Cross-Shell Excitation in F and Ne Isotopes around N = 20

Abstract

Within the framework of the configuration–interaction shell model, the present work applies three effective interactions to investigate the effects of the cross-shell excitation on F and Ne isotopes around N = 20, which are significantly proton–neutron asymmetric, and have different properties compared with the proton–neutron symmetric nuclei. It is shown that cross-shell excitation is necessary in order to reproduce separation energies, neutron drip lines, and low-energy levels of these isotopes. Furthermore, the cross-shell excitation of (0–5)ħω is suggested to be important in the description of ²⁹F and ³⁰Ne. However, the three interactions are insufficient in describing the bound structure of ^29,31Ne, and provide inconsistent shell structures and evolutions in the target nuclei. Their cross-shell interactions are suggested to be improved.

1. Introduction

Nuclei with neutron numbers close to 20 are of interest. According to the conventional shell model, the N = 20 shell is closed, defining the boundary between the sd shell and the pf shell [1,2]. As a result, nuclei with N = 20 should be magic, which is valid for nuclei near the β-stability line, where protons and neutrons are almost symmetrical in the light nuclear region. However, for lighter elements, such as F [3], Ne [4,5,6], Na [7], and Mg [8], it is known that the N = 20 shell gap vanishes in extremely proton–neutron asymmetric nuclei, leading to the existence of the “island of inversion” [7]. Moreover, the neutron drip lines of C, N, and O have 16 neutrons [9,10,11], while ³¹F and ³⁴Ne are the heaviest bound isotopes of F and Ne, respectively [12]. The abrupt change of the neutron drip line also manifests the evolution of the N = 20 shell. Since ²⁹F and ³⁰Ne are the two lightest bound nuclei with N = 20, a systematic study on the F and Ne isotopes around ²⁹F and ³⁰Ne is significant.

F and Ne isotopes with N~20 are challenging to study due to their extreme proton–neutron asymmetry. For instance, ³¹F has been known to be bound since 1999 [13], while the unbound character of ^32,33F was not determined until 2019 [3]. In fact, to date, only the masses of the F (Ne) isotopes with mass numbers less than 30 (32) have been experimentally determined [14]. On the other hand, some interesting phenomena in such nuclei have been experimentally discovered in recent years. For example, the “island of inversion” has been extended with several F and Ne isotopes [3,15], and exotic properties of F and Ne isotopes in the N~20 region have been discovered [16,17]. In 2020, the two-neutron halo structure in ²⁹F [18] was experimentally identified. In 2021, neutron-unbound states were experimentally explored in ³¹Ne [19]. Therefore, theoretical models can be helpful to investigate the extremely neutron-rich F and Ne nuclei.

There usually are three types of theoretical models of nuclear structure: the ab initio theories [20,21], mean-field approaches [22], and shell models [1,2,23]. All three have been employed for neutron-rich F and Ne nuclei [5,24,25,26]. In the past two years, various theoretical models considering the continuum coupling for nuclei near the drip line—including the coupled-cluster method [18], Gamow shell model [27], and Green’s function method in the complex-momentum representation [25]—have been used to investigate the halo structure in ^29,31F. Three-body systems have been built to analyze such exotic structures [28,29]. Nevertheless, a systematic study can be helpful to understand such unexpected phenomena.

The shell model is an essential tool for the systematic understanding of the nuclear structure. Recently, it has been employed in describing light, proton-rich, exotic nuclei and the related isospin symmetry breaking [30,31,32,33], interpreting the shell evolution in the medium mass region [34,35], reproducing isomeric states in the medium [36,37] and heavy nuclei [38,39,40], and investigating the α-decay properties of heavy-mass nuclei [41,42].

Within the shell model framework, it should be noticed that pure sd- or pf-shell model space is insufficient to study F and Ne isotopes around N = 20, due to the vanishing of the N = 20 shell gap. On this basis, the neutron excitation across the N = 20 shell is rather crucial. Fukunishi et al. employed an interaction in the model space, including the whole sd shell and the 0f_7/2 and 1p_3/2 orbits, and calculated some properties of ³⁰Ne, ensuring the necessity of cross-shell excitation [43]. The SDPF-M interaction, proposed by Utsuno et al., was presented with its results on the average neutron occupancy in the pf shell of even-mass ^26–34Ne [44]. Caurier et al. proposed the SDPF-U-MIX interaction, and showed the results of ^29,31F and ^30–32Ne, considering the contribution of the particle-hole states in [45].

In fact, cross-shell excitation is not a fresh idea; just as near the N = 20 shell, it can also be seen in the N = 8 and N = 28 shells. For example, shell model calculations with the YSOX interaction have succeeded in reproducing the properties of neutron-rich Be [46,47], B [48], C [49,50], and O [51] isotopes with the help of the cross-shell excitation beyond N = 8. In the sd region, the interpretation of the exotic β-γ-α decay mode in ²⁰Na [52] also requires the neutron cross-shell excitation. Moreover, shell model studies with the SDPF-MU [53] and SDPF-U-SI [54] interactions were performed to describe the exotic Mg [55], Si [56], and S [53,54] isotopes, considering the neutron excitation across the N = 28 shell.

