate r' ), respectively.They have the form: the neutron and proton densities are given by the following relation [14]: where k denotes the proton or neutron,   is the single-particle wave function of the state  and   represents the occupation probability of the state .
The charge form factor,   (), where q is the momentum transfer, is obtained from the nuclear charge density by the Fourier-Bessel transform [15]: The root mean square (rms) radii of the neutron, proton and charge distributions can be obtained from these densities as follows [2] The neutron skin thickness t, is defined as:

Results and Discussions
The ground state properties of Sn isotopes, such as charge, proton and mass densities and corresponding root mean square radius have been calculated using Skyrme-Hartree-Fock (SHF) method with certain Skyrme force parameters namely; SII, SIII, SV, SKXce, SLy4 and SKT are listed in table-1.For completeness, elastic charge form factors and the binding energies per nucleon are evaluated within the same framework.The calculated charge rms radii for considered nuclei along with the experimental data [16] and those of relativistic mean field theory (RMFT) [17] results are tabulated in table- One can notice from this comparison that the calculated proton rms radii with SKT parameter are more close to the RMFT results than other parameters, while the evaluated results of neutron rms radii with the all parameters are smaller than the results of RMFT.Furthermore, it can be seen from figs.Investigation of the ground state features for some Sn-isotopes in the framework of Skyrme-Hartree-Fock method SIII, SV, SKXce, SLy4, SKT ‫تشمل‬ ‫الخصائص‬ ‫هذه‬ ‫الشحنة‬ ‫من‬ ‫كل‬ ‫كثافة‬ ‫البروتون‬ ، ‫التش‬ ‫عوامل‬ ‫و‬ ‫النووية‬ ‫الربط‬ ‫طاقة‬ ‫معدل‬ ‫النيوتروني،‬ ‫السمك‬ ‫لها،‬ ‫المرافقة‬ ‫االقطار‬ ‫انصاف‬ ‫مع‬ ‫والكتلة‬ ‫كل‬ .‫مقارنة‬ ‫تمت‬ ‫لقد‬ ‫مع‬ ‫النظرية‬ ‫النتائج‬ ‫من‬ ‫نظيراتها‬ ‫المتاحة‬ ‫العملية‬ ‫القيم‬ ‫النسبي.‬‫المجال‬ ‫معدل‬ ‫نظرية‬ ‫باستخدام‬ ‫المحسوبة‬ ‫النظرية‬ ‫والنتائج‬ of the ground state features for some Sn-isotopes in the framework of Skyrme-Hartree-Fock method Vol: 13 No:3 , July 2017 DOI : P-ISSN: 2222-8373 E-ISSN: 2518-9255   () = 4 ∫  2  0 () 2 and shown in fig.1(a).It can be shown that the calculations by the SHF method reveal a good Investigation of the ground state features for some Sn-isotopes in the framework of experimental data especially, theoretical calculations with SKXce parameter.The calculated results based on the different parameters shown in this table, decrease with increasing the neutron number from 4.603-4.678fm for ( 116 Sn) to 4.648-4.735fm for ( 124 Sn).In addition, the obtained values of charge rms radii calculated with SKT parameter are in good accordance with the RMFT results.The proton, neutron and matter rms radii for Sn-isotopes obtained by different Skyrme parameters are given in tables 3-5 and demonstrated in figures 1(b) -1(d), respectively.The calculated results of the proton and neutron rms radii are compared with the relativistic mean field theory (RMFT) results [17].

Figs.1a,
Figs.1a,b,c and d: The charge, proton, neutron and matter rms radii for Sn isotopes calculated with different Skyrme parameters.

Figs.2a,
Figs.2a,b,c,d,e and f: The charge density distributions for Sn isotopes calculated by different Skyrme parameters along with measured data [18,19].The charge and proton density distributions of the consider isotopes have been calculated using SHF with SKT parameterization and presented in figs.3(a) and 3(b) as a function of r (fm).From these figures one can deduce that, at the center (r=0) the charge density is decreased approximately from (0.069 fm -3 ) for 116 Sn to (0.062 fm -3 ) for 124 Sn, while the

Figs. 3 aFig. 5 :
Figs.3 a and b: The density distributions for the Sn-isotopes calculated with SkT parameter; (a) charge and (b) proton.The matter density distributions for Sn-isotopes have been also investigated and depicted in figs.4(a) and 4(b).These profiles are calculated using SKT Skyrme parameter.The experimental matter densities of Sn-isotopes [18,19] are also shown in fig.4(a) for comparison.As seen from this figure, the calculations by the SHF method reveal a good accordance with the measured data.Besides, the matter densities shown in fig.4(b) are constant in the range of radii from 2 fm to 4 fm, while they decrease drastically to zero beyond (4 fm).Values close to zero are about in the vicinity of 9 fm to 10 fm.

Table - 2: Comparison of the calculated charge rms radii (in fm) Sn isotopes with the experimental data
1(b) -1(d) that these radii increase with increasing the nucleon number.In addition, the obtained neutron skin thickness t values with SLy4 parameter are given in table 4. Note that the neutron skin thickness t values have been increase from 0.098 fm for ( 116 Sn) to 0.175 fm for ( 124 Sn) by increasing the neutron number.