Low-lying energy levels of ²²⁹Th³⁵⁺ and the electronic bridge process

A specific feature of the ²²⁹Th nucleus is that the energy difference between the ground state and the first excited state is only several eV. An existence of such a low-lying level was established more than forty years ago [1], but a precise measurement of the isomeric state energy turned out to be very difficult. In 1994, Helmer and Reich [2] measured this excitation energy (ω_N) to be 3.5 ± 1 eV. In reference [3], it was obtained as 5.5 ± 1.0 eV. The experiments of Beck et al [4, 5] gave an even larger value with less error, ω_N = 7.8 ± 0.5 eV. Finally, in recent experiments of Seiferle et al [6] and Sikorsky et al [7], the values ω_N = 8.28 ± 0.17 and ω_N = 8.10 ± 0.17 eV were obtained. Both these results were used in an analysis carried out in reference [8], yielding the value ω_N = 8.19 ± 0.12 eV, which is, in fact, the average of the two results. Thus, the current most precise value differs from the result of 1994 by more than two times, but is in a good agreement with the result obtained in reference [5].

The nuclear transition between the ground and the low-energy isomeric state in the ²²⁹Th nucleus is of interest due to its high sensitivity to a hypothetical temporal variation of the fundamental constants [9]. For instance, a possible change of ω_N due to a variation of the fine structure constant α is given by δω_N/ω_N = Kδα/α. In reference [10], the sensitivity coefficient K was predicted to be as large as 8200. Local Lorentz invariance violating and Einstein equivalence principle violating effects are also strongly enhanced in the nuclear transition in the ²²⁹Th nucleus [11]. Another unique feature of this transition is a possibility to build a very precise nuclear clock [12]. It requires the precise knowledge of the nuclear clock transition frequency and, consequently, further investigations aiming to refine the value of ω_N.

An experimental progress in trapping and sympathetic cooling of highly charged ions (HCIs) (see review in reference [13] for details) using electron-beam ion traps (EBITs) opened new possibilities to use optical transitions of such ions for different applications. As it was discussed in reference [13], an interaction region of an electron beam with maximum magnetic field in EBITs is short enough. It reduces possible electron-beam instabilities and also allows for a high charge density. Both of these effects speed up the ionization process. If a HCI, considered as a clock candidate, does not have a transition suitable for laser cooling, sympathetic cooling can be done by trapping the HCI together with the cooling ion in the same trapping potential and using their mutual Coulomb interaction. In the recent paper [14], it was suggested to use the electronic bridge (EB) process in Th³⁵⁺ for an accurate determination of the nuclear isomer energy and the energies of the low-lying states, needed to estimate the EB process rate, were calculated.

Since the EB process rate depends drastically on the energies of the states involved in this process, we calculated the energies of the low-lying states of Th³⁵⁺ in different approximations of increasing complexity. The main configuration of the ground state of the ion is (1s², ..., 4d¹⁰4f⁹) and we used configuration interaction (CI) method for calculations. First, we performed a nine-electron CI calculation, including nine 4f electrons in the valence field while treating all other electrons as the core electrons. Then, we carried out 19- and 25-electron CI calculations, including the 4f, 4d and 4f, 4d, 4p electrons into the valence field, respectively. In such a way, we were able to estimate the role of core–valence correlations, which allowed us to determine the uncertainties of the energies.

To carry out these calculations for such a complicated open-shell system as Th³⁵⁺, we used a new parallel atomic structure code package developed and described in reference [15]. This package (i) allows us much quicker computations and (ii) enables us to work with a CI space of a much larger size than was possible previously when we used serial versions of the programs. Using a parallel version of the program for the CI calculation, we were able to use several hundreds of processors simultaneously and, consequently, to work with a large CI space, involving up to 120 millions of determinants.

Using the energies obtained and assuming a resonant character of the induced EB process, we estimated its rate for several possible values of the nuclear isomeric energy. Based on these calculations, we conclude that modern facilities of EBITs and available ultra-violet lasers allow us to determine more precisely the nuclear isomeric energy and reduce its uncertainty.

