Introduction

The T_1/2=2.54-min Jπ=19/2^- isomer at 3.0 MeV in ⁵³Fe is unique in atomic nuclei in that it has a direct decay branch to the Jπ=7/2^- ground state, requiring a hexacontatetrapole electric transition E6 [1-3]. This was recently experimentally confirmed with improved precision [4], providing a sensitive probe for nuclear structures [5, 6]. More generally, this scenario provides unique insights into the highest-order shapes and correlations in nuclei, and tests the applicability of nuclear models under extreme conditions. In the present work, we focus on ab initio calculations with bare nucleon charges, which can explain both ⁵³Fe excited-state energies and electromagnetic decay transition rates.

In nature, a free photon has an angular momentum of 1 ħ, whereas in a high multipole transition, a single photon carries away a large angular momentum. Naively, the probability of emitting a high-spin photon should be very small. In ⁵³Fe, a single E6 photon brings a spin change as large as 6ħ. No other isolated physical system is known to exhibit high multipole or single-photon emission. By comparison, infrared spectra have revealed 6ħ absorption in solid hydrogen [7], and magneto-optical studies of atomic ⁸⁵Rb have reported 6ħ spontaneous emissions [8]. Thus, there is a small group of physical environments where high multipole radiation provides access to novel and exceptional physics.

In a recent experimental study by Palazzo et al. [4], the existence of E6 decay in ⁵³Fe was firmly established using empirical shell model calculations. However, Ref. [4] does not provide calculations of high multipole electromagnetic transition probabilities. This is because that the empirical calculations require both proton and neutron effective charges that cannot be determined at present by only one available set of experimental E6 transition rate data. In empirical models, different effective charges are used for different multipolarities of the transitions. To improve such ad hoc features, ab initio calculations that use realistic interactions and bare nucleon charges are required. This provides the prospect of gaining insight into rare high-multipole transitions and testing the ab initio method itself. The ab initio nuclear theory is at the forefront of the current nuclear structure studies. Over the past two decades, significant progress has been made in ab initio many-body methods [9-13] using nuclear forces based on chiral effective field theory [14].

In this study, we employed the ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) [15-18] to derive the valence-space effective Hamiltonian and effective multipole transition operators for further shell model calculations of the observed excited states and their electromagnetic transitions in ⁵³Fe. Unique bare nucleon charges were used in the calculation of all the multipolar transitions. We aim to provide solid calculations to explain the experimental observations and provide deep insight into the structure of the Jπ=19/2^- isomer.

Theoretical framework

Our calculations start from the intrinsic Hamiltonian of the A-nucleon system [19, 20] $H={\displaystyle \sum _{i=1}^A}\left(1-\frac{1}{A}\right)\frac{p_i^2}{2m}+{\displaystyle \sum _{i<j}^A}\left(v_{ij}^{\text{NN}}-\frac{p_i\cdot p_j}{mA}\right)+{\displaystyle \sum _{i<j<k}^A}{v}_{ijk}^{3\text{N}}\mathrm{,}$ (1) where p_i is the nucleon momentum in the laboratory system, m is the nucleon mass, and $v_{ij}^{\text{NN}}$ and $v_{ijk}^{3\text{N}}$ represent the nucleon-nucleon (NN) and three-nucleon (3N) interactions, respectively. An explicit treatment of the three-nucleon force (3NF) in many-body calculations has not yet been accessible for nuclei as heavy as ⁵³Fe. Therefore, we obtain a normal-ordered approximation of the Hamiltonian (Eq. (1)), neglecting the residual 3NF [20-24].

The IMSRG evolves the normal-ordered Hamiltonian to a block diagonal, which gives the valence-space effective Hamiltonian. Using this technique, we can also derive valence-space effective operators for other observables, including multipolar transitions. The Magnus expansion [25] was used in the IMSRG evolution. Operators appearing in the IMSRG equations are truncated at the two-body level, which is referred to as IMSRG(2) [21]. The technique of using IMSRG to obtain a valence-space Hamiltonian for shell model calculations is called VS-IMSRG [21].