Despite the previous studies, a systematic investigation of the influences of cross-shell excitation on F and Ne isotopes around N = 20 is lacking. It would be interesting to know, for example, how many neutrons of cross-shell excitation should be considered if we wish to reproduce the neutron drip line of F and Ne. For this purpose, based on the widely used configuration–interaction shell model (CISM), this paper applied three different interactions—SDPF-M, SDPF-MU, and SDPF-U-SI—to investigate the role of the cross-shell excitation in the F and Ne isotopes with N~20.

The paper is organized as follows: in the next section, CISM is briefly introduced, and the details of the three interactions are given; then, the results from the shell model calculations are shown to analyze the effects of the cross-shell excitation on the F and Ne neutron-rich isotopes; finally, a summary is presented.

2. Theories

The nuclear structure problem is essentially a many-body quantum problem, where protons and neutrons interact in a complex manner. To this end, CISM assumes an inert core, and defines a model space where the remaining nucleons can occupy the orbits. In the truncated model space, the effective Hamiltonian—also called the effective interaction—can be built from the nuclear force. Usually, the nuclear force can be derived from either the nucleon–nucleon scattering experimental data [57] or the observed nuclear structure data [58], called realistic nuclear force and phenomenological nuclear force, respectively. The former starts in a more ab initio manner, which needs to deal with hard-core and in-medium effects before the utilization. On the other hand, the phenomenological nuclear force depends a lot on experiments. The three effective interactions applied in this work are all constructed in a phenomenological manner. They can partially include continuum-coupling effects and proton–neutron asymmetry via fitting nuclear structure data of nuclei near the drip line. Then, through diagonalizing the Hamiltonian matrix, which consists of single-particle energies (SPEs) and two-body matrix elements (TBMEs), the wavefunctions can be derived as eigenstates of the many-body Schrodinger equation. As a result, configuration-mixing is used to express each state.

There are normally three steps to perform CISM calculations in practice: The first step of CISM is to choose an appropriate model space. The second step, which is also the most critical step, is to construct the effective interaction in the model space. Finally, the shell model codes are used to diagonalize the Hamiltonian matrix. In this paper, the size of the model space is discussed by analyzing the effects of the cross-shell excitation. The input of the effective interaction is varied with three choices, while all of the calculations are performed with the code KSHELL [59,60,61].

To reproduce the structure of F and Ne isotopes around N = 20, the model space includes not only the sd-shell orbits, but also two or all pf-shell orbits. On this basis, the size of the model space can be further determined by restricting the number of neutrons of cross-shell excitation. Then, the interactions involving two major shells are divided into three parts: the sd-shell part, the pf-shell part, and the cross-shell part. The two former terms are based on well-known interactions in the sd- and pf-shell model space. Thus, the interactions constructed in the larger space can immediately describe the N~Z nuclei. As for F and Ne nuclei around N = 20, the cross-shell interactions can be crucial, since both the sd- and pf-shell configurations contribute significantly to these nuclei. In the following subsections, the details of the three interactions will be described.

2.1. SDPF-M

The SDPF-M interaction was proposed by Utsuno et al. for studying neutron-rich nuclei with nearly 20 neutrons [44]. The interaction takes ¹⁶O as the inert core, and considers the model space consisting of the full sd shell and the 0f_7/2 and 1p_3/2 orbits; it uses the USD interaction [62] and the KB interaction [63] for the sd-shell and pf-shell parts, respectively. More importantly, its cross-shell part starts from the MK interaction [64].

In detail, the MK interaction is close to a G-matrix interaction in character, except that MK has a repulsive triplet-odd central force. Warburton et al. [65] modified the eight TBMEs <0f_7/20d_3/2|V|0f_7/20d_3/2> _{J = 2–5,T = 0,1} of the MK interaction to build the cross-shell part of the SDPF interaction. The modified MK interaction is exactly the adopted cross-shell interaction of the SDPF-M interaction.

On the basis of the interactions—including the USD, KB, and modified MK interactions—two important modifications are performed: On the one hand, in order to reproduce the unbound property of ²⁶O, modifications are added to the monopole interactions [66]. On the other hand, the pairing matrix elements of the USD interaction are reduced, as it implicitly includes the effects of the pf shell.

The finally obtained SDPF-M was successfully applied to study the exotic properties of the O [44], F [67], Ne [4,16,44], Na [68], Mg [8,44], and Si [44,69] isotopes, such as the unbound structure of ^26,28O and the bound structure of ^32,34Ne. This interaction did well in describing the nuclei with N ~ 20, and was used to study the evolution of the N = 20 shell gap [44].

2.2. SDPF-MU

The SDPF-MU interaction [53] is based on the USD interaction, the GXPF1B interaction [70], and the V_MU interaction [71]. More precisely, its sd-shell part starts from the USD interaction, with the monopole part for the 0d_3/2 and 0d_5/2 orbits modified in the same way as that in the SDPF-M interaction. The GXPF1B interaction is used for the pf-shell part, whereas the TBMEs <0f_7/20f_7/2|V|0f_7/20f_7/2> _{J = 0,2} are those from the interaction KB3 [72], so as to better describe the nuclei around N = 22. Finally, the cross-shell part is composed of the V_MU and the M3Y spin-orbit force [73]. To be clearer, V_MU is the monopole-based universal force, including the Gaussian central force and the π + ρ tensor force. SDPF-MU has been used in the exotic F [3], Mg [55], Si [53,56], and S [53] isotopes [53,56].

