We apply the KIDS (Korea: IBS-Daegu-Sungkyunkwan) nuclear energy density ... (EDF) 1 at low density may be expanded by taking the Fermi momentum kF as ...
Proceedings of the Workshop on Quarks and Compact Stars 2017 (QCS2017) Downloaded from journals.jps.jp by 191.96.83.215 on 06/05/18 Proc. Workshop Quarks and Compact Stars 2017 (QCS2017) JPS Conf. Proc. 20, 011041 (2018) https://doi.org/10.7566/JPSCP.20.011041
KIDS Nuclear Energy Density Functional: 1st Application in Nuclei Hana Gil1 , Panagiota Papakonstantinou2 , Chang Ho Hyun3 and Yongseok Oh1 1
Department of Physics, Kyungpook National University, Daegu 41566, Korea Rare Isotope Science Project, Institute for Basic Science, Daejeon 34047, Korea 3 Department of Physics Education, Daegu University, Gyeongsan, Gyeongbuk 38453, Korea 2
(Received June 18, 2017) We apply the KIDS (Korea: IBS-Daegu-Sungkyunkwan) nuclear energy density functional model, which is based on the Fermi momentum expansion, to the study of properties of lj-closed nuclei. The parameters of the model are determined by the nuclear properties at the saturation density and theoretical calculations on pure neutron matter. For applying the model to the study of nuclei, we rely on the Skyrme force model, where the Skyrme force parameters are determined through the KIDS energy density functional. Solving HartreeFock equations, we obtain the energies per particle and charge radii of closed magic nuclei, namely, 16 O, 28 O, 40 Ca, 48 Ca, 60 Ca, 90 Zr, 132 Sn, and 208 Pb. The results are compared with the observed data and further improvement of the model is shortly mentioned.
Motivated by the Brueckner theory for a realistic potential in many fermion systems and effective field theory within a dilute fermion system, the nuclear energy density functional (EDF) 1 at low density may be expanded by taking the Fermi momentum kF as the expansion parameter in homogeneous nuclear matter. This idea was investigated in Ref. [2] and the EDF called KIDS is developed. In this model, the nuclear energy density functional is written as E(ρ, δ) = T (ρ, δ) +
3 ∑
ci (δ) ρ1+i/3 ,
(1)
i=0
where T is the kinetic energy part, the nuclear density is ρ = ρp + ρn and the asymmetry parameter is defined as δ = (ρn − ρp )/ρ, where ρp and ρn are the densities of protons and neutrons, respectively. The parameters of the nuclear EDF ci ’s are defined as ci (δ) = αi + βi δ 2 ,
(2)
so that ci = αi for the symmetric nuclear and ci = αi + βi for the pure neutron matter. These parameters are determined by the properties of symmetric nuclear matter at the nuclear saturation density and the results of a microscopic calculation in Ref. [3]. The details on the fitting process and the results can be found in Ref. [2]. To apply the obtained EDF to the study of the properties of nuclei, we need a model for nuclear forces and we adopt the Skyrme force model and perform the Skyrme-Hartree-Fock calculations [4, 5]. In our study, we write the Skyrme force as 1 2 v(ri , rj ) = (t0 + y0 Pσ )δ(ri − rj ) + (t1 + y1 Pσ )[k′ δ(ri − rj ) + δ(ri − rj )k2 ] 2 1 Although E(ρ) has the form of a function, it is, in the end, only a mapping, whose existence and uniqueness was put forth by Hohenberg and Kohn [1]. We therefore call it a functional as is common practice.
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JPS Conf. Proc. 20, 011041 (2018)
Table I. The Skyrme force parameters determined by the KIDS EDF. t0 (MeV fm3 ) −1772.044
t1 (MeV fm5 ) 2492.112 × k
t2 (MeV fm5 ) −1459.767 × k
t31 (MeV fm4 ) 12216.732
t32 (MeV fm5 ) 642.115 × (1 − k)
t33 (MeV fm6 ) 0.000
y0 (MeV fm3 ) −127.524
y1 (MeV fm5 ) 0.000
y2 (MeV fm5 ) 0.000
y31 (MeV fm4 ) −11969.990
y32 (MeV fm5 ) 33153.477 × (1 − k)
y33 (MeV fm6 ) −22955.280
1∑ (t3n + y3n Pσ )[ ρ(R) ]n/3 δ(ri − rj ) 6 3
+ (t2 + y2 Pσ )k′ δ(ri − rj )k +
n=1
′
+ iW0 (σ i − σ j ) · k × δ(ri − rj ) k,
(3)
where R = (ri + rj )/2, k = (∇i − ∇j )/(2i) and k′ = −(∇′i − ∇′j )/(2i). The spin-exchange operator is represented by Pσ and W0 is the strength of the spin-orbit coupling. Then we can obtain the Skyrme EDF for nuclei as E(ρ) =
∑ ρ2q } ℏ2 1{ 1 (∇ρ)2 τ+ (2t0 + y0 )ρ − (t0 + 2y0 ) + {(6t1 + 3y1 ) − (2t2 + y2 )} 2m 4 ρ 32 ρ q
