Introduction

Exotic nuclear structures in drip-line nuclei have become a subject of great interest in recent years, as they are characterized by unique properties that exhibit significant differences from those of stable nuclei [1-5]. One of the most significant phenomena observed in these systems is the Thomas-Ehrman shift (TES) [6, 7]. This effect is most pronounced in nuclei close to the proton drip lines, where the balance between the strong force and the Coulomb force is delicate. States exhibiting TES effects are often weakly bound or unbound, which is characteristic of open quantum systems, while their neutron-rich mirror counterparts remain deeply bound, resulting in a large mirror energy difference (MED) in their isobaric states [6-11]. The large MEDs are attributed to their proximity to near-threshold effects, in which the continuum effects must be well treated. Consequently, a thorough comprehension of the Thomas-Ehrman shift is pivotal for elucidating the dynamics of weakly bound and unbound nuclear systems and understanding the mechanisms underlying mirror-symmetry breaking in mirror nuclei.

Two possible reasons exist for the states with large MED: external or internal characteristics. If the extended single-particle wave functions of weakly or unbound s- or p-waves are significantly occupied in the considered states, the large MED is of an external nature, as in the TES states [9, 12, 13]. The second possibility is related to configuration mixing (see Refs.[14, 15]), is of internal nature. In this case, the extended wave function is given via strong configuration mixing, in which a few nodal states of s or p waves are included in the calculations. These two external and internal effects are different but can be intertwined in complex ways. For instance, the inversion of ground states in the ¹⁶F and ¹⁶N mirror nuclei is primarily due to the unbound proton 1s_1/2 orbital, which can also be well described in GSM calculations within the configuration mixing framework [13, 16].

The sd-shell nuclei, situated at the boundary between the light and heavy nuclei, exhibit a wide range of nuclear structure phenomena that remain somewhat mysterious [17]. In recent years, these nuclei have been extensively studied using various experimental techniques [18-21]. A wealth of information on the Thomas-Ehrman shift has been gleaned from sd-shell proton drip-line nuclei, where numerous states exhibiting significant TES effects have been identified [8-10, 22]. For instance, the mirror pairs ¹⁸Ne/¹⁸O [8, 23] and ¹⁹Na/¹⁹O [22] serve as notable examples. For sd-shell nuclei, the TES is mainly driven by s-waves. Indeed, the proton 1s_1/2 orbital is weakly bound or unbound in proton drip-line nuclei, whereas the neutron 1s_1/2 orbital is well bound in their mirror neutron-rich nuclei.

Several theoretical models have been developed to probe the mirror asymmetry for mirror nuclei, such as the standard shell model (SM) [24-28], mean-field calculations [29, 30], and ab initio approaches [23, 31-34]. Within the standard SM calculations, weakly bound and unbound wave functions on eigenenergies are indirectly considered by phenomenologically adjusting the matrix element related to 1s_1/2 orbit [9, 12]. Mean-field calculations, such as the Skyrme-Hartree-Fock, have also been extensively employed in MED studies [29, 30]. However, these models involve parameters constrained by data [24, 29, 35]. In recent years, ab initio approaches, such as the ab initio valence-space in-medium similarity renormalization group, have also been applied to study MEDs of sd-shell nuclei [10, 11, 23, 31-34, 36], in which the extended many-body wavefunctions are partially described using a large number of HO spaces. Moreover, current theoretical calculations have pointed out that the TES is caused by the repulsive Coulomb interaction and the occupations of weakly bound or unbound s- or p-waves for the valence protons. However, detailed studies on the TES mechanism are lacking. In recent shell model calculations [28], the TES was investigated using the calculated spectroscopic factors. Moreover, in our previous work [36], we compared the results of the MED calculated using the shell model with spectroscopic factors and the ab initio valence-space in-medium similarity renormalization ground approach. The two models yielded similar results.

