Abstract
In this work, the effect of incompressibility modulus KNM and symmetry energy density J on charge distribution and root-mean-square radii of neutron Rn and proton Rp has been investigated for light, medium and heavy closed-shell nuclei 40, 48Ca, 90Zr, 116Sn, 144Sm and 208Pb within the framework of self-consistent Hartree-Fock (HF) with 20 types of Skyrme interactions having wide range of nuclear properties such as incompressibility modulus KNM and symmetry energy density J. 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 nuclei. Also, the sensitivity of proton and neutron root-mean-square radii to nuclear matter properties has been examined by determining and discussion the statistical Pearson linear correlation coefficient.
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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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca, ^{48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca and ^{48}Ca. 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 ^{90}Zr, 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 ^{116}Sn, 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 ^{144}Sm 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 ^{208}Pb 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].
Symmetry Energy Density J (MeV) |
Incompressibility Modulus K_{NM} (MeV) |
Type |
30.03 |
216.610 |
SkM^{*} |
33.65 |
218.110 |
SQMC650 |
30.00 |
218.230 |
SV-K218 |
33.47 |
222.200 |
SQMC700 |
31.97 |
223.340 |
SkO |
32.78 |
225.700 |
NRAPR |
33.00 |
228.820 |
KDE0 |
32.00 |
230.010 |
T31 |
30.94 |
230.400 |
Skχm^{∗} |
28.00 |
231.290 |
v075 |
22.83 |
235.890 |
SkSC10 |
29.26 |
239.520 |
RATP |
28.00 |
245.000 |
SAMi |
37.40 |
254.930 |
SK255 |
30.81 |
267.154 |
SkD |
25.96 |
300.140 |
GS2 |
31.22 |
324.550 |
SIV |
28.70 |
330.100 |
QMC2 |
28.16 |
355.370 |
SIII |
26.96 |
366.440 |
SVII |
Fig.1 Our obtained charge density distribution (fm^{-3}) of ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb.
In Fig. 5. the incompressibility K_{NM} are shown as function of R_{n} and R_{p} for ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb nuclei, respectively, where it is small (weak) in ^{40}Ca and^{ 48}Ca. 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb nuclei, respectively, also it is small (weak) in ^{40}Ca and^{ 48}Ca. 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb nuclei, respectively and it is very small correlated except for ^{40}Ca 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 ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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 ^{40}Ca and ^{48}Ca (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 ^{40}Ca, ^{48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb nuclei as a function of (a) R_{n} and (b) R_{p.}
Fig. 6. Symmetry energy J of ^{40}Ca,^{ 48}Ca, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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, ^{90}Zr, ^{116}Sn, ^{144}Sm and ^{208}Pb 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.