II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Eﬀects of ﬁnite nucleon size, vacuum polarization, andelectromagnetic spin-orbit interaction on nuclear binding energies
and radii in spherical nuclei†
T. Naito,∗1,∗2 X. Roca-Maza,∗3,∗4 G. Colò,∗3,∗4 and H. Z. Liang∗1,∗2
In the density functional theory (DFT) for nuclear
physics, the ground-state energy is usually given byEgs = T0 + Enucl [ρp , ρn ] + ECd [ρch ] + ECx [ρch ], whereT0 is the Kohn-Sham kinetic energy and Enucl , ECd ,and ECx are the energy density functionals (EDFs) ofthe nuclear, Coulomb direct, and Coulomb exchange
parts, respectively. Here, ρp and ρn are the groundstate density distributions of protons and neutrons,
respectively, and ρch is the charge density distribution.
The Coulomb EDF EC = ECd + ECx is, in principle,
written in terms of the charge density ρch 1) because theCoulomb interaction involves the charge itself, insteadof the point protons. Nevertheless, the protons andneutrons are assumed to be point particles, i.e., ρch ≡ρp , in most self-consistent nuclear DFT calculations.
In this paper, the ﬁnite-size eﬀects of nucleons are implemented to the self-consistent steps of the Skyrme
Hartree-Fock calculation, where only the electric form
factors of nucleons are considered. The Coulomb direct term ECd holds the conventional form, while themodiﬁed Perdew-Burke-Ernzerhof GGA Coulomb exchange functional 2,3) is used instead of the exact Fockterm.
The charge density distribution ρch is written interms of ρp , ρn , and the electric form factors ofprotons( ) and neutrons,( G̃) Ep and G̃En , as ρ̃ch (q) =G̃Ep q 2 ρ̃p (q) + G̃En q 2 ρ̃n (q), where the quantities
with the tilde denote those in the momentum space.
Once the ﬁnite-size eﬀects are considered, i.e., ρch ̸≡ρp , the chain rule of the functional derivative should
be applied to derive the eﬀective potential of the nuδE[ρ ,ρ ]cleon τ , Veﬀτ (r) = δρτp(r)n . Finally, the Coulomb
potential for nucleons with the ﬁnite-size eﬀects reads
VCτ (r) = {VC [ρch ] ∗ GEτ } (r), where VC [ρch ] is theconventional form of the Coulomb potential and ∗ denotes the convolution. The Coulomb potential forthe neutrons does not vanish within the self-consistentﬁnite-size eﬀects, because GEn ̸≡ 0.
Other possible electromagnetic (EM) contributions,
i.e., the vacuum polarization and EM spin-orbit interaction, are also considered. We used eﬀective one-bodypotential of the vacuum polarization for a charged
particle under the Coulomb potential caused by ρch ,known as the Uehling potential.4) The EM spin-orbit
†
∗1
∗2
∗3
∗4

interaction, which originates from the interaction between the nucleon spin and VCτ , is also considered byusing the ﬁrst-order perturbation theory.
The isospin-symmetry breaking (ISB) terms of thenuclear force are also implemented to compare the contributions of the EM interaction with those of the ISB
terms of nuclear force. The SAMi functional5) is used
for the nuclear EDF for most calculations, and theSAMi-ISB functional6) is used instead when the ISB
originating from Enucl is considered explicitly.
The proton ﬁnite-size eﬀect makes the nuclei morebound; for example, the nucleon ﬁnite-size eﬀect decreases the binding energy by 8.2 MeV in 208 Pb. Incontrast, the neutron ﬁnite-size eﬀect makes the nuclei
less bound, and its contribution is almost one order ofmagnitude smaller than the proton contribution, forexample, by 1.2 MeV in 208 Pb, which is non-negligible
in heavy nuclei. The contribution of the vacuum polarization to the total energy is also non-negligible, and it
makes the nuclei less bound, for example, by 3.7 MeV
in 208 Pb. The contribution of the electromagnetic spinorbit interaction to the total energy is approximately
50 keV. Systematically, the contribution of the ISB
terms of the nuclear force to the total energy is comparable to that of the proton ﬁnite-size eﬀect in heavy
nuclei, while the former is more signiﬁcant than thelatter in light nuclei. The neutron ﬁnite-size eﬀect andvacuum polarization are also non-negligible. On theother hand, the contribution of the electromagnetic
spin-orbit interaction to the total energy depends onthe shell structure.
The mass diﬀerence between the mirror nuclei 48 Ca
and 48 Ni was also calculated. All the corrections to theCoulomb functional with the SAMi-ISB functional cooperate to reproduce the mirror-nuclei mass diﬀerence
within 300 keV accuracy, while the error is 1.5 MeV if
the conventional Coulomb functional is used.
References
1) A. Bulgac, V. R. Shaginyan, Nucl. Phys. A 601, 103(1996).
2) J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett.
77, 3865 (1996).
3) T. Naito, X. Roca-Maza, G. Colò, H. Liang, Phys. Rev.
C 99, 024309 (2019).
4) E. A. Uehling, Phys. Rev. 48, 55 (1935).
5) X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. C 86,031306 (2012).
6) X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. Lett.
120, 202501 (2018).

Condensed from the article in Phys. Rev. C 101, 064311(2020)
Department of Physics, The University of Tokyo
RIKEN Nishina Center
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano- 43 -1