Effect of Incompressibility and Symmetry Energy Density on Charge Distribution and Radii of Closed-Shell Nuclei

1. Introduction:

The main problem in physics is many-body problem [1], In nuclear physics, the many-body systems and the nature of the nuclear force [2] are the most important challenges facing physics the theoretical nuclear physics, therefore, theoretical researches in nuclear physics aim to describe the structure of the nuclei with a universal nuclear model.

The properties of ground-state of atomic nuclei can be studied by Hartree-Fock (HF) [3, 3] or Hartree-Fock-Bogoliubov [5, 6]. There are many models used to study the excited levels, and the Random Phase Approximation (RPA) is considered successful in studying the closed-shell nuclei [7].

Charge density distribution and radii are fundamental nuclear properties applied to probe the nucleus structure and describes the effect of effective interactions on nuclear structure [8, 9] which were measured by electron elastic scattering and muonic X-ray data [10] by methods based on the electromagnetic interaction between the nucleus and electrons or muons.

It has also been indicated by many research groups that the nuclear properties are closely related to the symmetry term of the equation of state (EOS) [11, 12]. Charge radii is a good indicator of nuclear properties and the nuclear symmetry energy that characterizes the equation of state of matter [13], also, it found correlates strongly with other observables in nuclei, such as nuclear charge and radii [14-16], or neutron skin thickness [17-19], and their relationship to the nuclear matter properties [20, 21].

The present work aims to study the effect of incompressibility modulus K_NM and symmetry energy density J on charge distribution and root-mean-square radii of neutron (R_n)and proton (R_p) for light, medium and heavy closed-shell nuclei ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb using the self-consistent Hartree-Fock (HF) with 20 types of Skyrme interactions. Having a large number of Skyrme-force parameterizations requires continuous search for the best to describe the experimental data, therefore, the nuclear charge densities to be obtained and compared with the experiment data to give us an imagination about the internal structure of the investigated nuclei, as well as, the dependence of proton and neutron root-mean-square radii to nuclear matter properties such as incompressibility modulus K_NM and symmetry energy density J to be examined by determining the statistical Pearson linear correlation coefficient.

2. Theoretical Framework:

The second quantization Hamiltonian of a many-body fermion system is a summation of the kinetic energy t is and the anti-symmetrized two body matrix-elements  as given below [22],

Where  are annihilation and creation operators, respectively with  The indices:  covers all degrees of freedom of all available single particle states.

The Hartree-Fock energy can be expressed as,

where single-particle density  is diagonal which has the eigenvalues 1 or 0 for occupied and non-occupied states, and the energy should be minimized to determine the HF-basis [22],

with the self-consistent field. 

The Skyrme effective interaction [23, 24] is one of the most convenient forces used in HF calculations to describe the nuclear probertite in ground-state, commonly contains central, non-local and density-dependent terms [25, 26]:

where   , is the spin-exchange operator and  is the Pauli spin operator.

The Skyrme energy-density functional  contains many terms and given by [27],

Where  is a kinetic-energyterm, a zero-range term,  is a density-dependent term,  is an effective-mass term,  is a finite-range  term, is a spin-orbit term,  is a tensor coupling with spin and gradient  term and  is a Coulomb  interaction  term.

The charge density distribution in ground-state (in terms of Skyrme HF radial wave function  of the state  and occupation probability ) can be obtained by [28, 29],

The root-mean-square (rms) radii is defined as, 

3- Result and Discussion:

In this work, The HF calculations have been investigated for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei by solving the equations numerically with 20 Skyrme-type interaction: SkM^* [30], SQMC650 [31], SV-K218 [32], SQMC700 [31], SkO [33], NRAPR [34], KDE0 [35], T31 [36], Skχm^∗ [37], v075 [38], SkSC10 [39], RATP [40], SAMi [41], SK255 [42], SkD [43], GS2 [44], SIV [45], QMC2 [46], SIII [44], SVII [47]) the meaning of the abbreviation for each type of interaction and its parameters can be found in the cited references). Nuclear matter properties of the used Skyrme interaction such as the incompressibility modulus K_NM (MeV) and the symmetry energy density J (MeV), are presented in Table 1. Our results are consistent with those of HFB calculations [5. 6].

The nuclear charge densities have been obtained and compared with the experiment data to give us a picture about the internal structure of the investigated. Figs 1-4 show the calculated HF charge distribution for ⁴⁰Ca and ⁴⁸Ca. Good agreements were obtained between our HF results for all of the used types of Skyrme interaction with the experimental data [8, 49] at the surface and interior regions, while our results underestimated the experimental data line inside the investigated nuclei SVII, SIII, QMC2 and GS2 sets.

For ⁹⁰Zr, our results of charge distribution for all interactions were in a good agreement with the experimental data [8, 50] in the center and at the surface of the nucleus but not so good inside the nucleus as GS2, QMC2 and SVII interactions underestimated the experimental data line inside the nucleus and SQMC650, SQMC700, NRAPR, Skχm^∗, SK255, T31, SAMi, RATP parameters in less overestimated the experimental data line inside the nucleus. There are many factors that control the success of the theoretical model, in Table 1, we notice that the value of K_MN for the first group is higher than that of the second group.

 For ¹¹⁶Sn, our results of charge distribution for all interactions were in a good agreement with the experimental data [8] in the center and at the surface of the nucleus but not so good inside the nucleus as in SQMC650, SKO, Skχm^∗ parameters in less overestimated the experimental data line inside the nucleus and GS2, QMC2, SIII and SVII interactions underestimated the experimental data line inside the nucleus. The differences in describing data may due to differences in nuclear properties such as K_NM, where its value the first group is lower than of the second group.