2.3. SDPF-U-SI

The SDPF-U/SDPF-U-SI interaction [54] proposed by Nowacki et al. in 2009 is mainly independent of the two interactions mentioned previously. Its proposition aims firstly at nuclei with 8 ≤ Z ≤ 20 and 20 ≤ N ≤ 40. The model space comprises the full sd shell for the protons and the full pf shell for the neutrons. Thus, nuclei with around 28 neutrons, along with the neutron excitation across the N = 28 shell gap, are the main focus. This is actually based on the SDPF-NR interaction [74,75], which takes the USD interaction, KB’ interaction [72], and G-matrix of Kahana, Lee, and Scott [76] as the sd-shell interaction, pf-shell interaction, and cross-shell interaction, respectively. Iterations are realized to derive the proper monopole changes with the help of the newly discovered experimental data. For instance, after iterations, SDPF-U-SI is able to reproduce the level of the 3/2₁⁻ state of ³⁵Si and the newly discovered spectrum of ⁴¹Ca [54].

When applying the three interactions involving the orbits of two major shells, the center-of-mass (c.m.) correction is also considered. This work uses the model of Gloeckner and Lawson [77], which writes the Hamiltonian as H^’ = H_SM + β_c.m.H_c.m. in order to decouple the original Hamiltonian (H_SM) and the c.m. Hamiltonian (H_c.m.). According to [51], β_c.m. = 10 is large enough in the calculations of low-lying states, and is used in this work.

In sum, the three interactions are proposed in different ways. The model space of SDPF-MU is the same as that of SDPF-U-SI, while it is larger than that of SDPF-M. Moreover, SDPF-MU and SDPF-U-SI are proposed primarily for nuclei with N ~ 28. The two interactions have been compared in the calculations of ⁴²Si. Results showed that CISM calculations with SDPF-MU agreed better with existing data [56]. Nevertheless, in the calculations of nuclei around N = 20, the best interaction is not evident. Despite the differences, all of the interactions showed good performance for nuclei in the sdpf region. With the three interactions, this paper focuses on the effects of the cross-shell excitation on describing F and Ne nuclei around N = 20.

3. Results and Discussions

In this work, we performed CISM calculations to derive the binding energies and energy spectra of F and Ne neutron-rich isotopes. We started with three interactions—SDPF-M, SDPF-MU, and SDPF-U-SI—separately, in order to examine the effects of the cross-shell excitation on the ground-state properties and excitation energies of F and Ne isotopes near N = 20. The CISM calculations were carried out in three cases: with cross-shell excitation of the least neutrons (no for positive states and one for negative states, labeled as (0–1)ħω), with cross-shell excitation of at most two neutrons beyond the first case (labeled as (0–3)ħω), and with cross-shell excitation of at most four neutrons beyond the first case (labeled as (0–5)ħω). There are a total of nine groups of theoretical results for all of the involved nuclei.

In this section, the nine groups of ground-state properties and excitation energies are compared with experimental results in order to analyze the contributions of the cross-shell excitation, and then the results of configuration-mixing and occupancy are given to illustrate the analysis; furthermore, the shell structure and evolution described by the three interactions are discussed in order to complete the analysis.

3.1. Ground-State Properties

The foremost task of the theoretical study on F and Ne isotopes around N = 20 is to reproduce and explain the location of neutron drip lines by analyzing the separation energies of one neutron (S_n) and two neutrons (S_2n). Normally, if the S_n (S_2n) of a nucleus is negative, it will be single (double)-neutron(s) unbound. Thus, both the S_n and S_2n of a bound nucleus should be positive. In addition, if adding one or two neutrons to a bound nucleus results in unbound nuclei, it can be speculated that such bound nuclei are located in the neutron drip line.

Table 1. The S_n and S_2n values (in MeV) of ^27–33F and ^28–36Ne nuclei calculated by CISM, in comparison with experimental data from the Atomic Mass Evaluation Table 2020 [14].

