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Theoretical evaluation of decay mode of $ {}^{229m} \mathrm{Th} $ in solid samples

Ryotaro Masuda, Tomoya Naito, Masashi Kaneko, Hiroyuki Kazama, So Hashiba, Kosuke Misawa, Yoshitaka Kasamatsu

TL;DR

This work addresses inconsistent gamma-decay half-lives of ${}^{229m}_{\text{Th}}$ across different chemical states by linking decay channels to the electronic structure, leveraging density functional theory (DFT) and natural bond orbital (NBO) analysis. By modeling Th in ion-trap-like and crystal environments (CaF$_2$, MgF$_2$, LiSrAlF$_6$) under two placement schemes (gap and replaced) and examining $E_{ ext{IS}} = 8.4$ eV against the Th HOMO binding energies, the authors classify possible decay modes into $\gamma$-ray decay, IC, and EB for each system. They find CaF$_2$ and MgF$_2$ (replaced) favor gamma-only decay, CaF$_2$ gap can allow IC, and LiSrAlF$_6$ can exhibit EB and/or IC depending on the site, consistent with some experimental half-life trends and suggesting that crystal environment critically influences the nuclear-clock suitability. The results offer a qualitative, first-principles route to predict decay channels in solids and point to future work on quantitative IC probabilities and refined refractive-index corrections to reconcile remaining discrepancies and guide material choice for a nuclear clock.

Abstract

The excitation energy of $ {}^{229m} \mathrm{Th} $ is extremely low at $ 8.4 \, \mathrm{eV} $; thus, this isotope exhibits changes in its decay modes depending on the chemical state, specifically the outermost electronic states. However, the reported half-lives of the $ γ$-ray transition are not consistent among the previous experiments. In this study, we investigate the chemical states of $ {}^{229m} \mathrm{Th} $ by density functional theory calculations. Based on these results, we evaluate the relationship between the experimental half-life of each sample and the electronic state of $ \mathrm{Th} $. The calculation results indicate that ion trap method, $ \mathrm{Ca} \mathrm{F}_2 $ model and $ \mathrm{Mg} \mathrm{F}_2 $ one decay only via the $ γ$-ray transition, whereas $ \mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6 $ one decays via the $ γ$-ray transition and has a possibility of decay via internal conversion and electron bridge.

Theoretical evaluation of decay mode of $ {}^{229m} \mathrm{Th} $ in solid samples

TL;DR

This work addresses inconsistent gamma-decay half-lives of across different chemical states by linking decay channels to the electronic structure, leveraging density functional theory (DFT) and natural bond orbital (NBO) analysis. By modeling Th in ion-trap-like and crystal environments (CaF, MgF, LiSrAlF) under two placement schemes (gap and replaced) and examining eV against the Th HOMO binding energies, the authors classify possible decay modes into -ray decay, IC, and EB for each system. They find CaF and MgF (replaced) favor gamma-only decay, CaF gap can allow IC, and LiSrAlF can exhibit EB and/or IC depending on the site, consistent with some experimental half-life trends and suggesting that crystal environment critically influences the nuclear-clock suitability. The results offer a qualitative, first-principles route to predict decay channels in solids and point to future work on quantitative IC probabilities and refined refractive-index corrections to reconcile remaining discrepancies and guide material choice for a nuclear clock.

Abstract

The excitation energy of is extremely low at ; thus, this isotope exhibits changes in its decay modes depending on the chemical state, specifically the outermost electronic states. However, the reported half-lives of the -ray transition are not consistent among the previous experiments. In this study, we investigate the chemical states of by density functional theory calculations. Based on these results, we evaluate the relationship between the experimental half-life of each sample and the electronic state of . The calculation results indicate that ion trap method, model and one decay only via the -ray transition, whereas one decays via the -ray transition and has a possibility of decay via internal conversion and electron bridge.

Paper Structure

This paper contains 10 sections, 5 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Identification of crystal structure
  3. Band analysis
  4. Calculation setup
  5. Result and discussion
  6. ${}^{229m}_{}\mathrm{Th}$ atom and ions
  7. $\mathrm{Ca} \mathrm{F}_2$
  8. $\mathrm{Mg} \mathrm{F}_2$
  9. $\mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6$
  10. Summary

Figures (6)

  • Figure 1: Crystal structure with $3 \times 3 \times 3$ supercell model of (a) $\mathrm{Ca}\mathrm{F}_2$, (b) $\mathrm{Mg}\mathrm{F}_2$ and, (c) $\mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6$. The white-filled sphericals indicate the gap spaces.
  • Figure 2: Considered structure for (a) a $\mathrm{Th}$ atom simulating ion trap method; (b) $\mathrm{Ca} \mathrm{F}_2$ gap model; (c) $\mathrm{Ca} \mathrm{F}_2$-$\mathrm{Ca}$ replaced model; (d) $\mathrm{Mg} \mathrm{F}_2$ gap model; (e) $\mathrm{Mg} \mathrm{F}_2$-$\mathrm{Mg}$ replaced model; (f) $\mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6$ gap model; (g) $\mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6$-$\mathrm{Al}$ replaced model; (h) $\mathrm{Li} \mathrm{Sr} \mathrm{Al} \mathrm{F}_6$-$\mathrm{Sr}$ replaced model. The same structures were adopted for all the initial charges of $\mathrm{Th}$. Except for $\mathrm{Th}$, the colors are the same as shown in Fig. \ref{['QE-model']}.
  • Figure 3: Feynman and schematic diagram of (a) the internal conversion and (b) electron bridge.
  • Figure 4: The number of electrons of $\mathrm{Th}$ that can be involved in IC process and their constituent orbital for each sample.
  • Figure 5: The total number of $s$ electrons in the levels that are potential initial states for EB.
  • ...and 1 more figures