Introduction

The development of radioactive ion beam facilities [1-3] has significantly advanced nuclear physics research, enabling studies of nuclei far from the β-stability line, where the evolution of the nuclear shell structure and the emergence of exotic excitation modes have garnered considerable attention [4-7]. A striking example is the evolution of the N=28 shell gap, a magic number that arises from strong spin-orbit coupling in the single-nucleon potential [8, 9], which drives the f_7/2 orbital significantly lower than the p_3/2 orbital. Experimental data reveal a gradual weakening of the N=28 shell gap in isotones lighter than ⁴⁸Ca. For instance, measurements of the β-decay half-lives of ⁴⁴S and ⁴⁵Cl revealed deviations from shell model predictions based on spherical configurations [10], indicating the weakening of the N=28 shell effect. Subsequent Coulomb excitation experiments observed low excitation energies of the states and enhanced electric quadrupole transition strengths in ^40,42,44S [11, 12]. Similar behavior has been reported in neighboring nuclei, such as ⁴²Si [13-15] and ⁴⁰Mg [16]. These observations indicate the onset of strong quadrupole collectivity in neutron-rich N=28 isotones with proton number Z<18, leading to the crossing of the ν1/2^-[321] component of the 2νp_3/2 orbital with the ν7/2^-[303] component of the 1νf_7/2 orbital. Consequently, several near-degenerate configurations with admixtures of 1f_7/2 and 2p_3/2 orbitals coexist at similar energies within these nuclei. In the case of ⁴⁴S, two coexisting low-lying 0⁺ states have been observed [17], suggesting a spherical-deformed shape coexistence. The onset of a deformed ground state in ⁴⁴S is supported by quantum-number-projected generator coordinate method (GCM) studies based on the Gogny force [18, 19] and a collective Hamiltonian study based on a relativistic energy density functional (EDF) [20]. These two theoretical studies reveal a general trend in shape evolution, that is, the predominant shape transitions from a γ-soft, moderately deformed configuration with to a strongly prolate shape with as the angular momentum increases up to J=4 in the ground-state band. This rotational band coexists with a strongly prolate-deformed state characterized by . The specific deformation parameter β₂ of the dominant configuration depends on the details of the employed EDF. The two-proton knockout reaction from ⁴⁶Ar identified the state as an isomeric state [21], which, in combination with shell-model calculations, was suggested to exhibit a strong prolate deformation. A half-life measurement revealed a hindered E2 transition with W.u. [22], supporting the interpretation of the state as a K=4 isomer. Shell-model studies further indicate that this state is predominantly characterized by the two-quasiparticle configuration [23].

The odd-mass neutron-rich sulfur isotope ⁴³S exhibits a more complex low-energy structure than ⁴⁴S owing to the interplay between the single-particle motion of the unpaired neutron and the collective excitations of the ⁴²S core. The mass measurements, combined with theoretical studies based on the shell model and relativistic mean-field (RMF) theory, suggested the coexistence of a prolate deformed ground state and an isomeric state in ⁴³S [24]. Subsequent g-factor measurements [25], along with shell-model calculations and the collective Hamiltonian approach based on the Gogny force, determined the spin-parity of the isomeric state as at an excitation energy of 320.5(5) keV. These studies also established the intruder nature of the ground state with K=1/2 [25], while initially suggesting that the isomeric state was quasispherical. However, later measurements of the spectroscopic quadrupole moment of the isomeric state yielded e fm² [26], which was significantly larger than the expected value for a single-particle state. This observation indicates the strong collective nature of the isomeric state, which is further supported by shell-model calculations [26]. The structure of ⁴³S was investigated using antisymmetrized molecular dynamics plus GCM (AMD+GCM), which predicted a prolate-deformed ground state, a triaxially deformed state, and an oblate-deformed excited band atop the state at low excitation energy [27]. In contrast, a recent shell-model study suggested that the prolate ground-state band coexists with a triaxial band built on the isomer and an excited prolate structure associated with the Kπ=5/2^- deformed orbital [28]. Meanwhile, the angular-momentum-projected variation-after-projection (AMP+VAP) approach suggests that the ground state and the state are dominated by K=1/2 and K=7/2, respectively, classifying the state as a high-K isomer [23]. Experimental lifetime measurements of excited states provide the first evidence of a doublet of states. Together with shell-model and AMD calculations, these results suggest the possible existence of three coexisting bands built upon the , , and states [29]. Furthermore, Coulomb excitation experiments have shown that the intraband B(E2) values for the transitions within the ground-state band and the isomeric band are large and nearly equal [30], which was also found in the study of the valence space in-medium similarity renormalization group [31].

