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

Kjeld Beeks1,2 Georgy A. Kazakov1 Fabian Schaden1 Ira Morawetz1 Luca Toscani
De Col1 Thomas Riebner1 Michael Bartokos1 Tomas Sikorsky1 Thorsten Schumm1
email<EMAIL_ADDRESS>1Vienna Center for Quantum Science and
Technology, Atominstitut, TU Wien, 1020 Vienna, Austria  2Laboratory for
Ultrafast Microscopy and Electron Scattering (LUMES), Institute of Physics,
École Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland
Chuankun Zhang, Tian Ooi, Jacob S. Higgins, Jack F. Doyle, Jun Ye  JILA, NIST
and University of Colorado, Department of Physics, University of Colorado,
Boulder, CO 80309  Marianna S. Safronova  Department of Physics and Astronomy,
University of Delaware, Newark, Delaware 19716, USA

###### Abstract

State-resolved laser spectroscopy at the 10-12 precision level recently
reported in $arXiv$:2406.18719 determined the fractional change in nuclear
quadrupole moment between the ground and isomeric state of ${}^{229}\rm{Th}$,
$\Delta Q_{0}/Q_{0}$=1.791(2) %. Assuming a prolate spheroid nucleus, this
allows to quantify the sensitivity of the nuclear transition frequency to
variations of the fine-structure constant $\alpha$ to $K=5900(2300)$, with the
uncertainty dominated by the experimentally measured charge radius difference
$\Delta\langle r^{2}\rangle$ between the ground and isomeric state. This
result indicates a three orders of magnitude enhancement over atomic clock
schemes based on electron shell transitions. We find that $\Delta Q_{0}$ is
highly sensitive to tiny changes in the nuclear volume, thus the constant
volume approximation cannot be used to accurately relate changes in $\langle
r^{2}\rangle$ and $Q_{0}$. The difference between the experimental and
estimated values in $\Delta Q_{0}/Q_{0}$ raises a further question on the
octupole contribution to the alpha-sensitivity.

Introduction.– The 229Th nucleus features a first excited metastable state,
229mTh, with an unusually low excitation energy of 8.4 eV. This near-
degeneracy of ground and isomeric state results from a fortuitous cancellation
of the (repulsive) Coulomb and (attractive) nuclear energy contributions to
the respective binding energies of the two states, which are each on the GeV
level individually. This nucleus represents a unique opportunity to build a
high-precision nuclear clock (see review [1] and references therein), as the
transition is very 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]. 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
realizing the Lamb-Dicke regime [3, 4].

Resonant laser excitation of the isomer was first demonstrated in Th-doped
CaF2 [5] and then in Th-doped LiSrAlF6 crystals [6] using tabletop broadband
tunable laser systems. Soon after, a frequency-stabilized VUV frequency comb
was used to improve frequency resolution by $\sim$$10^{6}$ and directly
resolve the nuclear quadrupole splitting in Th-doped CaF2 [7]. 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 [8]. Moreover, ultralight dark matter may
lead to oscillations of fundamental constants [9, 10, 11, 12, 13], 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 [14, 15, 16, 17, 18, 19]. Any
spacetime-dependent variation of clock frequency ratios would be an
unambiguous signal of new physics [20, 21].

Highly enhanced sensitivity of the 229Th nuclear clock transition frequency to
variation of the fine-structure constant $\alpha$ and the strong interaction
coupling constant $\alpha_{S}$ was predicted by Flambaum [22]. Such an
enhancement leads to increased discovery potential of any new physics that
manifests as a variation of the fundamental constants [17, 12]. However, the
nuclear clock enhancement factor $K$ for the variation of $\alpha$ defined as
$\frac{\delta\nu}{\nu}=K\frac{\delta\alpha}{\alpha}$ has yet to be precisely
determined, where $\nu$ is the clock transition frequency. To determine
$K=\Delta E_{C}/E$, where $E=h\nu$ is the isomer excitation energy, we need to
know the difference in the Coulomb energy of the isomer and the ground state
$\Delta E_{C}$ [22]. Since the Coulomb energies of both nuclear states are on
the order of GeV, and $\Delta E_{C}$ 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 example [23]).
Therefore, a method to determine $\Delta E_{C}$ from measured nuclear
properties of the ground and isomeric state using a geometric model was
proposed [24] but it hinges on a sufficiently precise measurement of the ratio
of quadrupole moments of the two states. Hartree-Fock-Bogolubov nuclear many-
body calculations [23] were used for validating the geometric model [24].
Here, we use the precision laser spectroscopy performed in [7] to extract the
nuclear quadrupole splitting and determine the sensitivity of the nuclear
clock to $\alpha$-variation.

