Excitation of the $^{229}$Th nucleus via a two-photon electronic transition

TL;DR

This paper explores a novel method to excite the $^{229}$Th nucleus using a two-photon electronic transition, combining advanced theoretical calculations to propose a feasible experimental approach for nuclear excitation.

Contribution

It extends the theory of nuclear excitation via two-photon electron transitions to the $^{229}$Th ion, providing detailed calculations and an experimental scenario for nuclear state excitation.

Findings

Calculated NETP probabilities for $^{229}$Th in resonance conditions.

Identified optimal initial states for different nuclear energy levels.

Proposed a multi-step experimental excitation scheme.

Abstract

We investigate the process of nuclear excitation via a two-photon electron transition (NETP) for the case of the doubly charged thorium. The theory of the NETP process has been devised originally for heavy helium-like ions. In this work, we study this process in the nuclear clock isotope $^{229}$Th in the $2+$ charge state. For this purpose, we employ a combination of configuration interaction and many-body perturbation theory to calculate the probability of NETP in resonance approximation. The experimental scenario we propose for the excitation of the low lying isomeric state in $^{229}$Th is a circular process starting with a two-step pumping stage followed by NETP. The ideal intermediate steps in this process depend on the supposed energy $\hbar\omega_N$ of the nuclear isomeric state. For each of these energies, the best initial state for NETP is calculated. Special focus is put on the most recent experimental results for $\hbar\omega_N$.

Equations

$$M_{fi}=$$

$$+\frac{⟨{\gamma }_f{J}_f{\mu }_f,I_e{M}_e⟩H_{int}{\gamma }_n{J}_n{\mu }_n,I_g{M}_g⟨{\gamma }_n{J}_n{\mu }_n,I_g{M}_g⟩\mathbf{\alpha }\cdot {\mathbf{u}}_{\lambda }{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{r}}{\gamma }_i{J}_i{\mu }_i,I_g{M}_g}{{ϵ}_f+{\omega }_N-{ϵ}_n-\mathrm{i}\frac{{\Gamma }_n}{2}}),$$

$${\mathcal{W}}_{fi}=\frac{{\alpha }^3\omega }{2\pi }\sum _{{\mu }_i{\mu }_f{M}_g{M}_e\lambda }[J_i,I_g{]}^{-1}\int |M_{fi}{|}^2\mathrm{d}{\Omega }_k,$$

$${\mathbf{u}}_{\lambda }{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{r}}=\sqrt{2\pi }\sum _{LMp}{\mathrm{i}}^L[L{]}^{\frac{1}{2}}(\mathrm{i}\lambda {)}^p{D}_{M\lambda }^L({\varphi }_k,{\theta }_k,0){\mathbf{a}}_{LM}^{(p)},$$

$${\hat{H}}_{int}=\sum _{qr}{\hat{T}}_{qr}{\hat{M}}_{qr}^{\ast },$$

$${\mathcal{W}}_{fi}^{(Lpq)}=$$

$$\times \left(G_1^{(Lpq)}+G_2^{(Lpq)}+G_{12}^{(Lpq)}\right),$$

$$G_1^{(Lpq)}=\sum _{J_n}\frac{1}{[J_n]}{\left|\sum _{{\gamma }_n}\frac{⟨{\gamma }_f{J}_f⟩\mathbf{\alpha }\cdot {\mathbf{a}}_L^{(p)}{\gamma }_n{J}_n⟨{\gamma }_n{J}_n⟩{\hat{T}}_q{\gamma }_i{J}_i}{{ϵ}_i-{\omega }_N-{ϵ}_n-\mathrm{i}\frac{{\Gamma }_n}{2}}\right|}^2,$$

$$G_2^{(Lpq)}=\sum _{J_n}\frac{1}{[J_n]}{\left|\sum _{{\gamma }_n}\frac{⟨{\gamma }_f{J}_f⟩{\hat{T}}_q{\gamma }_n{J}_n⟨{\gamma }_n{J}_n⟩\mathbf{\alpha }\cdot {\mathbf{a}}_L^{(p)}{\gamma }_i{J}_i}{{ϵ}_f+{\omega }_N-{ϵ}_n-\mathrm{i}\frac{{\Gamma }_n}{2}}\right|}^2,$$

