Simultaneous Determination of Multiple Nuclear Parameters of $^{229}$Th Using Highly Charged Ions

Technical Summary: Simultaneous Determination of Multiple Nuclear Parameters of $^{229}$Th Using Highly Charged Ions

Problem Statement
The development of a $^{229}$Th nuclear optical clock is hindered by the lack of precise, model-independent nuclear-structure parameters. Current knowledge of key parameters—the magnetic dipole moments ($\mu_g, \mu_m$), the bare-nucleus transition energy ($\omega_n$), the charge-radius difference ($\delta\langle r^2 \rangle_{gm}$), and the magnetic-dipole reduced transition probability ($B(M1)$ or matrix element $T_{M1}$)—suffers from limited accuracy and poor consistency.

• The transition energy $\omega_n$ inferred from doped crystals has an uncertainty of several THz.
• The charge-radius difference $\delta\langle r^2 \rangle_{gm}$ is known with only $\sim$12% precision.
• The matrix element $T_{M1}$, which governs the isomer's radiative lifetime, has experimental values spanning nearly an order of magnitude.
  Existing indirect methods using atomic spectroscopy on highly charged ions (HCIs) are limited because they rely on external nuclear inputs that are themselves imprecise. Furthermore, the Bohr–Weisskopf (BW) effect, arising from the finite spatial distribution of nuclear magnetization, introduces uncertainties up to $\sim$35% in calculated hyperfine matrix elements, constituting the dominant error source in current approaches.

Methodology
The authors propose a joint spectroscopic scheme utilizing two specific highly charged $^{229}$Th ions: $^{229}$Th$^{87+}$ (Li-like, $2s_{1/2}$ ground state) and $^{229}$Th$^{79+}$ (Na-like, $3s_{1/2}$ ground state). Both ions possess a $J=1/2$ electronic ground state.

• Physical Regime: In these ions, the first electronic excitation energy ($\gtrsim$7 keV) far exceeds the nuclear isomer energy ($\omega_n \approx 8.4$ eV). This hierarchy restricts the hyperfine interaction to a three-level system comprising the electronic ground state and the nuclear ground ($g$) and isomeric ($m$) states.
• Enhanced Nuclear Hyperfine Mixing (NHM): In this regime, the second-order hyperfine energy shift is dominated by the nuclear intermediate states rather than electronic ones. This results in an enhancement of the NHM effect by approximately four orders of magnitude compared to standard cases.
• Overdetermined System: The scheme measures six independent hyperfine transition frequencies (three for each ion). These frequencies depend on five unknown nuclear parameters ($\mu_g, \mu_m, \omega_n, \delta\langle r^2 \rangle_{gm}, T_{M1}$) and calculable atomic properties (electronic reduced matrix elements $T_{g(m),2s(3s)}$ and field-shift factors $F^{ved}_{2s(3s)}$).
• Analytical Solution: The system is overdetermined (6 equations, 5 unknowns). The authors derive a constraint equation (Eq. 7) that eliminates nuclear unknowns, allowing for an internal consistency check. The remaining five nuclear parameters are then expressed analytically as functions of the measured frequencies and atomic inputs (Eqs. 8–12). This approach eliminates the need for external nuclear inputs.
• Uncertainty Mitigation: To address the BW effect, the authors employ a ratio-based approach. They normalize BW-corrected matrix elements to a common hydrogen-like reference, exploiting the strong correlation of BW effects across different charge states and $ns$ configurations. This significantly reduces the uncertainty associated with the nuclear magnetization distribution.

Key Results
Using theoretical atomic structure calculations (MCDHF with GRASP2018 and RIS4) and assuming a conservative experimental frequency uncertainty of 5 GHz (based on precision demonstrated for Bi$^{80+}$), the authors evaluated the performance of their scheme:

• Simultaneous Extraction: The method successfully extracts all five nuclear parameters simultaneously without external nuclear data.
• Precision Improvements:
  • $\omega_n$: The uncertainty is reduced from 5320 GHz (current crystal-based value) to 1551 GHz, a factor of $\sim$3 improvement.
  • $\delta\langle r^2 \rangle_{gm}$: The uncertainty is reduced from 0.013 fm$^2$ to 0.0008 fm$^2$, a factor of $\sim$2 improvement.
  • Magnetic Moments ($\mu_g, \mu_m$): Predicted uncertainties are $\sim 6 \times 10^{-3} \mu_N$ and $\sim 8 \times 10^{-3} \mu_N$, respectively, consistent with current reference values.
  • $T_{M1}$: The uncertainty is estimated at 0.38 $\mu_N$. While this is larger than the uncertainty derived from current radiative lifetime measurements (0.11 $\mu_N$), the authors note this is dominated by uncorrected higher-order QED contributions in the electronic matrix elements, not the BW effect.
• Robustness: The scheme is robust against variations in the BW ratio factors between Li-like and Na-like ions. The dominant theoretical uncertainties for $\omega_n$, $\delta\langle r^2 \rangle_{gm}$, and $T_{M1}$ arise from QED corrections to the electronic matrix elements, whereas the magnetic moments are primarily limited by BW corrections.

Significance and Claims
The paper claims that this work provides a self-contained path to determining the nuclear parameters necessary for a $^{229}$Th nuclear clock.

• Benchmark for Theory: The simultaneous determination of $\omega_n$ and $\delta\langle r^2 \rangle_{gm}$ offers a high-precision benchmark for testing nuclear theories, free from the dominant external uncertainties plaguing previous methods.
• Foundation for Clocks: By reducing the uncertainties in key nuclear parameters, the scheme lays a robust foundation for the future realization of a $^{229}$Th-based nuclear optical clock.
• Scalability: The theoretical framework is extensible to ions with total angular momentum $J \ge 1$, which would allow for the simultaneous extraction of nuclear quadrupole moments ($Q_g, Q_m$) alongside other structural parameters.
• Feasibility: The authors argue that the scheme is experimentally feasible using existing technologies, including electron-beam ion traps (EBITs) for ion production, cryogenic Penning traps or storage rings for confinement, and frequency-comb Raman spectroscopy or VUV difference-frequency measurements to access the required THz-range transitions.

The authors emphasize that these improvements are achieved even under a conservative assumption of GHz-level spectral resolution, which is six orders of magnitude coarser than the kHz-level resolution already demonstrated in doped crystals. The primary gains stem from the simultaneous extraction of parameters rather than increased spectral resolution alone.