Energy of the $^{229}$Th nuclear clock transition

TL;DR

This paper reports the first direct measurement of the $^{229}$Th nuclear excited state energy, enabling the development of a highly precise nuclear optical clock with broad scientific applications.

Contribution

It provides the first direct excitation energy measurement of the $^{229}$Th nuclear state, constraining it to 8.28±0.17 eV, facilitating nuclear laser spectroscopy and clock development.

Findings

Nuclear excitation energy constrained to 8.28±0.17 eV.

Transition wavelength determined as 149.7±3.1 nm.

Measurement method merges nuclear and atomic physics techniques.

Abstract

The first nuclear excited state of $^{229}$Th offers the unique opportunity for laser-based optical control of a nucleus. Its exceptional properties allow for the development of a nuclear optical clock which offers a complementary technology and is expected to outperform current electronic-shell based atomic clocks. The development of a nuclear clock was so far impeded by an imprecise knowledge of the energy of the $^{229}$Th nuclear excited state. In this letter we report a direct excitation energy measurement of this elusive state and constrain this to 8.28$\pm$0.17 eV. The energy is determined by spectroscopy of the internal conversion electrons emitted in-flight during the decay of the excited nucleus in neutral $^{229}$Th atoms. The nuclear excitation energy is measured via the valence electronic shell, thereby merging the fields of nuclear- and atomic physics to advance precision metrology. The transition energy between ground and excited state corresponds to a wavelength of 149.7$\pm$3.1 nm. These findings set the starting point for high-resolution nuclear laser spectroscopy and thus the development of a nuclear optical clock of unprecedented accuracy. A nuclear clock is expected to have a large variety of applications, ranging from relativistic geodesy over dark matter research to the observation of potential temporal variation of fundamental constants.

Tables

Table 1. Table 1: Influence of different numbers of initially excited electronic states on the systematic uncertainty of the data analysis. While the centroid of the distribution ( E 0 subscript 𝐸 0 E_{0} ) stays almost constant, the systematic uncertainty ( σ 𝜎 \sigma ) gets narrower with increasing n 𝑛 n . The calculations were performed with an assumed energy of 8.3 eV.

Number of excited states ($n$)	$E_0$	Width of the distribution $\sigma$
2	6.56$\pm$0.02	0.22
4	6.517$\pm$0.003	0.18
5 $\ast$	6.506$\pm$0.001	0.16
10	6.504$\pm$0.001	0.12
20	6.492$\pm$0.001	0.09
30	6.494$\pm$0.001	0.06
40	6.493$\pm$0.001	0.05

Table 2. Table 2: List of all states in the Th atom below 20000 cm -1 that are considered in the analysis. Full data can be found in ref. 10 .

even states		odd states
State index	$E_i$ [cm-1]	State index	$E_i$ [cm-1]
1	0	1	7795.275
2	2558.057	2	8243.601
3	2869.259	3	10414.136
4	3687.987	4	10526.544
5	3865.475	5	10783.154
6	4961.659	6	11197.031
7	5563.142	7	11241.730
8	6362.396	8	11877.839
9	7280.124	9	12114.366
10	7502.288	10	13175.113
11	8111.005	11	13945.307
12	8800.251	12	14032.085
13	9804.807	13	14206.917
14	11601.031	14	14243.993
15	11802.934	15	14247.307
16	12847.971	16	14465.222
17	13088.563	17	14481.869
18	13297.434	18	15166.901
19	13847.771	19	15490.077
20	13962.522	20	15618.984
21	14204.264	21	15736.969
22	14226.822	22	16217.482
23	15493.221	23	16346.651
24	15863.891	24	16783.847
25	15970.095	25	17224.303
26	16351.943	26	17354.639
27	16554.245	27	17411.224
28	17073.811	28	17501.176
29	17166.108	29	17847.077
30	17398.398	30	18011.380
31	17959.898	31	18053.617
32	18431.686	32	18069.065
33	18549.405	33	18382.826
34	18574.608	34	18614.338
35	18699.623	35	18809.887
36	19273.279	36	18930.293
37	19532.419	37	19039.153
38	19713.031	38	19227.336
39	19832.116	39	19503.144
		40	19516.981
		41	19588.362
		42	19817.182
		43	19948.395
		44	19986.166

