Fine-structure constant sensitivity of the Th-229 nuclear clock transition

Abstract

Nuclear laser spectroscopy at the 10⁻¹² precision level (Nature 633.8028 (2024): 63–70) determined the fractional change in the nuclear quadrupole moment of ²²⁹Th upon excitation, ΔQ₀/Q₀ = 1.791(2)%. Such high-accuracy nuclear parameters enable stringent tests and refinement of ²²⁹Th nuclear models. Using a semi-classical prolate-spheroid model, we quantify the transition frequency’s sensitivity to fine-structure constant variations as K = 5900(2300), with uncertainty dominated by the measured charge-radius change Δ〈r²〉. This supports the predicted higher α-sensitivity of nuclear clocks over atomic clocks, important for new-physics searches. We find ΔQ₀ strongly dependent on nuclear volume, challenging the constant-volume approximation. The deviation between measured and predicted ΔQ₀/Q₀ underlines the need for improved modeling and measurement of additional nuclear parameters. We explicitly assess the octupole contribution to α-sensitivity.

Introduction

The ²²⁹Th nucleus features a first excited metastable state, ^229mTh, with an unusually low excitation energy of 8.4 eV. This nucleus represents a unique opportunity to build a high-precision nuclear clock (see review¹ and references therein), as the transition is narrow, insensitive to external perturbations, and within the range of tabletop narrow linewidth lasers.

A nuclear clock based on trapped Th³⁺ ions promises low systematic shifts, with an estimated total fractional inaccuracy of 10^−19 2,3. Another unique opportunity enabled by the Th isomer is the development of a solid-state clock based on ²²⁹Th doped into a VUV-transparent crystal, where atoms are confined to a lattice in a space that is much smaller than the excitation wavelength (Lamb-Dicke regime)^4,5.

Resonant laser excitation of the isomer was first demonstrated in Th-doped CaF₂⁶ and then in Th-doped LiSrAlF₆ crystals⁷ using tabletop broadband tunable laser systems. Soon after, a frequency-stabilized VUV frequency comb was used to improve frequency resolution by  ~10⁶ and directly resolve the nuclear quadrupole splitting in Th-doped CaF₂⁸. The nuclear transition frequency was determined to be 2 020 407 384 335(2) kHz.

A variation of fundamental constants is predicted by many theories beyond the standard model of particle physics^9,10. Moreover, ultralight dark matter may lead to oscillations of fundamental constants^{11,12,13,14,15}, and searches for such variation present a direct dark matter detection opportunity. If fundamental constants of nature change with space or time, so will atomic and nuclear energy levels, and consequently the associated clock frequencies. The amplitude of such changes, however, strongly depends on the particular clock transition under investigation^{16,17,18,19,20,21}. Any spacetime-dependent variation of clock frequency ratios would unambiguously point to new physics^22,23.

Highly enhanced sensitivity of the ²²⁹Th nuclear clock transition frequency to variation of the fine-structure constant α and the strong interaction coupling constant α_S was predicted by Flambaum²⁴. Such an enhancement leads to increased discovery potential of any new physics that manifests as a variation of the fundamental constants^14,19. However, the nuclear clock enhancement factor K for the variation of α defined as $\frac{\delta \nu }{\nu }=K\frac{\delta \alpha }{\alpha }$ has yet to be precisely determined, where ν is the clock transition frequency.

To determine K = ΔE_C /E, where E = hν is the isomer excitation energy, we need to know the difference in the Coulomb energy of the isomer and the ground state ΔE_C²⁴. Since the Coulomb energies of both nuclear states are on the order of GeV, and ΔE_C is expected to be on the order of MeV, direct nuclear computation of the isomer and ground state energy is not sufficiently accurate for the determination of K (see, for example²⁵). Therefore, a method to determine ΔE_C from measured nuclear properties of the ground and isomeric state using a geometric model, the liquid drop model, was proposed²⁶. The validity of this approach stems from the fact that the change in fine-structure constant modifies the total coulomb energy of the nucleus slightly. The change in coulomb energy is not enough to affect nuclear structure and to compute the coulomb energy of the nucleus on the MeV level precision a liquid drop model suffices²⁷. Currently, it is the only existing model relating experimental measurements and model parameters, as other microscopic models rely on intense numerical calculations to relate model parameters to experimental observables which complicates direct comparison. The geometrical model hinges on a sufficiently precise measurement of the ratio of quadrupole moments of the two states. Hartree-Fock-Bogolubov nuclear many-body calculations²⁵ were used for validating the geometric model²⁶. More elaborate collective quadrupole-octupole models were recently developed^28,29, as there is indication for ²²⁹Th being on the edge of the octupole deformation region of nuclei and possibly displaying octupole vibrational character³⁰. However, the new models could not yet provide an accurate determination of ΔE_C.

