Enhanced Yield Rate of \textsuperscript{229m}Th via Cascade Decay in Storage Rings and Electron Beam Ion Traps

This contribution proposes and demonstrates that the utilization of cascade decay pathways via nuclear excitation by inelastic electron scattering (NEIES) and nuclear excitation by electron capture (NEEC) in storage rings and electron beam ion traps can significantly increase the production yield of the $^{229\mathrm{m}}$Th isomer, thereby facilitating its application in nuclear clocks and precision metrology.

The Gist

The Big Picture: Catching a "Ghost" Atom

Imagine you are trying to catch a very specific, shy ghost (the 229mTh isomer) that possesses a unique superpower: it could become the most accurate clock in the universe, far superior to the atomic clocks we use today.

The problem is that this ghost is incredibly hard to catch. Normally, scientists try to produce it by waiting for other atoms to naturally decay into it (like waiting for a tree to drop a specific fruit), but this happens so rarely that it is like waiting for a single raindrop in a desert. Other methods involve hitting the atom directly with a laser, yet the laser is either too weak or has the wrong color to perform the task efficiently.

The New Idea: The "Staircase" Strategy

This paper proposes a clever new way to catch the ghost. Instead of trying to jump directly to the top of a tall building (the isomer state), the authors suggest taking the stairs.

1. The Setup: They propose using two high-tech machines: storage rings (like a giant racetrack for atoms) and electron-beam ion traps (like a high-tech cage for atoms).
2. The Method: Instead of gently hitting the atom to bring it into the isomer state, they hit it hard with a stream of electrons to kick the atom up to a very high energy level (the top of the stairs).
3. The Cascade: Once the atom is at the top, it naturally tumbles down the stairs. As it tumbles, it passes through several steps. The authors calculated that if you kick the atom high enough, it is much more likely to land on the specific "ghost" step (the isomer) on its way down than if you tried to aim directly at it.

The Two Tools: The "Slingshot" and the "Magnet"

The paper describes two ways to kick the atoms up the stairs using electrons:

• NEIES (The Slingshot): Imagine throwing a ball at a target. If the ball hits the target hard enough, it transfers energy and knocks the target upward. This happens whenever the electron moves fast enough. The paper finds that this "slingshot" method becomes incredibly powerful when using very fast electrons, particularly for kicking atoms to the higher steps.
• NEEC (The Magnet): This is more like a magnetic lock. An electron flies by, and if it has the exact right speed and energy, it gets "snapped" and coupled to the atom, and this specific snap pushes the nucleus up the stairs. This is very precise but requires perfect timing.

The Results: A Massive Boost

The authors ran the numbers (theoretical calculations) to see how well this "staircase" idea works compared to the old "direct jump" method.

• The Slingshot (NEIES): When they applied the "staircase" method with high-energy electrons, they found they could produce the isomer 10,000 times more frequently (four orders of magnitude) than before. It is like the transition from finding a single grain of sand on the beach to finding a whole bucket of it.
• The Magnet (NEEC): When they applied the precise "magnetic" method, they saw an additional boost that made the process a dozen times more efficient than direct methods.

Why This Matters (According to the Paper)

The paper concludes that by using these "staircase" paths in these high-tech machines, scientists can finally produce enough of these special atoms to actually build the nuclear clocks mentioned in the introduction.

The authors also point out that this method offers a clear path to test and confirm the "magnetic" capture process (NEEC), which has long been theorized but never fully confirmed in a laboratory.

In short: The paper says, "Stop trying to jump directly to the top. Kick the atoms all the way up the stairs and let them tumble down; it is a much faster and more reliable way to catch the specific atom we need for future super-clocks."

Technical Summary

Technical Summary: Enhanced Yield of 229mTh via Cascade Decay in Storage Rings and Electron Beam Ion Traps

Problem Statement
The low-energy nuclear isomer state of $^{229m}$Th (at 8.36 eV) is a critical resource for nuclear optical clocks, precision metrology, and tests of fundamental physics. However, the efficient and controllable production of $^{229m}$Th remains a significant experimental bottleneck. Current methods, such as the $\alpha$-decay of $^{233}$U or the $\beta$-decay of $^{229}$Ac, suffer from limited production rates. Direct laser excitation in crystals faces challenges due to low efficiency and the lack of powerful continuous-wave VUV sources, while indirect methods using synchrotron radiation require large facilities with limited tunability. Consequently, there is a need for a versatile mechanism to generate $^{229m}$Th with significantly higher yields.