Equations

$$E_I=8.28\pm 0.17\text{mbox}eV.$$

$${\hat{H}}_{\mathrm{magn}}=-\frac{1}{c}\vec{\alpha }\int d^3{r}_n\frac{{\vec{j}}_n({\vec{r}}_n)}{|{\vec{r}}_e-{\vec{r}}_n|},$$

$${\Gamma }_{\text{IC}}=\frac{8{\pi }^2}{9}\Lambda B_{\downarrow }^{M1}\sum _{\begin{array}{c}\kappa \\ \text{ (even }l)\end{array}}(2j+1)(\kappa -1{)}^2{\left(\begin{array}{ccc}\frac{1}{2}& j& 1\\ \frac{1}{2}& -\frac{1}{2}& 0\end{array}\right)}^2{\left|R_{\epsilon \kappa }\right|}^2,$$

$$R_{\varepsilon\kappa}=\int_{0}^{\infty}\frac{dr}{r^{2}}\Bigl{(}P_{n_{i}\kappa_{i}}(r)Q_{\varepsilon\kappa}(r)+P_{\varepsilon\kappa}(r)Q_{n_{i}\kappa_{i}}(r)\Bigr{)}\;,$$

$$|\gamma \kappa m⟩=\frac{1}{r}\left(\begin{array}{c}{P}_{\gamma \kappa }(r){\Omega }_{\kappa m}(\hat{r})\\ i{Q}_{\gamma \kappa }(r){\Omega }_{-\kappa m}(\hat{r})\end{array}\right),$$

$$\Lambda =(2J_f+1)(2S_f+1){\left\{\begin{array}{ccc}{S}_i& S_f& \frac{1}{2}\\ J_f& J_i& L_i\end{array}\right\}}^2$$

$$S_{\text{mbox}tot}(E_I,U)=\sum w_i{S}_i(E_I,U),$$

$$E_I=E_{\text{defl}}+E_0,$$

$$E=8.28\pm 0.03_{\text{mbox}stat.}\pm 0.16_{\text{mbox}syst.}\text{mbox}eV.$$

Full text

Energy of the ${}^{\text{\large 229}}$Th nuclear clock transition

Benedict Seiferle1, Lars von der Wense1, Pavlo V. Bilous2, Ines Amersdorffer1, Christoph Lemell3, Florian Libisch3, Simon Stellmer4, Thorsten Schumm5, Christoph E. Düllmann6,7,8, Adriana Pálffy2 & Peter G. Thirolf1 †† 1Ludwig-Maximilians-University Munich, 85748 Garching, Germany. 2Max-Planck-Institut für Kernphysik, Heidelberg, 69117 Heidelberg, Germany. 3 Inst. for Theoretical Physics, TU Wien, 1040 Vienna, Austria. 4 University of Bonn, 53105 Bonn, Germany. 5 Atominstitut, TU Wien, 1020 Vienna, Austria. 6 GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany. 7 Helmholtz Institute Mainz, 55099 Mainz, Germany. 8 Johannes Gutenberg University, 55099 Mainz, Germany.

**The first nuclear excited state of 229Th offers the unique opportunity for laser-based optical control of a nucleus1, 2. Its exceptional properties allow for the development of a nuclear optical clock3 which offers a complementary technology and is expected to outperform current electronic-shell based atomic clocks 4. The development of a nuclear clock was so far impeded by an imprecise knowledge of the energy of the 229Th nuclear excited state. In this letter we report a direct excitation energy measurement of this elusive state and constrain this to 8.28$\pm$0.17 eV. The energy is determined by spectroscopy of the internal conversion electrons emitted in-flight during the decay of the excited nucleus in neutral 229Th atoms. The nuclear excitation energy is measured via the valence electronic shell, thereby merging the fields of nuclear- and atomic physics to advance precision metrology. The transition energy between ground and excited state corresponds to a wavelength of 149.7$\pm$3.1 nm. These findings set the starting point for high-resolution nuclear laser spectroscopy and thus the development of a nuclear optical clock of unprecedented accuracy. A nuclear clock is expected to have a large variety of applications, ranging from relativistic geodesy5 over dark matter research6 to the observation of potential temporal variation of fundamental constants7.

**

The first excited isomeric state of 229Th, denoted by ${}^{229\text{m}}$Th, has the lowest excitation energy of all known nuclear states. While typical transition energies in nuclear physics range from several keV to MeV, the excitation energy of ${}^{229\text{m}}$Th is in the eV region8, 9, 10, 11. ${}^{229\text{m}}$Th possesses a long radiative lifetime of expectedly up to 104 s (refs. 12, 13) resulting in a narrow relative linewidth of $\Delta E/E\approx 10^{-20}$.