Over the past decades, covariant density functional theory (CDFT) has achieved remarkable success in various areas of nuclear physics [32-38]. A key advantage of the CDFT is that Lorentz invariance imposes strict constraints on the number of parameters in the EDF. Moreover, the relativistic framework naturally accounts for the spin-orbit interaction, whereas time-odd fields are incorporated without introducing additional free parameters. This characteristic is particularly crucial for accurately describing odd-mass nuclei and their rotating systems. To restore the missing quantum numbers, including particle numbers and angular momentum, in the solution of CDFT and to consider the shape-mixing effect, the multi-reference covariant density functional theory (MR-CDFT) was developed [39-41] and successfully applied to study low-lying spectra in even-even nuclei with either triaxial or octupole shapes [42-45]. The MR-CDFT has also been applied to the studies of neutrinoless double-beta decay [46-51] and the low-lying states of hypernuclei [52-54].

Recently, the MR-CDFT was successfully extended to describe low-lying states in odd-mass nuclei [55]. In this work, we present a new development in the MR-CDFT for the low-lying states of odd-mass nuclei by mixing the configurations not only with different intrinsic quadrupole shapes, but also with different K quantum numbers. All the configurations are projected onto good particle numbers and angular momenta. This newly-developed framework offers an alternative and computationally efficient approach to account for the triaxiality effects in nuclear low-lying states, avoiding the need for full three-dimensional angular momentum projection [39], which is numerically demanding. The success of our framework is demonstrated through its application to the low-energy structure of ⁴³S. It is worth noting that a similar idea has been implemented in the projected shell model (PSM) to study the effect of K-mixing on isomeric states in even-even isotopes with N = 104 [56], although the effect of shape mixing was not included in that study. Our extended MR-CDFT enables us to identify the dominant mechanism responsible for the formation of the isomeric state, shedding new light on the interplay among shape coexistence, K-mixing, and isomerism in the low-energy structure of odd-mass nuclei.

The remainder of this paper is organized as follows. In Sect. 2, we present the extended framework of MR-CDFT for odd-mass nuclei. The results of the calculations for ⁴³S are discussed in Sect. 3. The conclusions of this study are summarized in Sect. 4.

The MR-CDFT for odd-mass nuclei

The MR-CDFT theory for the low-lying states of odd-mass nuclei was introduced in detail in Ref. [55]. Here, we only present a brief description of the extension of this theory, in which the wave functions of low-lying states are constructed as a mixing of configurations with different deformation parameter q and quantum number K,(1)Here, c is a collective label for (K, q), and α distinguishes states with the same angular momentum J. The basis function with quantum numbers (NZJπ) is given by(2)where and are projection operators that select components with angular momentum J and neutron(proton) number N(Z) [57].

The mean-field configurations for odd-mass nuclei are chosen as 1qp states,(3)where denotes a quasiparticle vacuum state with even number parity obtained through the false quantum vacuum (FQV) scheme [55] in the single reference (SR)-CDFT calculation starting from a relativistic EDF [34, 58]. In this scheme, the time-reversal invariance is preserved [59]. The quasiparticle creation operator α^† switches the number parity to odd. The index κ distinguishes different quasiparticle states. In this study, axial symmetry was assumed. In this case, each configuration is labeled with the quantum numbers Kπ, determined by the quantum numbers Ω^π of the blocked orbital. In the configuration-mixing calculation, only the lowest-energy quasiparticle configuration with a given Kπ is included for each deformed configuration. In addition to the configurations employed in Ref. [55], we also mixed the configurations with different K values.