Here we consider the process of the excitation of the nucleus from the ground (g) to the isomeric state (m) by an EB process driven by a one-photon excitation of the electron shell. Such a process, relying on the absorption of an incident photon, can be represented by the Feynman diagram in figure 1. In this process, the electronic shell is promoted by a laser photon from its initial state t to an excited state k and then decays to a lower-lying state i. The energy transferred to the nucleus is used to excite it from the ground to isomeric state. Assuming a resonant character of the process, we take into account only one Feynman diagram that gives the main contribution to the probability of the process.

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In the following, we use the formalism developed in references [16–18], with the difference that an M1 photon (instead of a E1 photon) is absorbed by the electron shell. Such a photon is described by the magnetic moment operator

$$\begin{equation}\boldsymbol{\mu }=-{\mu }_{0}\left(\mathbf{J}+\mathbf{S}\right).\end{equation} \tag{ 1 }$$

Here J and S are the total and spin momentum operators and μ₀ is the Bohr magneton determined as μ₀ = |e|ℏ/(2mc), where e and m is the electron charge and mass, c is the speed of light, and ℏ is the Plank constant. (If not stated otherwise the atomic units ℏ = m = |e| = 1 and c ≈ 137 are used in the following).

We assume that the incident radiation with spectral intensity I_ω is isotropic and unpolarized. Following reference [19] the relation between the EB excitation rate, Γ_exc, and the rate of the inverse spontaneous EB process, Γ, that can be formally described by the mirror-image of figure 1 with an outgoing photon arrow, can be written as

$$\begin{equation}{{\Gamma}}_{\text{exc}}={\Gamma}\frac{4{\pi }^{3}{c}^{2}}{{\omega }^{3}}{g}_{r}{I}_{\omega },\end{equation} \tag{ 2 }$$

where ω is the frequency of the absorbed photon. The factor g_r ≡ (2I_m + 1)(2J_i + 1)/[(2I_g + 1)(2J_t + 1)] appears because we average over the initial projections and summing over the final projections of the electronic and nuclear total angular momenta. Here I_m (I_g ) is the nuclear spin of the isomeric (ground) state and J_i (J_t ) is the total angular momentum of the initial (final) electronic state in the spontaneous EB process.

The general expression for the probability of the EB process was derived in reference [17]. Assuming a resonance character of the spontaneous EB process, its transition rate can be written as,

$$\begin{equation}{\Gamma}\approx \frac{16\pi }{3\left(2{J}_{i}+1\right)}{\left(\frac{\omega }{c}\right)}^{3}\sum\limits _{K=1}^{2}\frac{B\left(\tau K\right){G}_{K}}{{\left(2K+1\right)}^{2}}.\end{equation} \tag{ 3 }$$

The reduced probabilities, B(M1) and B(E2), of the nuclear magnetic-dipole and electric-quadrupole m → g transition are determined as

$$\begin{equation}B\left(\tau K\right)=\frac{1}{2{I}_{m}+1}\frac{2K+1}{4\pi }\vert \langle {I}_{g}{\Vert}{\mathcal{M}}_{K}{\Vert}{I}_{m}\rangle {\vert }^{2},\end{equation} \tag{ 4 }$$

where τ ≡ M for K = 1 and τ ≡ E for K = 2; ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ are the magnetic-dipole and electric-quadrupole nuclear moment operators, and |I_g ⟩ = 5/2⁺ [633] and |I_m ⟩ = 3/2⁺ [631] are the ground and isomeric nuclear states, respectively, given in their Nilsson classification.

The explicit expression for the quantity G_K was derived in reference [17], and in our case, is reduced to

$$\begin{equation}{G}_{K}\approx \frac{1}{2{J}_{k}+1}{\left\vert \frac{\langle i{\Vert}{T}_{K}{\Vert}k\rangle \langle k{\Vert}\mu {\Vert}t\rangle }{{E}_{k}-{E}_{i}-{\omega }_{\text{N}}}\right\vert }^{2},\end{equation} \tag{ 5 }$$

where |i⟩, |k⟩, and |t⟩ are the electronic states, E_k and E_i are the energies of the respective states, and J_k is the electron total angular momentum of the state |k⟩. We assume that there is only one state giving the dominant contribution to G_K , while the contribution of all other intermediate states is negligible, i.e., the state |k⟩ is chosen so that E_k − E_i ≈ ω_N.