One of the major challenges in studying drip-line nuclei is accounting for the interplay between configuration mixing and continuum effects. The Gamow shell model (GSM) [14, 15, 23, 31, 37-42] has emerged as a powerful tool in this regard, as it provides a unified framework for describing the structure of nuclei close to the particle emission threshold and allows for an accurate understanding of the exotic properties of drip line nuclei. Based on the above situation, we employ the GSM to investigate the significant mirror symmetry breaking with large MED values and the mechanism behind it for the sd-shell nuclei, taking the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror partners as examples.

Method

The GSM is a multiconfiguration shell model framework that works in the picture of a core plus valence nucleons [39, 40, 43-45]. At the heart of the GSM lies the utilization of the one-body Berggren basis [46], which possesses bound, resonance, and scattering states generated by a finite-range potential, typically of Woods-Saxon (WS) type (see details in Ref. [39, 40, 46]). The GSM Hamiltonian matrix is characterized by complex symmetry [39, 40]. The overlap method, along with the Jacobi-Davidson method extended to complex-symmetric matrices, was adopted to diagonalize and identify many-body resonance eigenstates [39, 40, 47]. Consequently, the GSM calculation includes both the inter-nucleon correlations and continuum coupling [39, 40, 43].

The many-body Schrödinger equation of the GSM Hamiltonian can be solved within the so-called cluster orbital shell model (COSM) formalism [48] (see Refs. [40, 49, 50]). The GSM Hamiltonian in COSM coordinates reads [40, 49, 50]: (1)where Aval is the number of valence nucleons, μi is the effective mass of the nucleon, is represented by a one-body WS potential mimicking the inert core. is the residual inter-nucleon interaction, which is modeled by a pionless effective field theory (EFT) interaction [51, 52], in which only two-body contact terms up to next-to-next leading order are considered. The regularization approach adopted in Refs. [44, 53-56] is used. The last term embodies the recoil effects induced by the finite mass of the core M_c. The EFT interaction was optimized to reproduce the low-lying states of the selected nuclei.

In the present work, the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror partners are considered as examples. The doubly magic nucleus ¹⁶O was chosen as the inert core, and the s_1/2, p_1/2,3/2 and d_3/2,5/2 partial waves were represented by the Berggren basis, in which 40 discretization points were used in total for continuum states in each partial wave. The f_5/2,7/2 partial waves were treated using the HO basis, in which six HO states were adopted for each of the partial waves. To estimate the effects of higher orbitals, we compared the results after adding the g_7/2,9/2 partial waves under the HO basis. The difference in the binding and excitation energies of the states studied in the present work was less than 5 keV. Thus, higher partial waves are neglected. Only the Coulomb force is considered for the isospin non-conserving part of the GSM Hamiltonian. The contribution of the isospin-dependent part of the nuclear interaction to the TES is small, which is neglected in the present GSM calculations. The Hamiltonian used in Ref. [57] was adopted in this study. The calculated excitation energies of ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror partners are presented in Tables. 1 and 2, which show good agreements with experimental data [58]. In the following section, the mechanics of the mirror energies differences for the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror partners are investigated in detail.

Results

Our GSM calculations accurately describe the excitation energies of the low-lying states for the mirror partners ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O. To delve deeper into the significant mirror symmetry breaking observed in these mirror nuclei, we define the MED for a given state Jπ as , where and denote the negative and positive isospin projections , respectively, for a mirror pair. We have calculated MED values for mirror states in ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O, as presented in Tables 1 and 2, along with experimental data. It is observed that in most states, the GSM-calculated MED values for low-lying states in the mirror partners ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O align with the experimental data. Although the numerical differences in the excitation energies between the calculated and experimental values for some states are relatively large, they still exhibit the expected qualitative trend of MED. Meanwhile, the SM results are poor, demonstrating that coupling to continuum states plays an essential role in the study of mirror states bearing significant mirror symmetry breaking. However, an exception is noted for the state in the ¹⁸Ne/¹⁸O mirror nuclei, where our GSM calculations yield larger values than those of the experimental data. Both our GSM calculations and the experimental data highlight significant mirror symmetry breaking in the state of the ¹⁸Ne/¹⁸O mirror nuclei and the state of the ¹⁹Na/¹⁹O mirror nuclei, as evidenced by their large MED values.