$$G_{12}^{(Lpq)}=$$

$$\times \frac{⟨{\gamma }_f{J}_f⟩{\hat{T}}_q{{\gamma }_n^{\prime }{J}_n^{\prime }}^{\ast }⟨{\gamma }_n^{\prime }{J}_n^{\prime }⟩\mathbf{\alpha }\cdot {\mathbf{a}}_L^{(p)}{{\gamma }_i{J}_i}^{\ast }}{{ϵ}_f+{\omega }_N-{ϵ}_n^{\prime }+\mathrm{i}\frac{{\Gamma }_n^{\prime }}{2}}).$$

$$G_1^{(Lpq)}\approx$$

$$\times {\left|\frac{⟨{\gamma }_f{J}_f⟩\mathbf{\alpha }\cdot {\mathbf{a}}_L^{(p)}{\gamma }_n{J}_n⟨{\gamma }_n{J}_n⟩{\hat{T}}_q{\gamma }_i{J}_i}{{ϵ}_i-{\omega }_N-{ϵ}_n-\mathrm{i}\frac{{\Gamma }_n}{2}}\right|}^2,$$

$$G_2^{(Lpq)}\approx$$

$$\times {\left|\frac{⟨{\gamma }_f{J}_f⟩{\hat{T}}_q{\gamma }_n{J}_n⟨{\gamma }_n{J}_n⟩\mathbf{\alpha }\cdot {\mathbf{a}}_L^{(p)}{\gamma }_i{J}_i}{{ϵ}_f+{\omega }_N-{ϵ}_n-\mathrm{i}\frac{{\Gamma }_i+{\Gamma }_n}{2}}\right|}^2,$$

$${\beta }^{(Lpq)}=\frac{{\mathcal{W}}_{fi}^{(Lpq)}}{{\Gamma }_q},$$

$${\Gamma }_q$$

$${\beta }^{(111)}=\frac{3\omega }{2{\omega }_N^3}\frac{1}{[J_i]}\left(G_1^{(111)}+G_2^{(111)}\right),$$

$${\beta }^{(112)}=\frac{30\omega }{{\alpha }^2{\omega }_N^5}\frac{1}{[J_i]}\left(G_1^{(112)}+G_2^{(112)}\right).$$

Full text

I Introduction

Atomic clocks are amongst the most precise measurement instruments available to date Bloom et al. (2014); Huntemann et al. (2016). Accurate time measurements, clock comparisons offer the opportunity to investigate fundamental physics and possible physics beyond the standard model Kosteleckỳ and Lane (1999); Rosenband et al. (2008); Safronova et al. (2014). Fifteen years ago it has been proposed by Peik and coworkers to build a clock based on a nuclear transition Peik and Tamm (2003). The most suitable of such transitions is found in the thorium isotope with mass number $A=229$ between the nuclear ground and the first excited isomeric state, nowadays sometimes referred to as nuclear clock isomer. Therefore intense research, theoretically and experimentally, has been performed on 229Th and especially the nucleus in its first excited state, the isomer 229mTh von der Wense (2017); Tkalya (2018); Seiferle et al. (2017); Safranova et al. (2018). Recently, for example, the nuclear moments of 229mTh have been determined Thielking et al. (2018); Müller et al. (2018), which may give insight into the energy of the nuclear isomeric state Beloy (2014). Moreover the emission of internal conversion electrons from the 229mTh$\rightarrow^{229}$Th transition has been observed von der Wense et al. (2016). However a controlled excitation of the nuclear isomer has not been achieved yet Stellmer et al. (2018).

A large number of different processes have been proposed to produce the 229mTh nuclear isomer ranging from direct laser excitation to the interaction with hot plasmas Karpeshin (2002); Peik and Tamm (2003); Pálffy et al. (2006); Gunst et al. (2014); Andreev et al. (2019); von der Wense et al. (2017). Out of these the excitation of nuclei by the energy excess from electronic processes appears to be very efficient and is much stronger than e.g. direct laser excitation Gunst et al. (2014, 2015); Müller et al. (2018). However all such electronic bridge processes come with a major challenge: For the process to be sufficiently strong, the electronic transition needs to be in proximity to the transition between the nuclear ground and the low lying isomeric state of 229Th. In the classic electron bridge process the energy difference between the electronic and nuclear transition is accounted for by the absorption of a photon. Instead of absorbing a photon with an enery that needs to be precisely tuned we consider a two-photon decay in the electron shell Volotka et al. (2016). In such a transition one, virtual, photon excites the nucleus while the other is emitted as a real photon. The energy share between both photons is continuous and, thus, there is no scanning necessary to excite the nucleus. This so called nuclear excitation by a two-photon electron transition (NETP) has been introduced for heavy highly charged ions, to access nuclear excited states in the keV regime Volotka et al. (2016).

