*2.2. Monopole Interaction*

The change from the Mayer–Jensen scheme is discussed from the viewpoint of the nucleon-nucleon (*NN*) interaction. The Hamiltonian is written as,

$$
\hat{H} = \hat{H}\_0 + \hat{\mathcal{V}} \,. \tag{3}
$$

where *H*ˆ <sup>0</sup> denotes the one-body term given by

$$
\hat{H}\_0 = \Sigma\_j \epsilon^p\_{0;j} \hbar^p\_j + \Sigma\_j \epsilon^n\_{0;j} \hbar^n\_j \tag{4}
$$

and *V*ˆ stands for the *NN* interaction. Here, *n*ˆ *p*,*n <sup>j</sup>* means the proton- or neutron-number operator for the orbit *j*, and  *p*,*n* 0;*<sup>j</sup>* implies proton or neutron SPE of the orbit *j*. This SPE is composed of the kinetic energy of the orbit *j* and the binding energy on the orbit *j* generated by all nucleons in the inert core. We note that the interaction *V*ˆ in Equation (3) can be any interaction between two nucleons in the following discussions but actually refers to effective *NN* interactions between valence (i.e., active) nucleons.

The interaction *V*ˆ can be decomposed, in general, into the two components: monopole and multipole interactions [17], irrespectively of its origin, derivation, or parameters. The monopole interaction, denoted as *V*ˆ mono, is expressed in terms of the monopole matrix element, which is defined for single-particle orbits *j* and *j* as,

$$V^{\text{mono}}(j, j') = \frac{\Sigma\_{\text{(m,m')}} \left< j, m \; j', m' \middle| \hat{\mathcal{V}} \left| j, m \; j', m' \right>}{\Sigma\_{\text{(m,m')}} 1} \right> \tag{5}$$

where *m* and *m* are magnetic substates of *j* and *j* , respectively, and the summation over *m*, *m* is taken for all ordered pairs allowed by the Pauli principle. The monopole matrix element represents, as displayed schematically in Figure 3, an orientation average for two nucleons in the orbits *j* and *j* . See [15] for more detailed descriptions.

**Figure 3.** Schematic illustration of the monopole matrix element for a two-body interaction *v*. See text for details. Taken from Figure 7 of [15].

The monopole interaction between two neutrons is then given as

$$
\hat{V}^{\text{mono}}\_{nn} = \Sigma\_{\text{j}} V^{\text{mono}}\_{nn}(j,j) \frac{1}{2} \hbar^n\_{\text{j}} \left( \hbar^n\_{\text{j}} - 1 \right) + \Sigma\_{\text{j}<\text{j}'} V^{\text{mono}}\_{nn}(j,j') \, \hbar^n\_{\text{j}} \, \hbar^n\_{\text{j}'}.\tag{6}
$$

The monopole interaction between two protons is given similarly. The monopole interaction between a proton and a neutron can be given as

$$\begin{split} \mathcal{V}\_{\mathrm{pu}}^{\mathrm{moon}} &= \Sigma\_{j \neq j'} \frac{1}{2} \left\{ V\_{T=0}^{\mathrm{moon}}(j, j') + V\_{T=1}^{\mathrm{moon}}(j, j') \right\} \hbar\_{j}^{p} \,\hbar\_{j'}^{n} \\ &+ \Sigma\_{j} \, \frac{1}{2} \left\{ V\_{T=0}^{\mathrm{moon}}(j, j) \frac{2j + 2}{2j + 1} + V\_{T=1}^{\mathrm{moon}}(j, j) \frac{2j}{2j + 1} \right\} \hbar\_{j}^{p} \,\hbar\_{j}^{n} . \end{split} \tag{7}$$

where *V*mono *<sup>T</sup>*=0,1(*j*, *j* ) stands for the monopole matrix element for the isospin *T* = 0 or 1 channel, respectively, defined by Equation (5) including isospin-symmetry effects (see Sec. III A of Ref. [15] for details). Note that *V*mono *<sup>T</sup>*=<sup>1</sup> (*j*, *j* ) implies *V*mono *nn*,*pp* (*j*, *j* ). The second term on the right-hand-side (r.h.s.) of Equation (7) is slightly different from the first term on the r.h.s. of Equation (7) due to the special isospin property for the cases of *j* = *j* . Obviously, *V*ˆ mono *pn* can be rewritten as

