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# Electromagnetic character of the competitive γγ/γ-decay from 137mBa

## Abstract

Second-order processes in physics is a research topic focusing attention from several fields worldwide including, for example, non-linear quantum electrodynamics with high-power lasers, neutrinoless double-β decay, and stimulated atomic two-photon transitions. For the electromagnetic nuclear interaction, the observation of the competitive double-γ decay from 137mBa has opened up the nuclear structure field for detailed investigation of second-order processes through the manifestation of off-diagonal nuclear polarisability. Here, we confirm this observation with an 8.7σ significance, and an improved value on the double-photon versus single-photon branching ratio as 2.62 × 10−6(30). Our results, however, contradict the conclusions from the original experiment, where the decay was interpreted to be dominated by a quadrupole-quadrupole component. Here, we find a substantial enhancement in the energy distribution consistent with a dominating octupole-dipole character and a rather small quadrupole-quadrupole component in the decay, hindered due to an evolution of the internal nuclear structure. The implied strongly hindered double-photon branching in 137mBa opens up the possibility of the double-photon branching as a feasible tool for nuclear-structure studies on off-diagonal polarisability in nuclei where this hindrance is not present.

## Introduction

Polarisability is a fundamental concept in physics and chemistry defined from the principles of electromagnetic interaction. It describes how applied electric or magnetic fields induce an electric or magnetic dipole, or higher-order multipole, moment in the matter under investigation1. In nuclear physics, the simple concept of polarisability influences observables over a broad range of topics. For example, the static dipole polarisation of the shape of the ground and excited states in atomic nuclei is influenced by the coupling to high-energy collective modes like the giant dipole resonance (GDR) via virtual excitations. In this case, the nuclear static dipole polarisability, αd, is obtained2 from the photonuclear population of excited states,

$${\alpha }_{{\rm{d}};{\rm{E1}}}=2e\sum _{n}\frac{{\left|\langle {I}_{0}| | {\rm{E1}}| | {I}_{n}\rangle \right|}^{2}}{{E}_{n}-{E}_{0}},$$
(1)

where the transition matrix elements of the wave functions correspond to the electric dipole transition, E1, between the ground state, I0, and an excited state, In, with e the elementary unit charge and En the energy of the state.

By expanding the concept of polarisability beyond the scalar case, one can divide the polarisability tensor into separate components. Typically, these are either spatial components like the birefringence properties of crystals or electric and magnetic multipole components. Within the nuclear structure framework, this type of off-diagonal polarisabilities can appear in very weak second-order processes. In the electromagnetic case, the off-diagonal nuclear polarisability can be defined analogously to Eq. (1) in terms of either electric and magnetic components, or components of different multipolarities as

$${\alpha }_{{\rm{M2E2}}}=\sum _{n}\frac{\langle {I}_{{\rm{f}}}| | {\rm{E2}}| | {I}_{n}\rangle \langle {I}_{n}| | {\rm{M2}}| | {I}_{{\rm{i}}}\rangle }{{E}_{n}-\omega }$$
(2)

or,

$${\alpha }_{{\rm{E3M1}}}=\sum _{n}\frac{\langle {I}_{{\rm{f}}}| | {\rm{M1}}| | {I}_{n}\rangle \langle {I}_{n}| | {\rm{E3}}| | {I}_{{\rm{i}}}\rangle }{{E}_{n}-\omega }.$$
(3)

Due to the parity conserving properties of the strong force, these decays can only be observed between two different states, Ii and If. In the definition above, the denominator depends on the interference frequency, ω, of the emitted γ rays and is approximated to be half of the initial state energy. This type of second-order electromagnetic processes of atoms was discussed in the doctorate dissertation of Maria Göppert-Meyer3 where she estimated a probability for an atomic two-photon absorption process relative to the single-photon process to be approximately 10−7, later to be confirmed with the observation of this effect in CaF2:Eu2+ crystals4.

