# Nuclear Equation of State in the Relativistic Point-Coupling Model Constrained by Excitations in Finite Nuclei

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## Abstract

**:**

## 1. Introduction

## 2. Relativistic Point-Coupling Functionals for Studies of the Symmetry Energy

## 3. Results

#### 3.1. Nuclear Matter Properties and Equation of State

#### 3.2. Dipole Polarizability and Neutron Skin

#### 3.3. Sensitivity of Magnetic-Dipole Excitation to Symmetry Energy

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EDF | Energy density functional |

RHB | Relativistic Hartree–Bogoliubov |

QRPA | Quasiparticle random phase approximation |

EoS | Equation of state |

## References

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**Figure 1.**Equation of state of symmetric nuclear matter (

**a**) and pure neutron matter (

**b**) using the family of point-coupling functionals spanning the range of symmetry energy at saturation J = 29–36 MeV. The results are also presented using the DD-PC1 and DD-PCX functionals, and the symmetry energy values of the functionals are given within the square brackets in units of MeV. Panels (

**c**,

**d**) are the same, but the results are displayed up to the density $0.34$ fm${}^{-3}$ to compare with the predictions from chiral effective field theory ($\chi $EFT) (see Reference [51]).

**Figure 2.**(

**a**) Symmetry energy as a function of density using a family of point-coupling functionals with J = 29, 30, …, 36 MeV alongside the DD-PC1 and DD-PCX functionals. For the DD-PCX and DD-PC1 functionals, the symmetry energy values around the saturation densities are given within the square brackets in units of MeV. (

**b**) The same but for densities up to $0.34$ fm${}^{-3}$, with the results of the chiral effective field theory ($\chi $EFT) (see Reference [51]). The isobaric analog state (IAS) and neutron skin ($\mathsf{\Delta}{R}_{np}$) constraints on the symmetry energy (see Reference [22] for more details) are also presented.

**Figure 3.**The neutron skin thickness as a function of the neutron excess for the DD-PC1 and DD-PCX functionals. The experimental data are taken from References [53,54] in which the proton distributions were obtained from electron scattering data for Sn nuclei [61] and from muonic atom data for others [62,63,64].

**Figure 5.**Dipole polarizability as a function of neutron skin thickness for ${}^{48}$Ca, ${}^{68}$Ni, ${}^{120}$Sn and ${}^{208}$Pb using the relativistic point-coupling functionals. The experimental data for the dipole polarizability values are taken from References [55,56,57,58,59,60]. The recent experimental data for ${}^{120}$Sn is taken from Reference [58] and displayed (${\alpha}_{D}=8.08\pm 0.6$ fm${}^{3}$) with the violet shaded sphere. The band for the neutron radii represents the model-averaged results using many functionals (see Reference [60] for more information).

**Figure 6.**Comparison of the theoretical results for the dipole polarizability with the experimental data for ${}^{48}$Ca-${}^{208}$Pb, ${}^{68}$Ni-${}^{208}$Pb, and ${}^{120}$Sn-${}^{208}$Pb. The theoretical results for the KDE0-J, DD-ME, Skyrme, SAMI-J, NL3$\mathsf{\Lambda}$, and FSU functionals are taken from Reference [60]. The experimental data with the error bars are shown as the shaded regions over the graph. The experimental data for ${}^{48}$Ca (${\alpha}_{D}=2.07\pm 0.22$ fm${}^{3}$), ${}^{68}$Ni (${\alpha}_{D}=3.88\pm 0.31$ fm${}^{3}$), and ${}^{208}$Pb (${\alpha}_{D}=19.6\pm 0.6$ fm${}^{3}$) are taken from References [55], [56,60], and [59,60], respectively. For ${}^{120}$Sn, the experimental value from References [57,60] (${\alpha}_{D}=8.59\pm 0.37$ fm${}^{3}$) is also shown using the shaded regions, and the recent data from Reference [58] (${\alpha}_{D}=8.08\pm 0.6$ fm${}^{3}$) is displayed with a green shaded sphere.

**Figure 7.**The M1 transition strength distribution for ${}^{48}$Ca and ${}^{208}$Pb obtained by the family of DD-PC interactions with the symmetry energy at the saturation, J = 29, 30, …, 36 MeV.