$$
\hat{\mathcal{V}}\_{\text{pn}}^{\text{mono}} = \Sigma\_{j, \text{j}'} \hat{\mathcal{V}}\_{\text{pn}}^{\text{mono}}(j, j') \,\,\hat{n}\_{\text{j}}^{p} \,\hat{n}\_{\text{j}'}^{n} \,\, \tag{8}
$$

with *V*˜ mono *pn* (*j*, *j* ) defined so as to reproduce Equation (7).

The functional forms in Equations (6) and (8) appear to be in accordance with the intuition from the averaging over all orientations: no dependencies on angular properties (e.g., coupled *J* values) between the two interacting nucleons and the sole dependence on the number of particles in those orbits.

The (total) monopole interaction is written as

$$
\hat{V}^{\text{mono}} = \hat{V}^{\text{mono}}\_{pp} + \hat{V}^{\text{mono}}\_{nn} + \hat{V}^{\text{mono}}\_{pn} \tag{9}
$$

and the monopole Hamiltonian is defined as,

$$\hat{H}^{\text{mono}} = \hat{H}\_0 + \hat{V}^{\text{mono}} = \Sigma\_{\dot{\jmath}} \epsilon\_{0;\dot{\jmath}}^p \hbar\_{\dot{\jmath}}^p + \Sigma\_{\dot{\jmath}} \epsilon\_{0;\dot{\jmath}}^n \hbar\_{\dot{\jmath}}^n + \hat{V}^{\text{mono}}.\tag{10}$$

The multipole interaction is introduced as

$$
\mathcal{V}^{\text{multi}} = \mathcal{V} - \mathcal{V}^{\text{mono}},\tag{11}
$$

and the (total) Hamiltonian is written as *H*ˆ = *H*ˆ mono + *V*ˆ multi. The multipole interaction becomes crucial in many aspects of nuclear structure, for instance, the shape deformation, as touched upon in later sections of this article. The monopole interaction has been studied over decades with many works, for example, [17–20] (see [15] for more details).

We define the effective SPE (ESPE) of the proton (neutron) orbit *j*, denoted by ˆ *p <sup>j</sup>* (<sup>ˆ</sup>*<sup>n</sup> j* ), as the change of the monopole Hamiltonian, *H*ˆ mono in Equation (10), due to the addition of one proton (neutron) into the orbit *j*. This change is nothing but the difference, when *np*,*<sup>n</sup> j* is replaced by *np*,*<sup>n</sup> <sup>j</sup>* +1. For instance, the first term on the r.h.s. of Equation (10) contributes to ˆ *p <sup>j</sup>* by a constant,  *p* 0;*j* . As another example, the r.h.s. of Equation (8) contributes by Σ*j V*˜ mono *pn* (*j*, *j* ){(*n*ˆ *p <sup>j</sup>* + <sup>1</sup>) *<sup>n</sup>*<sup>ˆ</sup> *<sup>n</sup> <sup>j</sup>* − *n*ˆ *p <sup>j</sup> <sup>n</sup>*<sup>ˆ</sup> *<sup>n</sup> <sup>j</sup>*} <sup>=</sup> <sup>Σ</sup>*j <sup>V</sup>*˜ mono *pn* (*j*, *j* )*n*ˆ *<sup>n</sup> <sup>j</sup>* . Combining all terms, the ESPE of the proton orbit *j* is given as,

$$\mathfrak{e}\_{\dot{j}}^{p} = \mathfrak{e}\_{0; \dot{j}}^{p} + \Sigma\_{\dot{j}'} V\_{pp}^{\text{monno}}(\dot{j}, \dot{j}') \, \mathfrak{h}\_{\dot{j}'}^{p} + \Sigma\_{\dot{j}'} \tilde{V}\_{pn}^{\text{monno}}(\dot{j}, \dot{j}') \, \mathfrak{h}\_{\dot{j}'}^{n} \,. \tag{12}$$

