*5.2. Monopole–Quadrupole Interplay for the Driplines*

Figure 17 shows the ground-state energies of F, Ne, Na, and Mg isotopes as functions of the neutron number *N*. These energies are decomposed into several pieces according to their origins: SPE (on top of the 16O inert core), monopole, pairing, and rest terms. The Coulomb contribution is ignored in the following discussion, because it is of virtually no relevance. Here, the multipole interaction is divided into the pairing and rest terms. The pairing is the BCS-type pairing interaction acting on two neutrons coupled to *J<sup>π</sup>* = 0<sup>+</sup> and on two protons coupled to *J<sup>π</sup>* = 0+. The rest term means the multipole interaction subtracted by the pairing term. Although the rest term contains many different pieces, its major effects in the present discussion is simulated by the quadrupole interaction. This is the reason why the rest term is associated with "(quadrupole etc)" in the figure.

The lower edges of the red areas exhibit the ground-state energies as functions of the neutron number *N*, while only even *N* values are taken. These values show a good agreement with measured values shown by black dots. As long as the ground-state energy becomes lower as *N* increases, the isotope gains more binding energy by having more neutrons, and the isotope chain is stretched. However, if the ground-state energy is not lowered, there is no gain in the binding energy by having these extra neutrons; these extra neutrons are emitted, and the neutron dripline implies the nucleus with the lowest groundstate energy. The driplines obtained by the present calculation are shown by red arrows for each isotopic chain, reproducing experimental driplines for F, Ne, and Na isotopes [94].

We focus on the lower edge of the green areas in Figure 17. This represents the monopole contributions comprising the SPE and the monopole interaction. For Ne, Na, and Mg isotopes, this edge is lowered almost linearly as *N* increases from *N* = 16 to each dripline. We then fit the edge with pink dashed, purple dotted, and black solid lines for Ne, Na, and Mg isotopes, respectively. The lines of Ne and Na isotopes are copied to the panel for Mg, with their positions adjusted at a certain *N*. It is evident that the lines become steeper almost linearly as *Z* increases. This edge is almost flat for F isotopes for *N* ≥ 16, and this feature is discussed below.

Figure 17 indicates that the effect of the pairing term shows small variations. In contrast, the rest term changes more, which is largely due to the quadrupole interaction. Figure 18a schematically indicates the variation of the effect of the quadrupole interaction: The effect is small at the far-left position with a spherical shape. As some neutrons are added, the shape is deformed, and the ground-state energy is lowered due to the quadrupole interaction. This trend continues but becomes its maximum at a certain value of *N* (red object in the figure). However, the dripline is not determined just by this maximum point.

**Figure 17.** Ground-state energies of even-N isotopes of (**a**) F, (**b**) Ne, (**c**) Na and (**d**) Mg, relative to the 16O value. Colored segments exhibit decompositions into various effects from the monopole (green), pairing (blue) and rest (such as quadrupole) (red) components of the effective nucleon–nucleon interaction as well as those from Coulomb interaction (black) and single-particle energies (bare SPE; grey). The monopole effect grows steadily as a function of *N* in all cases, as highlighted by straight lines: dashed (Ne), dotted (Na) and solid (Mg). The experimental values are indicated by black circles [42]. The theoretical driplines indicated by red arrows. Modified from Figure 4 of [57].

**Figure 18.** (**a**) Presently proposed mechanism based on shape evolution and the resulting change in the ground-state energy. (**b**) The rest-term contribution to the ground-state energies for F, Ne, Na, and Mg isotopes. Dashed arrows indicate the monopole displacement. See text for more details. Modified from Figures 2 and 6 of [57].

Figure 18b depicts the actual effect of the rest term. It follows the trend illustrated in Figure 18a, with the maximum effect at *N* = 22 in all four chains. However, the driplines are different among these four. This is due to the monopole interaction. Let me explain it by taking the Mg isotopes as an example. The black straight line of the monopole effect in Figure 17d depicts about 3 MeV lowering per additional neutron, implying about 6 MeV for an additional two neutrons. After *N* = 22, the rest effect loses its magnitude. If the loss is less than the monopole gain (∼6 MeV), this loss is compensated by the monopole effect. However, the loss becomes larger for *N* larger, and at a certain point, the loss exceeds the monopole compensation. The dripline thus arises with the "monopole displacement" from *N* = 22 to *N* = 30 as shown in Figure 18b (and also in Figure 18a schematically).

The monopole effect depends directly on the number of protons, as visualized by three straight lines in Figure 17. Consequently, the monopole displacement is Δ*N* = 2 (6) for Ne (Na) isotopes. For F isotopes, the monopole effect is negligibly small for *N* ≥ 16, and the dripline is located at the maximum rest (quadrupole etc.) effect.

## *5.3. Stability of Spherical Isotopes and the Monopole-Quadrupole Interplay*

An immediate lemma of the present dripline mechanism is that the driplines of spherical nuclei, such as Ca, Sn, and Pb isotopes, can be further away from the stability line than other elements. One can assume a basically constant pairing contribution and a minor rest-term contribution. These two are thus irrelevant to the driplines of these isotopes. The remaining monopole effect gradually changes, pushing the driplines away.