**Figure 8.**(

**Top**) Energy-weighted M1-summation results for ${}^{48}$Ca obtained with the present DD-PC interactions. (

**Bottom**) Centroid energy of M1 transition, $\overline{E}\equiv {m}_{1}\left(\mathrm{M}1\right)/{m}_{0}\left(\mathrm{M}1\right)$.

**Table 1.**The experimental data for the binding energies (B.E.) (72 nuclei) [46,47] and charge radii ${r}_{c}$ (36 nuclei) [48] are used alongside with the neutron (15 nuclei) and proton (14 nuclei) mean gap values ${\mathsf{\Delta}}_{n\left(p\right)}$, diffraction radius ${r}_{diffr}$ (22 nuclei), and surface thickness $\sigma $ (18 nuclei) values for the selected nuclei [28] in the fitting of the DD-PC (J = 29, 30, …, 36 MeV) functionals. In the second line, we display the adopted errors for each observable. The adopted error is also multiplied by a further integer weight factor that is given in the parentheses next to each observable.

Nucleus | ${\mathbf{\Delta}}_{\mathit{n}}$ | ${\mathbf{\Delta}}_{\mathit{p}}$ | B.E. | ${\mathit{r}}_{\mathit{c}}$ | ${\mathit{r}}_{\mathbf{diffr}}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|