The expressions for the single-electron operators T₁ and T₂ can be written as

$$\begin{align}{T}_{1\lambda }\left(\mathbf{r}\right)& =-\frac{i\sqrt{2}\enspace \boldsymbol{\alpha }\cdot {\mathbf{C}}_{1\lambda }^{\left(0\right)}\left(\mathbf{n}\right)}{c{r}^{2}},\\ {T}_{2\lambda }\left(\mathbf{r}\right)& =-\frac{{C}_{2\lambda }\left(\mathbf{n}\right)}{{r}^{3}},\end{align} \tag{ 6 }$$

where α is the Dirac matrix, n ≡ r/r, and ${\mathbf{C}}_{K\lambda }^{\left(0\right)}$ is a normalized vector spherical harmonic defined by (see, e.g., reference [20])

$$\begin{equation}{\mathbf{C}}_{K\lambda }^{\left(0\right)}\left(\mathbf{n}\right)=\frac{\mathbf{L}}{\sqrt{K\left(K+1\right)}}{C}_{K\lambda }\left(\mathbf{n}\right).\end{equation} \tag{ 7 }$$

Here L is the orbital angular-momentum operator and C_Kλ is a spherical harmonic given by

$$\begin{equation}{C}_{K\lambda }\left(\mathbf{n}\right)=\sqrt{\frac{4\pi }{2K+1}}{Y}_{K\lambda }\left(\mathbf{n}\right).\end{equation} \tag{ 8 }$$

A most accurate determination of the nuclear transition frequency ω_N was recently done in reference [8] to be ω_N = 8.19(12) eV. The authors of reference [14] calculated the low-lying energy levels of Th³⁵⁺ and found the transition frequencies from the low-lying excited states with J = 13/2 and J = 11/2 to the ground state (J = 15/2) to be 8.40 eV (which is close to ω_N) and 4.2 eV, respectively. As we discuss in detail below, our calculations give the close values, 8.19 and 4.62 eV.

The state with J = 11/2 and E ≈ 4.6 eV can be used as the initial state in the induced EB process. Applying the designations in figure 1, we have |t⟩ ≡ |J = 11/2⟩, |k⟩ ≡ |J = 13/2⟩, and |i⟩ ≡ |J = 15/2⟩. A scheme of the low-lying electronic and nuclear states, involved in the EB process, is presented in figure 2.

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We carried out the calculation of the low-lying states in the framework of the CI method using the program package described in reference [21] and further developed in reference [15]. We included the Breit interaction correction and estimated the core–valence correlations. Our results, discussed in more detail in the following sections, confirmed the main conclusion of reference [14] about a possibility to determine accurately the nuclear isomer energy using the EB process with realistic laser parameters.

We start calculations of energy levels of Th³⁵⁺ belonging to the (1s², ..., 4f⁹) configuration from solutions of the Dirac–Hartree–Fock equations, carrying out a self-consistency procedure for the [1s², ..., 4f⁹] electrons. The Breit interaction was included in this procedure. The remaining virtual orbitals were formed using a recurrent procedure described in [21, 22], where the large component of the radial Dirac bispinor, f_n'l'j', was obtained from a previously constructed function f_nlj by multiplying it to r^l '^−l  sin(kr), where l' and l are the orbital quantum numbers of the new and old orbitals (l' ⩾ l) and the coefficient k is determined by the properties of the radial grid. The small component g_n'l'j' was found from the kinetic balance condition:

$$\begin{equation}{g}_{{n}^{\prime }{l}^{\prime }{j}^{\prime }}=\frac{\boldsymbol{\sigma }\boldsymbol{p}}{2mc}{f}_{{n}^{\prime }{l}^{\prime }{j}^{\prime }},\end{equation} \tag{ 9 }$$

where σ are the Pauli matrices, p and m are the electron momentum and mass, and c is the speed of light. The newly constructed functions were then orthonormalized with respect to the functions of the same symmetry. In constructing the virtual orbitals, we did not diagonalize the basis set.