Quadrupole structure.– In the nuclear laser spectroscopy performed in [7],
samples containing 229Th nuclei doped in a CaF2 matrix were used [25, 26, 27].
The interaction of the nuclear quadrupole moment ($Q_{0}$) in the lab frame
($Q_{\rm lab}$) with the crystal electric field gradient (EFG),
$V_{ij}={\partial^{2}V}/({\partial x_{i}\partial x_{j}})$, leads to the
emergence of a nuclear quadrupole splitting [4].

The intrinsic quadrupole moment $Q_{0}$ in the body-fixed frame is connected
with the spectroscopic quadrupole moment $Q_{\rm lab}$ in the lab frame via
the relation

$Q_{0}=\frac{Q_{\rm lab}}{q_{e}Z}\frac{(2I+3)(I+1)}{3k^{2}-I(I+1)},$ (1)

where $Z=90$ is the charge number of the thorium nucleus, $q_{e}$ is
elementary charge, $I$ is the nuclear spin ($I=5/2$ for ground, and $I=3/2$
for isomeric state), and $k$ is the projection of the nuclear spin on the
deformation axis. To determine the enhancement factor, $K$, we rely on the
ratio of quadrupole moments of the nuclear isomer and ground state
$Q_{0}^{\mathrm{m}}/Q_{0}$ from the experimentally determined nuclear electric
quadrupole splitting, see Ref. [7].

We first describe the interaction Hamiltonian between the EFG $V_{ij}$ and the
spectroscopic nuclear quadrupole moment $Q_{\rm lab}$ and derive the energy
levels and eigenstates. Here we define axes of the EFG such that $V_{ij}$ is
diagonal, $|V_{\rm zz}|$ is the maximum value of the EFG and $\eta=(V_{\rm
xx}-V_{\rm yy})/(V_{\rm zz}$) is the asymmetry of the EFG. We then calculate
the energies of the quadrupole states and excitation probabilities
$W(E_{2}-E_{1})$ 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 material.

Table 1: Observed lines and their interpretations according to [7], together with calculated relative intensities $W$. $j$aaa |  $\nu_{j}$ [THz] | $~{}m_{g}$ | $m_{e}$ | $W(E_{2}-E_{1})$
---|---|---|---|---
1 | 2020.407 283 847(4)a |  a $\pm 3/2$ a |  a $\pm 1/2$ a | 0.184
2 | 2020.407 298 727(4) | $\pm 5/2$ | $\pm 3/2$ | 0.328
3 | 2020.407 446 895(4) | $\pm 1/2$ | $\pm 1/2$ | 0.310
4 | 2020.407 530 918(4) | $\pm 3/2$ | $\pm 3/2$ | 0.149
5 | 2020.407 693 98(2) | $\pm 1/2$ | $\pm 3/2$ | 0.0233
6 | 2020.407 051 655111unobserved, asymptotically forbidden transition | $\pm 5/2$ | $\pm 1/2$ | 0.0058

Data Fitting.– In Ref. [7], 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 $\eta$, $Q_{\rm lab}V_{\rm zz}$ and $Q_{\rm lab}^{\mathrm{m}}V_{\rm
zz}$ thus obtaining the ratio of $Q_{\rm lab}^{\mathrm{m}}/Q_{\rm lab}$. The
frequency of the EFG-free isomer transition $\nu_{\rm Th}$ 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 $Q_{\rm lab}V_{\rm zz}$, $\eta$,
$Q_{0}^{\mathrm{m}}/Q_{0}$ and $\nu_{\rm Th}$ as free parameters. Details of
the employed statistical methods and in particular the estimation of
uncertainties can be found in the supplementary materials. We obtain $\nu_{\rm
Th}=2020\,407\,384\,335(2)~{}{\rm kHz}$,
$Q_{0}^{\mathrm{m}}/Q_{0}=1.01791(2)$, $\eta=0.59164(5)$ and $Q_{\rm
lab}V_{\rm zz}=339.263(7)~{}{\rm eb\,V/\r{A}^{2}}$. The fit results are fully
consistent with the results reported in Ref. [7].

The experimental line intensities reported in [7] qualitatively agree with the
theoretical expectations. The transition frequencies are arithmetically
consistent and stable over the measurement campaign; we therefore assume
fluctuations $V_{ij}$, 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 will be a
topic of further studies.