(±0.15 MeV) | (±0.15 MeV) | (±1 MeV) | (±0.02 fm) | (±0.05 fm) | (±0.05 fm) | |

${}^{16}$O | −127.619 (2) | 2.777 (1) | ||||

${}^{18}$O | −139.807 (1) | |||||

${}^{20}$O | −151.371 (1) | |||||

${}^{22}$O | −162.026 (1) | |||||

${}^{18}$Ne | −132.142 (1) | |||||

${}^{20}$Mg | −134.479 (1) | |||||

${}^{34}$Si | −283.428 (1) | |||||

${}^{36}$S | 1.52 (1) | −308.714 (1) | 3.299 (1) | 3.577 (1) | 0.994 (2) | |

${}^{38}$Ar | 1.44 (1) | −327.342 (1) | 3.404 (1) | |||

${}^{36}$Ca | −281.371 (1) | |||||

${}^{38}$Ca | −313.121 (1) | |||||

${}^{40}$Ca | −342.052 (2) | 3.485 (2) | 3.831 (2) | 0.978 (2) | ||

${}^{42}$Ca | 1.68 (1) | −361.895 (1) | 3.513 (2) | 3.876 (2) | 0.999 (2) | |

${}^{44}$Ca | 1.70 (1) | −380.959 (1) | 3.523 (3) | 3.912 (1) | 0.975 (2) | |

${}^{46}$Ca | 1.49 (1) | −398.772 (1) | 3.502 (1) | |||

${}^{48}$Ca | −416.001 (1) | 3.484 (1) | 3.936 (1) | 0.881 (1) | ||

${}^{50}$Ca | −427.508 (1) | |||||

${}^{42}$Ti | −346.888 (1) | |||||

${}^{50}$Ti | −437.784 (1) | |||||

${}^{52}$Cr | −456.350 (1) | 3.642 (1) | 4.173 (1) | 0.924 (1) | ||

${}^{54}$Fe | −471.763 (1) | 3.693 (5) | 4.258 (5) | 0.900 (5) | ||

${}^{56}$Ni | −483.994 (1) | |||||

${}^{68}$Ni | −590.407 (1) | |||||

${}^{72}$Ni | −613.455 (1) | |||||

${}^{84}$Se | −727.338 (1) | |||||

${}^{86}$Kr | −749.234 (1) | |||||

${}^{88}$Sr | −768.468 (1) | 4.220 (1) | 4.994 (1) | 0.923 (1) | ||

${}^{90}$Zr | −783.898 (1) | 4.272 (1) | 5.040 (1) | 0.957 (1) | ||

${}^{92}$Mo | 1.40 (1) | −796.510 (1) | 4.315 (1) | 5.104 (1) | 0.950 (1) | |

${}^{94}$Ru | −806.864 (1) | |||||

${}^{96}$Pd | −815.040 (1) | |||||

${}^{98}$Cd | −821.072 (1) | |||||

${}^{100}$Sn | −825.297 (2) | |||||

${}^{106}$Sn | −893.795 (1) | |||||

${}^{108}$Sn | −914.654 (1) | |||||

${}^{112}$Sn | 1.41 (1) | −953.526 (1) | 4.596 (1) | 5.477 (2) | ||

${}^{114}$Sn | 1.26 (3) | −971.570 (1) | 4.610 (1) | 5.509 (2) | 0.948 (2) | |

${}^{116}$Sn | 1.21 (3) | −988.681 (1) | 4.626 (1) | 5.541 (1) | 0.945 (1) | |

${}^{118}$Sn | 1.34 (1) | −1004.951 (1) | 4.640 (1) | 5.571 (1) | 0.931 (1) | |

${}^{120}$Sn | 1.39 (1) | −1020.539 (1) | 5.591 (1) | |||

${}^{122}$Sn | 1.37 (1) | −1035.524 (1) | ||||

${}^{124}$Sn | 1.31 (1) | −1049.960 (1) | 4.674 (1) | 5.640 (1) | ||

${}^{126}$Sn | 1.26 (1) | −1063.883 (1) | ||||

${}^{128}$Sn | 1.22 (1) | −1077.373 (1) | ||||

${}^{130}$Sn | 1.17 (3) | −1090.286 (1) | ||||

${}^{132}$Sn | −1102.840 (1) | |||||

${}^{134}$Sn | −1108.871 (1) | |||||

${}^{134}$Te | 0.81 (1) | −1123.410 (1) | ||||

${}^{136}$Xe | 0.98 (1) | −1141.881 (1) | ||||

${}^{138}$Ba | 1.12 (1) | −1158.292 (1) | 4.834 (1) | 5.868 (2) | 0.900 (1) | |

${}^{140}$Ce | 1.21 (1) | −1172.687 (1) | 4.877 (1) | |||

${}^{142}$Nd | 1.23 (1) | −1185.136 (1) | 4.915 (1) | 5.876 (3) | 0.989 (3) | |

${}^{144}$Sm | 1.25 (1) | −1195.729 (1) | ||||

${}^{146}$Gd | 1.42 (1) | −1204.427 (1) | 4.984 (1) | |||

${}^{148}$Dy | 1.49 (1) | −1210.773 (1) | 5.046 (1) | |||

${}^{150}$Er | −1215.329 (1) | 5.076 (2) | ||||

${}^{152}$Yb | −1218.396 (1) | |||||

${}^{206}$Hg | −1621.048 (1) | 5.485 (1) | ||||

${}^{198}$Pb | −1560.018 (1) | 5.450 (2) | ||||

${}^{200}$Pb | −1576.361 (1) | 5.459 (2) | ||||

${}^{202}$Pb | −1592.193 (1) | 5.474 (1) | ||||

${}^{204}$Pb | −1607.505 (1) | 5.483 (1) | 6.749 (1) | 0.918 (1) | ||

${}^{206}$Pb | 0.59 (1) | −1622.323 (1) | 5.494 (1) | 6.766 (1) | 0.921 (1) | |

${}^{208}$Pb | −1636.429 (1) | 5.505 (1) | 6.806 (1) | 0.900 (1) | ||

${}^{210}$Pb | 0.66 (1) | −1645.552 (1) | 5.523 (1) | |||

${}^{212}$Pb | −1654.514 (1) | 5.542 (1) | ||||

${}^{214}$Pb | −1663.290 (1) | 5.562 (1) | ||||

${}^{210}$Po | 0.81 (1) | −1645.212 (1) | 5.534 (1) | |||

${}^{212}$Rn | 0.88 (1) | −1652.496 (1) | 5.555 (2) | |||

${}^{214}$Ra | 0.96 (1) | −1658.322 (1) | 5.571 (3) | |||

${}^{216}$Th | −1662.694 (1) | |||||

${}^{218}$U | −1665.659 (1) |

**Table 2.**Parameters of the DD-PC interactions with different values of the symmetry energy at the saturation J, given in units of MeV.