For many years, double-γ decay was only observed in exceptional cases where both the ground state and the initial state have a spin-parity Jπ = 0+ character for the doubly magic nuclei 16O5,6, 40Ca7, and 90Zr7. Here single γ-emission is blocked, and only conversion-electron decay and double-γ decay are allowed. In these experiments, the obtained information consists of correlations between energies and angles of these γ-rays, used to determine the decay probabilities of electric and magnetic dipoles. For a generalisation of this phenomenon and the possibility to use it as a spectroscopic tool for a more fundamental understanding of the underlying physics, large state-of-the-art high-purity germanium (HPGe) detector systems8,9 have been used to search for the competitive double-γ (γγ/γ) decay, where also the single γ decay is allowed. Even though unsuccessful in that respect, these experiments successfully measured an E5 transition with the branching of 1.12(9) × 10−7. It is only with instrumentation developments of detector materials that can provide both the energy and time resolution required10 that the observation of the γγ/γ decay mode was announced11. The set-up used for that experiment consisted of five LaBr3:Ce detectors arranged in a planar configuration with relative angles of 72 between the detectors, providing angular distribution data points at 72 and 144. Thus, the collaboration could announce a γγ/γ decay signal with 5.5 σ (standard deviations) statistical significance, near but above the typical discovery limit of 5 σ. From the two angular data points as well as the energy spectrum of the individual γ rays at 72 angle, the off-diagonal polarisabilities αM2E2 = 33.9(2.8) e2fm4/MeV and αE3M1 = 10.1(4.2) e2fm4/MeV were favoured. While the observation of the peak associated with γγ/γ decay was statistically clear, the nature of this decay was more uncertain, having the two dominating multipolarity combinations separated only by a small statistical difference, favouring the αM2E2 component12. The decay diagram of this process is shown in Fig. 1.

## Results

### Experimental set-up

The experiment was performed using eleven 3 × 3 CeBr3 detectors from ELIGANT, shown in Fig. 2a. While ELIGANT consists of both LaBr3:Ce and CeBr3 detectors, the CeBr3 detectors were chosen to remove any possible source of background contribution from the natural radioactivity in lanthanum. The detector configuration was a circle with an inner radius to the front-face of the scintillators of 40 cm. This distance was enough to separate true coincidences from multiple Compton scattering of single γ rays using the photon time-of-flight (TOF), see Fig. 2b. The relative angles between the eleven detectors were 32.7, with an opening angle, given by the lead shielding, of  ±3.4. This gave five independent γγ-correlation angles centred at: 32.7, 65.5, 98.2, 130.9, and 163.6. The detectors were separated with a minimum of approximately 15 cm of effective lead shielding between two neighbouring detectors to remove any contribution from single Compton scattering between detector pairs at low angles. The set-up was characterised both with an in-house toolkit23 based on the GEANT4 framework24, and a 152Eu source with an activity of 460 kilo Becquerel (kBq) and a 60Co source with an activity of 60 kBq. For a comprehensive overview, see reference21. The γγ/γ-decay data on 137Ba were collected using a 137Cs source with an activity of 336  kBq for 49.5 days of active data taking. The source had a thin circular active area with a diameter of 3 mm, encapsulated in the centre of a cylindrical polymethylmethacrylate capsule with a diameter of 25 and 3 mm thickness, see Fig. 2.

### Energy spectra

From the data set obtained with the 137Cs source, a (γ1γ2) coincidence matrix was constructed where the γ rays were considered coincident if the time difference between them were less than one standard deviation from the prompt time distribution, Δt1,2 ≤ 655 ps. This condition was obtained from the coincident 444 kilo electron-volt (keV) and 245 keV γ rays from the $${2}_{2}^{+}\to {4}_{1}^{+}\to {2}_{1}^{+}$$ decay chain in 152Sm following the electron capture decay of 152Eu. Corrections for detector efficiencies were done on an event-by-event basis21. A time difference of 20 ns ≤ Δt1,2 ≤ 820 ns was used to estimate the uncorrelated background events with two detected γ rays and subtracted after applying an appropriate scaling factor. To remove the background contribution from electron–positron pairs produced by cosmic rays a multiplicity-two condition was assigned together with an additional energy condition that $$\left|{E}_{1}-{E}_{2}\right|<960-\left({E}_{1}+{E}_{2}\right)$$ keV. The full data set, as well as the different angular groups, were used to construct the summed energy spectra. The peaks were fitted assuming a quadratic background both with a Gaussian distribution as well as GEANT4 simulated data. Both fitting methods gave consistent results. The full summed spectrum of E1 + E2 with these conditions imposed is shown in Fig. 3.

### Branching

As experimental observable to evaluate the relative decay probability, we use the definition of the integrated differential branching ratio11,

$$\delta ({E}_{1},{E}_{2},{\theta }_{1,2})=\frac{{(4\pi )}^{2}}{{\Gamma }_{\gamma }}\mathop{\int}\nolimits_{{E}_{1}}^{{E}_{2}}{\rm{d}}\omega {\left.\frac{{\rm{d}}{\Gamma }_{\gamma \gamma }^{5}}{{\rm{d}}\omega {\rm{d}}\Omega {\rm{d}}\Omega ^{\prime} }\right|}_{{\theta }_{1,2}}.$$
(4)

In this definition, Γγ is the total single-gamma decay width, proportional to the size of the single-gamma peak. Given an angle, θ1,2, the differential decay is integrated over the frequency of the γ ray, ω. The frequency is proportional to the energy, and the integration limits are taken as the edges of the energy bin of interest. In the experimental spectrum, a natural low-energy limit comes from the low-energy threshold of the detectors around 120 keV. However, to reduce the contamination from the 511 keV γ-rays originating from electron–positron annihilation, the integration limits E1 = 180 keV and E2 = 331 keV were chosen. The upper limit was chosen as half of the total energy as we are not able to distinguish any relative ordering of the γ rays. This procedure was performed for all combinations of θ1,2 and δ was evaluated as a function of angle. The results of this evaluation are shown in Fig. 4.