The normal-ordering approximation of the Hamiltonian and the evolution of the VS-IMSRG were implemented with a closed-shell reference state in the Hartree-Fock basis. However, to reduce the effect of the residual 3NF, which is neglected, the fractional filling of open-shell orbitals in an open-shell nucleus has been suggested, called ensemble normal ordering (ENO) [17]. Using the ENO approximation of the VS-IMSRG, we can obtain the nucleus-dependent valence-space effective Hamiltonian and the effective operators of the other observables. The effective Hamiltonian obtained is diagonalized using the large-scale shell model code KSHELL [26], giving eigenenergies and one- and two-body transition densities (OBTD and TBTD, respectively) for further calculations of states.

The reduced transition probability is calculated by $B\left(\sigma \lambda ;{\psi }_i\to {\psi }_f\right)=\frac{{\left|\langle {\psi }_f\Vert {\mathcal{M}}_{\sigma \lambda }\Vert {\psi }_i\rangle \right|}^2}{2J_i+1}\mathrm{,}$ (2) where σ denotes the electric (σ=E) or magnetic (σ=M) transition and λ is the rank of the tensor operator ${\mathcal{M}}_{\sigma \lambda }$ [27]. ψi(ψf) represents the initial (final) state of the transition, where Ji denotes the angular momentum of the initial state. The free-space bare electric transition tensor operator is defined by ${\mathcal{M}}_{E\lambda }=Q_{\lambda }$ with the tensor components given by [28] $Q_{\lambda \mu }={\displaystyle \sum _{j=1}^A}{e}_j{r}_j^{\lambda }{Y}_{\lambda \mu }\left({\widehat{r}}_j\right)\mathrm{,}$ (3) where ej is the natural (bare) charge of the jth nucleon, that is, e=1 (0) for the proton (neutron). $Y_{\lambda \mu }$ is the spherical harmonic function. The free-space bare magnetic transition tensor operator is defined by ${\mathcal{M}}_{M\lambda }=M_{\lambda }$ with the tensor components given by [28] $M_{\lambda \mu }={\displaystyle \sum _{j=1}^A}\left[\frac{2}{\lambda +1}{g}_l^j{l}_j+g_s^j{s}_j\right]\cdot {\nabla }_j\left[r_j^{\lambda }{Y}_{\lambda \mu }\left({\widehat{r}}_j\right)\right]\mathrm{,}$ (4) where $l_j$ and $s_j$ are the orbital and intrinsic angular momenta of the jth nucleon, respectively. The values of g factors were taken from [28]. The free-space bare operators, ${\mathcal{M}}_{\sigma \lambda }$, must be renormalized into the valence space, which can be performed using the same ENO VS-IMSRG technique. After renormalization, the one-body operators of the free space bare ${\mathcal{M}}_{\sigma \lambda }$ become two-body operators in the valence space, defined by the corresponding two-body matrix elements. Using the valence-space OBTD and TBTD obtained in the diagonalization of the effective Hamiltonian, the reduced transition probability is calculated using Eq. (2).

Calculations and discussions

The ⁵³Fe Jπ=19/2^- isomer is the only case in which the highest multipole E6 transition is observed [1-4]. In a recent experimental study [4], shell model calculations using two empirical interactions, GFPX1A and KB3G, were performed with a model space of only the 0f_7/2 or the full fp shell. In empirical calculations of electromagnetic transitions, effective nucleon charges are used with different effective charges for different multipolarities of transitions, determined by fitting the data. The problem is that only one E6 transition has been observed to date, that is, from the 19/2^- isomer to the 7/2^- ground state in ⁵³Fe. One E6 transition data cannot simultaneously determine both the proton and neutron effective charges. In the present work, we performed ab initio calculations using the shell model with a full fp model space above the ⁴⁰Ca core. The valence-space shell model effective Hamiltonian, including the one-body single-particle energies of the valence particles, is obtained using the IMSRG evolution with the ENO approximation [17, 21]. In ab initio calculations of multipole transitions, unique bare nucleon charges are used, meaning that the same bare charges are used for different multipolarities of the transitions.

In our calculations, two different versions of the chiral NN plus 3N interaction were used to test its dependence on the interaction used. The potential labeled 1.8/2.0(EM) [29-31] takes the NN force (2NF) at N³LO [32] and 3NF at N²LO. This potential can globally reproduce the ground-state properties of nuclei, from light-to heavy-mass regions, with the harmonic-oscillator (HO) basis frequency optimized at $\hslash \text{Ω}=16$ MeV [33, 34]. We also used the recently developed interaction labeled by NN+3N(lnl) [35, 36] in which N⁴LO 2NF and N²LO 3NF were used. For both the 1.8/2.0(EM) and NN+3N(lnl) potentials, the induced 3NF was properly included [29-31, 35, 36]. One major difference between the two potentials was the treatment with 3NF. The 1.8/2.0(EM) potential uses a nonlocal 3N regulator, whereas the NN+3N(lnl) potential employs both local and nonlocal (lnl) 3N regulators.