The second and third terms on the r.h.s. are obviously contributions from valence protons and neutrons, respectively. The neutron ESPE is expressed similarly as

$$\mathfrak{A}\_{\dot{j}}^{\rm n} = \mathfrak{e}\_{0;\dot{j}}^{\rm n} + \Sigma\_{\dot{f}} \, V\_{\rm nn}^{\rm mono}(\dot{j}, \dot{f}') \, \mathfrak{H}\_{\dot{f}'}^{\rm n} + \Sigma\_{\dot{f}'} \, \tilde{V}\_{\rm pr}^{\rm mono}(\dot{f}', \dot{f}) \, \mathfrak{H}\_{\dot{f}'}^{\rm p} \,. \tag{13}$$

In many practical cases, an appropriate expectation value of the ESPE operator is also called the ESPE with an implicit reference to some state characterizing the structure, e.g., the ground state.

The ESPE as an expectation value is often discussed in terms of the difference between two states, e.g., Ψ and Ψ . The states Ψ and Ψ may belong to the same nucleus or to two different nuclei. We here show the formulas for this difference. First we introduce the symbol <sup>Δ</sup><sup>O</sup> for an operator <sup>O</sup><sup>ˆ</sup> implying the difference, <sup>Ψ</sup> |O| <sup>ˆ</sup> <sup>Ψ</sup>−Ψ |O| <sup>ˆ</sup> <sup>Ψ</sup> . Such differences of the ESPE values are expressed as,

$$
\Delta \epsilon\_{\dot{\jmath}}^{p} = \Sigma\_{\dot{\jmath}'} V\_{pp}^{\text{monno}}(\dot{j}, \dot{j}') \, \Delta n\_{\dot{\jmath}'}^{p} + \Sigma\_{\dot{\jmath}'} \hat{V}\_{pn}^{\text{monno}}(\dot{j}, \dot{j}') \, \Delta n\_{\dot{\jmath}'}^{n} \,. \tag{14}
$$

and

$$
\Delta \epsilon\_{\dot{j}}^{\eta} = \Sigma\_{\dot{j}'} V\_{nn}^{\text{macro}}(\dot{j}, \dot{j}') \, \Delta n\_{\dot{j}'}^{\eta} + \Sigma\_{\dot{j}'} \tilde{V}\_{pn}^{\text{macro}}(\dot{j}', \dot{j}) \, \Delta n\_{\dot{j}'}^{p} \,. \tag{15}
$$

If Ψ is a doubly closed shell and Ψ is an eigenstate with some valence protons and neutrons on top of this closed shell, these quantities stand for the evolution of ESPEs as functions of *Z* and *N*. One can thus see various physics cases represented by Ψ and Ψ . Such ESPEs can provide picturesque prospects and great help in intuitive understanding without resorting to complicated numerical calculations. The notion of the ESPE has been well utilized, for instance, in empirical studies in [6,21], in certain ways related to the present article.

The interaction *V*ˆ can be decomposed into several parts according to some classifications. The discussions in this subsection can then be applied to each part separately: the monopole interaction of a particular part of *V*ˆ can be extracted, and its resulting ESPEs can be evaluated. Examples are presented in the subsequent subsections.

We note that the definition of the ESPE can have certain variants with similar consequences, for instance, the combination of *np*,*<sup>n</sup> <sup>j</sup>* <sup>−</sup> 1/2 and *<sup>n</sup>p*,*<sup>n</sup> <sup>j</sup>* <sup>+</sup> 1/2 instead of *<sup>n</sup>p*,*<sup>n</sup> <sup>j</sup>* and *np*,*<sup>n</sup> <sup>j</sup>* + 1. Appendix A shows a note on the relation to Baranger's ESPE.

#### *2.3. Central, Two-Body Spin-Orbit and Tensor Parts of the NN Interaction*

With these formulations, we can discuss a variety of subjects ranging from the shell structure, to the collective bands, and to the driplines. Let us start with the shell structure. While the discussions in Section 2.1 are based on basic nuclear properties, some aspects are missing. One of them is the orbital dependencies of the monopole matrix element. This dependence generally appears but shows up more crucially in certain cases.