	Exp. (Error)	SDPF-M (0–1)ħω	SDPF-M (0–3)ħω	SDPF-M (0–5)ħω	SDPF-MU (0–1)ħω	SDPF-MU (0–3)ħω	SDPF-MU (0–5)ħω	SDPF-U-SI (0–1)ħω	SDPF-U-SI (0–3)ħω	SDPF-U-SI (0–5)ħω
Sn (27F)	1.610 (0.060)	0.886	2.437	2.659	1.114	3.200	3.934	2.402	3.027	3.154
Sn (28F)	−0.199 (0.006)	0.157	−0.479	−0.584	0.003	−0.796	−1.113	−0.090	−0.541	−0.622
Sn (29F)	1.320 (0.540)	−1.762	0.116	1.498	−0.980	0.908	2.619	0.152	1.465	2.016
Sn (30F)	/	0.830	−0.799	−1.217	1.075	−1.005	−1.774	0.240	−0.362	−0.652
Sn (31F)	/	−0.502	0.999	1.005	0.117	1.491	1.845	1.325	1.232	1.200
Sn (32F)	/	0.178	−1.423	−1.787	−0.222	−2.015	−2.689	−0.095	−0.860	−1.077
Sn (33F)	/	−0.851	0.535	0.564	−0.140	0.895	1.067	0.537	0.636	0.675
Sn (28Ne)	3.820 (0.160)	2.607	4.477	4.824	2.500	5.328	6.359	4.086	4.861	5.060
Sn (29Ne)	0.970 (0.200)	0.908	0.005	−0.091	1.111	0.137	−0.410	0.592	0.059	−0.081
Sn (30Ne)	3.190 (0.290)	−0.771	2.225	3.562	−0.274	1.757	3.987	1.032	2.532	3.663
Sn (31Ne)	0.170 (0.130)	2.864	0.304	−0.474	3.657	0.998	−1.179	2.441	0.681	−0.478
Sn (32Ne)	/	0.520	2.247	2.352	1.247	2.616	3.763	2.239	2.990	3.319
Sn (33Ne)	/	1.495	−0.969	−1.334	1.598	−1.216	−2.224	0.723	−0.127	−0.446
Sn (34Ne)	/	−0.085	1.483	1.540	0.970	2.315	2.468	2.264	2.173	2.186
Sn (35Ne)	/	0.067	−1.629	−1.902	−1.514	−2.470	−3.016	−0.512	−0.943	−1.032
Sn (36Ne)	/	−0.110	1.156	1.143	0.721	1.211	1.327	0.798	0.822	0.830
S2n (27F)	2.340 (0.150)	1.276	2.746	2.975	2.038	4.307	5.079	3.323	4.217	4.353
S2n (28F)	1.410 (0.060)	1.043	1.958	2.075	1.117	2.404	2.821	2.312	2.486	2.532
S2n (29F)	1.130 (0.540)	−1.605	−0.363	0.914	−0.977	0.112	1.506	0.062	0.924	1.394
S2n (30F)	/	−0.932	−0.683	0.281	0.095	−0.097	0.845	0.392	1.103	1.364
S2n (31F)	/	0.328	0.200	−0.212	1.192	0.486	0.071	1.565	0.870	0.548
S2n (32F)	/	−0.324	−0.424	−0.782	−0.105	−0.524	−0.844	1.230	0.372	0.123
S2n (33F)	/	−0.673	−0.888	−1.223	−0.362	−1.120	−1.622	0.442	−0.224	−0.402
S2n (28Ne)	5.320 (0.130)	3.766	5.367	5.719	4.415	6.404	7.378	5.382	6.183	6.375
S2n (29Ne)	4.790 (0.170)	3.515	4.482	4.733	3.611	5.465	5.949	4.678	4.920	4.979
S2n (30Ne)	4.160 (0.280)	0.137	2.230	3.471	0.837	1.894	3.577	1.624	2.591	3.582
S2n (31Ne)	3.360 (0.310)	2.093	2.529	3.088	3.383	2.755	2.808	3.473	3.213	3.185
S2n (32Ne)	/	3.384	2.551	1.878	4.904	3.614	2.584	4.680	3.671	2.841
S2n (33Ne)	/	2.015	1.278	1.018	2.845	1.400	1.539	2.962	2.863	2.873
S2n (34Ne)	/	1.410	0.514	0.206	2.568	1.099	0.244	2.987	2.046	1.740
S2n (35Ne)	/	−0.018	−0.146	−0.362	−0.544	−0.155	−0.548	1.752	1.230	1.154
S2n (36Ne)	/	−0.043	−0.473	−0.759	−0.793	−1.259	−1.689	0.286	−0.121	−0.202

shows the theoretical and experimental [14] separation energies of F and Ne nuclei around N = 20. For the nuclei with experimental separation energies listed in Table 1, the inclusion of the (2–3)ħω cross-shell excitation can globally improve the calculation accuracy. Figure 1 illustrates the root-mean-square errors (RMSEs) of the calculated S_n and S_2n values of nuclei ^27–29F and ^28–31Ne, compared to experimental data. It can be seen that the RMSEs are smaller after considering the (2–3)ħω cross-shell excitation. However, the cross-shell excitation part of the SDPF-MU interaction should be improved, as the RMSEs of S_n in the nuclei of ^27–29F and ^28–31Ne are larger in the (0–5)ħω case than those in the (0–3)ħω case.

It should be noted that the description improvement of the ground-state properties caused by the cross-shell excitation is rather evident in ²⁹F and ³⁰Ne. The S_n and S_2n values of the two nuclei with N = 20 are shown in Figure 2. In detail, CISM with the least cross-shell excitation underestimates the energies necessary to separate one or two neutrons from ²⁹F and ³⁰Ne. Then, the inclusion of higher ħω states leads to the increase in the S_n and S_2n values of ²⁹F and ³⁰Ne, typically making the theoretical values closer to the observation results. For instance, in the calculation of S_n (²⁹F) with SDPF-M, the further inclusion of (2–3)ħω states can take it from negative value to positive value, while that of (4–5)ħω states allows it to approach the experimental results further. In total, a model space including (0–5)ħω cross-shell excitation is suggested to sufficiently describe the ground-state properties of ²⁹F and ³⁰Ne within the framework of CISM.