The basis space is another important issue that must be addressed. The present calculation starts by using a spherical HO basis with $e=2n+l\le e_{\text{max}}$ single-particle shells. The 3NF is limited to $e_1+e_2+e_3\le E_{\text{3max}}$. For the 1.8/2.0(EM) and NN+3N(lnl) potentials, it has been demonstrated that basis truncations with both e_max and E_3max around 14 can provide good convergence of VS-IMSRG calculations, particularly for ground-state energies up to medium-mass regions, with an optimized frequency around $\hslash \text{Ω}=16$ MeV [31, 33, 35, 36]. However, spectroscopic calculations of heavier nuclei would require larger basis space. For example, a recent study using the 1.8/2.0(EM) potential showed that E_3max=24 is required to achieve convergence of the calculations of excited states in $A\approx 130$ mass region [34]. Another study on the neutrinoless double-beta decay (0νββ) of $A\approx 130$ candidates showed that E_3max=28 is required to converge all the operators [37]. However, both studies show that e_max=14 is sufficient for truncation of the single-particle basis. In the present work, we tested the convergence of the calculations of the observed highest-multipole electromagnetic transition probabilities for which no ab initio calculations have been performed previously. Figure 1 shows the convergence of the E4, M5 and E6 reduced transition probabilities as a function of e_max, E_3max and ℏΩ. We observe that at both potentials, the calculations converge well at $e_{\text{max}}\ge 12$, $E_{\text{3max}}\ge 16$ and $\hslash \text{Ω}=14-16$ MeV. Therefore, in the following calculations, we consider e_max=14, E_3max=24, and $\hslash \text{Ω}=16$ MeV. This choice is similar to that used in the calculations of the excitation spectra [34] and 0νββ decays [37] in the mass 130 region.

Fig. 1Full-screen View

Calculated reduced probabilities of the E6, M5 and E4 transitions from the 19/2^- isomer in ⁵³Fe, as functions of e_max, E_3max and ℏΩ. The 1.8/2.0(EM) and NN+3N(lnl) potentials were used

The present calculations reproduce the excitation spectrum of ⁵³Fe. However, for the sake of simplicity, as shown in Fig. 2 we show only a limited set of levels associated with the 19/2^- isomer. The calculations reproduced experimental data [38]. In particular, the calculation using the NN+3N(lnl) interaction was in better agreement with the data. Both calculations yield the correct energy ordering of the 19/2^- level, which is lower than the 13/2^- level. The lower energy of the 19/2^- level compared to that of the 17/2^- and 15/2^- levels makes it natural that the M1 and E2 transitions cannot occur from the 19/2^- state, which leads to the formation of the 19/2^- metastable isomeric state.

Fig. 2Full-screen View

Calculated levels involved in the observed highest-multipole E6, M5 and E4 electromagnetic transitions from the 19/2^- isomer in ⁵³Fe, compared with data [38]. The 1.8/2.0(EM) and NN+3N(lnl) potentials were used

Experiments [1=4] observed that the 19/2^- states decayed to lower 11/2^-, 9/2^- and 7/2^- states by high-multipole E4, M5 and E6 transitions, respectively. Higher multipole γ decays have lower transition probabilities, which can lead to long-lived isomers. Experimental observations [1-4] of the rare high multipole γ decay in ⁵³Fe provide a unique laboratory for the study of electromagnetic transitions with high multipolarity. Table 1 shows the calculated reduced probabilities of the observed highest-multipole transitions E6, M5 and E4 compared with the experimental data [4]. The NN+3N(lnl) potential provides an improved result for the M5 transition probability that is closer to the data. However, the calculated E6 transition probability differed from the experimental data by a factor of approximately three, although the experimental error was large for this transition. The present VS-IMSRG calculation is approximated with operators truncated at a two-body level by IMSRG(2) [21], whereas a recent study has developed the IMSRG with operators truncated at the three-body level labeled by IMSRG(3) [39, 40]. Another study indicated that the IMSRG combined with the generator coordinate method can capture more collective correlations in deformed nuclei [41]. In Refs. [42, 43], it was found that the inclusion of two-body currents can improve the calculation of magnetic dipole moments (transitions). It is clear that the consideration of these factors can improve calculations, but such an investigation goes beyond the scope of the present work and will be addressed separately. In empirical shell model calculations, effective nucleon charges are used to compensate for missing correlations. The values of the proton and neutron effective charges were determined by fitting the corresponding experimental data to the electromagnetic transitions. Different effective charges of protons and neutrons are used for different multipolarities of the transitions. Unfortunately, only one E6 experimental data point is available at present, which is not sufficient to determine both the proton and neutron effective charges of the E6 transition.

