In order to construct a nuclear periodic table, we first arrange the elements with the proton magic numbers in the same column. Those are: He (Z = 2), O (Z = 8), Ca (Z = 20), Ni (Z = 28), Sn (Z = 50), and Pb (Z = 82). Zr (Z = 40) often shows behavior similar to the magic nuclei due to the sub-shell closure at Z = 40 (see Fig. 1; Garcia-Ramos and Heyde 2019), and we also include it in the same column. Even though the magic numbers may change in neutron-rich nuclei when the number of neutrons is much larger than the number of protons (Steppenbeck et al. 2013; Otsuka et al. 2020), in this paper we consider only those nuclei which are close to the beta-stability line and do not consider such effect. Though the heaviest element discovered so far is Oganessson (Z = 118), the proton magic number after Z = 82 is currently unknown. Theoretical calculations based on the so called macroscopic–microscopic approach carried out in the 1960s have predicted that the proton shell closure deviates from Z = 126 due to the Coulomb interaction among protons, and appears at Z = 114 together with the neutron shell closure at N = 184 (Sobiczewski et al. 1966; Nilsson et al. 1968). Here, the proton shell closure at Z = 114 is obtained by filling the 2f7/2, 1h9/2, and 1i13/2 orbitals above the Z = 82 magic number (see Fig. 1). The region around this nucleus, 298114
Fl184, has been called the island of stability, providing an important motivation to explore superheavy elements. More recent calculations have predicted different proton shell closures (Z = 114, 120, or 126) depending on a theoretical model employed, even though the neutron shell closure at N = 184 is more robust (Bender et al. 1999). In this paper, we choose the traditional proton magic number, Z = 114, for superheavy elements and arrange Fl below Pb in the nuclear periodic table.
After we set up the column for the magic and semi-magic nuclei, we next arrange other nuclei according to the nuclear shell structure shown in Fig. 1. The ordering of each single-particle level within shells depends on the number of neutrons. Moreover, for open shell nuclei, those single-particle levels are occupied only partially due to the pairing correlation. In mid-shell nuclei, nuclei may even be deformed, yielding a deformed mean-field potential. In that situation, the single-particle levels shown in Fig. 1, which are based on a spherical mean-field potential, lose their clear physical meaning. We therefore consider a group of single-particle levels within each shell, instead of treating each single-particle level individually.
Figure 2 shows the nuclear periodic table so constructed. The elements shown in round-corner boxes are the ones in which the nucleus is statically deformed in the ground state. Here, we regard that a nucleus is deformed when the absolute value of the quadrupole deformation parameter, β2, is larger than 0.15. The deformation parameter β2 is related to the angle dependent radius of a nucleus given by
$$R\left( \theta \right) \, = R_{0} \left( {1 + \beta_{2} Y_{20} \left( \theta \right)} \right),$$
where R0 is the radius of the sphere and Y20 is the spherical harmonics. Here, we have assumed that a nucleus has axial symmetric shape and took the symmetric axis to be the z-axis (θ = 0). For each element, we choose the most abundant nucleus and estimate the deformation parameter using the theoretical calculations by Möller et al. (2016). We note that the resultant periodic table will be almost the same even if we choose the deepest bound isotopes. For the elements lighter than N, we regard the elements Li, Be, B, and C as deformed due to the well-known alpha-particle structure of atomic nuclei. The elements with the white symbols are unstable elements, that is, elements for which all the isotopes are unstable.
In Fig. 2, one can immediately see that the elements in the vicinity of the shell closures are all spherical, while the deformation is developed in the mid-shell regions. The former elements can be interpreted in terms of one or two proton holes outside the shell closures, and it may be meaningful to arrange them in the same columns. In Appendix, we show another representation of this table with coloring for different orbitals l based on the energy levels in Fig. 1.
Because of particle-hole symmetry, it may be more appropriate to rearrange the elements symmetrically around the shell closures. This is done in Fig. 3, in which the magic and the semi-magic nuclei are placed in the center of the periodic table. Its paper model, inspired by a similar spiral model of the atomic periodic table “Elementouch” (Maeno 2002), is shown in Fig. 4. One can see that many elements with one additional proton to the shell closures are spherical in the ground state, even though Li and Cu show deformation, the latter of which is caused by a correlation between the neutrons in Cu and the valence proton. For the nuclei with two valence protons outside the closed shells, the nuclei tend to be deformed due either to the alpha-particle structure (Be and Ne) or to the fact that the neutron is in the mid-shell (Zn and Mo).