We use the geometric model²⁶ in combination with the new precision laser spectroscopy performed in⁸ to extract the nuclear quadrupole splitting (see “methods”), and then determine the sensitivity of the nuclear clock to α-variation. Within this model, the spectroscopy data quantitatively confirms a non-zero K. The constant volume assumption, used in previous work²⁰, is found to be unreliable in extracting the mean-square charge radius. We also estimate the effect of including octupole deformation into the geometric model and find that it may have a significant effect on K, depending on the yet unknown octupole moment of ²²⁹Th in ground and isomeric state. Therefore we emphasize the need for the development of advanced nuclear models relating experimental measurements to model parameters.

Results

Nuclear theory

Following^20,26,31, we describe the nucleus as a prolate spheroid with uniform charge density. Although ground and excited state have different charge density functions, we assume the uniformity is not affected^32,33. If a generalized Fermi function or for example a folded Yukawa-plus-exponential potential is used for the charge density profile, the Coulomb energy is corrected with a factor $\propto \left(1+{(\frac{z}{R})}^{2}\right)$ for a spherical distribution where z is the skin thickness³⁴. Although this factor might seem small, it needs to be tested for a prolate spheroid as well. This further refinement was tested for in³⁵ and found that for typical values (z ≈ 0.5 fm), K is between 10³ − 10⁵, in line with the following calculations. The nuclear quadrupole moment Q₀ and the mean-square charge radius 〈r²〉 are expressed via semi-minor and semi-major axes a and c as

$$\langle {r}^{2}\rangle=\frac{1}{5}\left(2{a}^{2}+{c}^{2}\right),\quad {Q}_{0}=\frac{2}{5}\left({c}^{2}-{a}^{2}\right).$$

(1)

Very coarsely, upon nuclear excitation, the distribution of a valence neutron is modified and polarizes the proton distribution via the strong interaction, altering 〈r²〉 as well as the nuclear deformation and in consequence Q₀. These two quantities, therefore, are essential for the determination of the Coulomb energies of the nuclear ground and isomeric state. The Coulomb energy (E_C)³⁶ is given by Eq. (5.11) in ref. ³⁷,

$${E}_{C}=\frac{3{q}_{e}^{2}{Z}^{2}}{5{R}_{0}}\frac{{(1-{{{{\rm{e}}}}}^{2})}^{1/3}}{2{{{\rm{e}}}}}\ln \frac{1+{{{\rm{e}}}}}{1-{{{\rm{e}}}}},$$

(2)

where e² = 1 − (a²)/(c²) is the eccentricity, Z the proton number, and ${R}_{0}^{3}={a}^{2}c$ is the equivalent sharp spherical radius, see Eq. (6). The difference of the Coulomb energy (ΔE_C) of the isomer and the ground state is computed as

$$\Delta {E}_{C}=\langle {r}^{2}\rangle \frac{\partial {E}_{C}}{\partial \langle {r}^{2}\rangle }\frac{\Delta \langle {r}^{2}\rangle }{\langle {r}^{2}\rangle }+{Q}_{0}\frac{\partial {E}_{C}}{\partial {Q}_{0}}\frac{\Delta {Q}_{0}}{{Q}_{0}}$$

(3)

$$=-485\,{{{\rm{MeV}}}}\frac{\Delta \langle {r}^{2}\rangle }{\langle {r}^{2}\rangle }+11.3\,{{{\rm{MeV}}}}\left(\frac{{Q}_{0}^{{{{\rm{m}}}}}}{{Q}_{0}}-1\right),$$