The weight function in Eq. (1) is determined by the variational principles that lead to the following Hill-Wheeler-Griffin (HWG) equation [57, 60],(4)where the Hamiltonian kernel and norm kernel are defined by(5)with the operator representing and 1, respectively. In the present study, based on a covariant EDF, the mixed-density prescription is employed in the evaluation of the Hamiltonian kernel. Details of the calculation of the kernels in Eq. (5) can be found in Ref. [55].

The HWG equation (4) for a given set of quantum numbers (NZJπ) is solved in the standard way, as discussed in Refs. [40, 57]. This is done by first diagonalizing the norm kernel . A new set of basis functions is then constructed using the eigenfunctions of the norm kernel with eigenvalues larger than a pre-chosen cutoff value, which removes the possible redundancy in the original basis. The Hamiltonian is diagonalized using this new basis. In this way, the energies and the mixing weights of the nuclear states can be obtained. Since the basis functions are nonorthogonal to each other, one usually introduces the collective wave function as below(6)which fulfills the normalization condition. The distribution of over K and q reflects the contribution of each basis function to the nuclear state . With the mixing weight , it is straightforward to determine the observables of nuclear low-lying states, including the electric quadrupole moment Qs, magnetic dipole moment μ, and the E2 and M1 transition strengths. The strength of the Eλ (Mλ) transition from the initial state to the final state is determined by(7)where the configuration-dependent reduced matrix element is simplified as follows,(8)where represents either an electric (L=λ) or magnetic (L=λ+1) multipole operator and . The detailed formulas can be found in Ref. [55].

Results and discussion

In the calculation of the mean-field configurations, Dirac spinors for single nucleons were solved using a harmonic oscillator (HO) basis with a major shell number of N_sh = 10, with frequency MeV. Pairing correlations between nucleons are treated within the BCS approximation using a density-independent δ force with a smooth cutoff [39, 61]. The PC-PK1 parameterization was employed for the relativistic EDF [58]. Even though PC-PK1 is not the best EDF, it is very successful in various nuclear structure studies [34]. Therefore, it is natural to continue using this EDF parameter set in the illustrative study of the low-lying states of ⁴³S. In the calculation of the projected kernels , the number of mesh points in the interval [0, π] for the rotation angle β and the gauge angle φ were chosen as Nβ = 12 and Nφ = 7, respectively. These values are sufficient to achieve convergent results for ⁴³S.

Figure 1(a) presents the energies of mean-field states from the SR-CDFT calculation based on the FQV scheme [55] as a function of the quadrupole deformation parameter . A pronounced energy minimum appears on the prolate side with , with a second minimum on the oblate side with , suggesting that ⁴³S may exhibit coexisting prolate and oblate shapes in its low-lying states. This phenomenon can be understood from the Nilsson diagram of neutrons in ⁴³S, as shown in Fig. 1(b). One can observe large shell gaps or low level densities on both prolate and oblate sides around the Fermi energy. Moreover, the downward ν1/2^-[321] component of the 2νp_3/2 orbital crosses with the upward ν7/2^-[303] component of the 1νf_7/2 orbital at around the Fermi energy. This crossing leads to the population of valence neutrons from ν7/2^-[303] to ν1/2^-[321]. This result is consistent with the AMD+GCM calculation [27]. On the oblate side, the N=28 shell gap increases with |β₂|.