To investigate the significant mirror symmetry breaking and the associated large MEDs, we began by calculating the average occupations of the low-lying states through the GSM. The focus is particularly on the s_1/2 and d_5/2 partial waves above the ¹⁶O core for the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror nuclei, as shown in Figs. 1 and 2. Notably, other partial waves such as d_3/2 and f_5/2,7/2 exhibit negligible occupations and are, therefore, excluded from these figures. The calculated average occupations reveal almost identical patterns for the mirror states within the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O pairs. Our GSM calculations further indicate that states exhibiting significant mirror symmetry breaking with large MED values also show significant occupancy in the s_1/2 partial waves, which are markedly higher than in their respective ground states. For instance, the occupations of the s_1/2 partial wave for the and states in the ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror pairs, respectively, are substantially greater than those of their ground states. Additionally, our calculations show that the of ¹⁸Ne/¹⁸O demonstrates notable s_1/2 partial wave occupations compared to the ground states, resulting in a large MED. In contrast, experimental data provide a smaller MED value, hinting at a complex structure of the states in ¹⁸Ne/¹⁸O that might not be fully captured by ¹⁶O plus valence particle picture.

Fig. 1Full-screen View

(Color online) The average occupation numbers for the s_1/2 and d_5/2 partial waves in the low-lying states of the ¹⁸Ne/¹⁸O mirror pair, calculated using the GSM above the ¹⁶O core

Fig. 2Full-screen View

(Color online) Similar to Fig. 1, but for low-lying states in ¹⁹Na/¹⁹O

Aligned with results from other theoretical frameworks, such as the standard shell model and the ab initio VS-IMSRG approach, our results indicate that the significant mirror symmetry breaking with large MEDs observed in mirror states stems primarily from the extensive occupation of the s_1/2 partial waves, which are weakly bound or unbound in the proton-rich nucleus but deeply bound in its mirror neutron-rich nucleus, called TES. However, a deeper understanding of the mirror state bearing significant mirror symmetry breaking and a large MED value is lacking. The GSM is a suitable model that properly treats both the inter-nucleon correlations and continuum coupling to describe the properties of dripline nuclei, including a precise description of the many-body wave function in the asymptotic regions [44, 57, 59, 60].

To elucidate the underlying mechanism of large MEDs, we conducted a detailed analysis of the radial density distributions of mirror states in ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O pairs using the GSM. The results allow us to systematically compare the radial distributions of valence protons in proton-rich nuclei and those of valence neutrons in their neutron-rich mirror counterparts. The results are shown in Figs. 3 and 4 for ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror partners, respectively. Our GSM results reveal that the states characterized by minor mirror symmetry breaking with small MEDs exhibit almost identical radial density distributions, which decline sharply in the asymptotic regions, such as the ground states of both ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O. This phenomenon is largely attributed to the dominance of d_5/2 partial waves, which are constrained within the nuclear region by high centrifugal and Coulomb barriers, despite the state being unbound. Conversely, GSM calculations depict the radial density distributions of the state of ¹⁸Ne and the state of ¹⁹Na as more extended in the asymptotic region than their neutron-rich counterparts ¹⁸O and ¹⁹O, respectively. This disparity stems from the non-existent centrifugal barrier for the s_1/2 partial wave, leading to a more pronounced distribution in the proton-rich nucleus owing to the weakly bound or unbound nature of the s_1/2 partial wave. A similar mechanism underlies the formation of halo nuclei, where the valence nucleons occupy weakly bound s- or p- partial waves, resulting in an extended density distribution owing to the minimal or absent centrifugal barrier [14, 49, 61]. In previous works, these exotic phenomena that result from the mechanism for the extension of the s- partial wave have been widely recognized. However, in reality, most theoretical models cannot directly provide the properties of this extension and its impact on physical phenomena when performing calculations, such as the results of our SM calculations within the single sd shell model space in Table 1, where the MEDs of all states for ¹⁸Ne/¹⁸O are quite small.