In this work we want to investigate NETP in 229mTh. In contrast to other nuclear levels, the 229mTh isomeric state is found only about $8\text{\,}\mathrm{e}\mathrm{V}$ above the 229Th ground state. Therefore the electronic transition needs to be in the same energy range. Consequently lower charge states, especially 229Th2+, are promising candidates to observe NETP in thorium.

In contrast to the scenario discussed in Ref. Volotka et al. (2016) for helium like ions, Th2+ has many real intermediate resonances between the upper and the final state of the NETP process, provided by the rich level structure of the thorium ion. Ideally such a resonance is close to the nuclear excitation energy, thus enhancing the probability of the NETP process. The location and number of the resonances, however, strongly depends on the initially pumped upper state. Therefore the upper state which offers the highest probability for NETP depends on the energy of the nuclear isomeric state. In this paper we therefore provide detailed calculations for NETP in 229Th2+ and give clear recommendations for the levels to excite, depending on the energy range in which the isomer is searched.

Hartree atomic units ($\hbar=m_{e}=e=1$) are used throughout this paper unless stated otherwise.

II Scenario

A sketch of the scheme we propose for the excitation of the low lying isomeric state in 229Th can be seen in Fig. 1. First, starting from the $5f6d(J=4):$0\text{,}\mathrm{c}\mathrm{m}^{-1} ground state, the electron shell of the thorium ion is excited to an upper state with odd parity. From this upper state the NETP process occurs, where the nuclear excitation energy either corresponds to the energy splitting between the upper and the intermediate (left panel) or the intermediate and the lower state. The NETP decay is either of E1+M1 or E1+E2 type, therefore the final state of the process is of even parity. In this work we consider the $6d^{2}(J=2):$63\text{\,}\mathrm{c}\mathrm{m}^{-1} state as the final state, which is almost degenerate with the $5f6d(J=4):$0\text{,}\mathrm{c}\mathrm{m}^{-1} ground state. In previous experiments on Th+ and Th*2+* with a buffer gas quenched sample no significant population of dark states has been observed. Thus, we can safely assume that the $6d^{2}(J=2):$63\text{\,}\mathrm{c}\mathrm{m}^{-1} state decays quickly to the ground state due to collisional coupling Knoop et al. (1998); Herrera-Sancho et al. (2012); Thielking et al. (2018); Müller et al. (2018); Thielking (2019).

III Theory

III.1 NETP Transition Amplitudes and Rates

In the previous section we have described the process we propose for the excitation of 229Th. Now we will derive the probability of NETP in doubly charged thorium below. To simplify our considerations we will assume that the pumping of the upper state (cf. 1) is very efficient so that it is always populated. Therefore the probability of the process is given by the last two deexcitation steps which resemble the NETP process as discussed in Ref. Volotka et al. (2016).

In this work we will identify each many-electron state by its total angular momentum $J$, the projection $\mu$ of $J$ onto the quantization ($z$-) axis and a set of additional quantum numbers summarized by $\gamma$. The nuclear states are labelled by the nuclear spin $I$ and its projection $M$. The theoretical description of NETP consists of two interfering channels. As seen also in Fig. 1 either the first or the second photon can excite the nucleus. Consequently the NETP matrix element $M_{fi}$ consists of two terms:

[TABLE]

where $\omega_{N}$ is the frequency of the nuclear transition and $\boldsymbol{\alpha}$ denotes the vector of Dirac matrices. State energies and widths are denoted by $\epsilon$ and $\Gamma$, respectively, while the subscripts $i$, $n$ and $f$ specify the initial, intermediate and final states. Generally the intermediate state can be virtual and, thus, we have to sum over the entire spectrum $\ket{\gamma_{n}J_{n}\mu_{n}}$, where we assume that the continuous spectrum can be neglected. Note that in Eq. (1) we have omitted the width of the nuclear excited state, since it is much narrower than the electronic states.