Using bare nucleon charges, we also calculated the ⁵³Fe E2 and M1 reduced transition probabilities that were available experimentally, as shown in Table 2. We observe that the observed E2 and M1 transition probabilities can be reproduced reasonably well (all are well within $2\sigma$ of the experimental values). Nevertheless, improvements in the IMSRG and the inclusion of two-body currents should also improve the calculations of low-multipole transitions.

To better understand the long-lived 19/2^- isomer of ⁵³Fe, we discuss the state configurations. Figure 3 shows the configurations of the 7/2^-, 9/2^-, 11/2^- and 19/2^- states obtained from the calculation. We see that the 0f_7/2 component dominates the states, which can be understood by the simple three-hole (3h) coupling ${\left(\pi f_{7/2}\right)}^{-2}{\left(\nu f_{7/2}\right)}^{-1}$ with $J=19/2$ being the highest spin that can be obtained in 3h coupling. As shown in Fig. 3, the 19/2^- isomer has a purer 0f_7/2 component, with fewer occupations in the higher 0f_5/2 and 1p_3/2,1/2 orbitals. The pronounced dominance of the 0f_7/2 component for both neutron and proton orbits arises from the $N(Z)=28$ shell gaps on either side of the f_7/2 orbit. However, we find that the effects of the higher 0f_5/2 and 1p_3/2 orbitals are not negligible.

Fig. 3Full-screen View

(Color online) Configurations of the 7/2^-, 9/2^-, 11/2^- and 19/2^- states, obtained in the present calculations with the NN+3N(lnl) interaction

Summary

High-multipole electromagnetic transitions are of particular interest in the study of nuclear structures, providing a sensitive test of the theories used. To date, the highest multipole transition is the electric hexacontatetrapole E6 transition from the Jπ=19/2^- isomer to the 7/2^- ground state in ⁵³Fe, accompanied by M5 and E4 transitions to the first 9/2^- and 11/2^- excited states, respectively. A recent experiment [4] provided clear detection of the highest-multipole transitions with greater precision for the transition rates. Shell-model calculations with empirical interactions were performed in an experimental study [4]. However, empirical calculations cannot definitively determine the E6 transition probability. The empirical shell model uses the effective charges determined by fitting the experimental data. To date, only one E6 transition was observed, i.e., that from the Jπ=19/2^- isomer to the Jπ=7/2^- ground state in ⁵³Fe. An experimental E6 transition rate cannot simultaneously determine both the proton and neutron effective charges.

In this study, we performed ab initio calculations of the highest multipole electromagnetic transitions observed from the 19/2^- isomer of ⁵³Fe. Two different versions of the chiral two- plus three-nucleon interactions were used in the calculations. Using the valence-space in-medium similarity renormalization group, the free-space interaction matrix was decoupled to form a low-momentum valence space, obtaining the valence-space effective interaction for the shell-model calculation. The calculation convergence of the excitation spectrum and the electromagnetic transitions of ⁵³Fe has been well tested. The present ab initio calculations reproduce the experimental spectrum well. The 19/2^-isomerism is due to its lower energy compared to those of the 17/2^- and 15/2^- levels, which leads to the absence of natural E2 and M1 transitions from the 19/2^- state. Using the same decoupling method, free-space bare operators of electromagnetic transitions evolve into the valence space. With the unique bare nucleon charges, we can reproduce the measured highest multipole E6, M5 and E4 transition strengths. The low multipole E2 and M1 transitions observed in this nucleus can also be reproduced using bare nucleon charges. Moreover, we provide the configurations of the 7/2^-, 9/2^-, 11/2^- and 19/2^- states, and reveal the structure of the Jπ=19/2^- isomer.