Fig. 1Full-screen View

(Color online) (a) Total energy of the mean-field states for ⁴³S from the SR-CDFT calculation based on the FQV scheme, as a function of the quadrupole deformation parameter β₂. (b) Nilsson diagram for neutrons in ⁴³S, with the Fermi energies indicated by the pink squares. The numbers around the lines are the Nilsson quantum numbers [57]

The wave functions of the mean-field states in Fig. 1(a) do not preserve the particle numbers or total angular momentum. By applying quantum-number projection operators onto the 1qp configurations with Kπ=1/2^-, 3/2^-, 5/2^-, and 7/2^-, c.f. Eqs. (2) and (3), one obtains the energies of the symmetry-conserved 1qp states in ⁴³S, as displayed in Fig. 2. The low-lying states from the shape-mixing calculation are plotted at their mean quadrupole deformation , which is defined as(9)The energies of symmetry-conserving states with angular momentum J increasing from 1/2 to 9/2, projected out from the configurations with Kπ=1/2^- as a function of the quadrupole deformation β₂ are displayed in Fig. 2(a). Similar to the energy curve of the mean-field calculation shown in Fig. 1(a), all the projected energy curves present two energy minima on the prolate and oblate sides, with . It is interesting to note the change in the energy ordering of states with different angular momenta as a function of the deformation, as shown in Fig. 2(a). For configurations with β₂>0.4, the energies of the projected states with ΔJ=1 follow the ordering of (1/2^-, 3/2^-, 5/2^-, 7/2^-, 9/2^-), which are consistent with the strong coupling limit of the particle-rotor model (PRM) [57]. In contrast, for weakly deformed configurations with β₂ < 0.3, the 1/2^- state rises rapidly from the bottom as β₂ decreases toward zero. When only the configurations of the oblate energy minima are considered, the energy ordering is (7/2^-, 3/2^-, 1/2^-, 9/2^-, 5/2^-). After mixing the configurations with different shapes but the same quantum numbers Kπ=1/2^-, the yrast states with the energy ordering of (3/2^-, 1/2^-, 5/2^-, 7/2^-) are mainly located around β₂=0.3.

Fig. 2Full-screen View

(Color online) Energies of symmetry-conserving 1qp states with different angular momenta J in ⁴³S for (a) Kπ = 1/2^-, (b) Kπ = 3/2^-, (c) Kπ = 5/2^-, and (d) Kπ = 7/2^-, as functions of the quadrupole deformation parameter β₂. The discrete low-lying states obtained by mixing configurations of different shapes, but with fixed Kπ values, within the GCM framework are shown at their mean quadrupole deformation , as defined in Eq. (9)

Figure 2(b), (c), and (d) display the energies of symmetry-conserving states with , projected out from the configurations with Kπ=3/2^-, 5/2^-, and 7/2^-, as well as the two lowest energy states from the shape-mixing calculation. It can be seen that the energy level 7/2^- dominated by the prolate configuration with Kπ=7/2^- is the yrast state among all the Jπ=7/2^- states. All states with the same quantum numbers Jπ, but with different Kπ can be mixed. The final states from the MR-CDFT calculation by mixing 1qp configurations with different deformations and Kπ values are plotted in Fig. 3(b). The distributions of the collective wave functions for the first two 3/2^-, 5/2^-, and 7/2^- states are shown in the bottom panels of Fig. 3. It is seen that the main features of the measured low-lying states of ⁴³S, cf. Fig. 3(a), are reasonably well reproduced only in the MR-CDFT calculations with the mixing of these two types of configurations.

