Charge state regulation of nuclear excitation by electron capture in $^{229}$Th ions

TL;DR

This work analyzes nuclear excitation by electron capture (NEEC) in $^{229}$Th$^{q+}$ across charge states $q=1^+$ to $90^+$, incorporating the electronic quantum numbers $(n,l,j)$ and focusing on the isomeric state (IS, $E_{ ext{IS}}=8.36$ eV) and the second-excited state (SE, $E_{ ext{SE}}=29.19$ keV). A Dirac-Hartree-Fock-Slater framework is used to compute NEEC cross sections, resonance strengths, and widths, with explicit treatment of multipole transitions and radial matrix elements $T_{fi}^{ ext{E(M)} ext{λ}}$. The results reveal charge-state–dependent mechanisms: IS channels exhibit threshold migration with the dominant $n$ following $n \,\approx\, 1.28 q + 4.23$ and a relatively stable $S$ due to compensatory coupling, while SE channels show negligible channel screening and a monotonic rise of $S_{ ext{total}}$ with $q$, eventually surpassing the IS around $q\geq87^+$. These insights suggest an experimentally viable route to indirectly populate the IS by regulating the SE NEEC, offering a practical parameter window for nuclear optical clock development and precise control of $^{229}$Th nuclear states.

Abstract

Nuclear excitation by electron capture (NEEC) in $^{229}$Th holds significant potential for precise nuclear state manipulation. In this study, we thoroughly investigate NEEC in $^{229}\text{Th}^{q+}$ ions by integrating quantum numbers ($n, l, j$) effects and analyzing key parameters (e.g., resonance energy $E_r$, cross section $σ$, resonance strength $S$, and NEEC transition width $Γ_{\text{NEEC}}$) influences across charge state from $q=1^+$ to $90^+$. Especially, we focus on the charge-state regulation of the isomeric state (IS, 8.36 eV) and second-excited state (SE, 29.19 keV). Our calculations uncover critical charge-state-dependent behaviors of NEEC in $^{229}\text{Th}$ ions: (1) For the IS, valid NEEC channels exhibit threshold migration, where the dominant principal quantum number $n$ increases linearly with $q$ following the relation $n \approx 1.28q + 4.23$; meanwhile, single-$n$-channel $S$ stabilizes between $10^{-2}$ to $10^0$ barn eV via compensatory nucleus-electron coupling, ensuring the total resonance $S$ constant. (2) For the SE, its excitation energy far exceeds nearly all electron binding energies, leading to negligible channel screening and causing the total $S$ to increase monotonically with $q$. This research clarifies the intrinsic mechanisms of charge-state-driven nuclear-electronic interactions in $^{229}\text{Th}^{q+}$ NEEC and provides a critical reference for future experimental efforts to manipulate $^{229}\text{Th}$ nuclear states, particularly via indirect regulation of the SE.

Figures (10)

• Figure 1: Three-dimensional visualization of NEEC parameters $\sigma$, $\Gamma_{\text{NEEC}}$, $S$ ($z$-axis) as functions of $E_i$ ($x$-axis) and final-state $n$ ($y$-axis), with projections on the $y-z$ plane for clarity.
• Figure 2: Radial components of Dirac spinors ($g(r)$, $f(r)$) and transition matrix elements for different final states.
• Figure 3: Three-dimensional visualization of NEEC parameters for ground-state $\text{Th}^{1+}$: influence of $l$ and $j$ in $n$=8 transitions (with side-plane projections; $x$/$y$-axes: initial/final states; $z$-axis: $\sigma$, $S$, and $\Gamma_{\text{NEEC}}$).
• Figure 4: Binding energy spectrum of electronic holes in $\text{Th}^{q+}$ across low charge states ($q=1^{+}\to8^{+}$). The pink region indicates valid NEEC channels with $|E_{\text{bound}}| < 8.36\ \text{eV}$.
• Figure 5: NEEC parameters ($E_r$, $\sigma$, $S$, $\Gamma_{\text{NEEC}}$) for the $f_{5/2} \to 11f_{5/2}$ transition as a function of charge state $q$.