Using the value $Q_{\rm lab}=3.11(2)$ eb for the 229Th ground state quadrupole
moment derived in [28] yields $V_{\rm zz}=109.1(7)\,{\rm V/\r{A}^{2}}$. The
rather high asymmetry parameter $\eta$ indicates a non-rotationally symmetric
microscopic charge compensation configuration involving multiple atoms, such
as two fluoride interstitials [29, 30]. The microscopic atomic and electronic
structure of the 229Th defect in the CaF2 crystal will be subject of an
upcoming publication. Here we focus on the consequences of the fractional
change of $Q_{0}^{\mathrm{m}}/Q_{0}$ for the nuclear properties of 229Th.

Nuclear Theory.– Following [24, 31, 18], we describe the nucleus as a prolate
spheroid with uniform volume charge. The mean-square radius $\langle
r^{2}\rangle$ and the intrinsic quadrupole moment $Q_{0}$ are expressed via
semi-minor and semi-major axes $a$ and $c$ as

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

Very coarsely, upon nuclear excitation, the distribution of a valence neutron
is modified and polarizes the proton distribution via the strong interaction,
altering the mean-square charge radius $\langle r^{2}\rangle$ as well as the
nuclear deformation and in consequence the electric quadrupole moment $Q_{0}$.
These two quantities, therefore, are essential for the determination of the
Coulomb energies of the nuclear ground and isomeric state. The Coulomb energy
[32] is given by Eq. (5.11) in Ref. [33],

$E_{C}=\frac{3q_{e}^{2}Z^{2}}{5R_{0}}\frac{(1-{\rm e}^{2})^{-1/3}}{2{\rm
e}}{\rm ln}\frac{1+{\rm e}}{1-{\rm e}},$ (3)

where ${\rm e}^{2}=1-(a^{2})/(c^{2})$ is the eccentricity and
$R_{0}^{3}=a^{2}c$ is the equivalent sharp spherical radius, see Eq. (7). The
difference of the Coulomb energy of the isomer and the ground state is
computed as

$\displaystyle\Delta E_{C}$ $\displaystyle=$ $\displaystyle\langle
r^{2}\rangle\frac{\partial E_{C}}{\partial\langle
r^{2}\rangle}\frac{\Delta\langle r^{2}\rangle}{\langle
r^{2}\rangle}+Q_{0}\frac{\partial E_{C}}{\partial Q_{0}}\frac{\Delta
Q_{0}}{Q_{0}}$ (4) $\displaystyle=$ $\displaystyle-485\
\textrm{MeV}\frac{\Delta\langle r^{2}\rangle}{\langle r^{2}\rangle}+11.3\
\textrm{MeV}\left(\frac{Q_{0}^{\mathrm{m}}}{Q_{0}}-1\right),$ (5)

where we used $\langle r^{2}\rangle=5.756(14)$ fm2 [34], and the spectroscopic
quadrupole moment $Q_{\rm lab}=3.11(2)$ eb [28] for the ground state. The
ground state spectroscopic quadrupole moment $Q_{\rm lab}$ was recently
extracted from experimental measurements of the electronic hyperfine-structure
[35] using the coupled cluster method with single, double, and triple
excitations for both valence and core electrons, improving the accuracy by a
factor of 3 [28]. This newer value accounts for a small difference in the
second coefficient 11.3 MeV compared to [31, 18]. We express Eq. (5) in terms
of the difference of $\langle r^{2}\rangle$ and the ratio of quadrupole
moments as these are the quantities which are determined from the experiments,
see discussion below.

Results and Discussions.– We start with the discussion of the experimental
results used to compute the $\Delta E_{C}$ from Eq. (5). The ratio, $\zeta$,
of the isomer and isotope shifts

$\zeta=\frac{\langle r^{2}_{229m}\rangle-\langle r^{2}_{229}\rangle}{\langle
r^{2}_{232}\rangle-\langle r^{2}_{229}\rangle}=0.035(4)$ (6)

was measured in [31] for two transitions in Th2+. New isotope shift
measurements and more advanced theoretical calculations [36] yielded an
improved value $\langle r^{2}_{232}\rangle-\langle
r^{2}_{229}\rangle=0.299(15)$ fm2, reducing the uncertainty in this value by a
factor of 3, from 15 % to 5 %. As a result, the uncertainty in the
$\Delta\langle r^{2}_{229}\rangle=\langle r^{2}_{229m}\rangle-\langle
r^{2}_{229}\rangle=0.0105(13)$ fm2 was reduced from 17% to 12 % [36]. We note
that this improvement in the accuracy is important for the determination of
the $\alpha$-sensitivity $K$ due to a strong cancellation of the $\langle
r^{2}\rangle$ and $Q_{0}$ contribution, as discussed below.