In total, the basis set consisted of six partial waves (l ⩽ 5), including the orbitals up to 10s, 10p, 10d, 10f, 10g, and 10h. The configuration space grows very rapidly with an increase of the basis set; for this reason the basis set is rather short.

For an accurate calculation of energies, we need to take into account valence–valence and core–valence correlations. The former can be treated explicitly in the framework of the CI method. A CI many-electron wave function of a given angular momentum J and parity can be represented by a linear combination of Slater determinants [23]:

$$\begin{equation}{{\Psi}}_{J}=\sum\limits _{k}{a}_{k}{{\Phi}}_{k}.\end{equation} \tag{ 10 }$$

A way to account for core–valence correlations was suggested in reference [23], but in practice, it is applicable when the number of valence electrons does not exceed 4–5. In our consideration, we include nine 4f electrons into the valence field, so such a method is impractical. To estimate the role of different core shells, we carried out several calculations of increasing complexity in the framework of the CI method by sequentially adding the core shells into the valence field. In this way, we carried out the nine-, 19- and 25-electron CI calculations when the 4f, (4f, 4d), and (4f, 4d, 4p), electrons, respectively, were included in the CI space. Below we discuss these calculations in more detail.

3.1. Nine-electron CI

We start from the most simple case of the nine-electron CI. To check the convergence of the CI method, we calculated the low-lying energy levels for five cases. In the first case, we included the single and double excitations of the electrons from the main configuration 4f⁹ to the 6s, 6p, 6d, 6f, 6g and 6h orbitals (we designate it as [6spdfgh]). In the other four calculations, we included the single and double excitations to [nspdfgh], where n = 7–10. We checked that an inclusion of the triple excitations change the energies only by few cm⁻¹. Thus, this contribution can be neglected.

The results are presented in table 1. The terms are determined by their total angular momentum J. The energies of four lowest-lying excited states counted from the ground state and found for different n (when the single and double excitations were allowed to [nspdfgh]) are presented in the columns labeled 'E' in cm⁻¹. The energy differences Δ(n) ≡ E(nspdfgh) − E(n − 1spdfgh) and the ratios δ_n ≡ Δ(n)/Δ(n − 1) are given in the columns labeled 'Δ(n)' (in cm⁻¹) and 'δ_n '. The total values are given in the last row.

Table 1. Nine-electron CI. The energies of four lowest-lying excited states counted from the ground state and found for different n (when the single and double excitations were allowed to [nspdfgh]) are presented in the columns labeled 'E' in cm⁻¹. The energy differences E(nspdfgh) − E(n − 1spdfgh) are given in the columns labeled 'Δ(n)' (in cm⁻¹). The ratios Δ(n)/Δ(n − 1) are listed in the columns labeled 'δ_n '. The total values, obtained as E(n = 10) + Δ(n > 10), are given in the last row.

	E	J = 9/2Δ(n)	δn	E	J = 11/2 Δ(n)	δn	E	J = 3/2 Δ(n)	δn	E	J = 13/2 Δ(n)	δn
n = 6	30 503			36 728			55 006			67 200
n = 7	30 279	−224		36 411	−316		54 441	−565		67 255	54.6
n = 8	30 171	−108	0.483	36 266	−145	0.459	54 175	−266	0.471	67 290	35.6	0.652
n = 9	30 118	−52.8	0.487	36 205	−61.2	0.421	54 061	−114	0.430	67 313	23.1	0.648
n = 10	30 090	−28.0	0.529	36 178	−26.4	0.432	54 010	−51.1	0.447	67 329	15.9	0.689
q a			0.500			0.437			0.449			0.663
n > 10 b		−35			−13			−29			33
Total	30 055			36 165			53 980			67 362

^a.See explanation in the text. ^b.The contribution to the energies from the configurations containing shells with n > 10.