This data can be directly fitted to the generalised polarisability functions of Eq. (6) discussed in the methods section, using only αM2E2 and αE3M1 as free parameters. Other components like αE2M2 or αM3E1 could, in principle, also contribute. However, the general polarisability functions are linearly dependent in the exchange of terms, weighted by the coefficients given by the Wigner 6j symbols, and this experiment is not sensitive to this ordering. These additional components are, furthermore, expected to be small. Thus, we restrict the discussion to the αM2E2 and αE3M1 polarisabilities from here on.

### Energy-sharing distributions

The angular distributions themselves are not enough to completely distinguish between the contribution from the different polarisabilities. When calculating the goodness-of-fit (χ2), two local minima corresponding to either a large αM2E2 component or a large αE3M1 component appear. Instead, it is necessary to study the energy-sharing distributions between the two individual γ-rays. From Eqs. (6) and (7) in the methods section it is clear that the energy dependence of the decay for the two different cases follows $$\frac{{\rm{d}}{\Gamma }_{\gamma \gamma }}{{\rm{d}}\omega }\propto {\omega }^{5}\omega {^{\prime} }^{5}$$ for M2E2 and as $$\frac{{\rm{d}}{\Gamma }_{\gamma \gamma }}{{\rm{d}}\omega }\propto {\omega }^{3}\omega {^{\prime} }^{7}$$ for E3M1 with $$\omega ^{\prime} =662-\omega$$. It is clear from these relations that the energy-sharing distributions are expected to have a maximum at $${E}_{\gamma }=E^{\prime} =331$$ keV for the M2E2 type transitions, while an asymmetric maximum is expected at Eγ = 200 keV and $$E^{\prime} =442$$ keV for the E3M1 type transitions.

For this purpose, δ from Eq. (4) was evaluated in separate slices of 30 keV energy difference between the low- and high-energy limit of Eγ. Figure 5a shows the results of these evaluations. A χ2 value was then calculated based on (4) for different values of αM2E2 and αE3M1 simultaneously using the energy-integrated angular data points and the angle-summed energy data points as

$${\chi }^{2}=\sum_{{{{\theta_{1,2}^{i}}\atop {{E}_{1}=181}}\atop {{E}_{2}=331}}}\frac{{\delta }^{2}({E}_{1},{E}_{2},{\theta }_{1,2}^{i})}{{\sigma }_{\delta }^{2}({E}_{1},{E}_{2},{\theta }_{1,2}^{i})}+\sum_{{{{E}_{1}={E}^{i}_{{\rm{low}}}}\atop {{E}_{2}={E}^{i}_{{\rm{high}}}}}}\frac{{\delta }^{2}({E}_{1},{E}_{2},\Sigma {\theta }_{1,2})}{{\sigma }_{\delta }^{2}({E}_{1},{E}_{2},\Sigma {\theta }_{1,2})},$$
(5)

where σδ is the statistical uncertainty in each data point, including both the signal and the subtracted background. The systematic uncertainty mainly originates from uncertainties in the intrinsic and geometric efficiencies of the set-up and is expected to be on the order of a few %, much smaller than the statistical uncertainties, and have been neglected in this expression. When including the data from Walz et al., only the energy distribution of the 72 data was included in the second part of Eq. (5), and the lower energy summation limit for the 144 data point was set to 206 keV in the first part of Eq. (5). The resulting χ2 surface is shown in Fig. 5b. As seen here, the χ2 analysis from this data favours a large αE3M1 component, in contradiction with both the experimental interpretation and theoretical conclusions reported in ref. 11.