Using the spectroscopic quadrupole moment for the ground state $Q_{\rm
lab}=3.11(2)$ eb [28] and the ratio of the intrinsic quadrupole moments
$Q_{0}^{\mathrm{m}}/Q_{0}=1.01791(2)$, we find
$Q_{0}^{\mathrm{m}}=9.85(6)\>$fm2 and $Q_{\rm lab}^{\mathrm{m}}=1.77(1)$ eb
222Note that we follow $Q_{0}$ designations of [18], which differs from [31],
where the value of with $Q_{0}$ was defined without the factor $q_{e}Z$. Using
designations and units of [31], we get $Q_{0}^{\mathrm{m}}=8.86(6)\>e$b..

The measured value of $\Delta{Q_{0}}/Q_{0}=0.01791(2)$, where
$\Delta{Q_{0}}=Q_{0}^{\mathrm{m}}-Q_{0}$, differs by a factor of 2.4 from the
prediction of $\Delta{Q_{0}}/{Q_{0}}=0.0075(20)$ obtained assuming a constant
nuclear volume for the ground and the isomeric state [18]. Using this value,
the authors arrived at a negative $K=-8200(2500)$ [18].

We computed the respective volumes using

$V=\frac{4\pi}{3}R_{0}^{3}=\frac{4\pi}{3}a^{2}c$ (7)

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 $Q_{0}$ contribute with opposite signs, +0.05 %
and -0.105 %, respectively, leading to some cancellation of the overall
effect. We tested the dependence of $\Delta{Q_{0}}/{Q_{0}}$ 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 $\Delta{Q_{0}}/{Q_{0}}$, making the
constant volume approximation unreliable for extracting this quantity from the
difference in the $\langle r^{2}\rangle$.

Substituting experimental values into the expression for the Coulomb energy
difference given by Eq. (5) gives

$\displaystyle\Delta E_{C}$ $\displaystyle=$
$\displaystyle-0.154(19)\,\textrm{MeV}+0.203(4)\,\textrm{MeV}$ (8)
$\displaystyle=$ $\displaystyle 0.049(19)\,\textrm{MeV},$ (9)

demonstrating strong cancellation between $\langle r^{2}\rangle$ and $Q_{0}$
terms, making the $\Delta E_{C}$ and $K$ values extremely sensitive to
$\Delta{Q_{0}}/Q_{0}$. The enhancement factor is

$K=-18400(2300)+24300(400)=5900(2300),$ (10)

where the uncertainty is overwhelmingly due to the uncertainty in
$\Delta\langle r^{2}\rangle$. An improved measurement of $\zeta$ 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 Yb E3 transitions [38]). 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. This effect
was evaluated in [18], where the axially symmetric surface of the nucleus
$r(\theta)$ was expanded as

$r(\theta)=R_{s}\left[1+\sum_{n=1}^{N}\left(\beta_{n}{Y_{n}^{0}}(\theta)\right)\right],$
(11)

where $\beta_{n}$ 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 $R_{0}$. The
normalization condition can be expressed as
$\int\rho_{q}(\bm{r})d^{3}\bm{r}=1$, where in the present approximation of
constant charge density, $\rho_{q}=1/V$ is the charge density divided by the
total charge $q_{e}Z$ and $V=(4\pi/3)R^{3}_{0}$:

$\frac{2\pi}{3}\int_{0}^{\pi}r^{3}(\theta)\>\textrm{sin}(\theta)\>d\theta=V.$
(12)

For a pear-shaped, axially symmetric nucleus with quadrupole and octupole
deformation, $N=3$. The coefficient $\beta_{1}$ is determined from the
condition that the center of mass of the shape is at the origin of the
coordinate system, $\int_{V}\bm{r}d^{3}\bm{r}=0$ [39] . For a pure quadrupole
deformation (i.e. prolate spheroid), $N=2$ and $\beta_{1}=0$. The nuclear
properties are related to the $\beta$ coefficients via

$\displaystyle\langle r^{2}\rangle$ $\displaystyle=$ $\displaystyle\int
r^{2}\rho_{q}(\bm{r})d^{3}\bm{r}$ (13) $\displaystyle Q_{0}$ $\displaystyle=$
$\displaystyle 2\int r^{2}P_{2}({\cos}\>\theta)\rho_{q}(\bm{r})d^{3}\bm{r},$
(14) $\displaystyle Q_{30}$ $\displaystyle=$ $\displaystyle 2\int
r^{3}P_{3}({\cos}\>\theta)\rho_{q}(\bm{r})d^{3}\bm{r},$ (15)

where $P_{2}$ and $P_{3}$ are the Legendre polynomials, and $Q_{30}$ is the
intrinsic charge octupole moment [39].