These properties of the excited state make ${}^{229\text{m}}$Th the only candidate for a new type of optical clock that uses a nuclear transition instead of an electronic transition for time measurement3, 4. While posing a complementary technology to existing atomic clocks, a single-ion nuclear optical clock is expected to achieve a systematic frequency uncertainty of 1.5$\times 10^{-19}$ (ref. 4), thereby reaching and even surpassing existing atomic clock technology. Moreover, the possibility to dope 229Th nuclei into a vacuum ultra-violet (VUV) transparent host crystal could allow for the operation of a solid-state optical clock, which profits from the high density of nuclei that can be addressed14. During the past years ${}^{229\text{m}}$Th has been subject to vivid research. This includes the theoretical prediction of the isomeric properties 12, 13, 15, as well as experimental work aiming for further characterisation of the isomer16, 17, 18, 19, 20, 21, 22, 24, 23, 25, 26.

Until today the energy of ${}^{229\text{m}}$Th has been exclusively inferred from indirect measurements1, 8, 9, 10, 11 probing the gamma emission from higher lying excited states populating the ground and isomeric state. The latest of these measurements constrained the energy to 7.8$\pm$0.5 eV11. The uncertainty of this result is due to the detectors’ energy resolution. A direct detection of the isomeric decay in the internal conversion decay channel2 already led to the determination of the isomeric half-life in neutral, surface-bound, ${}^{229\text{m}}$Th atoms27. Additionally, laser spectroscopic characterisation of the isomeric state has been achieved recently28.

In this letter we report the first direct energy measurement of the 229Th nuclear isomer. We use the internal conversion (IC) decay of the isomeric excited state in a 229Th atom. In this decay channel the nuclear energy is transferred to the electronic shell with ionisation of the atom. The emitted electron’s kinetic energy can be measured, which allows to deduce the energy of the isomer. This approach offers the advantage that it relies on the atomic structure of the Th atom which is on the same energy scale as the isomer’s energy. Our results are sufficiently precise to develop a laser for the direct excitation of the isomeric state in 229Th and paves the way for a nuclear clock.

In our experimental setup, ${}^{229\text{m}}$Th is generated by a 2% decay branch in the 233U $\alpha$-decay. The experimental setup for thorium ion-beam and -bunch formation is described in refs. 2, 27 and shown in Fig. 1a. Bunches containing 400 ${}^{229\text{m}}$Th3+ ions are generated at a 10 Hz repetition rate. The ions are guided by four focusing electrodes onto two layers of graphene set to $-$300 V (graphene layers are supported by lacey-carbon on a copper transmission electron microscopy (TEM) grid, 300 Mesh, with 50 % transmission). In passing these foils, the ions are neutralised and continue their flight as neutral atoms (Fig. 1b). The extraction and neutralisation is monitored with a multi-channel plate (MCP) detector placed in the central beam axis (MCP detector II in Fig. 1a). Contrary to the long lifetime of the isomeric state in the Th3+ ions, the lifetime in neutral thorium is about 109 times shorter as the IC channel opens up energetically via the availability of more loosely bound valence electrons. Therefore, the isomer decays within microseconds27 by emitting an electron. The electron’s kinetic energy is measured free from any surface influences with a magnetic bottle-type retarding field electron spectrometer29 which is placed 90∘ off-axis behind the graphene (see Fig. 1). Secondary particles, such as electrons generated as the ions are passing the graphene or ions which were not fully neutralised, are removed by bending electrodes placed between the point of neutralisation and the spectrometer (see Fig. 1b). The detected electrons can be unambiguously attributed to the nuclear decay of ${}^{229\text{m}}$Th. Comparative measurements that were performed under identical conditions with 230Th, where no such isomer exists, do not show any comparable signal behaviour (see Extended Data Figure 6). Therefore, 229Th atoms in the nuclear ground-state, secondary electrons or auto-ionising states populated in the neutralisation process can be safely excluded as signal origin.

The spectrometer30 consists of a strong permanent magnet which generates an inhomogeneous magnetic field ($\approx$ 200 mT in the region above the magnet) and a solenoid coil that generates a weak homogeneous field (typically 2 mT). Electrons which are emitted in a spherical region of $\approx$1 mm radius above the permanent magnet are collected by the magnetic field and directed towards the solenoid coil. In this way a collimated electron beam is generated. The kinetic energy of the electrons is analysed by retarding fields, applied to a gold grid (electroformed gold mesh, transmission 90 %) that is surrounded by ring electrodes to ensure a smooth gradient, and terminated by additional gold grids (Ext. Data Figure 2). Electrons whose kinetic energies are sufficient to overcome the applied retarding voltage are counted with an MCP detector (MCP detector I in Fig. 1a). This results in an integrated spectrum which is monotonically decreasing with increasing retarding voltage and in which transition lines are visible as edges. The spectrometer reaches a FWHM resolution $(\Delta E/E)$ of about 3% and its performance and calibration was verified regularly during the measurements. The integrated spectrum of IC electrons as measured by the electron spectrometer is shown in Fig. 2a. Data was collected for 3 days in a continuous measurement.