The qualitative description of the ground states in the nuclei near the F and Ne drip line requires the consideration of the neutron cross-shell excitation beyond the (0–1)ħω case. Based on the S_n and S_2n values, the neutron drip lines are finally determined. In the (0–1)ħω case, CISM calculations using SDPF-M, SDPF-MU, and SDPF-U-SI predict 28, 31, and 33, respectively, as the largest atomic number of bound F isotopes. However, according to recent experiments, the heaviest bound F isotope is ³¹F [12]. In the (0–3)ħω case, both SDPF-M and SDPF-U-SI succeed in reproducing the drip line of F; the results of Ne are rather similar. Only with (0–1)ħω do CISM calculations with SDPF-M predict an inner drip line of Ne, while those with SDPF-U-SI predict outer. Then, in the case of (0–3)ħω, the results of the two interactions are consistent with the observations. Though SDPF-MU with (0–1)ħω can reproduce the location of the drip line of F and Ne, it wrongly gives unbound results of ²⁹F and ³⁰Ne, while the neutron cross-shell excitation corrects the descriptions.

Nevertheless, whether or not CISM with (0–5)ħω can give a significantly better theoretical description of F and Ne chains than that with (0–3)ħω requires further study. For instance, the three interactions with (0–5)ħω all incorrectly result in the unbound properties of ^29,31Ne, while those with (0–3)ħω do not, which should not happen in principle. Improvements are needed in the cross-shell excitation parts of the three interactions.

The cross-shell excitation is also fundamental to reproducing the ground-state spin of F and Ne nuclei around N = 20.

Table 2. The experimental and theoretical ground-state spin of ^27–31F and ^28–34Ne (experimental results are from [78]).

Nucleus	Exp.	SDPF-M (0–1)ħω	SDPF-M (0–3)ħω	SDPF-M (0–5)ħω	SDPF-MU (0–1)ħω	SDPF-MU (0–3)ħω	SDPF-MU (0–5)ħω	SDPF-U-SI (0–1)ħω	SDPF-U-SI (0–3)ħω	SDPF-U-SI (0–5)ħω
27F	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+
28F	4−	6–	4–	4–	4–	4–	4–	4–	4–	4–
29F	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+
30F	/	3+	4–	4–	3+	3+	4–	3+	4–	4–
31F	/	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+	5/2+
28Ne	0+	0+	0+	0+	0+	0+	0+	0+	0+	0+
29Ne	3/2–	7/2–	7/2–	3/2+	3/2–	3/2–	3/2–	3/2–	3/2–	3/2–
30Ne	0+	0+	0+	0+	0+	0+	0+	0+	0+	0+
31Ne	3/2–	3/2+	3/2–	3/2–	3/2+	3/2+	3/2–	3/2+	3/2+	3/2+
32Ne	0+	3–	0+	0+	0+	0+	0+	0+	0+	0+
33Ne	/	3/2+	3/2–	3/2-	3/2+	3/2+	3/2–	3/2–	3/2–	3/2–
34Ne	/	0+	0+	0+	0+	0+	0+	0+	0+	0+

shows the ground-state spin of ^27–31F and ^28–34Ne, given by experiments or CISM calculations. In the (0–1)ħω case, the CISM calculation with the SDPF-M interaction cannot reproduce the ground-state spin of ²⁸F and ^29,31,32Ne, while those with SDPF-MU or with SDPF-U-SI fail, in ³¹Ne. In the (0–3)ħω case, the CISM calculation with SDPF-M corrects the ground-state spin of ²⁸F and ^31,32Ne. Then, in the (0–5)ħω case, the CISM calculations using the SDPF-MU interaction correct the ground-state spin of ³¹Ne calculated with (0–1)ħω and (0–3)ħω. Therefore, the cross-shell excitation is necessary in order to calculate the ground-state spin in F and Ne nuclei around N = 20, and more than (0–3)ħω should be considered. Furthermore, it is suggested that 4^– and 3/2⁻ should be the ground-state spin of ³⁰F and ³³Ne, respectively.

In total, the cross-shell excitation is crucial to describing the ground states of F and Ne isotopes around N = 20. Furthermore, (0–5)ħω excitation could be necessary to the ²⁹F and ³⁰Ne nuclei. The cross-shell excitation parts of the three interactions may not be well fixed yet.

3.2. Excitation Energies

Since many of the F and Ne nuclei around N = 20 are weakly bound, the experimental results of the excited states are rather rare. To date, there is no information about the excited states of the F (Ne) isotopes heavier than 29 (32) in the Nudat2 database [79]. Therefore, this paper focused on selected low-lying states.

Table 3. The experimental and theoretical energy levels (in MeV) of the excited state in F and Ne nuclei (the experimental results are from [79], except those for ^27,29F, which are from [67]).

