*2.1. Mayer–Jensen's Shell Model and Observed Magic Numbers*

Mayer [1] and Jensen [2] proposed, in 1949, the model of the shell structure and magic numbers of atomic nuclei. This model provided major guides for a deeper and wider understanding of the structure of atomic nuclei. While this is a similar situation to electrons in atoms, there are some differences. Figure 1 depicts the basic idea and consequences of the Mayer–Jensen's scheme. We start with the nuclear matter composed of protons and neutrons. This matter shows an almost constant density of nucleons (collective name of protons and neutrons) inside the surface, which is a sphere as a natural assumption (see Figure 1a). Because of the short-range character of nuclear forces, this constant density results in a mean potential with a constant depth inside the surface, as shown in Figure 1b. Let us assume that the density distribution is isotropic, producing an isotropic mean potential. Figure 1b also suggests that the Harmonic Oscillator (HO) potential is a good

approximation to this mean potential as long as the mean potential shows negative values as a function of *r*, the radius from the center of the nucleus. We then switch from the mean potential to the HO potential, which is analytically more tractable. Thus, the HO potential can be introduced from the constant density (sometimes referred to as "density saturation") and the short-range attraction due to nuclear forces.

The eigenstates of the HO potential are single-particle states shown in the far-left column of Figure 1c with associated magic numbers and HO quanta, N. These HO magic numbers do not change by adding the minor correction of the -<sup>2</sup> term, the scalar product of the orbital angular momentum*l* (see the second column from left in Figure 1c; for details see [12]).

**Figure 1.** Schematic illustration of (**a**) density distribution of nucleons in atomic nuclei, (**b**) a mean potential (solid line) produced by nucleons in atomic nuclei and an approximation by a Harmonic Oscillator (HO) potential (dashed line). The abscissa, *r*, implies the radius from the center of the nucleus. (**c**) The shell structure produced with resulting magic numbers in circles. Left column: only the HO potential is taken with HO quanta shown as N = 0, N = 1, . . . (N here does not mean the neutron number, *N*.) Middle column: the -<sup>2</sup> term is aded to the HO potential, where the magic gaps are shown in circles. The single-particle orbits are labeled in the standard way to the left. Right column: the spin-orbit term, ( *<sup>l</sup>* ·*s*), is included further, and magic gaps emerging from this term are shown in red. The single-particle orbits are labeled to the right, including*j* =*l* +*s*. The magic gaps are classified as "HO" and "SO" for the HO potential and spin-orbit origins, respectively. Taken from Figure 2 of [15], which was based on [16].

The crucial factor introduced by Mayer and Jensen was the spin-orbit (SO) term, ( *<sup>l</sup>* ·*s*), the effect of which is shown in the third column from the left in Figure 1c. The two orbits with the same orbital angular momentum, -, and the same HO quanta are denoted as,

$$j\_{\succeq} = \ell + 1/2 \text{ and } j\_{\prec} = \ell - 1/2,\tag{1}$$

where 1/2 is due to the spin, *s* = 1/2. The notation of *j*> and *j*< is used frequently in this paper. The spin-orbit term,

$$
v\_{\rm lss} = f(\vec{l} \cdot \vec{s}),\tag{2}$$

is added to the HO + -<sup>2</sup> potential, where *f* is the strength parameter. With *f* < 0 as is the case for nuclear forces, the *j*> state is lowered in energy, whereas the *j*< state is raised. The value of *<sup>f</sup>* is known empirically to be about −20*A*−2/3 MeV (see Equation (2-132) of [12]).

The final pattern of the single-particle energies (SPE) is shown schematically in Figure 1c. The single-particle states are labeled in the standard way up to their *j* values, and both HO and spin-orbit magic gaps are indicated in black and red, respectively. The magic numbers have been considered to be *Z*, *N* = 2, 8, 20, 28, 50, 82, and 126, because the effect of the spin-orbit term becomes stronger as *j* becomes larger. In fact, the magic numbers 28, 50, 82, and 126 are all due to this effect. Instead, the HO magic numbers beyond 20 were considered to be absent or show only minor effects. We shall look back on them, from modern views of the nuclear structure covering stable and exotic nuclei.

We now investigate to what extent magic gaps in Figure 1c have been observed. Figure 2 displays the observed excitation energies of the first 2<sup>+</sup> states of even-even nuclei as a function of *N*, where even-even stands for even-*Z*-even-*N*. These excitation energies tend to be high at the magic numbers, because excitations across the relevant magic gap are needed. The conventional magic numbers of Mayer and Jensen, *N* = 2, 8, 20, 28, . . . 126 are expected to arise, and we indeed see sharp spikes at these magic numbers in Figure 2a where the excitation energies are shown for stable and long-lived (i.e., meta stable) nuclei. Figure 2b includes all measured first 2<sup>+</sup> excitation energies as of 2016. In addition to the spikes in Figure 2a, one sees some new ones. One of them is at *N* = 40, which corresponds to 68Ni40, representing a HO magic gap at *N* = 40. There are three others corresponding to the nuclei, 24O16, 52Ca32, and 54Ca34, as marked in red. The 2<sup>+</sup> excitation energies of these nuclei are about a factor of two higher than the overall trend, suggesting that *N* = 16, 32 and 34 can be magic numbers, although none of them is present in Figure 1c.

These new possible magic numbers are consequences of what are missing in the argument for deriving magic gaps in Figure 1c. We now turn to follow some passages along which this subject has been studied.

**Figure 2.** Systematics of the first 2<sup>+</sup> excitation energies (E*x*(2<sup>+</sup> <sup>1</sup> ), for (**a**) stable and long-lived nuclei and (**b**) all nuclei measured up to 2016, as functions of the neutron number. Peaks in (**a**) are labelled by the neutron number (*N*), while the names of the nuclei are displayed for some new points in (**b**). Taken from Figure 4 of [15].