The expression for the Coulomb energy [33], Eq. (6.47)

$E_{C}=\frac{3q_{e}^{2}Z^{2}}{5R_{0}}\left(1-\frac{1}{4\pi}\beta_{2}^{2}-\frac{5}{14\pi}\beta_{3}^{2}\right),$
(16)

where $\beta_{2}$ and $\beta_{3}$ are the quadrupole and octupole
deformations, respectively, and where we omit $O(\beta_{n}^{3})$ terms.

The change in the Coulomb energy was described in [18] using the expression

$\displaystyle\Delta E_{C}$ $\displaystyle=$ $\displaystyle\frac{\partial
E_{C}}{\partial\beta_{2}^{2}}\Delta\beta_{2}^{2}+\frac{\partial
E_{C}}{\partial\beta_{3}^{2}}\Delta\beta_{3}^{2}$ (17) $\displaystyle=$
$\displaystyle\frac{3q_{e}^{2}Z^{2}}{5R_{0}}\left(-\frac{1}{4\pi}\Delta\beta_{2}^{2}-\frac{5}{14\pi}\Delta\beta_{3}^{2}\right).$
(18)

However, this expression utilizes the constant volume approximation, while
$R_{0}$ depends on $\beta_{2}$ and $\beta_{3}$ via the volume normalization
above. As in the ellipsoid model above, this leads to a wrong result for
$\Delta E_{C}$, when using the experimental values for $\langle r^{2}\rangle$
and $Q_{0}$.

To evaluate the effect of a possible octupole deformation described by the
change in $\beta_{3}$ upon nuclear excitation, we calculate $\Delta E_{C}$
using experimentally available numbers. We use the definitions for $\langle
r^{2}\rangle$ and $Q_{0}$ and the normalization condition above, the
experimental values for $\langle r^{2}\rangle$ and $Q_{0}$, to calculate
$R_{s}$, $\beta_{2}$, and $R_{0}$ for both ground and isomeric state.

First, we take $\beta_{3}=0$ for both the ground and isomeric state. The
resulting values of $\beta_{2}=0.220$ and $\beta_{2}^{\rm{m}}=0.223$ are
consistent with 228Th value from [40]. We compute $\Delta E_{C}=0.052$ MeV
giving $K=6300(2300)$ using Eq. (16), in agreement with the result obtained
for the ellipsoid model (we note that Eq. (16) is approximate). Then, we
repeat the same computation but use $\beta_{3}=\beta_{3}^{\rm{m}}=0.115$ from
[41], which yields essentially the same result for $K$, as expected. However,
when we vary $\beta_{3}$ differentially between the ground and excited state,
while keeping all other parameters constant, we find that 1 %, 3 %, and 5 %
differential change in $\beta_{3}$ lead to $\Delta K=2850$, $\Delta K=8600$,
and $\Delta K=14500$, respectively. A larger $\beta_{3}$ of the isomer yields
a positive change in $K$.

For reference, the measured change in $Q_{0}$, and correspondingly
$\beta_{2}$, is 1.8 %. This computation shows that even 1 % variation of the
octupole deformation changes $K$ by more than $1\sigma$. Therefore, it is
important to estimate the sign and constrain the magnitude of the octupole
deformation change between the ground and isomeric state.

We can estimate the approximate size of the intrinsic charge octupole moment
in 229Th ground state and isomer. Using the $\beta_{2}$ value obtained above
and $\beta_{3}$ within the range used in [41], 0.11 to 0.145, we get
$Q_{30}=35-44$ fm3 [Eq. (15)]. This is consistent with estimates [40] for the
octupole moment of 228Th 333Note that [40] gives the values of $Zq_{e}Q$.. E3
matrix elements of nuclear Coulomb excitation can be measured to determine the
octupole moment, as has been demonstrated [43]. 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 [40, 44].

Conclusion.– The precise measurement of the change in nuclear quadrupole
moment $\Delta Q_{0}/Q_{0}$ [7] between the 229Th ground and isomeric state
together with previous measurements of the mean-square charge radius $\langle
r^{2}\rangle$ allow us to extract the change in Coulomb contribution $\Delta
E_{C}$ to the nuclear energies. We quantify the sensitivity of the nuclear
transition to changes of the fine-structure constant $\alpha$, with the factor
$K=5900(2300)$. Future theoretical and experimental work will focus on
determining the octupole contribution to the nuclear energies, as we have
shown that small octupole deformations strongly affect the alpha sensitivity.
In addition, an improved measurement of the ratio $\zeta$ of the isomer and
isotope shifts is required. The 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.