∑ (∇ρq )2 1 { } 1 {(3t1 + 6y1 ) + (t2 + 2y2 )} + (2t1 + y1 ) + (2t2 + y2 ) τ 32 ρ 8 q ∑ ∑ ρq τ q 1 1 1 − {(t1 + 2y1 ) − (t2 + 2y2 )} + (2t31 + y31 )ρ4/3 − (t31 + 2y31 )ρ−2/3 ρ2q 8 ρ 24 24 q q ∑ 1 1 ρ2q + (2t32 + y32 )ρ5/3 − (t32 + 2y32 )ρ−1/3 24 24 q ∑ 1 1 1 ( J · ∇ρ ∑ Jq · ∇ρq ) + (2t33 + y33 )ρ2 − (t33 + 2y33 ) ρ2q + W0 + , (4) 24 24 2 ρ ρ q q −
where q denotes proton and neutron. The spin density J is defined by J = Jp +Jn . Comparing with the KIDS EDF, we can relate the parameters (ti , yi ) to the parameters ci (δ). We leave the details to Refs. [4, 5] and the obtained Skyrme force parameters are presented in Table I. We have two additional parameters k and W0 that are absent in the KIDS EDF for infinite matter. The energies per nucleon and charge radii of l-closed nuclei, 16 O and 40 Ca, are used for determining k-value, which gives k = 0.11. To determine the W0 value, the calculated energies per nucleon and charge radii of 48 Ca and 208 Pb are compared with the experimental data, which leads to W0 = 110 MeV·fm5 . We then calculate the energies per nucleon, charge radii, and neutron skin of magic nuclei and compare our results with other Skyrme force model results. As shown in Fig. 1, the KIDS results of energies per nucleon and charge radii are similar to the other model results and experimental data for 48 Ca and 208 Pb. For exotic nuclei 28 O and 60 Ca, the predicted results show model dependence. For example, the binding energies of 28 O are 6.0757 MeV (KIDS), 6.1925 MeV (SLy4), and 6.4114 MeV (SkM*), and 5.9883 MeV (AME2016, [6]). In summary, we applied the newly developed KIDS nuclear energy density functional to calculate the properties of lj-closed nuclei. The only parameters that we adjust to nuclei are those which do not affect homogeneous matter: namely, the portion of momentum vs density dependence (k) and the spin-orbit interaction strength (W0 ). This model is found to successfully reproduce the experimental data for observed nuclei, and we present predictions 2 ■■■
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40
E/A [MeV]
9
Ca
16
48
O
8
90
Ca
Zr
132
Sn
208
60
Ca
7 6
Pb
28
O
5 6
0
20
40
60
80
100
120
140
132
Rc [fm]
200 Pb
Sn
90
Zr
4
60
Ca
40
Ca
16
3
28
O
O
0
20
KIDS SLy4 SkM* Exp.
48
Ca
40
60
80
100
120
140
160
180
200
28
O
60
0.6 ∆rnp [fm]
180
208
5
0.8
160
Ca
0.4
132
0.2 0
Pb
40
O
0
208
Zr
Ca
16
Sn
90
48
Ca
20
40
60
80
100
120
140
160
180
200
Mass number A
Fig. 1. Calculated energies per nucleon (top), charge radii (medium), and neutron skin (bottom) of l-closed nuclei. Compared are the results of other Skyrme force models and experiments [6–8].
28
MODEL SLy4 SkM* KIDS AME2016 Table II.
E/A [MeV] 6.1925 6.4114 6.0757 5.988
O Rc [fm] 2.8656 2.8646 2.8353
60
∆rnp 0.58476 0.61631 0.66398
E/A [MeV] 7.7030 7.7857 7.6652
Ca Rc [fm] 3.6734 3.6713 3.6452
∆rnp 0.4435 0.4685 0.4960
Properties of exotic nuclei calculated by using KIDS and other Skyrme force models.
for 28 O and 60 Ca. Next we plan to apply the model to an RPA calculation of giant resonances and exotic modes of nuclear excitations. References [1] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964). [2] P. Papakonstantinou, T.-S. Park, Y. Lim, and C. H. Hyun, Nuclear energy density functional inspired by an effective field theory, arXiv:1606.04219. [3] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, The equation of state of nuclear matter and neutron star structure, Phys. Rev. C 58, 1804 (1998). [4] H. Gil, P. Papakonstantinou, C. H. Hyun, T.-S. Park, and Y. Oh, Nuclear energy density functional for KIDS, Acta Phys. Pol. B 48, 305 (2017). [5] H. Gil, Y. Oh, C. H. Hyun, and P. Papakonstantinou, Skyrme-type nuclear force for the KIDS energy density functional, New Physics (Sae Mulli) 67, 456 (2017). [6] M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, and X. Xu, The Ame2016 atomic mass evaluation (II). Tables, graphs and references, Chin. Phys. C 41, 030003 (2017). [7] I. Angeli and K. P. Marinova, Table of experimental nuclear ground state charge radii: An update, Atom. Data Nucl. Data Tabl. 99, 69 (2013). [8] A. Trzci´ nska, J. Jastrz¸ebski, P. Lubi´ nski, F. J. Hartmann, R. Schmidt, T. von Egidy, and B. Klos, Neutron density distributions deduced from antiprotonic atoms, Phys. Rev. Lett. 87, 082501 (2001). 3 ■■■