(4)

where we used 〈r²〉 = 5.756(14) fm^2 38, the spectroscopic quadrupole moment Q_lab = 3.11(2) eb³⁹ for the ground state and $\Delta {Q}_{0}={Q}_{0}^{{{{\rm{m}}}}}-{Q}_{0}$. The spectroscopic quadrupole moment in the laboratory frame (Q_lab), can be related to the quadrupole moment in the nuclear frame (see supplemental). Q_lab was recently extracted by modeling experimental measurements of the electronic hyperfine-structure⁴⁰ using the coupled cluster method with single, double, and triple excitations for both valence and core electrons, improving the accuracy³⁹. This newer value accounts for a small difference in the second coefficient 11.3 MeV compared to ref. ^20,31. We express Eq. (4) in terms of the difference of 〈r²〉 and the ratio of quadrupole moments as these are the quantities which are determined from experiments.

The experimental results used to compute the ΔE_C from Eq. (4), can be derived as follows. The ratio, ζ, of the isomer and isotope shifts

$$\zeta=\frac{\langle {r}_{229m}^{2}\rangle -\langle {r}_{229}^{2}\rangle }{\langle {r}_{232}^{2}\rangle -\langle {r}_{229}^{2}\rangle }=0.035(4)$$

(5)

was measured in ref. ³¹ for two transitions in Th²⁺. Using this ratio, we can derive Δ〈r²〉. New isotope shift measurements and more advanced theoretical calculations⁴¹ yielded an improved value $\langle {r}_{232}^{2}\rangle -\langle {r}_{229}^{2}\rangle=0.299(15)$ fm², reducing the uncertainty in this value from 15% to 5%. As a result, the uncertainty in the $\Delta \langle {r}_{229}^{2}\rangle=\langle {r}_{229m}^{2}\rangle -\langle {r}_{229}^{2}\rangle=0.0105(13)$ fm² was reduced from 17% to 12%⁴¹. We note that this improvement in the accuracy is important for the determination of the α-sensitivity K due to a strong cancellation of the 〈r²〉 and Q₀ contribution, as discussed below.

Using the spectroscopic quadrupole moment (in the lab frame, see “methods”) for the ground state Q_lab = 3.11(2) eb³⁹ and the ratio of the intrinsic quadrupole moments ${Q}_{0}^{{{{\rm{m}}}}}/{Q}_{0}=1.01791(2)$, we find ${Q}_{0}^{{{{\rm{m}}}}}=9.85(6)$ fm² and ${Q}_{{{{\rm{lab}}}}}^{{{{\rm{m}}}}}=1.77(1)$ eb. Note that we follow the designations of ²⁰ for Q₀, which differs from³¹, where Q₀ was defined without the factor q_e Z. Using designations and units of ³¹, we get ${Q}_{0}^{{{{\rm{m}}}}}=8.86(6)\,e{{{\rm{b}}}}$. The measured value⁸ of $\Delta {Q}_{0}/{Q}_{0}={Q}_{0}^{m}/{Q}_{0}-1=0.01791(2)$ (see “methods”) differs by a factor of 2.4 from the prediction of ΔQ₀/Q₀ = 0.0075(20) obtained assuming a constant nuclear volume for the ground and the isomeric state²⁰. Using the predicted value in ref. ²⁰, the authors arrived at a negative K = − 8200(2500)²⁰. To test the constant volume approximation, we computed the respective volumes using

$$V=\frac{4\pi }{3}{R}_{0}^{3}=\frac{4\pi }{3}{a}^{2}c$$

(6)

and find a small but non-zero −0.055% difference, with the isomer volume being smaller. This change in the volume is well within the 0.8% experimental uncertainty. However, it is interesting to note that volume changes due to the change in the rms radius and Q₀ contribute with opposite signs, +0.05% and −0.105%, respectively, leading to some cancellation of the overall effect. We tested the dependence of ΔQ₀/Q₀ on the volume change and find that even a small −0.055% change leads to a difference by over a factor of two in the resulting ΔQ₀/Q₀, making the constant volume approximation unreliable for extracting this quantity from the difference in the 〈r²〉.