Fig. 3Full-screen View

(Color online) Energy spectra and collective wave functions of low-lying states in ⁴³S from MR-CDFT calculations. Experimental data [25, 30] are displayed in panel (a). Panel (b) shows the results including configuration mixing among states with different shapes and K values, while panel (c) presents the results of calculations by mixing only the configurations with different quadrupole deformations β₂, where the blocked orbitals correspond to Kπ = 1/2^- and 7/2^-, respectively. The black and blue arrows indicate the B(E2) values (in units of e² fm⁴) and B(M1) values (in units of ), respectively. The states dominated by a prolate (oblate) configuration, with a small admixture of oblate (prolate) components, are labeled as “(Prolate)” and “(Oblate)”, respectively. The distributions of the collective wave functions for the first two states with J = 3/2, 5/2, and 7/2 are shown in panels (d, e), (f, g), and (h, i), respectively

Here, we summarize the main findings of the MR-CDFT study based on Fig. 3. The ground state () of ⁴³S is predominantly characterized by a prolate-deformed configuration, ν1/2^-[321], with Kπ = 1/2^-. This can be understood from Fig. 2(a), which shows the potential energy surfaces with Jπ=3/2^- projected from the configurations with Kπ=1/2^- present a pronounced global energy minimum around β₂=0.35 and a second minimum around β₂=-0.35. These energies are much lower than those with Jπ=3/2^- but projected from the configurations with Kπ=3/2^-. Thus, it is expected that the lowest Jπ=3/2^- state is dominated by configurations with Kπ=1/2^- instead of those with Kπ=3/2^-. In contrast, the state is dominated by an oblate configuration with and Kπ = 1/2^-, with a small admixture of a prolate configuration having Kπ = 3/2^-. In the AMD+GCM calculation [27], the state is dominated by an oblate configuration with Kπ = 3/2^-, followed by the and states, forming an oblate rotational band. As shown in Fig. 3(b), we predict a strong E2 transition between the and states, whereas the E2 transition from the to the state is much weaker. This behavior can be understood from Fig. 3(g), which shows that the state is dominated by a prolate-deformed configuration with Kπ = 5/2^-, in contrast to the oblate structure of the state.

The first excited state of ⁴³S, , shares a similar predominant configuration with the ground state and exhibits a strong E2 transition to the ground state. The MR-CDFT predicts e² fm⁴, as well as a sizable M1 transition, with , as shown in Fig. 3(b). In the AMD+GCM [27], these numbers are 180 e² fm⁴ and , respectively. The value of has not yet been measured, but experimental data are available for [29], which is underestimated by both methods. We find that this underestimation is due to the extension of the wave function of the state into the region of large prolate deformation with β₂ > 0.4, as shown in Fig. 3(d). Moreover, we observe two ΔJ = 2 rotational bands, namely (, , ) and (, , ), which are connected by strong E2 transitions. The predicted e² fm⁴ is slightly larger than the data 46(9) e² fm⁴ [30]. All these states are predominantly characterized by the prolate-deformed configuration ν1/2^-[321], originating from the spherical νp_3/2 orbital, with deformation and Kπ = 1/2^-, as shown in Fig. 1. The energy ordering of the (, , ) sequence follows the weak coupling limit of the PRM [57], which predicts a parabolic energy pattern with a minimum at .

Figure 3(h) shows that the state is dominated by the prolate deformed configuration ν7/2^-[303] with and Kπ = 7/2^-. Consequently, the decay of the state to the ground state (3/2^-) is strongly quenched because of ΔK = 3. Quantitatively, the predicted e² fm⁴, compared to the data 0.41 e² fm⁴ [25]. Consequently, the state was classified as a high-K isomer state, consistent with the conclusions of previous studies [23, 28]. A comparison between Figs. 3(b) and (c) reveals that the excitation energy of the state decreases significantly and becomes closer to the experimental value after incorporating 1qp configuration mixing with different Kπ values. In contrast, the oblate nature in Fig. 3(c) is pushed very high and is not shown in Fig. 3(b). As shown in Fig. 3(h), the isomeric state also contains a small admixture of an oblate configuration with Kπ = 1/2^-. The remaining discrepancy is expected to be reduced by explicitly including the triaxial deformation effect, as suggested by the AMD+GCM calculation [27].