Fig. 3Full-screen View

(Color online) The calculated radial density distribution of valence protons and valence neutrons for low-lying mirror states in ¹⁸Ne/¹⁸O, respectively, above ¹⁶O inner core, using GSM

Fig. 4Full-screen View

(Color online) Similar to Fig. 3, but for low-lying states in ¹⁹Na/¹⁹O

However, mirror symmetry breaking can be calculated using the traditional SM by introducing further approximations and modifications. For example, traditional SM calculations using the USDC interaction give the MED of the mirror state of ¹⁹Na/¹⁹O as -225 keV, which is significantly smaller than the experimental data of -727 keV. Moreover, the MED of the mirror state of ¹⁹Na/¹⁹O is mainly caused by the difference in the optimized single-particle energy for valence protons and neutrons in the USDC interaction [28]. Nevertheless, since the calculated MED is significantly smaller than the experimental data, they simulated the TES arising from continuum coupling through , where the is spectroscopic factors of proton s_1/2 orbital and TES_sp is single-particle TES [28]. The approximate treatment of the TES provides the MED value of the mirror states close to the experimental data. Based on this, the relationship between TES and the extended s_1/2 partial wave in the mirror partner was determined indirectly. However, SM calculations cannot provide a direct description of the TES. Moreover, phenomenological adjustments to the matrix elements related to the proton s_1/2 orbital have also been employed in the SM to describe the indirect effects of weakly bound and unbound wave functions on eigenenergies. For nuclei, the strength of the effective SM interactions involving the loosely bound proton s_1/2 orbit is significantly reduced in comparison with their mirror nuclei to calculate the significant mirror symmetry [12], in which the reduction factors of the two-body matrix elements related to the proton s_1/2 orbital are evaluated using the Woods-Saxon potential. The large MEDs in ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O are well reproduced.

Unlike the SM calculation introduced above with the aforementioned corrections, our GSM calculations provide a more self-consistent direct calculation of the radial density distribution, yielding the expected MEDs without relying on other corrections and modifications. Moreover, our calculations directly reveal that the mirror states demonstrating significant mirror symmetry breaking with large MEDs possess many-body wave functions in proton-rich nuclei that are more extended than those in their neutron-rich mirror states, which further helps us understand the role of mirror symmetry breaking in shaping their properties.

The GSM Hamiltonian, as shown in Eq. (1) can be divided into nuclear interactions (encompassing core-nucleons and nucleon-nucleon interactions) and Coulomb interactions (including one-body Coulomb (1BC) interactions between the inner core and valence protons and two-body Coulomb (2BC) interactions between valence protons). We performed further calculations to dissect the contributions from different parts of the Hamiltonian, aiming to shed light on the underlying mechanisms in mirror states exhibiting significant mirror symmetry breaking with a large MED.

The computed energies for the low-lying mirror states in the pairs ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O, along with the experimental data [58], are shown in Figs. 5 and 6, respectively. To gain deeper insights, we also present the energy minus 2BC contribution (GSM-2BC) and energy minus 1BC and 2BC contributions (GSM-1BC-2BC) in proton-rich nuclei ¹⁸Ne and ¹⁹Na. Indeed, GSM-1BC-2BC also corresponds to the contribution of nuclear interactions. Within the isospin symmetry picture, the difference in mirror state energies should solely stem from Coulomb interactions, implying that the GSM-1BC-2BC values for a state in a proton-rich nucleus are the same as those for its mirror state in a neutron-rich nucleus.

Fig. 5Full-screen View

(Color online) Upper panel: calculated energies (GSM), energies minus the two-body Coulomb contribution (GSM-2BC), and energies minus one- and two-body Coulomb total contributions (GSM-1BC-2BC) of low-lying states of ¹⁸Ne using GSM, along with the energies of mirror states in ¹⁸O, with respect to the ¹⁶O inner core. The GSM results are also compared with the experimental data. Lower panel: the contribution for the calculated energy difference between the mirror states