Both terms in Eq. (1) each split into two matrix elements of the operators $H_{int}$ and $\boldsymbol{\alpha}\cdot\boldsymbol{u}_{\lambda}\,\mathrm{e}^{i\boldsymbol{k}\cdot\boldsymbol{r}}$. The latter is the usual interaction of the electron shell with a plane-wave photon with momenteum $\boldsymbol{k}$ polarized along $\boldsymbol{u}_{\lambda}$, where $\lambda$ is the helicity. The interaction Hamiltonian $H_{int}$ mediates the interaction between the electron shell and the nucleus, thus acting on both electronic and nuclear degrees of freedom.

To obtain the probability of the NETP process we can use Fermi’s golden rule:

[TABLE]

where $[k]=2k+1$, $\alpha$ is the fine structure constant, $\mathrm{d}\Omega_{k}$ the differential emission angle of the real photon and $\omega$ the frequency of the real, emitted, photon. In Eq. (2) we average over $M_{g}$ and $\mu_{i}$, assuming that the initial electronic and nuclear states are unpolarized. Moreover neither $\mu_{f}$ and $M_{e}$ nor the emission direction of the real photon is observed, thus we sum over the magnetic quantum numbers of the final states and integrate over $\Omega_{k}$.

To express Eq. (2) in a more convenient way the photon emission operator $\boldsymbol{\alpha}\cdot\boldsymbol{u}_{\lambda}\,\mathrm{e}^{i\boldsymbol{k}\cdot\boldsymbol{r}}$ is readily expanded into electric ($p=1$) and magnetic ($p=0$) multipoles $L$ with magnetic quantum number $M$Eisenberg and Greiner (1976); Rose (1995):

[TABLE]

where $D_{M\lambda}^{L}(\phi_{k},\theta_{k},0)$ is the Wigner-D matrix and $\boldsymbol{a}^{(p)}_{LM}$ are irreducible tensors of rank $L$ resembling the multipole fields.

Similar to the photon interaction operator, the electron-nucleus interaction $\hat{H}_{int}$ can be expanded into multipoles Porsev et al. (2010); Porsev and Flambaum (2010):

[TABLE]

where it is important to note that for each multipole the operator $\hat{H}_{int}$ splits into the hyperfine interaction operators $\hat{T}_{qr}$ acting only on electronic degrees of freedom and $\hat{M}_{qr}$ interacting with the nuclear part of the wave function. That way we can find the NETP probability for each multipolarity $q$ of the nuclear transition and electronic transitions $L$ and $p$.

[TABLE]

where the total probability of the process would be the sum over all possible $L$, $p$ and $q$ and

[TABLE]

The equations above show that the NETP probability for each multipole (5) splits into three parts proportional to the amplitudes $G^{(Lpq)}_{i}$. The first two amplitudes $G^{(Lpq)}_{1}$ and $G^{(Lpq)}_{2}$ correspond here to the cases illustrated in Fig. 1, where the real photon is emitted either due to the transition between the initial and the intermediate or the intermediate and the final state. The last amplitude $G^{(Lpq)}_{12}$ covers the interference between these two coherent processes.

III.2 Resonance Approximation

For our specific case the probability (5) can be further simplified. In contrast to the very simple electronic structure of helium-like systems, for which NETP has been first discussed (Volotka et al., 2016), Th2+ has a rich and dense level structure. Therefore it is safe to assume that only the closest resonance will contribute to the NETP probability. This allows for the application of the so-called resonance approximation. In this approximation all interference terms vanish, thus, $G^{(Lpq)}_{12}$ can be neglected and the terms $G^{(Lpq)}_{1}$ and $G^{(Lpq)}_{2}$ become:

[TABLE]

where we incorporated the width of the initial state in resonance approximation following Ref. Shabaev et al. (2010).

Now, the remaining task to calculate the NETP probability (5) in resonance approximation is the evaluation of the reduced nuclear and electronic matrix elements. The nuclear transition amplitudes $\braket{I_{e}}{\hat{M}_{q}}{I_{g}}$ are known from elaborate nuclear calculations, e.g. by Minkov and Pálffy Minkov and Pálffy (2017), where previous estimates by Tkalya et al. Tkalya et al. (2015) have been refined.