## Discussion

To understand these results, we performed theoretical calculations of the polarisation functions from Eq. (2), using the QPM25 approach. The application of the QPM in the case of odd-mass spherical nuclei is discussed in detail in reference26. In particular, the nuclear structure of 137Ba was studied within the framework of this model in references27,28 and in reference11. In the work presented here, the calculations were built on the EDF theory coupled with the QPM29 to obtain magnetic and electric spectral distributions. The model parameters of the EDF + QPM approach are firmly determined from nuclear structure data or derived fully microscopically30,31,32. The theoretical results are shown in Table 1 and agree with the data in terms of absolute branching strength, Γγγ/Γγ. In addition, the EDF + QPM used here predicts a significantly larger αE3M1 than the value reported in ref. 11 from the QPM, close to our experimental observations. However, the relative magnitude of the αM2E2 and αE3M1 coefficients obtained from the EDF + QPM theory, as well as from reference11, are different than the experimental results obtained in this work. In particular, the present measurement indicates that the αM2E2 coefficient is significantly smaller than previously reported, and at this level of complexity EDF + QPM it is not able to account for the apparent discrepancy with the experimental data.

To understand the origin of this discrepancy, the properties of the dominant, low-lying, states were investigated from another perspective using the state-of-the-art nuclear MCSM33,34. These calculations were used to extract information from the three lowest-energy Jπ = 7/2+ states, the five lowest-energy Jπ = 5/2+ states, as well as the ground Jπ = 3/2+ and isomeric Jπ = 11/2 states. The neutron component in the MCSM wave function of the isomeric Jπ = 11/2 state is dominated by a single neutron hole in νh11/2. The Jπ = 7/2+ state is different, however, with most of the neutron hole occupation is in νd3/2, coupled to a 2+ state of six valence protons. The νg7/2 orbital itself is almost full. This is in contrast with the EDF + QPM results where the 2+ νd3/2 contribution is 38.7% and the νg7/2 single-particle component is 51.3%. Thus, the odd-neutron contribution to the M2 transition rate in the MCSM would require a highly hindered transition between νh11/2 and νd3/2, or by utilising a minor νg7/2 vacancy. This, gives rise to a strongly hindered M2 transition within the MCSM, with a reduced transition probability, B(M2) = 13.5 × 10−3 μ2fm2, three orders of magnitudes less than predicted by the EDF + QPM model where B(M2) = 14.9 μ2fm2. This can explain the observed suppression of αE2M2. It is interesting to note that with increasing excitation energy, the MCSM predict a smooth change in orbital occupation from νd3/2 to νd5/2, constructively adding to the M2 transition strength for all the calculated 7/2+ transitions in contrast to the EDF + QPM where all higher-lying states act destructively. Table 2 lists the contributing low-lying matrix elements discussed here.

Regarding the αE3M1 component of the decay, the main components obtained from the EDF + QPM calculations are from the coupling of the single-particle mode with the surface vibrations of the even-even core. As a consequence, due to the exchange of the collective $${3}_{1}^{-}$$ octupole phonon, we obtain a rather strong E3 transition, consistent with our experimental observations. For these states, the EDF + QPM and the MCSM give a consistent picture with a constructive addition to the strength for each successive state among the first three excited states with the main difference that in the EDF + QPM, the main contribution comes from the $$5/{2}_{1}^{+}$$ state while the MCSM predicts that the $$5/{2}_{2,3}^{+}$$ states are dominating.

## Methods

### Experimental set-up

The set-up consisted of eleven 3” × 3” CeBr3 detectors coupled with Hamamatsu R6233 photomultiplier tubes and built-in voltage dividers. The high voltage for the photomultiplier tubes was provided by a CAEN SY4527 power supply. The signals were read out using one CAEN V1730 digitiser operating with a 14-bit resolution at a 500 MS/s sampling rate and a dynamic range of 0.5 Vpp, running PSD firmware. The digitisers were controlled using the Multi Instance Data Acquisition System software and triggered individually. Each event consisted of the energy, the time-stamp, and the the digitised voltage pulse from the detector. The sub-nanosecond time information was obtained from the value of the time-stamp corrected by a digital interpolation of the sampling points in the recorded pulse, at half of the maximum value of the pulse and interpolated using a quadratic polynomial.

### Polarisation functions

To obtain the nuclear polarisabilites, $${\alpha }_{S^{\prime} L^{\prime} SL}$$, from the differential decay probability we follow the theoretical treatment in refs. 11,35. Here the differential decay probability can be expressed in terms of generalised polarisation functions, $${P}_{J}^{\prime}(S,L,S^{\prime} ,L^{\prime} )$$, and Legendre polynomials, $${P}_{l}(\cos \theta )$$, as

$$\frac{{{\rm{d}}}^{5}{\Gamma }_{\gamma \gamma }}{{\rm{d}}\omega \Omega \Omega ^{\prime} }=\frac{\omega \omega ^{\prime} }{96\pi }\sum {P}_{J}^{\prime}({S}_{1}^{\prime}{L}_{1}^{\prime}{S}_{1}{L}_{1}){P}_{J}^{\prime}({S}_{2}^{\prime}{L}_{2}^{\prime}{S}_{2}{L}_{2})\sum {a}_{l}^{\xi }{P}_{l}(\cos \theta ),$$
(6)