Substituting the experimental values into the expression for the Coulomb energy difference given by Eq. (4) gives

$$\Delta {E}_{C}=\, -0.154(19)\,{{{\rm{MeV}}}}+0.203(4)\,{{{\rm{MeV}}}}\\=\, 0.049(19)\,{{{\rm{MeV}}}}$$

(7)

demonstrating strong cancellation between 〈r²〉 and Q₀ terms, making the ΔE_C and K values extremely sensitive to ΔQ₀/Q₀. The enhancement factor is

$$K=-18400(2300)+24300(400)=5900(2300),$$

(8)

where the uncertainty is overwhelmingly due to the uncertainty in Δ〈r²〉. An improved measurement of ζ is needed for further improvement in the accuracy of K.

Such a high value of K drastically increases the sensitivity of a nuclear clock to dark matter and other related new physics searches in comparison with current optical atomic clocks. The highest sensitivity in atomic clocks is K = − 6 (for ¹⁷¹Yb E3 transitions⁴²). Using a nuclear clock thus gives three orders of magnitude improvement for the same clock accuracy. This improvement translates into being able to probe dark matter with three orders of magnitude smaller couplings.

Effect of the octupole deformation

We point out that the discussion above does not include uncertainties connected with the present theoretical model, but only uncertainties of the experimentally determined quantities. In particular, a contribution of the currently unknown nuclear octupole deformation to the Coulomb energy E_C is not accounted for. In order to gauge the effect, we approximate the vibrational quadrupole-octupole character of the ²²⁹Th nucleus^28,29 by a static, axially symmetric octupole deformation. This approach is purely demonstrative to qualitatively study the effect of the octupole deformation on the coulomb energy and therefore the fine-structure constant sensitivity. Combined with experimental observations of β₃, these models can be used to estimate K as the validity of using macroscopic geometric nuclear models to relate macroscopic characteristics (β_n, Q, r, ΔE_C) does not change. Following²⁰, the surface of the nucleus r(θ) is expanded as

$$r(\theta )={R}_{s}\left[1+{\sum}_{n=1}^{N}\left({\beta }_{n}{Y}_{n}^{0}(\theta )\right)\right],$$

(9)

where β_n are deformation parameters, ${Y}_{n}^{0}(\theta )$ are spherical harmonics, and R_s is defined by normalizing the volume to that of the spherical nucleus with equivalent sharp spherical radius R₀. The normalization condition can be expressed as ∫ρ_q(r)d³r = 1, where in the present approximation of constant charge density, ρ_q = 1/V is the charge density divided by the total charge q_e Z and $V=(4\pi /3){R}_{0}^{3}$:

$$\frac{2\pi }{3}\int_{0}^{\pi }{r}^{3}(\theta )\,\sin (\theta )\,d\theta=V.$$

(10)

For a pear-shaped, axially symmetric nucleus with quadrupole and octupole deformation, N = 3. The coefficient β₁ is determined from the condition that the center of mass of the shape is at the origin of the coordinate system, ∫_V rd ³r = 0⁴³. For a pure quadrupole deformation (i.e., prolate spheroid), N = 2 and β₁ = 0. The nuclear properties are related to the β coefficients via

$$\langle {r}^{2}\rangle=\int{r}^{2}{\rho }_{q}({{{\boldsymbol{r}}}}){d}^{3}{{{\boldsymbol{r}}}}$$

(11)

$${Q}_{0}=2\int{r}^{2}{P}_{2}(\cos \theta ){\rho }_{q}({{{\boldsymbol{r}}}}){d}^{3}{{{\boldsymbol{r}}}},$$

(12)

$${Q}_{30}=2\int{r}^{3}{P}_{3}(\cos \theta ){\rho }_{q}({{{\boldsymbol{r}}}}){d}^{3}{{{\boldsymbol{r}}}},$$

(13)

where P₂ and P₃ are the Legendre polynomials, and Q₃₀ is the intrinsic charge octupole moment⁴³.

The expression for the Coulomb energy was given in ref. ³⁷, Eqs. (6.47),

$${E}_{C}=\frac{3{q}_{e}^{2}{Z}^{2}}{5{R}_{0}}\left(1-\frac{1}{4\pi }{\beta }_{2}^{2}-\frac{5}{14\pi }{\beta }_{3}^{2}\right),$$

(14)

β₂ and β₃ are the quadrupole and octupole deformations, respectively. We keep the first in two terms in Eq. (6.47) to omit $O({\beta }_{n}^{3})$ terms, negligible at the present level of accuracy. We use a substitution of Eq. (6.51) to account for a representation of R(θ) with axially symmetric spherical harmonics instead of Legendre polynomials.