To estimate the contribution to the energies from the configurations containing shells with n > 10, we note that δ_n for n = 8, 9, 10 are numerically close to each other for all terms. Assuming that the same is true and for n > 10 we are able to use the formula for a geometric progression to estimate the respective contribution.

Putting the first term of the geometric progression to be b₁ ≡ Δ(7), determining q as an average over δ_n for n = 8, 9, 10, i.e., q ≡ (δ₈ + δ₉ + δ₁₀)/3, and using the equation for the sum of k terms in the geometric progression,

$$\begin{equation}{S}_{k}=\frac{{b}_{1}\left(1-{q}^{k}\right)}{1-q},\end{equation} \tag{ 11 }$$

we are able to calculate S_k for any k. Then we can estimate Δ(n > 10), as

$$\begin{equation}{\Delta}\left(n{ >}10\right)\approx {S}_{k}-\left({{\Delta}}_{8}+{{\Delta}}_{9}+{{\Delta}}_{10}\right).\end{equation} \tag{ 12 }$$

For instance, for the J = 9/2 term we have b₁ = −224 cm⁻¹, q = 0.50, and putting k to be equal to 100, we find S₁₀₀ ≈ −448 cm⁻¹ and Δ(n > 10) ≈ −35 cm⁻¹. Carrying out similar calculations for other terms, we find the values listed in the row labeled 'n > 10' of table 1. The total values, obtained as E(n = 10) + Δ(n > 10), are given in the last row of the table.

3.2. Core–valence correlations

3.3. EB process excitation rate

Using equations (2)–(5), we are able to estimate the EB process probability. To find the probability of the spontaneous process given by equation (3) we need to calculate the matrix elements of the operators T₁, T₂, and μ in equation (5).

We carried out this calculation for the largest 25-electron CI when single and double excitations were allowed to [7spdfgh] and obtained

$$\begin{align}\vert \langle J=15/2{\Vert}{T}_{1}{\Vert}J=13/2\rangle \vert & \approx 2.0\enspace \mathrm{a}.\mathrm{u}.,\\ \vert \langle J=15/2{\Vert}{T}_{2}{\Vert}J=13/2\rangle \vert & \approx 39\enspace \mathrm{a}.\mathrm{u}.,\\ \vert \langle J=13/2{\Vert}\mu {\Vert}J=11/2\rangle \vert & \approx 2.28{\mu }_{0}.\end{align} \tag{ 13 }$$

The calculation performed in the framework of the nine-electron CI with allowing single and double excitations to [10spdfgh] led to the values that differ by less than 1% from the results given above.

The values of the reduced probabilities, B(M1) and B(E2), of the nuclear m → g transition available in the literature are contradictory. In reference [25], the value of B(M1) was found to be 0.048 Weisskopf units (W.u.). The calculation of Ruchowska et al [26] led to the value 0.014 W.u., while the recent model calculation of Minkov and Pálffy [27] predicted the B(M1) value in the limits of 0.005–0.008 W.u., an order of magnitude smaller than the result reported in reference [25].

A similar situation arises for B(E2). Strizhov and Tkalya [28], referring to reference [29], cited the value of several W.u., while in reference [27], it was found an order of magnitude larger value, in the limits of 29–43 W.u..

We note that in references [16–18], the values B(M1) = 0.048 W.u. [25] and B(E2) ∼ 1 W.u. [28, 29] were used for the calculation. As a result, the contribution of the E2 channel was small and this channel was disregarded. In this work, we use for an estimate of the EB transition rate the recent values, B(M1) = 0.005 W.u. and B(E2) = 29 W.u., obtained in reference [27]. In this case the E2 channel should be included into consideration.

For the M1 and E2 transitions rates from the nuclear excited state to the ground state, Weisskopf units can be expressed in usual units as

$$\begin{equation}\begin{gathered}M1 :1\enspace \mathrm{W}.\mathrm{u}.=1.790{\left(\frac{e\hslash }{2{m}_{\text{p}}c}\right)}^{2},\\ E2 :1\enspace \mathrm{W}.\mathrm{u}.=5.940{\times}1{0}^{-6}{A}^{4/3}{\left(eb\right)}^{2},\end{gathered}\end{equation} \tag{ 14 }$$

where m_p is the proton mass and A is the number of nucleons in the nucleus. Using these definitions, we can express B(M1) and B(E2) in a.u. as B(M1) ≈ 3.5 × 10⁻¹⁴ a.u. and B(E2) ≈ 3.1 × 10⁻¹⁶ a.u.

