Shake-up and shake-off spectra in the electron capture decay of atomic $^7$Be

This study employs multiconfiguration Dirac-Fock calculations to model electron shake-up and shake-off spectra in the electron capture decay of atomic $^7$Be, revealing that while the models explain some spectral features, material-induced wavefunction modifications remain a challenge, and providing a revised L/K electron capture ratio of 0.0756(20) that improves constraints on sub-MeV sterile neutrinos.

The Gist

The Big Picture: Hunting for Ghost Particles

Imagine the universe is a giant puzzle, and scientists have a picture of how it works called the "Standard Model." But there are missing pieces. One of the biggest mysteries is dark matter and why there is more matter than antimatter.

To find the missing pieces, scientists are hunting for a "ghost particle" called a sterile neutrino. These are invisible, weighty particles that don't interact with normal matter, making them incredibly hard to catch.

The BeEST experiment is one of the most sensitive traps set to catch these ghosts. It uses a radioactive atom called Beryllium-7 (7Be). When this atom decays, it usually spits out a neutrino and turns into a Lithium atom. By measuring the tiny "kick" (recoil) the Lithium atom gets, scientists can calculate the mass of the neutrino. If the neutrino is heavy (like a sterile neutrino), the kick will be smaller than expected.

The Problem: The "Shake" Effect

The paper focuses on a major source of confusion in this experiment: Electron Shake-up and Shake-off.

Think of the atom like a house with furniture (electrons) arranged in specific rooms (shells).

1. The Event: Suddenly, the owner of the house (the nucleus) changes. An electron is captured, and the house instantly becomes a different type of house (Lithium instead of Beryllium).
2. The Shock: Because the house changed so suddenly, the furniture doesn't just sit there. It gets shaken.
   • Shake-up: Some furniture gets bumped up to a higher shelf (an excited state).
   • Shake-off: Some furniture gets thrown out the window entirely (ionization).

In the past, scientists used rough, old maps to predict how much the furniture would shake. These maps were like "cartoon drawings"—they didn't account for the fact that the furniture pieces bump into each other (electron correlations) or the effects of high-speed physics (relativity). Because these maps were inaccurate, the "background noise" in the experiment was messy, making it hard to spot the ghost particle signal.

What This Paper Did: A High-Definition Remodel

The authors of this paper decided to build a 3D, high-definition simulation of this shaking process from scratch.

• The Tool: They used a super-advanced mathematical method called Multiconfiguration Dirac-Fock. Imagine this as a physics engine that simulates every single electron bumping into every other electron, taking into account the rules of relativity (Einstein's speed limits).
• The Calculation: They calculated exactly how likely it is for an electron to be shaken up to a higher shelf or shaken off the house entirely, for both the "K-shell" (inner room) and "L-shell" (outer room) captures.
• The Result: They found that the shaking is much more violent and complex than previously thought. Specifically, when the atom captures an electron from the outer "L" shell, the remaining electrons shake much harder than when it captures from the inner "K" shell.

The "Ta" Factor: Why the Simulation Isn't Perfect

The paper makes a crucial distinction: their perfect simulation was done for an isolated atom floating in empty space. However, in the real experiment, the Beryllium atoms are embedded inside a block of Tantalum (Ta) metal (the sensor).

• The Analogy: Imagine simulating how a drum sounds in a vacuum, but then hitting it inside a crowded, noisy subway station. The metal walls of the sensor change how the electrons behave.
• The Discrepancy: The authors found that their perfect "vacuum" simulation didn't match the real "subway" data perfectly. The real peaks were wider and shifted. They suspect the metal sensor is distorting the electron waves, a phenomenon they call "matrix effects."

The Main Discovery: A Better Measurement

Even though the simulation didn't perfectly match the messy real-world data, it was good enough to fix a specific measurement that had been slightly wrong.

• The Old Value: Scientists previously thought that for every 100 times the atom captured an inner "K" electron, it captured an outer "L" electron 7 times (a ratio of 0.070).
• The New Value: Using their new, more accurate shake-models, they recalculated this ratio. They found the old models were underestimating the "L" captures. The new, more accurate ratio is 0.0756.

Why This Matters

This might sound like a tiny number, but in the world of hunting ghost particles, it's huge.

1. Clearer Signal: By understanding exactly how the "furniture" shakes, scientists can subtract the background noise more accurately. This makes the "ghost particle" signal stand out more clearly.
2. No False Alarms: The paper confirms that the complex shaking of electrons does not create fake signals that look like sterile neutrinos in the energy range scientists are looking for (60–108 eV). This gives them confidence that if they see a signal there, it's real.
3. Future Proofing: The authors admit their simulation is for isolated atoms. The next step is to figure out how to simulate the atoms inside the metal sensor to get even closer to reality.