III.3 Enhancement Factor $\beta$

Due to the complexity of nuclear calculations, the nuclear amplitudes provided e.g. in Ref. Minkov and Pálffy (2017) are a major source of uncertainty in our calculations of the NETP probability (5). To circumvent these uncertainties one can define the enhancement factor $\beta$ (cf. Porsev and Flambaum (2010); Porsev et al. (2010)), which is independent on the nuclear transition probability:

[TABLE]

where the nuclear decay width $\Gamma_{q}$ is defined by:

[TABLE]

The enhancement factor (8) is defined in analogy to Refs. Porsev and Flambaum (2010); Porsev et al. (2010) and given here mainly to make a connection to these works and to test our theory with respect to effects coming from the electronic structure of Th2+.

Specifically for the case of 229Th, the leading multipoles of the nuclear transition are $M1$ and $E2$, so $q$ is either $1$ or $2$. From now on we will assume that all radiative electronic transitions are of $E1$ type, so that $L=1$ and $p=1$. Therefore, in resonance approximation, the enhancement factors of interest are

[TABLE]

IV Numerical details

Up to now we have shown how the NETP process may be discussed by taking the nuclear transition amplitude from the literature or by investigating the enhancement factor instead. Now we will briefly sketch the evaluation of the electronic matrix elements. To calculate these, we apply a combination of configuration interaction (CI) and many-body perturbation theory (MBPT), that has been described in detail in Refs. Dzuba et al. (1996); Dzuba (2005); Dzuba and Flambaum (2007). In particular we used the package assembled by Kozlov et al. Kozlov et al. (2015). The CI configuration state functions have been set up using Dirac-Hartree-Fock wave functions for the core orbitals and the $5f$, $6d$, $7s$ and $7p$ valence orbitals. For the higher lying orbitals we use a $b$-splines and orbitals constructed using the method described e.g. in Ref. Kozlov et al. (1996). The CI basis is constructed by virtual excitations from the $[Rn]+6d^{2}$ and $[Rn]+5f6d$ configuration.

The CI+MBPT method is a powerful method to calculate reliable transition matrix elements. Level energies, however, especially for complicated systems like Th2+ are determined more accurately in experiments. Because the exact position of the resonances is important to determine the NETP probability accurately, we take the experimental values Kramida et al. (2018) for all level energies instead of the theoretical ones. We will discuss the importance of this step in the section below.

V Results and Discussion

Before we discuss the probability of the NETP process in Th2+, we will have a brief look on the enhancement factor $\beta^{(111)}$ [cf. Eq. (8)]. In particular we want to investigate how the replacement of the calculated level energies by the experimental ones influences the results. Therefore we performed calculations for $\beta^{(111)}$ as a function of the nuclear excitation energy $\omega_{N}$ using both. The results of these calculations are shown in Fig. 2, the theoretical (black solid line) and the experimental (red dashed line) level energies. The first feature we notice in Fig. 2 is the different number of resonance peaks for different $J_{i}$ of the upper (initial) state. This can be explained by the sheer number of available decay paths to the $6d^{2}(J=2):$63\text{,}\mathrm{c}\mathrm{m}^{-1}$$ state from each of these upper states. While for $J_{i}=4$ and a $E1$ radiative transition, the intermediate state must have $J_{n}=3$, for $J_{i}=2$ there are three possible $J_{n}$ and, therefore, more intermediate resonances available. But there are two more important things to notice. Foremost we see that the high energy cutoff of $\beta$ is reduced for the case of the experimental level energies. Therefore we note, that it is very important to take the energy splitting between the initial and final electronic state accurately into account. Moreover we see that the replacement of the energies of the intermediate states to their experimental values does not change the qualitative behaviour of $\beta^{(111)}$ and, thus, can be safely done to achieve accurate results.