Acknowledgements.– We thank Victor Flambaum, David DeMille, Julian Berengut,
Adriana Palffy, Nikolay Minkov, Ronald Fernando Garcia Ruiz, Martin Pimon,
Andreas Grüneis, Logan Hillbery, and Dahyeon Lee for stimulating discussions.

This work has been funded by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
Agreement No. 856415) and the Austrian Science Fund (FWF) [Grant DOI:
10.55776/F1004, 10.55776/J4834, 10.55776/ PIN9526523]. The project 23FUN03
HIOC [Grant DOI: 10.13039/100019599] has received funding from the European
Partnership on Metrology, co-financed from the European Union’s Horizon Europe
Research and Innovation Program and by the Participating States. K.B.
acknowledges support from the Schweizerischer Nationalfonds (SNF), fund 514788
“Wavefunction engineering for controlled nuclear decays”.

We also acknowledge funding support from the Army Research Office
(W911NF2010182); Air Force Office of Scientific Research (FA9550-19-1-0148);
National Science Foundation QLCI OMA-2016244; National Science Foundation
PHY-2317149; and National Institute of Standards and Technology.

## References

  * [1] E. Peik, T. Schumm, M. S. Safronova, A. Pálffy, J. Weitenberg, and P. G. Thirolf, “Nuclear clocks for testing fundamental physics,” Quantum Science and Technology, vol. 6, p. 034002, July 2021.
  * [2] C. J. Campbell, A. G. Radnaev, A. Kuzmich, V. A. Dzuba, V. V. Flambaum, and A. Derevianko, “Single-ion nuclear clock for metrology at the 19th decimal place,” Phys. Rev. Lett., vol. 108, p. 120802, Mar 2012.
  * [3] W. G. Rellergert, D. DeMille, R. R. Greco, M. P. Hehlen, J. R. Torgerson, and E. R. Hudson, “Constraining the evolution of the fundamental constants with a solid-state optical frequency reference based on the ${}^{229}\mathrm{Th}$ nucleus,” Phys. Rev. Lett., vol. 104, p. 200802, May 2010.
  * [4] G. A. Kazakov, A. N. Litvinov, V. I. Romanenko, L. P. Yatsenko, A. V. Romanenko, M. Schreitl, G. Winkler, and T. Schumm, “Performance of a 229thorium solid-state nuclear clock,” New Journal of Physics, vol. 14, p. 083019, 2012.
  * [5] J. Tiedau, M. V. Okhapkin, K. Zhang, J. Thielking, G. Zitzer, E. Peik, F. Schaden, T. Pronebner, I. Morawetz, L. T. De Col, F. Schneider, A. Leitner, M. Pressler, G. A. Kazakov, K. Beeks, T. Sikorsky, and T. Schumm, “Laser excitation of the th-229 nucleus,” Phys. Rev. Lett., vol. 132, p. 182501, Apr 2024.
  * [6] R. Elwell, C. Schneider, J. Jeet, J. E. S. Terhune, H. W. T. Morgan, A. N. Alexandrova, H. B. Tran Tan, A. Derevianko, and E. R. Hudson, “Laser excitation of the ${}^{229}\mathrm{Th}$ nuclear isomeric transition in a solid-state host,” Phys. Rev. Lett., vol. 133, p. 013201, Jul 2024.
  * [7] C. Zhang, T. Ooi, J. S. Higgins, J. F. Doyle, L. von der Wense, K. Beeks, A. Leitner, G. Kazakov, P. Li, P. G. Thirolf, T. Schumm, and J. Ye, “Frequency ratio of the 229mTh isomeric transition and the 87Sr atomic clock,” 2024.
  * [8] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for New Physics with Atoms and Molecules,” Rev. Mod. Phys, vol. 90, p. 025008, 2018.
  * [9] A. Arvanitaki, J. Huang, and K. Van Tilburg, “Searching for dilaton dark matter with atomic clocks,” Phys. Rev. D, vol. 91, p. 015015, Jan 2015.
  * [10] Y. V. Stadnik and V. V. Flambaum, “Can dark matter induce cosmological evolution of the fundamental constants of nature?,” Phys. Rev. Lett., vol. 115, p. 201301, Nov 2015.
  * [11] Y. V. Stadnik and V. V. Flambaum, “Improved limits on interactions of low-mass spin-0 dark matter from atomic clock spectroscopy,” Phys. Rev. A, vol. 94, p. 022111, Aug 2016.