where the generalised polarisation functions are defined as

$${P}_{J}^{\prime}(S^{\prime} L^{\prime} SL)= \, {(-1)}^{S+S^{\prime} }2\pi {(-1)}^{{I}_{i}+{I}_{f}}{\omega }^{L}\omega {^{\prime} }^{L^{\prime} }\cdot \\ \sqrt{\frac{L + 1}{L}}\sqrt{\frac{L^{\prime} + 1}{L^{\prime} }}\frac{\sqrt{2L + 1}\sqrt{2L^{\prime} + 1}}{(2L+1)!!(2L^{\prime} +1)!!}\cdot \\ \left(\left\{\begin{array}{lll}L&L^{\prime} &J\\ {I}_{f}&{I}_{i}&I\end{array}\right\}{\alpha }_{S^{\prime} L^{\prime} SL}+{(-1)}^{S+S^{\prime} }\left\{\begin{array}{lll}L^{\prime} &L&J\\ {I}_{f}&{I}_{i}&I\end{array}\right\}{\alpha }_{SLS^{\prime} L^{\prime} }\right).$$
(7)

The sums in Eq. (6) run over all the permutations of electric, S = 0, and magnetic, S = 1, combinations with multipolarity, L, allowed in the decay, and over all Legendre polynomials with non-zero coefficients. The general polarisability functions in Eq. (7) consist of a linear combination of the off-diagonal polarisabilities of the nucleus weighted by coefficients determined by the corresponding angular momentum algebra of the decay.

### The quasiparticle-phonon model

The QPM Hamiltonian includes mean field, pairing interaction and separable multipole and spin-multipole interactions25. The mean field for protons and neutrons is defined as a Woods-Saxon potential with parameter sets derived self-consistently from a fully microscopic Hartree-Fock-Bogoljubov (HFB) calculations described in29,36. The method assures a good description of nuclear ground-state properties by enforcing that measured separation energies and nuclear radii are reproduced as close as possible36. The pairing and residual interaction parameters are fitted to reproduce the odd-even mass differences of neighbouring nuclei as well as the experimental values of the excitation energies and reduced transition probabilities of low-lying collective and non-collective states in the even-even core nucleus25. Of particular importance in these studies is the determination of the isovector spin-dipole coupling constant which is extracted from comparison to data from ref. 31 and fully self-consistent quasiparticle random phase approximation (QRPA) calculations using the microscopic EDF of ref. 30. Single-particle (s.p.) energies of the lowest-lying excited states in 137Ba are fine-tuned to experimental values to achieve the highest accuracy in the description of the experimental data. We point out that the s.p. energies problem is not a matter of the interaction parameters but originate in the quasiparticle spectrum, which indicates the necessity to go beyond the static mean-field formalism37,38.

In the QPM the wave functions of the excited states of an even-odd nucleus are constructed from a combination of quasiparticles originating from the single-particle orbitals and excitation phonons that are constructed from the excited states in the neighbouring even-even core nucleus :

$${\Psi }_{\nu }(JM)={C}_{J}^{\nu }\left\{{\alpha }_{JM}^{+}+\sum _{\lambda \mu i}{D}_{j}^{\lambda i}(J\nu ){[{\alpha }_{jm}^{+}{Q}_{\lambda \mu i}^{+}]} \, {JM}\right\}{\Psi }_{0}$$
(8)

The notation $${\alpha }_{jm}^{+}$$ is the quasiparticle creation operator with shell quantum numbers j ≡ [(nlj)] and projection m; $${Q}_{\lambda \mu i}^{+}$$ denotes the phonon creation operator with the angular momentum λ, projection m and QRPA root number i; Ψ0 is the ground state of the neighboring even-even nucleus and ν stands for the number within a sequence of states of given angular momentum Jπ and projection M. The coefficients $${C}_{J}^{\nu }$$ and $${D}_{j}^{\lambda i}(J\nu )$$ are the quasiparticle and ’quasiparticle phonon’ amplitudes for the ν state. The components $${[{\alpha }_{jm}^{+}{Q}_{\lambda \mu i}^{+}]}_{JM}$$ of the wave function (8) may violate the Pauli principle. The exact commutation relations between quasiparticle and phonon operators are used to solve this problem. The properties of the phonons are determined by solving QRPA equations from refs. 25,26. The model basis includes one-phonon states with spin and parity Jπ = 1±, 2±, 3±, 4±, 5± and excitation energies up to Ex = 20 MeV. The calculations of the α-coefficients of the double-γ decay probability of 137Ba include all low-energy excited states with spin and parity Jπ = 1/2±, 3/2±, 5/2±, 7/2±, 9/2± and excitation energies up to Ex = 10 MeV.