The primary aim of this paper is to provide information about the most promising excitation paths to observe the NETP process in 229Th2+. Therefore we assume according to available experimental setups that the exciting lasers are tunable between $3.45\text{\,}\mathrm{e}\mathrm{V}$ and $5.25\text{\,}\mathrm{e}\mathrm{V}$ Meier (2018) (cf. Fig. 1). With such lasers $19$ possible upper states $\ket{\gamma_{i}J_{i}}$ can be pumped. This number reduces to $16$, if we fix the final state to be the level $6d^{2}(J=2):$63\text{,}\mathrm{c}\mathrm{m}^{-1}$$, in order to be able to cycle through the process multiple times. For each these $16$ possible upper states we calculated the NETP probability (5) summing over $1\leq q\leq 2$ in order to account for both the $M1$ and $E2$ nuclear transition channels. This step is necessary because it has been shown recently that both, the $M1$ and the $E2$ channel, may contribute equally to the NETP probability Bilous et al. (2018). Similar to Fig. 2, we display the NETP probability $\mathcal{W}_{fi}=\sum_{q=1}^{2}\mathcal{W}^{(E1,q)}_{fi}$ as a function of the nuclear excitation energy $\omega_{N}$. This data, however, is not very conclusive. Thus it needed to be processed, which is illustrated in Fig. 3. In the upper panel of this figure we display the NETP probability for four upper states as a function of the nuclear excitation energy $\omega_{N}$. To get our final result we take the envelope of this family of curves as shown in the bottom panel of Fig. 3. Moreover we omit resonance peaks narrower than $0.1\text{\,}\mathrm{e}\mathrm{V}$ for it would make the figure impractical to use, especially at higher $\omega_{N}$, where the resonances get more dense. Note also that we do not show the NETP probability for those of the $16$ possible upper states that do not contribute to the envelope. The vertical lines in Fig. 3 denote the most recent values for the energy of the nuclear isomeric state, ranging from $7.1\text{\,}\mathrm{e}\mathrm{V}$ to $7.6\text{\,}\mathrm{e}\mathrm{V}$ and $7.8\text{\,}\mathrm{e}\mathrm{V}$ Beck et al. (2007); Tkalya et al. (2015); Borisyuk et al. (2018). The grey shaded area denotes the combined uncertainties of all three measurements and, thus, a recommended initial search area for the nuclear isomer.

With the preparation of the data explained above we are now able to generate the main result of this work. In Fig. 4 we show, which is the ideal upper state to observe the NETP process in 229Th2+ as a function of $\omega_{N}$. It can be seen that also for the entire energy range between $6.5\text{\,}\mathrm{e}\mathrm{V}$ and $10.5\text{\,}\mathrm{e}\mathrm{V}$ only $5$ of the possible $16$ upper states need to be considered for a possible experiment. Again the vertical lines and the grey area in Fig. 4 mark the recommended initial search area for the nuclear isomeric state.

Let us finally discuss how the excitation of the nucleus could be monitored in the experiment we propose. Recently the hyperfine structure of the electronic levels in Th2+ has proven to be a good indicator of whether the nucleus is in its ground or first excited state Thielking et al. (2018). This would be as well possible in the scenario proposed in the present work by either applying an additional laser or observing the fluorescence from one of the pumping stages. Another common option would be to observe the time delayed photoemission from the nuclear decay. This, however, would not be recommended for the scenario proposed here, because we could cycle through the process, no matter if the upper state decayed via NETP or the more likely two-photon cascade. This allows for a good statistics and does not require a shot-by-shot analysis of the data with accurate timing.

VI Concluding remarks

The NETP process has been shown to be a promising candidate to investigate the nuclear structure of highly charged ions Volotka et al. (2016). In the present work this process is discussed for many electron systems within the resonance approximation. To excite the 229Th nucleus, we propose a combination of a two-step pumping of an upper state from which the NETP process occurs. To overcome the difficulty of a small branching ratio between NETP and a generic radiative two-step decay of the upper state, the proposed process can be cycled independent on the way the ion decays.

With several promising experiments at the horizon that aim for a precise determination of $\hbar\omega_{N}$ von der Wense et al. (2017); Seiferle et al. (2018), the scenario described in this work aims for a controlled excitation of the 229Th. Therefore a challenge that comes with many proposed electronic bridge processes for the excitation of the 229Th nucleus, the requirement of a continuous scanning with a tunable laser, does not apply in the scenario described in this paper. In the proposed experiment the lasers are adjusted only once to ensure the most efficient pumping of the upper state. For a first test of our theory we recommend to pump the $5f7d(J=2):$83,237\text{,}\mathrm{c}\mathrm{m}^{-1} and the $5f7d(J=3):$84\,374\text{\,}\mathrm{c}\mathrm{m}^{-1} states, which both have resonances close to the currently assumed value $7.8\text{\,}\mathrm{e}\mathrm{V}$ of the nuclear excitation energy.

Acknowledgements.

RAM acknowledges support form the RS-APS and many useful discussions with David-Marcel Meier and Johannes Thielking.