Methodology
The authors propose a theoretical concept to produce $^{229m}$Th in storage rings (SRs) and electron beam ion traps (EBITs) using highly charged ions (HCIs) and a cascade decay pathway. Instead of direct excitation to the isomer, the concept involves exciting the nucleus to higher-lying states (specifically levels up to 125.44 keV) via electron-mediated processes, followed by radiative or internal conversion cascades that populate the isomeric state.

The study employs two primary excitation mechanisms:

1. Nuclear Excitation by Inelastic Electron Scattering (NEIES): Free electrons transfer kinetic energy to the nucleus via inelastic collisions.
2. Nuclear Excitation by Electron Capture (NEEC): A free electron is captured into a bound electronic state of the ion, simultaneously exciting the nucleus.

Theoretical calculations were performed within the framework of the Dirac-Hartree-Fock-Slater (DHFS) method to compute electronic wave functions and transition matrix elements. NEIES cross-sections and NEEC resonance strengths were calculated for various ion charge states (focusing on $q_i = 90$ for SRs and a distribution peaking at $q_i = 87+$ for EBITs). Effective excitation rates were determined by convolving these cross-sections with electron beam energy distributions and weighting them with the branching ratios of subsequent cascade decays to the isomer. EBIT simulations utilized the ebitsim package to model charge state evolution under electron beam irradiation.

Main Contributions and Results
The work provides quantitative calculations of cross-sections and production rates for both SR and EBIT environments, demonstrating that indirect excitation via higher-lying states significantly outperforms direct excitation.

• Characteristics of Cross-Sections:

  • NEIES: The cross-sections exhibit strong energy dependence. At high electron energies (above ~200 keV), the cross-sections for exciting higher states (particularly the 5th and 6th excited states) are substantially larger than those for direct excitation to the first excited state.
  • NEEC: Resonant structures were identified. For highly excited states, the maximum NEEC resonance strength reaches approximately 10 b·eV, which is nearly two orders of magnitude higher than for direct excitation to the isomer.

• Production Rates in Storage Rings (SRs):

  • Under typical SR conditions ($10^8$ ions, 200 mA electron beam), the total isomer yield rate via cascade decay is significantly enhanced.
  • At high electron energies, the NEIES mechanism alone offers an increase of up to four orders of magnitude compared to direct excitation.
  • When NEEC is considered at resonance energies, the yield rate increases by an additional factor of up to 80 (specifically for the cascade path of the 6th excited state).

• Production Rates in Electron Beam Ion Traps (EBITs):

  • Using a configuration with two electron beams (one for charge breeding, one for resonant excitation), the study simulated a charge state distribution peaking at $q_i = 87+$.
  • The cascade path of the 6th excited state ($0 \to 6 \to 1$) proved most favorable and is dominated by NEEC.
  • This path yields a total production rate of approximately 13.5 s$^{-1}$, corresponding to an increase by a factor of 29 compared to direct excitation.
  • The study notes that the paths of the 3rd and 4th excited states are dominated by NEIES, while others show mixed contributions, offering potential avenues to experimentally distinguish between NEEC and NEIES mechanisms.

Significance and Claims
The authors claim that their proposed concept offers a significant and practical route to overcome the production bottleneck for $^{229m}$Th. By leveraging the large excitation cross-sections of HCIs and the efficiency of cascade decays, the yield rate can be increased by up to four orders of magnitude via NEIES and by a multiplicative factor via NEEC.

The work affirms that these results provide quantitative guidelines for optimizing isomer production in existing and future facilities. Specifically, the proposed concepts enable:

1. The generation of sufficient $^{229m}$Th ions for applications in nuclear clocks and precision metrology.
2. A promising experimental pathway to verify the NEEC process, which has not yet been experimentally confirmed.
3. The indication that the cascade path of the 6th excited state in EBITs is particularly suitable for maximizing yield and potentially isolating NEEC signals.

The work concludes that electron-induced excitation concepts in SRs and EBITs represent a versatile and controllable alternative to current production methods, potentially enabling the development of atomic nuclear clocks and advancing research in nuclear photonics.