Parameters | J = 29 | J = 30 | J = 31 | J = 32 | J = 33 | J = 34 | J = 35 | J = 36 |
---|---|---|---|---|---|---|---|---|

${a}_{S}$ (fm${}^{2}$) | −10.418334630 | −10.384498639 | −10.390657606 | −10.389488800 | −10.387142700 | −10.386956740 | −10.384414509 | −10.387781497 |

${b}_{S}$ (fm${}^{2}$) | −9.163612956 | −9.202629009 | −9.189174805 | −9.197715907 | −9.215314920 | −9.208074838 | −9.215072973 | −9.218990127 |

${c}_{S}$ (fm${}^{2}$) | −4.968064171 | −5.014345965 | −5.031326770 | −5.030638408 | −5.021597419 | −5.023220520 | −5.012403311 | −5.019048957 |

${d}_{S}$ | 1.348374834 | 1.351382727 | 1.352967739 | 1.352329476 | 1.351064617 | 1.352262229 | 1.350836047 | 1.352173708 |

${a}_{V}$ (fm${}^{2}$) | 6.591051202 | 6.583857817 | 6.584374904 | 6.584425799 | 6.584907238 | 6.587404662 | 6.583350394 | 6.585576629 |

${b}_{V}$ (fm${}^{2}$) | 8.366192199 | 8.380745134 | 8.363054275 | 8.358949669 | 8.366666526 | 8.350225226 | 8.361167422 | 8.359267286 |

${d}_{V}$ | 0.737531721 | 0.742903470 | 0.740865993 | 0.739885318 | 0.740096759 | 0.740274812 | 0.740322161 | 0.740857140 |

${b}_{TV}$ (fm${}^{2}$) | 4.370433886 | 3.630253620 | 3.017651110 | 2.581533741 | 2.200070930 | 1.913161650 | 1.654675062 | 1.474609051 |

${d}_{TV}$ | 1.845561631 | 1.570350651 | 1.301516924 | 1.067071385 | 0.836496499 | 0.629569428 | 0.426269288 | 0.252072712 |

${\delta}_{S}$ (fm${}^{4}$) | −0.823980938 | −0.828843122 | −0.835583928 | −0.841884906 | −0.846057419 | −0.849721556 | −0.849999864 | −0.859856552 |

${G}_{n}$ (MeV.fm${}^{3}$) | −829.99300 | −829.32779 | −826.75466 | −825.97419 | −820.16793 | −820.80162 | −818.64718 | −818.82887 |

${G}_{p}$ (MeV.fm${}^{3}$) | −770.15586 | −769.51970 | −769.92950 | −769.75826 | −771.71259 | −771.63793 | −772.34643 | −773.77497 |

E/A (MeV) | ${\mathit{K}}_{0}$ (MeV) | J (MeV) | L (MeV) | |
---|---|---|---|---|

DD-PC-J29 | −16.019 | 230.0 | 29.0 | 29.0 |

DD-PC-J30 | −16.043 | 230.0 | 30.0 | 35.6 |

DD-PC-J31 | −16.055 | 230.0 | 31.0 | 43.8 |

DD-PC-J32 | −16.067 | 230.0 | 32.0 | 52.3 |

DD-PC-J33 | −16.076 | 230.0 | 33.0 | 62.0 |

DD-PC-J34 | −16.087 | 230.0 | 34.0 | 72.1 |

DD-PC-J35 | −16.096 | 230.0 | 35.0 | 83.2 |

DD-PC-J36 | −16.123 | 230.0 | 36.0 | 94.1 |

DD-PC1 | −16.061 | 230.0 | 33.0 | 70.0 |

DD-PCX | −16.026 | 213.0 | 31.1 | 46.3 |

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**MDPI and ACS Style**

Yüksel, E.; Oishi, T.; Paar, N.
Nuclear Equation of State in the Relativistic Point-Coupling Model Constrained by Excitations in Finite Nuclei. *Universe* **2021**, *7*, 71.
https://doi.org/10.3390/universe7030071

**AMA Style**

Yüksel E, Oishi T, Paar N.
Nuclear Equation of State in the Relativistic Point-Coupling Model Constrained by Excitations in Finite Nuclei. *Universe*. 2021; 7(3):71.
https://doi.org/10.3390/universe7030071

**Chicago/Turabian Style**

Yüksel, Esra, Tomohiro Oishi, and Nils Paar.
2021. "Nuclear Equation of State in the Relativistic Point-Coupling Model Constrained by Excitations in Finite Nuclei" *Universe* 7, no. 3: 71.
https://doi.org/10.3390/universe7030071