In the case of the E1 transitions, we have used effective charges $${e}_{{\rm{p}}}^{{\rm{eff}}}=(N/A)e$$ (for protons) and $${e}_{{\rm{n}}}^{{\rm{eff}}}=-(Z/A)e$$ (for neutrons) to separate the centre of mass motion and ’bare’ values for E2 and E3 transitions ep = e (for protons) and en = 0 (for neutrons), where e is the electron charge. Following previous QPM calculations32, the magnetic transitions are calculated with a quenched effective spin-magnetic factor $${g}_{{\rm{s}}}^{{\rm{eff}}}$$. The influence of the $${g}_{{\rm{s}}}^{{\rm{eff}}}$$ parameter on the experimental observables related to electromagnetic transitions of lowest-lying states and double-γ decay probability coefficients was investigated by carrying out EDF + QPM calculations for several choices of this parameter between 0.6 and 1 of the value of the ‘bare’ spin-magnetic moment, $${g}_{{\rm{s}}}^{{\rm{bare}}}$$. The theoretical observations indicate that the values $${g}_{{\rm{s}}}^{{\rm{eff}}}=0.6-0.7{g}_{{\rm{s}}}^{{\rm{bare}}}$$ which are in agreement with our previous findings27,28,32 reproduce quite well the experimental data on M1 and M2 transition strengths and the angular distribution of the two photons of the double-γ decay.

### Monte Carlo shell model

In the MCSM, the approximated wave functions, $$|{\Psi }_{{N}_{b}}\rangle$$, are obtained as a superposition of spin (I) and parity (π) projected Slater determinant basis states, $$|{\phi }_{n}\rangle$$,

$$\left|{\Psi }_{{N}_{b}}\right\rangle =\mathop{\sum }\limits_{n=1}^{{N}_{b}}\mathop{\sum }\limits_{K=-I}^{I}{f}_{n,K}^{{N}_{b}}{P}_{MK}^{I\pi }\left|{\phi }_{n}\right\rangle ,$$
(9)

where Nb is the number of basis states, $${P}_{MK}^{I\pi }$$ is the spin-parity projection operator, and the $${f}_{n,K}^{{N}_{b}}$$ coefficients are obtained from diagonalising the Hamiltonian. The set of basis states are selected by Monte Carlo methods and iteratively refined to minimise the ground-state energy. The model space for these calculations included the 1g9/2, 1g7/2, 2d5/2, 2d3/2, and 3s1/2 even-parity orbitals, as well as the 1h11/2, 2f7/2, and 3p3/2 odd-parity orbitals. The two-body matrix elements were obtained from the JUN45 and SNBG3 data sets39,40, and the VMU interaction41. To obtain the transition matrix elements effective proton and neutron charges ep = 1.25 and en = 0.75, and gyromagnetic factors g,p = 1, g,n = 0, gs,p = 5.586, and gs,n = − 3.826 was used. The calculations followed the procedure for the tin isotope chain closely42. Said reference and references within contains a detailed description of the procedure.

## Data availability

Raw data were obtained at the Extreme Light Infrastructure – Nuclear Physics facility, Romania. All the data used to support the findings of this study are available from the authors upon reasonable request. The final data points can be obtained from https://doi.org/10.17632/skhmjshxdj.

## Code availability

Sorting codes were developed at the Extreme Light Infrastructure – Nuclear Physics facility, Romania. All the codes for the experimental data used in this study are available from the authors upon reasonable request.

## References

1. 1.

Jackson,  J. D.  Classical Electrodynamics 3rd edn. (John Wiley, Sons, Inc., 1999).

2. 2.

Birkhan, J. et al. Electric dipole polarizability of 48Ca and implications for the neutron skin. Phys. Rev. Lett. 118, 252501 (2017).

3. 3.

Göppert-Mayer, M. Über elementarakte mit zwei quantensprüngen. Göttinger Dissertation, Georg-August-Universität Göttingen. Ann. Phys. 9, 273 (1931).

4. 4.

Kaiser, W. & Garrett, C. G. B. Two-photon excitation in CaF2:Eu2+. Phys. Rev. Lett. 7, 229 (1961).

5. 5.

Watson, B. A., Bardin, T. T., Becker, J. A. & Fisher, T. R. Two-photon decay of the 6.05-MeV state of 16O. Phys. Rev. Lett. 35, 1333 (1975).

6. 6.

