**1. Introduction**

One of the most important results obtained by investigating exotic nuclei (those from the *β* stability) is the evolution of the shell structure, which is often called the shell evolution [1]. The evolution sometimes occurs in a more drastic way than as predicted by the standard Woods–Saxon potential model: some of the conventional neutron magic numbers, such as *N* = 8, 20, and 28, disappear, and new magic numbers, such as *N* = 16 and 34, appear.

These phenomena indicate a mechanism of shell evolution beyond the potential models, and the role of effective interactions has recently received much attention. Historically, this idea was developed in the context of the shell model, dating back to 1960 when Talmi and Unna accounted for the inversion of single-particle levels in the *p*-shell nuclei [2]. Later, a similar expression was derived in Ref. [3], in which the effect of two-body interactions was formulated with what is now called the monopole interaction [4].

The impact of the monopole interaction on nuclear structure has been investigated with the development of large-scale shell-model calculations [4–6], in which *p f*-shell nuclei are very successfully described by using Kuo–Brown interactions with a few monopole matrix elements appropriately modified. The single-particle energy that includes the effect of the monopole interaction is often referred to as the effective single-particle energy [7,8].

One of the remaining issues concerning shell evolution is the general properties of the monopole interaction and their origin. One of the earliest attempts in this direction was carried out by Federman and Pittel [9], who indicated that the central force causes a sharp drop of the neutron 1*g*7/2 orbital with the proton 1*g*9/2 orbital occupied. With more data on exotic nuclei accumulated in the 1990s, the spin-isospin dependence of the effective interaction was highlighted in Ref. [10]. This property well accounts for the monopole interaction that was phenomenologically introduced in Ref. [11] to describe the shifting magic number from *N* = 16 to 20. Finally, Otsuka et al. demonstrated [12] that the tensor force significantly increases or decreases spin–orbit splitting depending on the relative direction of the spin and orbital angular momenta that the last nucleons have.

For a unified description of the shell evolution, in [13], it was proposed that the central and tensor forces are the major sources of shell evolution, whereas the two-body spin–orbit

**Citation:** Utsuno, Y. Probing Different Characteristics of Shell Evolution Driven by Central, Spin-Orbit, and Tensor Forces. *Physics* **2022**, *4*, 185–201. https:// doi.org/10.3390/physics4010014

Received: 29 November 2021 Accepted: 18 January 2022 Published: 9 February 2022

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force plays a unique role in the monopole matrix elements between specific orbitals [14]. The same conclusion was drawn from the spin-tensor decomposition of an effective interaction fitted to the experimental data [15]. In Ref. [13], shell evolution is described by an interaction that consists of a simple Gaussian central force and a *π* + *ρ* meson exchange tensor force, whose choice is supported by "renormalization persistency" [16]. This interaction, named the monopole-based universal interaction, *V*MU, and its variant were successfully applied to constructing effective interactions for shell-model calculations [17,18], whose focuses were placed on many-body properties, such as the onset of deformation due to the tensor force.

The aim of the present study is to quantitatively examine to what extent the shell evolution is described by such a simple scheme. To this end, the SDPF-MU interaction [18] is employed here whose cross-shell part is made of a variant of the *V*MU interaction with the two-body spin–orbit force included, and the validity of its shell evolution is carefully examined by comparing with the relevant experimental data.

In this paper, neutron-rich nuclei with the atomic mass number 25 - *A* - 55 are considered, where several doubly-closed-shell nuclei are known, including 24O, 34Si, 36S, 40Ca, 48Ca, 52Ca, and 54Ca. Hence, configuration mixing within the major shell is relatively suppressed along the atomic number, *Z*, and *N* = 20 chains, for instance, which makes easier to identify the monopole matrix element most relevant to the shell evolution under debate. Here, a rather complete survey that covers both the proton and neutron shell evolution is conducted, thus, enabling to separate the unique roles of the central, spin–orbit, and tensor forces.

This paper is organized as follows. In Section 2, the *V*MU interaction is introduced as used in the SDPF-MU interaction, and the different characteristics of the central, spin–orbit, and tensor forces are quantitatively presented with regard to the monopole matrix element. Section 3 dicusses how the shell evolution, caused by this interaction, can be validated by experimental data. Sections 3.1 and 3.2 are devoted to proton shell evolution with varying neutron number and neutron shell evolution with varying proton numbers, respectively. Section 4 gives conclusions of the study.
