II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 55 (2022)

RIKEN Accel. Prog. Rep. 55 (2022)

On the deformability of atoms†
T. Naito,∗1,∗2 S. Endo,∗3,∗4 K. Hagino,∗5 and Y. Tanimura∗3,∗6†
∗1
∗2
∗3
∗4
∗5
∗6

Condensed from the article in J. Phys. B 54, 165201 (2021)
Department of Physics, University of Tokyo
RIKEN Nishina Center
Department of Physics, Tohoku University
Frontier Research Institute for Interdisciplinary Science, Tohoku University
Department of Physics, Kyoto University
Graduate Program on Physics for the Universe, Tohoku University- 33 -Noble gasHalf-closed shells-block
p-blockd-block

0.3
0.2
0.1

β

Atoms and atomic nuclei share common features as
quantum many-body systems of interacting fermions.
In atoms, the inter-particle interaction is the repulsive
Coulomb interaction, and there exists the spherical external Coulomb potential due to the nucleus. In contrast, in atomic nuclei, the inter-particle interaction isthe attractive nuclear force, with no external potential.
Despite the diﬀerences in the fundamental interactions,many similar properties have been observed in both systems, such as the shell structure and the associatedmagic numbers.
One of the most important properties of atomic nuclei
is collective nuclear deformation. It was discussed that
this collective deformation originates from the strong attractive interaction between protons and neutrons.1,2) A
natural question then arises: Can atoms be deformed
collectively as in atomic nuclei? Calculations of theatomic structure have been frequently conducted by assuming spherical symmetry, and the non-sphericity hasbeen addressed only in a few studies. Thus, it is widelybelieved that atoms are rather spherical, in stark contrast to deformed atomic nuclei. What is the physicalorigin of this diﬀerence? To what extent can electron
density distributions in atoms be deformed collectively?
To answer these questions, we calculate electron density distributions of atoms and their deformation parameters without assuming spherical symmetry. By estimating the change in the energy induced by the deformation,in nuclear systems, we ﬁnd that the spin-up and spindown particles tend to be deformed in the same manner
due to the attractive inter-particle nuclear interaction.
In contrast, in atomic systems, we ﬁnd that the spinup and spin-down particles prefer to be deformed in the
opposite manner, cancelling each other and resulting ina small deformation as a whole owing to the repulsive
inter-particle Coulomb interaction. Thus, the diﬀerence
between atoms and atomic nuclei in their deformability primarily originates from the diﬀerent natures of theinter-particle interactions.
Figure 1 shows the deformation parameters β foratoms from Li to Kr calculated using the unrestricted
Hartree-Fock method with the 6-31+G basis.3) All thenoble-gas atoms are spherical (β = 0). For the otheratoms, the electronic conﬁguration of their cores is thesame as that of the noble-gas atoms. Hence, as longas the core density is not deformed owing to non-trivial

0.0
−0.1
−0.2
−0.3051015202530

3540ZFig. 1. Deformation parameters β calculated using the unrestricted Hartree-Fock method with the 6-31+G basis.

many-body eﬀects, the deformation of the atom is expected to result only from the valence electrons. The
valence electrons of the s-block atoms only occupy an sorbital, and the many-body eﬀects between the valence
electrons and spherically symmetric core necessarily result in a spherical electron density distribution. This isalso true for the half-closed-shell atoms.
For the other open-shell atoms, the deformation parameters β are generally non-zero. The deformation parameters for the atoms with the same group exhibit similar tendencies, i.e., |β|  0.1–0.3 for the p-block atoms
and |β|  0.01 for the d-block atoms. This indicatesthat the nature of the few valence open-shell electrons
contributes signiﬁcantly to determining β of the wholeatom. This deformation originates from single-particlevalence orbitals and thus it is misleading to regard it
as collective deformation. For example, the deformationof the p-block atoms can be understood as the valence
p orbitals that primarily contribute to the deformation.
Indeed, |β|  0.1–0.3 of the p-block atoms are similarvalues in order of magnitude to that of a single p orbital, although |β|  0.3 might appear a large collective
deformation in nuclear physics sense. We conclude inthis paper that the p-block atoms can be regarded as analmost inert core plus valence p-electrons, while manybody eﬀects between the core and the valence electrons
are relevant in the d-block atoms. In the d-block atoms,
the core electrons tend to collectively cancel the deformation of the valence d orbitals.
References
1) P. Federman, S. Pittel, Phys. Rev. C 20, 820 (1979).
2) J. Dobaczewski et al., Phys. Rev. Lett. 60, 2254 (1988).
3) W. J. Hehre et al., J. Chem. Phys. 56, 2257 (1972).

1