Nucleus	State	Exp.	SDPF-M (0–1)ħω	SDPF-M (0–3)ħω	SDPF-M (0–5)ħω	SDPF-MU (0–1)ħω	SDPF-MU (0–3)ħω	SDPF-MU (0–5)ħω	SDPF-U-SI (0–1)ħω	SDPF-U-SI (0–3)ħω	SDPF-U-SI (0–5)ħω
27F	1/21+	0.915 (0.012)	1.980	1.297	1.184	2.176	1.952	1.482	1.923	1.770	1.715
	3/21+	/	2.626	2.732	2.675	2.847	3.571	3.452	2.872	3.218	3.293
29F	1/21+	1.080 (0.018)	3.553	0.952	0.797	3.669	2.833	1.515	2.736	2.116	1.294
	3/21+	/	6.407	2.246	2.294	6.349	5.047	3.442	6.660	3.829	2.816
31F	1/21+	/	0.648	0.542	0.628	0.311	0.680	0.796	0.176	0.340	0.390
	3/21+	/	1.868	1.701	2.019	1.821	2.342	2.618	1.447	1.681	1.764
28Ne	21+	1.304 (0.003)	1.553	1.401	1.362	1.830	2.110	2.196	1.804	1.947	1.983
	41+	3.010 (0.006)	2.952	2.770	2.757	3.302	3.512	3.779	3.342	3.458	3.510
30Ne	21+	0.792 (0.004)	1.531	0.873	1.028	1.912	1.997	1.808	1.901	1.760	1.201
	41+	2.235 (0.012)	2.444	1.945	2.364	2.817	2.954	3.359	2.879	2.935	2.535
32Ne	21+	0.722 (0.009)	0.768	0.682	0.841	0.857	1.262	1.482	0.659	0.858	0.932
	41+	/	1.763	1.778	2.086	2.120	2.784	3.253	1.762	2.092	2.241
34Ne	21+	/	0.521	0.838	0.969	0.690	1.110	1.403	0.476	0.616	0.655
	41+	/	1.387	2.028	2.267	1.922	2.670	3.158	1.424	1.679	1.747

shows the experimental and theoretical energy levels of the excited states in F and Ne nuclei around N = 20.

For F isotopes, the odd-A nuclei are our first focused. According to the shell model, such nuclei should have a 5/2⁺ ground state caused by a valence proton on the π0d_5/2 orbit. When the valence proton moves to the π1s_1/2 (π0d_3/2) orbit, the spin of the excited states should be 1/2⁺ (3/2⁺). Although a state is rarely a pure, single-particle state in the view of CISM, the order 5/2⁺, 1/2⁺, and 3/2⁺ holds. This is because the π0d_5/2 configuration dominates on the 5/2⁺ ground state, and similar explanations work for the other cases.

In all cases discussed in this paper, the theoretical ground-state spins of odd-A F isotopes are 5/2⁺. For the first 1/2⁺ state, the cross-shell excitation is shown to be necessary to the CISM calculation. It can be seen from Table 3 that the theoretical energy levels of 1/2₁⁺ of ^27,29F are much higher than the existing experimental levels. From (0–1)ħω to (0–3)ħω, those energy levels become lower; then, from (0–3)ħω to (0–5)ħω, they decrease again. It can be noted that the latter decreases are less significant than the former. Overall, the inclusion of cross-shell excitation beyond the (0–1)ħω states makes the theoretical levels of the 1/2₁⁺ of odd-A ^27,29F approach the experimental results significantly. Considering the experimental errors and the possible deficiency of the interactions involved, it is suggested that (0–3)ħω is insufficient to reproduce those 1/2₁⁺ states. In particular, the results of the 1/2₁⁺ state in ²⁹F are illustrated in Figure 3; according to this figure, the necessity of (0–5)ħω is indicated in the interpretation of the energy spectrum of ²⁹F.

Theoretically, the three lowest lying states of even-mass Ne isotopes should be 0⁺, 2⁺, and 4⁺ by orders. The CISM with SDPF-M wrongly gives 3⁻ as the ground-state spin in the (0–1)ħω case, and corrects the result with (0–3)ħω. The levels of the 2₁⁺ and 4₁⁺ states in even-A ^28–32Ne can be seen in Table 3. CISM calculations of these levels with the three interactions are consistent with experimental results. It can be seen that the SDPF-M interaction reasonably reproduces these levels, with RMSE values less than 0.4 MeV. The levels of the 2₁⁺ state in ³⁰Ne are also presented in Figure 3. Including higher ħω states results in slight variations in the energy spectra of ²⁸Ne, while the inclusion of (2–3)ħω states helps SDPF-M to better describe the 2₁⁺ state in ³⁰Ne. For SDPF-U-SI, theoretical results with (0–5)ħω are the most consistent with the experimental results, both for the 2₁⁺ state in ³⁰Ne, and for the overall levels in even-A ^28–32Ne. However, it should also be noted that the global description accuracy of levels of SDPU-MU deteriorates with the enlargement of the model space. In sum, it is believed that the cross-shell excitation plays an important role in CISM calculations of Ne isotopes around N = 20.

The level of the first intruder state is also of interest. It was found that the levels of the first negative state of odd-A ^27–31F and even-A ^28–34Ne increase with the addition of ħω. Moreover, the level variation is more significant from the (0–1)ħω case to the (0–3)ħω case than from the (0–3)ħω case to the (0–5)ħω case.

Overall, the cross-shell excitation is necessary to well describe the excited states of F and Ne nuclei around N = 20, and it is suggested to consider (0–5)ħω in the calculations of ²⁹F and ³⁰Ne.

3.3. Configuration Occupancies

From Section 3.1 and Section 3.2, it is known that the cross-shell excitation is crucial to describe F and Ne nuclei around N = 20 within the CISM framework. This means that the N = 20 shell gap is reduced or vanished, and neutrons occupy the pf-shell orbits. This phenomenon can be illustrated with the configuration occupancy.