As we shall see, some parts of the *NN* interaction, *V*ˆ , show characteristic and substantial orbital dependencies. Such parts can be specified in terms of their spin properties, as the *NN* interaction involves a spin operator, an axial vector *σ* of nucleon. We first take the part where no spin operator is included or spin operators are coupled to scalar terms, like (*σ*<sup>1</sup> ·*σ*2) with*σ*1,2 denoting the spin operator of the nucleon 1 or 2, and (·) being a scalar product. This part is called the *central force*, and its effects are discussed in Section 2.4. In the second part, spin operators are coupled to axial vectors. Such axial vectors must be coupled with other axial vectors such as the orbital angular momentum. The *two-body spin-orbit force* belongs to this case, and its effects are discussed in Section 2.8, while the effects remain quite modest except for special orbital combinations. As presented in Section 2.5, significant contributions arise from the *tensor force*, where spin operators are coupled to a (rank-2) tensor, [*σ*<sup>1</sup> × *σ*2] (2), where the last superscript means rank 2. This is a very complicated coupling, and this term must be coupled, in the interaction, with another (rank-2) tensor of the coordinates, in order to form a scalar. Similar terms appear in the electromagnetic interaction, but their effects are minor. The tensor force is, however, crucial in the nuclear case, because the pion exchange process produces it as its primary source. Section 2.5 presents monopole properties of the lowest-order contribution of the tensor force, while higher-order contributions are largely included in the central force of the effective *NN* interaction mentioned above.

#### *2.4. Monopole Interaction of the Central Force*

We now discuss the monopole interaction of the central-force component of *NN* interactions. Because the *NN* interaction is characterized by intermediate-range (∼1 fm) attraction after modifications or renormalizations, the monopole matrix elements gain large magnitudes with a negative sign (i.e., attractive), if radial wave functions of the single-particle orbits, *j* and *j* in Equation (5), are similar to each other. This similarity is visible, if these orbits are spin-orbit partners (*j* = *j*> and *j* = *j*<) with the identical radial wave functions (see Equation (1)), for instance 1 *f*7/2 and 1 *f*5/2. Another example is the coupling between unique-parity orbits, such as 1*g*9/2 and 1*h*11/2, for which the radial wave functions are similar because of no radial node. These types of strong correlations were pointed out by Federman and Pittel in [22], where the total effect of the <sup>3</sup>*S*<sup>1</sup> channel of the *NN* interaction was discussed without the reference to the monopole interaction.

## *2.5. Monopole Interaction of the Tensor Force*

Another important source of the monopole interaction with strong orbital dependences is the tensor force. The tensor force produces very unique effects on the ESPE. This is shown in Figure 4: the intuitive argument in [15,23] proves that the monopole interaction of the tensor force is attractive between a nucleon in an orbit *j*< and another nucleon in an orbit *j* <sup>&</sup>gt;, whereas it becomes repulsive for combinations, (*j*>, *j* <sup>&</sup>gt;) or (*j*<, *j* <sup>&</sup>lt;). The magnitude of such monopole interaction varies also. For example, it is strong in magnitude between spin-orbit partners or between unique-parity orbits, etc. [15].

The ESPE is shifted in very specific ways as exemplified in Figure 5b: if neutrons occupy a *j* <sup>&</sup>gt; orbit, the ESPE of the proton orbit *j*<sup>&</sup>gt; is raised, whereas that of the proton orbit *j*< is lowered. This is nothing but a reduction in a proton spin-orbit splitting due to a specific neutron configuration. The amount of the shift is proportional to the number of neutrons in this configuration, as shown in Equation (14) and in Figure 5c. Other cases follow the same rule shown in Figure 4. These general features have been pointed out in [23] with an analytic formula and an intuitive description of its origin.

**Figure 5.** Schematic picture of the effective single-paticle energy (ESPE) change (i.e., shell evolution) due to the monopole interaction of the tensor force. (**a**) Single-particle energies (SPE) with no neutrons in the orbit *j* >. (**b**) The shifts in the proton ESPEs due to two (valence) neutrons in the orbit *j* >. (**c**) Same as (**b**) except for four neutrons. (**d**,**e**) Type-II shell evolution due to neutron particle–hole excitations. See text for more details. Taken from [24].