Table 1 lists the spectroscopic quadrupole moments Q_s and magnetic dipole moments μ for ⁴³S obtained from the MR-CDFT calculations with 1qp configuration mixing involving different quadrupole deformations β₂ and Kπ values, in comparison with the results of AMD+GCM calculations [27] and available data for the isomer state with e fm² and μN. Overall, the results from MR-CDFT and AMD+GCM were similar, with both models reasonably reproducing the experimental data. Quantitatively, the Q_s values of the states in the ground-state band predicted by MR-CDFT are slightly smaller than those from AMD+GCM. Notably, the magnetic moments of the and states obtained using MR-CDFT are only approximately half of the values predicted by AMD+GCM. In particular, a significant discrepancy was found in the prediction for the state. Although the AMD+GCM calculation suggests that this state is dominated by an oblate configuration, the MR-CDFT result favors a prolate deformation, consistent with the latest shell-model calculations [28]. According to the MR-CDFT calculations, the state primarily originates from blocking the Ω^π = 5/2^- component of the spherical νf7/2 state at a deformation of β₂ = 0.3, as evidenced by the collective wave function shown in Fig. 3(g). This state serves as the bandhead of a rotational band built on a prolate shape, as illustrated in the low-lying spectrum of ⁴³S in Fig. 3(b). In contrast, AMD+GCM calculations [27] indicate that this excited band is dominated by the oblate component Ω^π = 3/2^- of the spherical νf_7/2 state. Interestingly, the MR-CDFT calculations also predict an oblate arising from blocking the Ω^π = 3/2^- orbital of the νf_7/2 state with an excitation energy of approximately 2.5 MeV. This state has a different structure from the oblate state, which is dominated by the configuration of Kπ=1/2^-, see Fig. 3(e). Moreover, as seen from Table 1, the spectroscopic quadrupole moment Qs and magnetic dipole moment μ of our state are close to those of the state predicted by the AMD+GCM approach. Whether the second state is predominantly prolate or oblate can be clarified by future measurements of its electromagnetic properties.

Summary

In this study, we extended the multireference covariant density functional theory (MR-CDFT) for odd-mass nuclei by incorporating particle-number and angular-momentum projections, along with the simultaneous mixing of quasiparticle configurations characterized by different quadrupole deformations and K quantum numbers. The effectiveness of this extended framework is demonstrated through its application to the low-lying states of ⁴³S, where the available experimental data on the energy spectra, electric quadrupole and magnetic dipole transition strengths, and electromagnetic moments are reproduced with reasonable accuracy.

Our calculations reveal a pair of prolate rotational bands with ΔJ = 2, built on the ground-state configuration ν1/2^-[321] (Kπ = 1/2^-), consistent with the weak-coupling limit of the particle-rotor model. A rotational band is also found based on the ν7/2^-[303] (Kπ = 7/2^-) configuration, corresponding to the isomeric state, which is identified as a high-K isomer. These findings are generally consistent with those from AMD+GCM and shell-model calculations, supporting the erosion of the N=28 shell gap. We also identify a state dominated by an oblate configuration with Kπ = 1/2^-, where the valence neutron occupies the ν1/2^-[330] orbital. Furthermore, we predict and states, which are dominated by a prolate configuration with Kπ = 5/2^- and an oblate configuration with Kπ = 3/2^-, respectively. This result is slightly different from the predictions of AMD+GCM with explicit inclusion of triaxiality, which suggests that the state is dominated by the oblate configuration with Kπ = 3/2^-. This discrepancy calls for further clarification through additional lifetime measurements.

It is worth emphasizing that the present framework is based on axially deformed quasiparticle configurations, with triaxial effects partially incorporated through explicit K-mixing. This makes it a computationally efficient alternative to previous approaches that fully account for triaxiality and require a three-dimensional angular-momentum projection. The improved efficiency enables systematic beyond-mean-field studies of shape coexistence and isomeric states in heavy deformed odd-mass nuclei, such as ²²⁹Th.