Table 4. The average neutron occupancies of F and Ne nuclei in the pf shell with N~20.

Nucleus	State	(0–1)ħω	(0–3)ħω SDPF-M	(0–3)ħω SDPF-MU	(0–3)ħω SDPF-U-SI	(0–5)ħω SDPF-M	(0–5)ħω SDPF-MU	(0–5)ħω SDPF-U-SI
27F	5/21+	0	0.795	0.543	0.236	0.951	0.877	0.300
29F	5/21+	0	1.497	0.646	0.457	2.354	1.660	0.930
29F	1/21+	0	1.972	1.003	1.226	2.716	2.302	1.967
29F	3/21+	0	1.994	1.356	1.971	2.730	2.330	2.127
31F	5/21+	2	3.413	2.689	2.346	4.030	3.452	2.519
33F	5/21+	4	5.337	4.669	4.234	5.745	5.214	4.298
28Ne	01+	0	1.090	0.570	0.261	1.307	1.089	0.377
30Ne	01+	0	1.857	0.700	0.669	2.479	1.958	1.619
30Ne	21+	0	1.965	0.690	1.330	2.533	2.192	2.053
30Ne	41+	0	1.981	0.589	1.016	2.404	1.953	2.050
32Ne	01+	2	3.705	2.693	2.385	4.038	3.515	2.660
34Ne	01+	4	5.344	4.566	4.204	5.654	5.101	4.268

shows the average neutrons that occupy the pf-shell orbits in the odd-F and even-Ne isotopes. For the N = 18 nuclei, there are large possibilities of neutron excitation across the N = 20 shell gap; thus, the interactions without cross-shell excitation are insufficient to reproduce the ground-state properties of ²⁷F and ²⁸Ne. Similar explanations can be found in the nuclei with N = 22 or 24. Thus, the overall description of the ground states in F and Ne nuclei near the neutron drip lines—including the determination of the neutron drip lines—requires the cross-shell excitation.

Among the nuclei investigated in this paper, ²⁹F and ³⁰Ne are the most special, as the sd shell is fully occupied, without cross-shell excitation. However, results in a larger model space show that neutrons in ²⁹F and ³⁰Ne occupy the pf-shell orbits. CISM calculations using the SDPF-M interaction with the (0–5)ħω case indicate that more than two neutrons in ²⁹F and ³⁰Ne excite across the N = 20 shell gap on average. This explains the necessity of a model space including (0–5)ħω excitation in describing ²⁹F and ³⁰Ne.

In detail,

Table 5. Percentages (in %) of the configurations in the ground states.

Nucleus	Configuration	(0–1)ħω	(0–3)ħω SDPF-M	(0–3)ħω SDPF-MU	(0–3)ħω SDPF-U-SI	(0–5)ħω SDPF-M	(0–5)ħω SDPF-MU	(0–5)ħω SDPF-U-SI
29F	π0d5/2ν(0d3/2)4	1	24.98	65.81	76.24	8.53	30.65	56.14
	π0d5/2ν(0d3/2)2 (0f7/2)2	0	43.65	13.83	9.53	36.16	20.56	15.91
	π0d5/2ν(0d3/2)2 (1p3/2)2	0	15.41	4.42	2.93	15.14	14.28	7.92
30Ne	π(0d5/2)2 ν(0d3/2)4	1	6.55	57.88	59.73	2.98	17.45	22.58
	π(0d5/2)2ν(0d3/2)2 (0f7/2)2	0	45.4	12.88	11.08	32.53	17.94	17.49
	π(0d5/2)2ν(0d3/2)2 (1p3/2)2	0	10.27	3.41	3.74	8.56	10.33	10.21

gives the configuration occupancies of the ground states in ²⁹F and ³⁰Ne. In the (0–1)ħω case, ²⁹F and ³⁰Ne are pure π0d_5/2 ν(0d_3/2)⁴ and π(0d_5/2)² ν(0d_3/2)⁴ configurations, respectively. However, results considering the cross-shell excitation show that the π0d_5/2 ν(0d_3/2)² (0f_7/2)² configuration is strongly mixed with π0d_5/2 ν(0d_3/2)⁴ in ²⁹F. CISM calculations with SDPF-M indicate that the π0d_5/2 ν(0d_3/2)² (0f_7/2)² configuration is actually dominant in the ground state of ²⁹F. In such a configuration, two neutrons of ²⁹F (³⁰Ne also) excite from the ν0d_3/2 orbit to the ν0f_7/2 orbit. The configurations π0d_5/2 ν(0d_3/2)² (1p_3/2)² and π(0d_5/2)²ν(0d_3/2)² (1p_3/2)², which represent the excitation of a pair of neutrons to the ν1p_3/2 orbit, are also important in ²⁹F and ³⁰Ne, respectively. Therefore, it is necessary to consider the three kinds of configurations in ²⁹F and ³⁰Ne.

In conclusion, neutrons of F and Ne nuclei excite from sd-shell orbits to pf-shell orbits when the neutron number approaches 20. This phenomenon makes it necessary for F and Ne nuclei around N = 20 to include the cross-shell excitation. Taking ²⁹F and ³⁰Ne, for example, in both the ground state and excited states, the neutrons exciting to the pf-shell are in significant numbers, which explains why CISM calculations with (0–1)ħω or (0–3)ħω fail in describing ²⁹F and ³⁰Ne.