As follows from equations (3)–(5), the EB transition rate depends substantially from the magnitude of the nuclear transition frequency ω_N determined as 8.19(12) eV in reference [8]. To illustrate it we calculated these rates for three values of ω_N: its central value 8.19 eV and two edge values of the uncertainty interval 8.31 and 8.07 eV. We note that the frequency of the absorbed photon ω is determined as ω = ω_N − E_{J = 11/2} and assume that the time-averaged spectral intensity is I_ω ≃ 10⁻³ (W m⁻²) s [14, 16].

The results are presented in table 3. The possible values of the nuclear transition frequency ω_N are listed in the first column. The quantities G₁, G₂, and Γ, given by equations (5) and (3), are listed in columns 2–4. We note that for the used values of B(M1) and B(E2), both terms in equation (3) give approximately the same contribution. The values of the EB excitation rate per ion, Γ_exc, are presented in the fifth column for different ω_N.

Table 3. Possible values of the nuclear transition frequency ω_N are listed in the first column. The quantities G₁, G₂, and Γ are given by equations (5) and (3). The values of the EB excitation rate, Γ_exc, are presented in the fifth column.

ωN (eV)	G1 (a.u.)	G2 (a.u.)	Γ (s−1)	Γexc (s−1)
8.31	1570	5.93 × 105	5.70 × 10−4	0.3
8.19	1.1	404	3.50 × 10−7	2.10 × 10−4
8.07	0.3	99	7.70 × 10−8	5.0 × 10−5

As seen from table 3, the largest Γ_exc, obtained in the case of ω_N = 8.31 eV, reaches 0.3 s⁻¹. This is due to this value of ω_N being very close to E_{J = 13/2} ≈ 8.307 eV, leading to a very small denominator in equation (5) and, respectively, very large G_1,2 and Γ_exc. For the other two considered values of the nuclear transition frequency, 8.19 and 8.07 eV, there is no such a resonant enhancement of the effect and the EB excitation rates are several orders of magnitude smaller. Our results are in a reasonable agreement with those obtained in reference [14].

We carried out the calculation of the low-lying energy levels for such a complicated multivalent ion as Th³⁵⁺. To determine the contribution of the valence–valence and core–valence correlations, we performed the nine-, 19-, and 25-electron CI calculations, including 4f, 4f, 4d, and 4f, 4d, 4p shells, respectively, into the valence field. Our calculation showed that the transition energy from the J = 15/2 state to the ground state, 8.31 eV, is close to the central value of the experimentally determined nuclear isomer energy, 8.19 eV, and practically coincides with the upper edge value, 8.31 eV. It opens new possibilities for a more precise measurement of the nuclear isomer energy using an EB process.

We studied an EB process scheme and estimated the excitation rates of the spontaneous and (inverse) induced EB processes for possible values of the g → m nuclear transition frequency ω_N. We found the EB excitation rate per ion to be 0.3 s⁻¹ in the case of ω_N = 8.31 eV. For other considered values of the nuclear transition frequency, 8.19 and 8.07 eV, this rate is 3–4 orders of magnitude smaller. Based on these results and on the study of reference [14] where typical EBIT conditions were considered, we conclude that an efficient population of the nuclear isomer state and a precise determination of its energy using an EBIT and available ultra-violet lasers is already attainable. The Th³⁵⁺ ion is a very promising candidate for such an experiment.

We are grateful to P Bilous and A Pálffy for valuable discussion and useful remarks. This work is a part of the 'Thorium Nuclear Clock' project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 856415). SP acknowledges support by the Russian Science Foundation under Grant No. 19-12-00157. This research was supported in part through the use of the Caviness community cluster at the University of Delaware.

The data that support the findings of this study are available upon reasonable request from the authors.