For ¹⁴⁴Sm our results of charge distribution for all interactions were in a good agreement with the experimental data [8] at the surface of the nucleus but not so good inside the nucleus.       

For ²⁰⁸Pb our results of charge distribution for all interactions were in a good agreement with the experimental data [8] in the center and at the surface of the nucleus but not so good inside the nucleus as GS2, QMC2 and SVII and less for SIV and SIII interactions underestimated the experimental data line inside the nucleus and SQMC650, KDE0 parameters in less overestimated the experimental data line inside the nucleus. This may be due to differences in the nuclear matter properties of the investigated types of Skyrme interaction.

Table 1: Nuclear matter properties of the used Skyrme interactions [48].

Fig.1 Our obtained charge density distribution (fm^-3) of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei with Skyrme interactions Skm^*, SQMC650, SV-K218, SQMC700, SKO. Charge densities are plotted by black solid line. The red dotted line shows experimental data taken from Ref. [8]. 

Fig. 2 Our obtained charge density distribution (fm^-3) of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei with Skyrme interactions NRAPER, KDE0, T31 Skχm^∗ and v075. Charge densities are plotted by black solid line. The red dotted line shows experimental data taken from Ref. [8].

Fig. 3 Our obtained charge density distribution (fm^-3) of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei with Skyrme interactions SKSC10, RATP, SAMi, SK255 and SKD. Charge densities are plotted by black solid line. The red dotted line shows experimental data taken from Ref. [8].

Fig. 4 Our obtained charge density distribution (fm^-3) of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei with Skyrme interactions GS2, SIV, QMC2, SIII and SVII. Charge densities are plotted by black solid line. The red dotted line shows experimental data taken from Ref. [8].

By considering a light, medium and heavy nuclei, we can illustrate the impact of the effect of incompressibility modulus K_NM and symmetry energy density J on root-mean-square radii of neutron R_n and proton R_p for light, medium and heavy closed-shell nuclei ^{40, 48}Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb.

In Fig. 5. the incompressibility K_NM are shown as function of R_n and R_p for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei respectively. the linear correlation coefficients of K_NM versus R_n, within the Skyrme forces were 0.395, 0.369, 0.512^*, 0.515^*, 0.605^** and 0.544^* for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei, respectively, where it is small (weak) in ⁴⁰Ca and ⁴⁸Ca. The linear correlation coefficients of K_NM versus R_p within the Skyrme forces reaches 0.378, 0.489^*, 0.552^*, 0.562^*, 0.591^* and 0.606^** for   ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei, respectively, also it is small (weak) in ⁴⁰Ca and ⁴⁸Ca. From Fig. 5, we conclude there is an impact of the incompressibility K_NM on R_n and R_p.

In Fig. 6, the symmetry energy density J are shown as function of R_n and R_p for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei respectively within the Skyrme forces, the linear correlation coefficients of J versus R_n were -0.433-, -0.179-, -0.308-, -0.254-, -0.212- and -0.032- for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei, respectively and it is very small correlated except for ⁴⁰Ca it is rather better than other. The linear correlation coefficients of J versus R_p were -0.462^*-, -0.497^*-, -0.496^*-, -0.499^*-, -0.475^*- and -0.476^*- for ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei, respectively are small (weak). From Fig. 6, It can be concluded that the impact of the symmetry energy density J on neutron radii (J _↔ R_n) is weaker than that of R_p (J _↔ R_p) or nonexistent.

In general, from Figs. 5 and 6., there is an influence of two nuclear matter properties (J and K_NM) on proton and neutron radii in a space of Skyrme functional, we demonstrate the existence of a relation between (K_NM↔R_n andR_p) for all the investigated nuclei except for ⁴⁰Ca and ⁴⁸Ca (K_NM↔R_n) it is weak. This illustrates that these quantities are coupled by the skyrme force and higher than (J_↔R_{n ,} R_p), where the relation (J_↔R_n) is smaller (weaker) than (J_↔R_p).

Fig. 5.  Incompressibility K_NM of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei as a function of (a) R_n and (b) R_p.

Fig. 6.  Symmetry energy J of ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb nuclei as a function of (a) R_n and (b) R_p.

4. Conclusions:

In this study, the effect of incompressibility modulus K_NM and symmetry energy density J on charge distribution and root-mean-square radii of neutron R_n and proton R_p has been investigated for light, medium and heavy closed-shell nuclei ^{40, 48}Ca, ⁹⁰Zr, ¹¹⁶Sn, ¹⁴⁴Sm and ²⁰⁸Pb within the framework of self-consistent Hartree-Fock (HF) with 20 types of Skyrme interaction. Good agreements were obtained between our HF results of charge density distribution for all of the used types of Skyrme interaction with the experimental data at the surface and interior regions, while our results underestimated the experimental data line inside the investigated nuclei for SVII, SIII, QMC2 and GS2 sets. Concerning, the sensitivity of proton and neutron root-mean-square radii to nuclear matter properties, we conclude there is an impact of the incompressibility K_NM on R_n and R_p. Also, it can be concluded that the impact of the symmetry energy density J on neutron radii (J _↔ R_n) is weaker than that of R_p (J _↔ R_p) or nonexistent.

Funding: None.

Data Availability Statement: All of the data supporting the findings of the presented study are available from corresponding author on request.

Declarations: Conflict of interest: The authors declare that they have no conflict of interest.

Ethical approval: The manuscript has not been published or submitted to another journal, nor is it under review.