The change in the Coulomb energy was described in ref. ²⁰ using the expression

$$\Delta {E}_{C}=\, \frac{\partial {E}_{C}}{\partial {\beta }_{2}^{2}}\Delta {\beta }_{2}^{2}+\frac{\partial {E}_{C}}{\partial {\beta }_{3}^{2}}\Delta {\beta }_{3}^{2}\\=\, \frac{3{q}_{e}^{2}{Z}^{2}}{5{R}_{0}}\left(-\frac{1}{4\pi }\Delta {\beta }_{2}^{2}-\frac{5}{14\pi }\Delta {\beta }_{3}^{2}\right).$$

(15)

However, this expression utilizes the constant volume approximation and assumes intrinsic axial symmetry, while R₀ depends on β₂ and β₃ via the volume normalization above. As in the ellipsoid model above, this leads to a wrong result for ΔE_C, when using the experimental values for 〈r²〉 and Q₀.

To evaluate the effect of a possible octupole deformation described by the change in β₃ upon nuclear excitation, we calculate ΔE_C using experimentally available numbers and the geometric model described above. We use the definitions for 〈r²〉 and Q₀ and the normalization condition above, the experimental values for 〈r²〉 and Q₀, to calculate R_s, β₂, and R₀ for both ground and isomeric state.

First, we take β₃ = 0 for both the ground and isomeric state. The resulting values of β₂ = 0.220 and ${\beta }_{2}^{{{{\rm{m}}}}}=0.223$ are consistent with ²²⁸Th value from ref. ⁴⁴. We compute ΔE_C = 0.052 MeV giving K = 6300(2300) using Eq. (14), in agreement with the result obtained for the ellipsoid model (we note that Eq. (14) is approximate). Then, we repeat the same computation but use ${\beta }_{3}={\beta }_{3}^{{{{\rm{m}}}}}=0.115$ from ref. ²⁸, which yields essentially the same result for K, as expected. However, when we vary β₃ differentially between the ground and excited state, while keeping all other parameters constant, we find that 1%, 3%, and 5% differential change in β₃ lead to ΔK = 2850, ΔK = 8600, and ΔK = 14500, respectively. A larger β₃ of the isomer yields a positive change in K and inversely, a smaller β₃ yields a negative change in K, possibly jeopardizing the fine-structure constant sensitivity. However, it was concluded in³⁵ that this is unlikely.

For reference, the measured change in Q₀, and correspondingly β₂, is 1.8%. This computation shows that even 1% variation of the octupole deformation changes K by more than 1σ. Therefore, it is important to estimate the sign and constrain the magnitude of the octupole deformation change between the ground and isomeric state and model the effect on the change in intrinsic quadrupole moment when including the octupole deformation, for which more sophisticated nuclear models are needed.

We can estimate the approximate size of the intrinsic charge octupole moment in ²²⁹Th ground state and isomer. Using the β₂ value obtained above and β₃ within the range used in²⁸, 0.11 to 0.145, we get Q₃₀ = 35 −  44 fm³ (Eq. (13)). This is consistent with estimates⁴⁴ for the octupole moment of ²²⁸Th. Note that⁴⁴ gives the values of Zq_eQ. E3 matrix elements of nuclear Coulomb excitation should be measured to determine the octupole moment, as has been demonstrated on other isotopes^45,46. We also note that even higher moments might contribute to the nuclear Coulomb energies. Octupole deformation in ²²⁹Th allows for the search of permanent electric-dipole moments^44,47.