The neutralisation process of the original Th ion in the graphene layer and after separation from the graphene has to be considered in order to determine the isomeric energy. Accordingly, all electronic states of Th which have come into resonance with the graphene valence band during and after transit may be occupied31 (for details on the neutralisation process see Methods). Thus, IC occurs from excited electronic states: the kinetic energy of an electron, $E_{\text{kin}}$, which is emitted during IC is connected to the energy of the isomer, $E_{I}$, via $E_{\text{kin}}=E_{I}-\text{IP}+E_{i}-E_{f}$, with the thorium ionisation potential IP (6.31 eV32), $E_{i}$ the excitation energy of the Th atom undergoing IC and $E_{f}$ the final electronic state energy of the Th ion generated during the IC process. Initial-final state pairs are subject to selection rules and each initial state generates a spectrum with a set of lines (see Ext. Data Fig. 9). The sum of these spectra weighted according to the population of the respective initial electronic state results in the measured spectrum.

The energy is determined by fitting a complementary error function to the measured data ($f(U)=a(1-\mbox{erf}[(U-E_{\mbox{\scriptsize defl}})/b)])$, with $U$ being the applied retarding voltage, see Fig. 2a). The deflection point of this error function $E_{\text{defl}}$ is determined to $E_{\text{defl}}~{}=~{}$1.77$\pm$0.03  eV. The deflection point shifts according to the isomeric energy, to which it can be related by adding a reference energy $E_{0}$ such that $E_{I}=E_{\text{defl}}+E_{0}$. The error-function fit is applied to simulated spectra to predict $E_{0}$.

As the statistical distribution of initial electronic excited states is unknown, we quantify its influence by simulations determining the systematic error. Assuming random distributions of initial electronic states, we find that $E_{0}$ scatters around a central value of $E_{0}=6.51$ eV and follows a Gaussian distribution (Fig. 2c). The width of this Gaussian gets narrower (and thus the uncertainty of the approach improves) with an increasing number of random initial states that make a contribution to the simulation. As clear edges resulting from individual lines from a single or only very few initial electronic states are not observed in our experiment, we conclude that IC electrons from at least 5 initial states contribute to the measured spectrum. Therefore, to find an upper bound for the systematic error, we set the number of initial states in the simulation to $N=5$. A collection of simulated spectra is shown in Fig. 2b (thin lines), the distribution of deviations from $E_{0}$ in Fig. 2c. The uncertainty of 0.16 eV is given by the standard deviation of the normal distribution shown in Fig. 2c and serves as a conservative upper bound. Together with the central value $E_{0}$ this translates to an isomeric energy of $E_{I}=$8.28$\pm$0.03${}_{\mbox{\scriptsize~{}stat.}}\pm 0.16_{\mbox{\scriptsize~{}syst.}}$ eV. In total the uncertainties sum up quadratically to give

[TABLE]

Density functional theory calculations indicate that a subset of excited states is populated more dominantly in the neutralisation process. This would shift the energy to the lower end of the error bar (see Methods).

This measurement represents the first direct energy determination of the lowest nuclear excited state in 229Th. The energy corresponds to a wavelength of 149.7$\pm$3.1 nm that is required for a direct optical excitation of the isomeric state. The wavelength lies in the transparency window of commonly used VUV-transparent crystals (e.g. MgF2 or CaF2). The measured energy value agrees with the former value of 7.8$\pm$0.5 eV11 within its 1$\sigma$ statistical uncertainty. Based on our findings the laser technology required for precision laser spectroscopy can be determined. The wavelength can be reached by high-harmonic generation (e.g. the 7${}^{\text{th}}$ harmonic of an Yb-doped fibre laser). Our improved precision reduces the time required for laser-based scanning in search for the nuclear excitation, e.g., to less than one day using the scheme proposed in ref. 26. This paves the way for nuclear precision spectroscopy and the development of a nuclear optical clock that is expected to have major implications for future frequency metrology.