Fig. 6Full-screen View

(Color online) Similar to Fig. 5, but for low-lying states in ¹⁹Na/¹⁹O

Our GSM calculation shows that the GSM-1BC-2BC values for the ground states of ¹⁸Ne and ¹⁹Na closely align with the computed ground-state energies of their neutron-rich counterparts, ¹⁸O and ¹⁹O, respectively. The results indicate the preservation of mirror symmetry in these ground states. Conversely, for the excited state in ¹⁸Ne/¹⁸O and the state in ¹⁹Na/¹⁹O, our GSM calculations showcase a deviation from this symmetry. To quantitatively examine this discrepancy, we introduced ΔE as the differential metric for significant mirror-symmetry breaking. ΔE encapsulates the disparity between the GSM-1BC-2BC values in the state of the proton-rich nucleus and the energy calculated for the corresponding state in the neutron-rich mirror nucleus, which read as . Here, Ψ_proton and Ψ_neutron correspond to the many-body wave functions of proton-rich and neutron-rich nuclei, respectively. The ΔE corresponds to the difference in the contribution of the nuclear interactions in the mirror state.

Our GSM calculations show that both ΔE and Coulomb interactions, including 1BC and 2BC, significantly influence the energy discrepancies observed in the mirror states. Predominantly, Coulomb interactions emerged as the dominant factor contributing to these differences. This is illustrated in the lower panels of Figs. 5 and 6, we detail the ΔE, 1BC, and 2BC contributions to the energy differences in the low-lying mirror states of ¹⁸Ne/¹⁸O and ¹⁹Na/¹⁹O mirror pairs. Our GSM results indicate that the energy differences in the ground states of ¹⁸Ne/¹⁸O primarily stem from Coulomb interactions, with ΔE contributing minimally. Furthermore, for ¹⁹Na/¹⁹O, the ΔE contribution was approximately 100 keV. Interestingly, we find varying contributions of ΔE, 1BC, and 2BC across different mirror states within each state. For instance, the states in ¹⁸Ne/¹⁸O exhibit a higher ΔE contribution and lower Coulomb interactions relative to their ground states. The heightened ΔE values underscore the distinct contributions of nuclear interactions to mirror-symmetry breaking in these systems, showcasing the complex interplay of forces that shape the energy landscapes of mirror nuclei.

In evaluating the MED, the ground-state energy of the mirror nuclei serves as the baseline, with the MED being determined by the discrepancy in the excitation energies of the corresponding mirror states. The energy difference of the ground states of the mirror nuclei was adopted as a reference, as illustrated by the red dashed lines in Figs. 5 and 6. The difference between the values of the ground and excited mirror states corresponds to the MED, highlighted by red arrows in these figures. The results revealed that both the ΔE values and Coulomb interaction exhibited significant variations across different mirror states, both contributing to the MED.

Furthermore, to validate our conclusion, we performed Gamow shell model calculations using an optimized Hamiltonian fitted to reproduce a series of selected experimental data for sd shell nuclei. Moreover, the optimized Hamiltonian has been employed to investigate the low-lying states in ²¹Al [62]and ²²Si [63], as well as isospin symmetry breaking in these nuclei. Although the calculated excitation energies exhibited slight differences, the MED results aligned with both the current GSM calculations and the established conclusions regarding the MED mechanism.

Summary

Based on the GSM calculations, in which both the inter-nucleon correlation and continuum coupling are properly treated, we deduce that significant mirror symmetry breaking in mirror states, leading to large MED values, arises from the occupation of weakly bound or unbound s_1/2 partial waves in the proton-rich nucleus, while its counterpart in the neutron-rich nucleus remains deeply bound. This dichotomy culminates in a more expansive radial density distribution for states within the proton-rich nucleus than for their mirror counterparts. Additionally, the difference in the radial density distributions between the mirror states implies disparate contributions from nuclear interactions, underscored by significant ΔE values, which further highlights the presence of mirror symmetry breaking. Moreover, states with an extended radial density distribution tend to yield smaller Coulomb contributions than ground states characterized by more localized distributions. This factor chiefly accounts for the reduced excitation energies in states influenced by the Thomas-Ehrman shift effect, thereby engendering substantial negative MED values in mirror states. Our GSM calculations corroborate that both nuclear and Coulomb interactions play crucial roles in manifesting the significant mirror-symmetry breaking associated with significant MED values.