#### *2.6. Monopole-Interaction Effects from the Central and Tensor Forces Combined*

The combined effects of the central and tensor forces were discussed in [25] in terms of realistic shell–model interactions, USD [26], and GXPF1A [27]. These interactions were obtained in two steps: the starting point was given by microscopic G-matrix *NN* interactions proposed initially by Kuo and Brown [28,29], and as the second step, certain phenomenological improvements were made by the fit to large numbers of experimental energy levels. It is mentioned that some main features, for instance, the tensor-force component, remain unchanged by this fit [25]. Many other valuable shell–model interactions, for instance, KB3 [17], Kuo-Herling [30], sn100pn [31], and LNPS [32] interactions, have been constructed from the G-matrix interactions sometimes with refinements like monopole adjustments. It should be noticed that these shell–model interactions are derived microscopically to a large extent and that they should be distinguished from purely phenomenological interactions in earlier times, e.g., [33]. The M3Y interaction [34] is related to the G-matrix, too. We appreciate the original contribution of the G-matrix approach to the effective *NN* interaction [28,29].

The VMU interaction was then introduced as a general and simple shell–model *NN* interaction. Its central part consists of Gaussian interactions with spin/isospin dependencies, and their strength parameters are determined so as to simulate the overall features of the monopole matrix elements of the central part of USD [26] and GXPF1A [27] interactions. Its tensor part is taken from the standard *π*- and *ρ*-meson exchange potentials [23,35,36]. Thus, the VMU interaction is defined as a function of the relative distance of two nucleons with spin/isospin dependences, which enables us to use it in a variety of regions of the nuclear chart, as we shall see. A wide model space, typically a HO shell or more, is required in order to obtain reasonable results, though.

Figure 6 depicts some examples: Figure 6a displays the transition from a standard (à la Mayer–Jensen) *<sup>N</sup>* = 20 magic gap to an exotic *<sup>N</sup>* = 16 magic gap by plotting <sup>ˆ</sup>*<sup>n</sup> <sup>j</sup>* within the filling scheme (see Equation (13)), as *Z* decreases from 20 to 8. The tensor monopole interaction between the proton *d*5/2 and the neutron *d*3/2 orbits plays an important role. The small *N* = 20 magic gap for *Z* = 8–12 is consistent with the island of inversion picture (see reviews, e.g., [4,15]). Figure 6b depicts the inversion between the proton *f*5/2 and *p*3/2 orbits as *N* increases in Ni isotopes, by showing ˆ *p <sup>j</sup>* (see Equation (12)). The figure exhibits exotically ordered single-particle orbits for *N* > 44. The tensor monopole interactions between the proton *f*7/2,5/2 and the neutron *g*9/2 orbits produce crucial effects. Figure 6c shows significant changes in the neutron single-particle levels from 90Zr to 100Sn, in terms of <sup>ˆ</sup>*<sup>n</sup> <sup>j</sup>* . Without the tensor force, the approximate degeneracy of *g*7/2 and *d*5/2 orbits in 100Sn does not show up.

**Figure 6.** ESPEs calculated by the VMU interaction. The dashed lines are obtained by the central force only, while the solid lines include both the central-force and the tensor-force contributions. See text for more details. Taken from [25].

These changes in the shell structure as a function of *Z* and/or *N* were collectively called *shell evolution* in [23]. The splitting between proton *g*7/2 and *h*11/2 in Sb isotopes shows a substantial widening as *N* increases from 64 to 82 as pointed out by Schiffer et al. [37], which was one of the first experimental supports to the shell evolution partly because this was not explained otherwise. Note that while the origin of the shell evolution can be any part of the *NN* interaction, its appearance is exemplified graphically in Figure 5a–c for the tensor force. The shell-evolution trend depicted in Figure 6 appears to be consistent with experiment [15,25,38–42]. The monopole properties discussed in this subsection are consistent with the results shown by Smirnova et al. [43] obtained through the spin-tensor decomposition (see e.g., [15] for some account) for the "well-fitted realistic interaction for the *sdpf* shell–model space" [43].