  * [12] D. Antypas et al., “New horizons: Scalar and vector ultralight dark matter,” 2022.
  * [13] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye, “Precision metrology meets cosmology: Improved constraints on ultralight dark matter from atom-cavity frequency comparisons,” Phys. Rev. Lett., vol. 125, p. 201302, Nov 2020.
  * [14] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, “Space-time variation of physical constants and relativistic corrections in atoms,” Phys. Rev. Lett., vol. 82, pp. 888–891, Feb 1999.
  * [15] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, “Calculations of the relativistic effects in many-electron atoms and space-time variation of fundamental constants,” Phys. Rev. A, vol. 59, pp. 230–237, Jan 1999.
  * [16] M. Kozlov, M. Safronova, J. Crespo López-Urrutia, and P. Schmidt, “Highly charged ions: Optical clocks and applications in fundamental physics,” Reviews of Modern Physics, vol. 90, no. 4, p. 045005, 2018.
  * [17] M. S. Safronova, “The search for variation of fundamental constants with clocks,” Annalen der Physik, vol. 531, no. 5, p. 1800364, 2019.
  * [18] P. Fadeev, J. C. Berengut, and V. V. Flambaum, “Sensitivity of ${}^{229}\mathrm{Th}$ nuclear clock transition to variation of the fine-structure constant,” Phys. Rev. A, vol. 102, p. 052833, Nov 2020.
  * [19] G. Barontini, L. Blackburn, V. Boyer, F. Butuc-Mayer, X. Calmet, J. C. López-Urrutia, E. Curtis, B. Darquie, J. Dunningham, N. Fitch, et al., “Measuring the stability of fundamental constants with a network of clocks,” EPJ Quantum Technology, vol. 9, no. 1, p. 12, 2022.
  * [20] K. Beloy, M. I. Bodine, T. Bothwell, S. M. Brewer, S. L. Bromley, J.-S. Chen, J.-D. Deschênes, S. A. Diddams, R. J. Fasano, T. M. Fortier, Y. S. Hassan, D. B. Hume, D. Kedar, C. J. Kennedy, I. Khader, A. Koepke, D. R. Leibrandt, H. Leopardi, A. D. Ludlow, W. F. McGrew, W. R. Milner, N. R. Newbury, D. Nicolodi, E. Oelker, T. E. Parker, J. M. Robinson, S. Romisch, S. A. Schäffer, J. A. Sherman, L. C. Sinclair, L. Sonderhouse, W. C. Swann, J. Yao, J. Ye, X. Zhang, and Boulder Atomic Clock Optical Network (BACON) Collaboration*, “Frequency ratio measurements at 18-digit accuracy using an optical clock network,” Nature, vol. 591, pp. 564–569, Mar. 2021.  Publisher: Nature Publishing Group.
  * [21] M. Filzinger, S. Dörscher, R. Lange, J. Klose, M. Steinel, E. Benkler, E. Peik, C. Lisdat, and N. Huntemann, “Improved limits on the coupling of ultralight bosonic dark matter to photons from optical atomic clock comparisons,” Phys. Rev. Lett., vol. 130, p. 253001, Jun 2023.
  * [22] V. V. Flambaum, “Enhanced effect of temporal variation of the fine structure constant and the strong interaction in ${}^{229}\mathrm{Th}$,” Phys. Rev. Lett., vol. 97, p. 092502, Aug 2006.
  * [23] E. Litvinova, H. Feldmeier, J. Dobaczewski, and V. Flambaum, “Nuclear structure of lowest ${}^{229}\mathrm{Th}$ states and time-dependent fundamental constants,” Phys. Rev. C, vol. 79, p. 064303, Jun 2009.
  * [24] J. C. Berengut, V. A. Dzuba, V. V. Flambaum, and S. G. Porsev, “Proposed experimental method to determine $\alpha$ sensitivity of splitting between ground and 7.6 ev isomeric states in ${}^{229}\mathrm{Th}$,” Phys. Rev. Lett., vol. 102, p. 210801, May 2009.
  * [25] K. Beeks, The nuclear excitation of Thorium-229 in the CaF2 environment: Development of a crystalline nuclear clock.  PhD thesis, Technische Universität Wien, 2022.