In summary: This paper built a super-accurate computer model of how atoms "shake" when they decay. While the model showed that the real-world sensor material complicates things, the new math allowed scientists to correct a long-standing measurement error, giving them a sharper tool to hunt for the universe's missing ghost particles.

Technical Summary

Technical Summary: Shake-up and Shake-off Spectra in the Electron Capture Decay of Atomic 7Be

Problem Statement
The BeEST (Beryllium Electron capture in Superconducting Tunnel junctions) experiment seeks to set stringent limits on the existence of sub-MeV sterile neutrinos by measuring the recoil energy spectrum of $^7$Li following the electron capture (EC) decay of $^7$Be. The sensitivity of this search is currently limited by systematic uncertainties in modeling the electron shake-up (SU) and shake-off (SO) spectra. These effects arise from the sudden change in nuclear potential during EC, causing remaining atomic electrons to be excited (SU) or ionized (SO). Previous models relied on non-relativistic approximations and lacked state-of-the-art electron correlation effects, leading to inaccuracies in describing the spectral tails and peak broadening observed in experimental data. Furthermore, the L/K electron capture ratio, a critical parameter for both the sterile neutrino search and astrophysical $^7$Li production models, had been determined with significant systematic uncertainty due to these modeling limitations.

Methodology
The authors employed the multiconfiguration Dirac-Fock (MCDF) formalism within the ab initio MCDGME code to compute the atomic structure of $^7$Be and the resulting $^7$Li daughter atom and its ions. Key methodological features include:

• Relativistic and Correlated Wavefunctions: Calculations utilized the no-pair Hamiltonian, including Coulomb, Gaunt, and retardation interactions, as well as Quantum ElectroDynamics (QED) radiative corrections (self-energy and vacuum polarization).
• Configuration Interaction: To account for electron correlations, the basis space was augmented up to the 4s orbital using single and double excitations.
• Sudden Approximation (SA): SU and SO probabilities were calculated based on the overlap between the initial $^7$Be wavefunctions and the final $^7$Li (and ionized) wavefunctions. The formalism explicitly calculated single and double shake processes (both SU and SO) for K and L shell captures.
• Analytical Approximations: To facilitate fitting to experimental data, the numerically computed SO spectra were approximated using analytical functions (power laws for K-SO and log-normal distributions for L-SO) that matched the ab initio results with low reduced chi-squared values.

Key Results

1. Shake Probabilities: The calculations reveal that shake probabilities are significantly higher following L-shell capture compared to K-shell capture. Specifically, shake effects contribute to approximately 30% of all L-capture events.
2. Double Shake Processes: The study computed probabilities for double SU and double SO events. While these are currently below the detection threshold of the BeEST experiment (contributing <2.5% to L-capture and <0.2% to K-capture spectra), they are non-negligible and produce relative spectral differences of ~24% around 136 eV and ~12% near 225 eV.
3. Spectral Features:
   • Shake-up: The L-capture K-SU transition ($1s^2 2s \to 1s^2 2s^2$) produces a peak at the same energy as the K-GS peak, explaining the need for a secondary high-energy component in the K-GS fit.
   • Shake-off: The L-SO spectra extend to higher energies than predicted by previous Levinger-based models. The K-SO spectrum following L-capture is found to be non-negligible, contributing to the spectrum between 160 and 190 eV.
4. Comparison with Experiment: When compared to BeEST Phase-III data, the ab initio simulations explain some but not all spectral features. Discrepancies in peak centroids and widths are attributed to solid-state effects (matrix interactions with the Tantalum sensor) and potential detector non-linearities. Specifically, the Ta lattice environment likely modifies wavefunctions, affecting transition probabilities near ionization thresholds.
5. Revised L/K Capture Ratio: By incorporating the new, more accurate SU and SO models and removing background from Si substrate gamma interactions (via coincidence vetoing in Phase-III), the authors revised the L/K electron capture ratio for $^7$Be in Tantalum. The value was updated from 0.070(7) to 0.0756(20).

Significance and Claims
The paper claims that the new ab initio models provide a more rigorous theoretical foundation for analyzing the BeEST spectrum, reducing systematic uncertainties in the search for sterile neutrinos. The revised L/K ratio is a significant outcome, as it impacts calculations of $^7$Be decay in stellar environments relevant to cosmological $^7$Li production. The authors note that while their calculations are limited to isolated atoms, they successfully identify specific spectral components (such as the high-energy shoulder of the K-GS peak) that were previously unexplained. They emphasize that future work must incorporate matrix effects (e.g., via density functional theory) to fully resolve discrepancies between calculated and measured probabilities, particularly near ionization thresholds. The study concludes that no predicted spectral features mimic a sterile neutrino signal in the region of interest (60–108 eV), reinforcing the validity of the background modeling for the current search limits.