3.4. Shell Structure and Evolution

This section tries to explain why some CISM calculations, including (4–5)ħω states, are less consistent with experimental results. The shell structure and its evolution are compared among the three interactions via their effective single-particle energies (ESPEs) [80,81] and a monopole-frozen analysis [82].

For an orbit j, the ESPE can be expressed as [80,81]:

$${\epsilon }_j={\epsilon }_j^{core}+{\displaystyle \sum }_{j^{\prime }}{V}_{j{j}^{\prime }}\langle \psi \left|{\widehat{N}}_{j^{\prime }}\right|\psi \rangle$$

(1)

where ${\epsilon }_j^{core}$ represents the SPE of the j orbit regarding the inert core, $V_{j{j}^{\prime }}$ represents the monopole interaction [66] between the j and j^’ orbits, and $\langle \psi \left|{\widehat{N}}_{j^{\prime }}\right|\psi \rangle$ is the occupancy number of particles in the j^’ orbit. The monopole interactions are derived from the average of the diagonal TBMEs. In detail, $V_{j{j}^{\prime }}$ for a given isospin T is defined as:

$$V_{j{j}^{\prime },T}={\displaystyle \sum }_J\left[1-{\left(-1\right)}^{\left(2j-J-T+1\right)}\delta \left(j{j}^{\prime }\right)\right]\frac{2J+1}{\left(2j+1)(2j^{\prime }+1\right)}\langle j{j}^{\prime }\left|V\right|j{j}^{\prime }{\rangle }_{JT}$$

(2)

with J as the angular momentum and $\delta$ as Kronecker’s delta. When valence particles are added to orbits, the monopole interactions will change the SPEs.

To illustrate the shell evolution due to the addition of neutrons, Figure 4 shows the ESPEs in the ground states of Ne isotopes (the results of F isotopes are fairly similar), derived using the three interactions in the (0–5)ħω model space. It can be deduced that the three interactions give different interpretations of the shell structure and evolution. Firstly, the SDPF-M interaction indicates a rather strong mixing of the three orbits ν0d_3/2, ν1p_3/2, and ν0f_7/2, while ν1p_3/2, ν1p_1/2, and ν0f_7/2 orbits are strongly mixed in the other two interactions. With the increase in mass number, the relative position of ν0f_7/2 (ν1p_3/2) to ν1p_1/2 varies more slowly in SDPF-MU than in SDPF-U-SI. Such differences in shell structure suggest the necessity of improvements in the cross-shell interactions of at least two among the three interactions.

For a given nucleus, the shell evolution can also result from the excitation of valence particles [35,83], which is called the type Ⅱ shell evolution [83]. We employed monopole-frozen analysis to examine the effects of such shell evolution, as described by the three interactions. In detail, we substituted ESPEs for SPEs, removed the monopole interaction from the diagonal TBMEs, and performed CISM calculations for ²⁹F and ³⁰Ne in the (0–5)ħω model space. If the type Ⅱ shell evolution is not significant in the two nuclei, their excitation energies should not be significantly changed by the monopole-frozen calculations.

Figure 5 presents the energy spectra of the two nuclei derived before and after the monopole-frozen modification. Despite the distinct energy spectra, it can be seen that the monopole-frozen modification generates different effects on the selected states for different interactions. For instance, positive yrast states of ²⁹F and ³⁰Ne from all three interactions change little with monopole-frozen calculations, which means that they share similar shell structures to the ground states. After monopole-frozen modification, the negative states of ³⁰Ne from SDPF-MU and SDPF-U-SI interactions and the second 0⁺ state of ³⁰Ne from SDPF-M have changed a lot. Therefore, the three interactions provide different illustrations of the excitation of the ground state for both ²⁹F and ³⁰Ne. Furthermore, a more reliable and effective interaction to describe N~20 nuclei with extreme proton–neutron asymmetry is required.

The ESPEs and monopole-frozen effects of the three interactions provide various shell structures and evolutions for F and Ne nuclei near N = 20, indicating that they can hardly be consistent with one another. Therefore, the cross-shell parts of the three interactions are suggested to be improved in order to describe the neutron-rich nuclei with N~20 better.

4. Conclusions

In this work, three interactions—SDPF-M, SDPF-MU, and SDPF-U-SI—were applied to calculate the structure of neutron-rich F and Ne isotopes within the framework of the configuration–interaction shell model. This paper focused on the cross-shell excitation of F and Ne nuclei near N = 20, which are among the systems with the most proton–neutron asymmetry. As a result, the cross-shell excitation was shown to be necessary for the overall description of F and Ne isotopes around N = 20. For ²⁹F and ³⁰Ne, it is important to include the cross-shell excitation of (0–5)ħω. Considering the insufficiency of the three interactions in the (0–5)ħω model space to reproduce properties such as the bound structure of ^29,31Ne, it is also suggested to optimize the three interactions in order to better describe the cross-shell excitation near ²⁹F and ³⁰Ne in future works. This conclusion is supported by the investigation of ESPEs and a monopole-frozen analysis, which indicate the different illustrations of the three interactions on the shell structure and evolution in the target nuclear region.