Structure of 29F in the Rotation-aligned Coupling Scheme of the Particle-Rotor Model

Recent results from RIKEN/RIBF on the low-lying level structure of 29F are interpreted within the Particle-Rotor Model. We show that the experimental data can be understood in the Rotation-aligned Coupling Scheme, with the 5/2+ ground state as the bandhead of a decoupled band. In this picture, the energy of the observed 1/2+1 state correlates strongly with the rotational energy of the core and provides an estimate of the 2+ energy in 28O. Our analysis suggest a moderate deformation, epsilon_2 ~ 0.17, and places the 2+ in 28O at ~ 2.4 MeV.

The structure of exotic neutron-rich nuclei is one of the main science drivers in contemporary nuclear physics research. Our current knowledge of nuclear structure, towards the driplines, has clearly established that the paradigm of magic numbers and doubly magic nuclei as we know it near stability changes across the nuclear landscape [1]. It is expected that these changes in the underlying single-particle structure are intimately related to specific aspects of the effective nuclear force, specifically to its tensor and three-body (or higher) components.
Thus, a detailed mapping of shell evolution and collectivity at the limits of isospin becomes a key element to understand the atomic nucleus and all of its many-body intricacies. The Islands of Inversion at N=8, 20, and 40 provide dramatic examples with the underlying physics mechanism driven by the important role of the neutron-proton force [1][2][3][4]. The effect of isospin on the monopole average of the central and tensor components of the force affects the neutron effective single-particle energies (ES-PEs) in such a way that expected shell closures are quenched, opening the door for the collective degrees of freedom to become relevant in the low-lying excitation spectra of these systems, where single-particle excitations were anticipated.
The study of odd-A and odd-odd nuclei has traditionally been a tool of choice to disentangle the competition of single-particle and collective motion insofar as they can be regarded, at least a priori, as one or two valence nucleon(s) coupled to a core.
As nicely discussed in Ref. [5], a very appealing region to study is that near 28 O, which is today accessible experimentally and also theoretically with state-of-the-art large scale Shell Model calculations. The well known magic numbers, Z=8 and N=20, would anticipate 28 O as a doubly magic nucleus, however, the recent study of 29 F carried out at RIKEN/RIBF suggests otherwise [5]. As noted, the relatively low transition energy of the 1/2 + 1 state in 29 F (1.08 MeV) largely disagrees with Shell Model predictions restricted to the sd model space, ≈ 3.5 MeV. Based on their calculations using the SDPF-M effective interaction, the authors determined that the N= 20 shell gap is quenched for 29 F and concluded: "... extending the Island of Inversion to isotopes with proton number Z= 9." In this work we follow-up on this conclusion and ask whether the observed structure in 29 F is amenable to a description in terms of a collective picture [6], within the framework of the Particle-Rotor Model (PRM) [7,8].
The Hamiltonian of the system can be written as [7,8]: Here E Ω relates to the intrinsic level energies, I is the core moment-of-inertia, and H C the Coriolis coupling term given by where I ± and j ± the ladder operators for the total and single particle angular momenta respectively.
Given the conditions above and for small to moderate deformations ( 2 ∼ 0.15) the Coriolis matrix elements 1 (∼ 2 I j/I ) dominate over the intrinsic level spacings, (∆E Ω,Ω±1 ∼ ω 0 2 ), and a rotation-aligned coupling limit is anticipated [10,11]. In this case, the lowest-lying (yrast) band has spins I = j, j + 2, j + 4, ..., and energy spacings equal to that of the core; this type of band is referred to as a decoupled band. Specific to our case, the ground state is then predicted to be 5/2 + , for which a geometrical picture is shown on the left panel of Fig. 1. Figure 1: Schematic diagram to illustrate the vector coupling for the lowest states in 29 F. The black arrows represent the total momentum, I, while the blue arrows represent j and the red arrows the core rotation R.