Hayes, A. C., Friar, J. L. & Strottman, D. Two-photon decay of the first excited 0+ state in 16O. Phys. Rev. C 41, 1727 (1990).

7. 7.

Schirmer, J. et al. Double gamma decay in 40Ca and 90Zr. Phys. Rev. Lett. 53, 1897 (1984).

8. 8.

Moran, K. et al. E5 decay from the J π = 11/2 isomer in 137Ba. Phys. Rev. C 90, 041303(R) (2014).

9. 9.

Merchán, E. et al. 137Ba double gamma decay measurement with GAMMASPHERE. EPJ Web Conf. 93, 01033 (2015).

10. 10.

Higgins, W. et al. Crystal growth of large diameter LaBr3:Ce and CeBr3. J. Cryst. Growth 210, 2085 (2008).

11. 11.

Walz, C. et al. Observation of the competitive double-gamma nuclear decay. Nature 526, 406–409 (2015).

12. 12.

Walz,  C.  The Two-Photon Decay of the 11/2 Isomer of 137Ba and Mixed-Symmetry States of 92,94Zr and 94Mo. Springer Theses. Springer International Publishing (2016).

13. 13.

Brugnara, D. et al. Perspectives on the measurement of competitive double gamma decay with the AGATA tracking array. Nuovo Cimento C 42, 92 (2019).

14. 14.

Akkoyun, S. et al. AGATA - Advanced Gamma Tracking Array. Nucl. Instrum. Methods Phys. Res. A 688, 26 (2012).

15. 15.

Leach, K. G. et al. Electroweak decay studies of highly charged radioactive ions with TITAN at TRIUMF. Atoms 5, 14 (2017).

16. 16.

Camera, F. et al. Gamma above the neutron threshold experiments at ELI-NP. Rom. Rep. Phys. 68, S539 (2016).

17. 17.

Krzysiek, M. et al. Simulation of the ELIGANT-GN array performances at ELI-NP for gamma beam energies larger than neutron threshold. Nucl. Instrum. Methods Phys. Res. A 916, 257 (2019).

18. 18.

Gales, S. et al. New frontiers in nuclear physics with high-power lasers and brilliant monochromatic gamma beams. Phys. Scr. 91, 093004 (2016).

19. 19.

Gales, S. et al. The extreme light infrastructure-nuclear physics (ELI-NP) facility: new horizons in physics with 10 PW ultra-intense lasers and 20 MeV brilliant gamma beams. Rep. Prog. Phys. 81, 094301 (2018).

20. 20.

Tanaka, K. A. et al. Current status and highlights of the ELI-NP research program. Matter Radiat. Extremes 5, 024402 (2020).

21. 21.

Söderström, P.-A. et al. Source commissioning of the ELIGANT-GG setup for γ -ray coincidence measurements at ELI-NP. Rom. Rep. Phys. 71, 206 (2019).

22. 22.

Gonçalves, J. A. C. & Saxena, R. N. Directional correlations of γ transitions in 135Xe following the decay of 135I. Phys. Rev. C 43, 2586 (1991).

23. 23.

Lattuada, D. et al. A fast and complete GEANT4 and ROOT Object-Oriented Toolkit: GROOT. EPJ Web Conf. 165, 01034 (2017).

24. 24.

Agostinelli, S. et al. Geant4–a simulation toolkit. Nucl. Instrum. Methods Phys. Res. A 506, 250–303 (2003).

25. 25.

Soloviev, V. G. Theory of Complex Nuclei. (Pergamon Press, Oxford, 1976).

26. 26.

Gales, S., Stoyanov, C. & Vdovin, A. I. Damping of high-lying single-particle modes in heavy nuclei. Phys. Rep. 166, 125–193 (1988).

27. 27.

Tsoneva, N. et al. Population of isomers in the decay of the giant dipole resonance. Phys. Rev. C 61, 044303 (2000).

28. 28.

Scheck, M. et al. Dipole strength distributions in the stable Ba isotopes 134−136Ba: a study in the mass region of a nuclear shape transition. Phys. Rev. C 70, 044319 (2004).

29. 29.

Tsoneva, N. & Lenske, H. Energy-density functional plus quasiparticle-phonon model theory as a powerful tool for nuclear structure and astrophysics. Phys. At. Nucl. 79, 885–903 (2016).

30. 30.

Hofmann, F. & Lenske, H. Hartree-Fock calculations in the density matrix expansion approach. Phys. Rev. C 57, 2281 (1998).

31. 31.

Tonchev, A. P. et al. Spectral structure of the Pygmy dipole resonance. Phys. Rev. Lett. 104, 072501 (2010).

32. 32.