Discussion

The precise measurement of the change in nuclear quadrupole moment ΔQ₀/Q₀⁸ between the ²²⁹Th ground and isomeric state together with previous measurements of the mean-square charge radius 〈r²〉 allow us to extract the change in Coulomb contribution ΔE_C to the nuclear energies. We quantify the sensitivity of the nuclear transition to changes of the fine-structure constant α, within the presented model, with the factor K =  5900(2300) which is for the first time inconsistent with zero by 2σ. This result shows the potential for using the nuclear clock for measurements in fundamental physics. Future theoretical and experimental work will focus on developing and verifying a model that bridges experimental observables and the intrinsic quadrupole moment in a microscopic description of the ²²⁹Th isomeric excitation, taking into account the quadrupole-octupole deformed shell, as we have shown that small octupole deformations strongly affect the fine-structure sensitivity. In addition, an improved measurement of the ratio ζ of the isomer and isotope shifts is required. The measured quantitative deviation from the constant volume approximation illustrates the power of the emerging precision laser spectroscopy on ²²⁹Th for testing fundamental models in nuclear physics. Recently we learned about a new work that uses a theoretical d-wave halo model to estimate K ≈ 10⁴³⁵.

Methods

Quadrupole structure

In the nuclear laser spectroscopy performed in⁸, samples containing ²²⁹Th nuclei doped in a CaF₂ matrix were used^48,49,50. The interaction of the nuclear quadrupole moment (Q₀) in the lab frame (Q_lab) with the crystal electric field gradient (EFG), V_ij = ∂²V/(∂x_i∂x_j), leads to the emergence of a nuclear quadrupole splitting⁵. We note that any octupole moment interaction with the EFG that would show a measurable influence on the quadrupole level structure would need to be a CP violating effect⁵¹.

We first describe the interaction Hamiltonian (see supplementary information) between the EFG V_ij and the spectroscopic nuclear quadrupole moment Q_lab and derive the energy levels and eigenstates. Here we define axes of the EFG such that V_ij is diagonal, ∣V_zz∣ is the maximum value of the EFG and η = (V_xx − V_yy)/(V_zz) is the asymmetry of the EFG. We then calculate the energies of the quadrupole states and excitation probabilities W(E₂ − E₁) by calculating the transition matrix elements of the interaction Hamiltonian between the eigenstates and the driving laser field. The results of these calculations can be found in Table 1. A detailed derivation can be found in the supplementary information.

Table 1 Observed lines and their interpretations according to⁸, together with calculated relative intensities W

Fitted data

In ref. ⁸, an absolute frequency measurement of 5 lines was performed, which were identified as transitions between certain quadrupole sublevels of the ground and the isomeric state of the ²²⁹Th nucleus, see Table 1. The transition energies were fit to analytical solutions of the Hamiltonian to yield η, Q_labV_zz and ${Q}_{{{{\rm{lab}}}}}^{{{{\rm{m}}}}}{V}_{{{{\rm{zz}}}}}$ thus obtaining the ratio of ${Q}_{{{{\rm{lab}}}}}^{{{{\rm{m}}}}}/{Q}_{{{{\rm{lab}}}}}$. The frequency of the EFG-free isomer transition ν_Th was determined by a weighted averaging of the line frequencies.

Here, we independently verify the data fitting using the numerical solutions of the interaction Hamiltonian with Q_labV_zz, η, ${Q}_{0}^{{{{\rm{m}}}}}/{Q}_{0}$ and ν_Th as free parameters. Details of the employed statistical models and in particular the estimation of uncertainties can be found in the supplementary information, see figures 1 and 2 in there. We obtain ν_Th = 2020 407 384 335(2) kHz, ${Q}_{0}^{{{{\rm{m}}}}}/{Q}_{0}=1.01791(2)$, η = 0.59164(5) and Q_labV_zz = 339.263(7) eb V/Ų. The fit results are fully consistent with the results reported in ref. ⁸.

The experimental line intensities reported in ref. ⁸ qualitatively agree with the theoretical expectations. The transition frequencies are arithmetically consistent and stable over the measurement campaign; we therefore assume fluctuations in V_ij, i.e., due to changes in temperature or pressure, to be negligible. The temperature of the crystal in the measurement was 150(1) K. A detailed study of the above parameters with crystal temperature is presented in ref. ⁵².

Using the value Q_lab = 3.11(2) eb for the ²²⁹Th ground state quadrupole moment derived in ref. ³⁹ yields V_zz = 109.1(7) V/Ų. The rather high asymmetry parameter η indicates a non-rotationally symmetric microscopic charge compensation configuration involving multiple atoms, such as two fluoride interstitials⁵³. The microscopic atomic and electronic structure of the ²²⁹Th defect in the CaF₂ crystal will be subject of an upcoming publication.