  * [26] K. Beeks, T. Sikorsky, V. Rosecker, M. Pressler, F. Schaden, D. Werban, N. Hosseini, L. Rudischer, F. Schneider, P. Berwian, et al., “Growth and characterization of thorium-doped calcium fluoride single crystals,” Scientific Reports, vol. 13, no. 1, p. 3897, 2023.
  * [27] K. Beeks, T. Sikorsky, F. Schaden, M. Pressler, F. Schneider, B. N. Koch, T. Pronebner, D. Werban, N. Hosseini, G. Kazakov, et al., “Optical transmission enhancement of ionic crystals via superionic fluoride transfer: Growing vuv-transparent radioactive crystals,” Physical Review B, vol. 109, no. 9, p. 094111, 2024.
  * [28] S. G. Porsev, M. S. Safronova, and M. G. Kozlov, “Precision Calculation of Hyperfine Constants for Extracting Nuclear Moments of 229Th,” Phys. Rev. Lett. , vol. 127, p. 253001, Dec. 2021.
  * [29] P. Dessovic, P. Mohn, R. Jackson, G. Winkler, M. Schreitl, G. Kazakov, and T. Schumm, “229thorium-doped calcium fluoride for nuclear laser spectroscopy,” Journal of Physics: Condensed Matter, vol. 26, no. 10, p. 105402, 2014.
  * [30] M. Pimon, A. Grüneis, P. Mohn, and T. Schumm, “Ab-initio study of calcium fluoride doped with heavy isotopes,” Crystals, vol. 12, no. 8, p. 1128, 2022.
  * [31] J. Thielking, M. V. Okhapkin, P. Głowacki, D. M. Meier, L. von der Wense abd Benedict Seiferle, C. E. Düllmann, P. G. Thirolf, and E. Peik, “Laser spectroscopic characterization of the nuclear-clock isomer 229mth,” Nature, vol. 556, p. 321, 2018.
  * [32] R. Beringer and W. J. Knox, “Liquid-drop nuclear model with high angular momentum,” Phys. Rev., vol. 121, pp. 1195–1200, Feb 1961.
  * [33] R. Hasse and W. D. Myers, Geometrical Relationships of Macroscopic Nuclear Physics.  Springer-Verlag, Heidelberg, 1988.
  * [34] I. Angeli and K. Marinova, “Table of experimental nuclear ground state charge radii: An update,” At. Data Nucl. Data Tables, vol. 99, p. 69, 2013.
  * [35] C. J. Campbell, A. G. Radnaev, and A. Kuzmich, “Wigner crystals of ${}^{229}\mathrm{Th}$ for optical excitation of the nuclear isomer,” Phys. Rev. Lett., vol. 106, p. 223001, Jun 2011.
  * [36] M. S. Safronova, S. G. Porsev, M. G. Kozlov, J. Thielking, M. V. Okhapkin, P. Głowacki, D. M. Meier, and E. Peik, “Nuclear charge radii of ${}^{229}\mathrm{Th}$ from isotope and isomer shifts,” Phys. Rev. Lett., vol. 121, p. 213001, Nov 2018.
  * [37] Note that we follow $Q_{0}$ designations of [18], which differs from [31], where the value of with $Q_{0}$ was defined without the factor $q_{e}Z$. Using designations and units of [31], we get $Q_{0}^{\mathrm{m}}=8.86(6)\>e$b.
  * [38] V. V. Flambaum and V. A. Dzuba Canadian Journal of Physics, vol. 87, p. 1, 2009.
  * [39] P. A. Butler and W. Nazarewicz, “Intrinsic reflection asymmetry in atomic nuclei,” Rev. Mod. Phys., vol. 68, pp. 349–421, Apr 1996.
  * [40] M. M. R. Chishti, D. O’Donnell, G. Battaglia, M. Bowry, D. A. Jaroszynski, B. S. N. Singh, M. Scheck, P. Spagnoletti, and J. F. Smith, “Direct measurement of the intrinsic electric dipole moment in pear-shaped thorium-228,” Nature Physics, vol. 16, p. 853, 2020.
  * [41] N. Minkov and A. Pálffy, “${}^{229m}\mathrm{Th}$ isomer from a nuclear model perspective,” Phys. Rev. C, vol. 103, p. 014313, Jan 2021.
  * [42] Note that [40] gives the values of $Zq_{e}Q$.
  * [43] L. P. Gaffney, P. A. Butler, M. Scheck, A. B. Hayes, F. Wenander, M. Albers, B. Bastin, C. Bauer, A. Blazhev, S. Bönig, et al., “Studies of pear-shaped nuclei using accelerated radioactive beams,” Nature, vol. 497, no. 7448, pp. 199–204, 2013.
  * [44] V. V. Flambaum, “Enhanced nuclear schiff moment and time-reversal violation in ${}^{229}\mathrm{Th}$-containing molecules,” Phys. Rev. C, vol. 99, p. 035501, Mar 2019.