Rusev, G. et al. Fine structure of the giant M1 resonance in 90Zr. Phys. Rev. Lett. 110, 022503 (2013).

33. 33.

Otsuka, T., Honma, M., Mizusaki, T., Shimizu, N. & Utsuno, Y. Monte Carlo shell model for atomic nuclei. Prog. Part. Nucl. Phys. 47, 319–400 (2001).

34. 34.

Shimizu, N. et al. New-generation Monte Carlo shell model for the K computer era. Prog. Theor. Exp. Phys. 2012, 01A205 (2012).

35. 35.

Kramp, J. et al. Nuclear two-photon decay in 0+ → 0+ transitions. Nucl. Phys. A 474, 412–450 (1987).

36. 36.

Tsoneva, N. & Lenske, H. Pygmy dipole resonances in the tin region. Phys. Rev. C 77, 024321 (2008).

37. 37.

Lenske, H. & Tsoneva, N. Dissolution of shell structures and the polarizability of dripline nuclei. Eur. Phys. J. A 55, 238 (2019).

38. 38.

Tsoneva, N., Spieker, M., Lenske, H. & Zilges, A. Fine structure of the Pygmy quadrupole resonance in 112,114Sn. Nucl. Phys. A 990, 183–198 (2019).

39. 39.

Honma, M., Otsuka, T., Mizusaki, T. & Hjorth-Jensen, M. New effective interaction for f 5pg 9-shell nuclei. Phys. Rev. C 80, 064323 (2009).

40. 40.

Honma, M., Otsuka, T., Mizusaki, T. & Hjorth-Jensen, M. Shell-model fits for Sn isotopes. RIKEN Accel. Prog. Rep. 45, 35 (2012).

41. 41.

Otsuka, T. et al. Novel features of nuclear forces and shell evolution in exotic nuclei. Phys. Rev. Lett. 104, 012501 (2010).

42. 42.

Togashi, T., Tsunoda, Y., Otsuka, T., Shimizu, N. & Honma, M. Novel shape evolution in Sn isotopes from magic numbers 50 to 82. Phys. Rev. Lett. 121, 062501 (2018).

## Acknowledgements

We acknowledge A. Imreh from ELI-NP for the CAD drawings of the detector system used for the GEANT4 simulations and fig. 1. The authors P.-A.S. and N.T. acknowledge the support of the Romanian Ministry of Research and Innovation under research contract 10N / PN 19 06 01 05. L.C., E.A. D.L.B., G.L.G., D.L., C.M., D.N., A.P., and T.P. would like to acknowledge the support from the Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund - the Competitiveness Operational Programme (1/07.07.2016, COP, ID 1334). N.P. thanks the DFG for support under grant No. SFB 1245 (Project ID 279384907), the BMBF under grant No. 05P18RDEN9, and the state of Hesse for support of the "Nuclear Photonics” project within the LOEWE initiative. The MCSM calculations were performed on the K computer at RIKEN AICS, Project Number hp190160, and Oakforest-PACS operated by JCAHPC. This work was supported in part by the Post-K Computer Priority Issue "Elucidation of the Fundamental Laws and Evolution of the Universe” from MEXT and JICFuS, and the Multidisciplinary Cooperative Research Program at the CCS, University of Tsukuba. We also would like to thank Dr. D. Gambacurta at ELI-NP for many valuable discussions.

## Author information

Authors

### Contributions

P.-A.S., L.C., E.A., D.L.B., C.M., and A.P. designed the experimental set-up. P.-A.S., L.C., E.A., G.L.G., D.L., D.N., and T.P. collected the data. L.C. and D.N. wrote the software for data conversion. P.-A.S. wrote the software for data analysis and analysed the data. T.O., N.T., Y.T., and H.L. performed the theoretical calculations. P.-A.S. and D.L. performed the GEANT4 simulations. P.-A.S., L.C., T.O., N.T., D.L.B., and N.P. discussed the interpretation of the experimental and theoretical results. P-A.S. and N.T. prepared the manuscript draft, and all authors read and contributed to the discussion of the final manuscript.

### Corresponding author

Correspondence to P.-A. Söderström.

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### Competing interests

The authors declare no competing interests.

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Söderström, PA., Capponi, L., Açıksöz, E. et al. Electromagnetic character of the competitive γγ/γ-decay from 137mBa. Nat Commun 11, 3242 (2020). https://doi.org/10.1038/s41467-020-16787-4

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• ### Nuclear Matrix Elements for Heavy Ion Sequential Double Charge Exchange Reactions

• Horst Lenske
• , Jessica Bellone
• , Maria Colonna
•  & Danilo Gambacurta

Universe (2021)