As illustrated on the right panel, the excited 1/2 + 1 state must have, by necessity, anti-parallel coupling of j with the core rotation, R. 2 Therefore it follows that in the decoupled limit ( 2 → 0) the energy of the 1/2 + 1 state with respect to the ground state is proportional to the rotational energy of the core, E 2 + (core), and provides, in the case of 29 F, a proxy for the 2 + energy in 28 O. Since the splitting of the Nilsson multiplet is proportional to the quadrupole deformation, ∼ ω 0 2 , we expect a trade-off between E 2 + (core) vs. 2 , and thus a range of possible solutions matching the experimental results.
In Fig. 2 we compare PRM solutions as a function of the deformation, to a calculation of the core energy given by E 2 + = 3 2 /I with the moment of inertia calculated using the Migdal formula [12][13][14], using an isospin dependent expression of the pairing gap, ∆, from Ref. [15], adjusted to this region of the nuclear chart.
The values E 2 + and 2 where the two curves intersect defines a consistent solution to the problem. To estimate an uncertainty in the adopted solution we take into account an uncertainty of ∼ ± 13% in ∆ entering the calculation of I [16], and obtain the blue shaded band, from which E 2 + ≈ 2.4 ± 0.2 MeV and 2 ≈ 0.17 +0. 15 −0.2 .
A given state, |I, α , of angular momentum I and projection α onto the rotation axis (x-axis) has a wavefunction of the form: It can be shown [10,11], that in the rotation-aligned coupling limit, the amplitudes are given by the Wigner d-function evaluated at π/2, the angle between the symmetry and rotation axes: 2 The angle can be estimated from the semi-classical expression: cos θ =  For our adopted PRM solution, the square amplitudes for the lowest two states, 5/2 + and 9/2 + of the yrast band (α = j = 5/2) are given in Fig. 3 where they are compared to the limit above showing that the structure can be interpreted, indeed, as a decoupled band. Some geometrical and spectroscopic properties of the calculated low-lying levels are summarized in Table 1. If the conditions for a rotational aligned coupling scheme persist also in 30 F, the odd-neutron will occupy the 1 2 [330] level of the f 7/2 Nilsson multiplet. In complete analogy with the odd-A case, we expect a doubly-decoupled band structure [17] with spins I = ( j π + j ν ), ( j π + j ν )+2, ...., also following the core spacings. Thus, we predict the ground state of 30 F to be 6 − . Of course, it is also possible that the occupation of the deformation driving 1 2 [330] neutron level may polarize the core to a larger 2 and a strong coupling scheme be realized, with the Ω π + Ω ν configuration being favored and the |Ω π − Ω ν | nearby. In this limit the ground state will be 2 − . With the substantial difference in spin predicted for the ground state in these two coupling schemes, a measurement of the (unbound) ground-state resonance in 30 F will be interesting to illuminate our understanding.
In summary, the recent experimental results of Ref. [5] and Shell Model calculations suggest the extension of the N = 20 Island of Inversion to the Fluorine isotopes. We have shown that the low-lying excitation spectrum of 29 F can be understood in terms of a collective picture, with a level structure corresponding to the rotation-aligned coupling limit of the PRM. The Coriolis coupling effects on the proton d 5/2 Nilsson multiplet give rise to a (favored) decoupled band. Thus, the 5/2 + bandhead naturally emerges as the ground state, and the 1/2 + as the first excited state, with its excitation energy depending directly on the E 2 + (core). We have found a consistent solution at a deformation of 2 ≈ 0.17 +0. 15 −0.2 that suggests an excitation energy of the 2 + in 28 O at E 2 + ≈ 2.4 ± 0.2 MeV in line with the conclusions reached in Ref. [5] based on the SDPF-M effective interaction. PRM predictions for some spectroscopic observables were also presented. Similar conditions in 30 F would give rise to a πd 5/2 ⊗ ν f 7/2 double-decoupled structure with a predicted 6 − ground state.