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Effect of the near-proton-emission threshold resonance in $^{11}$ B on the branching ratio of beta-delayed proton emission from $^{11}$ Be

This study employs Skyrme Hartree-Fock calculations within a potential model to demonstrate that the branching ratio of beta-delayed proton emission from $^{11}$ Be is highly sensitive to the precise energy of a narrow resonance near the proton-emission threshold in $^{11}$ B, reaching up to $10^{-5}$ and necessitating experimental confirmation of whether this resonance lies below 200 keV.

Original authors: Nguyen Le Anh, Bui Minh Loc

Published 2026-07-20

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CC BY 4.0

Original authors: Nguyen Le Anh, Bui Minh Loc

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the atomic nucleus not as a solid marble, but as a bustling, chaotic dance floor where protons and neutrons hold hands in a tight, energetic waltz. Usually, these particles stick together so tightly that they never let go. But sometimes, in the exotic corners of the universe where atoms have weird ratios of protons to neutrons, the dance gets a little sloppy. A particle might get pushed to the very edge of the floor, barely holding on, forming a "halo" that pokes out into space. This is the world of nuclear physics, a field dedicated to understanding the tiny, powerful forces that hold matter together.

One of the most fascinating moves on this dance floor is called "beta-delayed proton emission." It's a bit of a mouthful, but here's the gist: an unstable atom (let's call it the "dancer") suddenly decides to change its identity by spitting out a tiny electron (a beta particle). This change leaves the remaining nucleus in a state of high excitement. Usually, this excitement just cools down by releasing energy as light or heat. But in rare, special cases, the nucleus is so excited that it kicks out a proton (a positively charged particle) instead. It's like the dancer, after spinning wildly, accidentally flings a shoe across the room. Scientists have been puzzled by a specific dancer, an atom called 11Be, which seems to fling its proton shoe much more often than anyone expected. The big question is: why is this happening so frequently?

This paper dives into that mystery, acting like a detective trying to figure out exactly where the "shoe" is being thrown from. The researchers, Nguyen Le Anh and Bui Minh Loc, used a sophisticated computer model called the "Skyrme Hartree-Fock" method. Think of this model as a super-accurate simulation of the nuclear dance floor, calculating how every particle interacts with every other particle. They focused on a specific "resonance"—a special, narrow energy level in the nucleus of the daughter atom (11B) that acts like a perfect trap or a sweet spot for the proton to escape through.

The team found that the answer to the mystery lies in the precise location of this trap. They discovered that the branching ratio (the fancy term for "how often the proton shoe gets thrown") is incredibly sensitive to where this energy trap sits. If the trap is located just a tiny bit lower in energy—specifically below 200 keV—the probability of the proton escaping skyrockets. In their simulations, when they tuned the resonance to sit at 182 keV (a value based on the average of current experimental data), the calculated branching ratio came out to be 8.98 × 10⁻⁶. This number is huge in the world of nuclear physics and matches what some experiments have seen.

However, the paper also highlights a major "what if." If the resonance were just a little bit higher, say above 200 keV (around 217 keV), the probability would drop dramatically to about 2.2 × 10⁻⁶. This explains why different experiments have reported different numbers; they might be measuring slightly different energy levels. The authors suggest that the "sweet spot" for this high probability is a very narrow window, less than 300 keV above the proton-emission threshold. They also checked if other factors, like the width of the resonance (how "fuzzy" the energy level is) or the presence of other decay paths (like alpha particles), would change the result. They found that while these factors matter, the exact energy position of the resonance is the boss; it has a much stronger effect on the outcome than the width of the trap.

Ultimately, the paper doesn't claim to have found the single, final, perfect answer to the puzzle. Instead, it provides a powerful tool and a clear rule: the branching ratio is a mirror that reflects the exact energy of the resonance. If the resonance is below 200 keV, the proton emission is frequent; if it's above, it's rare. The authors conclude that to solve the mystery once and for all, we need better experiments to pinpoint that resonance energy with extreme precision. Until then, their calculations suggest that the high probability of this rare event is real, provided the nuclear dance floor has that specific, low-lying sweet spot.

Technical Summary: Effect of the Near-Proton-Emission Threshold Resonance in $^{11}$ B on the Branching Ratio of Beta-Delayed Proton Emission from $^{11}$ Be

Problem Statement
The beta-delayed proton emission from the neutron-halo nucleus $^{11}$ Be is a rare decay process ( $^{11}\text{Be} \to {}^{10}\text{Be} + p + e^- + \bar{\nu}_e$ ) characterized by an unexpectedly high branching ratio. Experimental observations indicate a branching ratio around $8.6 \times 10^{-6}$ , significantly higher than many theoretical predictions. This enhancement is attributed to the existence of a narrow $1/2^+$ resonance in the daughter nucleus $^{11}$ B located near the proton-emission threshold. However, the precise energy of this resonance remains uncertain, with experimental reports varying between 171 keV, 197 keV, and 211 keV. This uncertainty creates a discrepancy between experimental upper limits (e.g., $2.2 \times 10^{-6}$ ) and direct measurements. Furthermore, theoretical models, including shell model and ab initio calculations, have struggled to accurately reproduce the resonance energy without phenomenological adjustments, often predicting states far above the threshold. The core problem is quantifying how sensitive the branching ratio is to the exact position of this near-threshold resonance within a self-consistent microscopic framework.

Methodology
The authors employ a potential model based on the Skyrme Hartree-Fock (SHF) formalism to calculate the branching ratio. The approach involves the following key steps:

Mean-Field Potentials: Single-particle potentials for the halo neutron (in $^{11}$ Be) and the emitting proton (in $^{11}$ B) are derived simultaneously using the SHF formalism. The study utilizes four Skyrme interactions: SkM*, SAMi-ISB, SLy4, and SAMi.
Calibration: Two global scaling parameters ( $N_i$ and $N_f$ ) are applied to the potentials. $N_i$ is tuned to reproduce the neutron separation energy ( $S_n = 501$ keV) of the $^{11}$ Be halo state. $N_f$ is adjusted to constrain the proton resonance energy ( $E_R$ ) to specific experimental values (171 keV, 182 keV, and 211 keV). This allows for a systematic investigation of the resonance's impact without imposing the state artificially.
Decay Calculation: The branching ratio ( $b_{\beta p}$ ) is calculated by integrating the differential decay probability over the available energy window ( $Q = 281$ keV). The calculation includes Fermi and Gamow-Teller transitions, utilizing a spectroscopic factor ( $S_F = 0.51 \pm 0.06$ ) for the neutron halo configuration.
Simplifications: The analysis is restricted to a single-channel calculation (proton emission), omitting the alpha channel ( $^7$ Li + $\alpha$ ). The authors justify this by noting the poor constraints on alpha-nucleus optical potentials at low energies and the predominantly single-particle nature of the resonance.

Key Contributions

Microscopic Interpretation: The paper provides a microscopic mean-field interpretation of the near-threshold resonance, demonstrating that the emergence of an s-wave state close to the proton-emission threshold is a natural consequence of the Skyrme mean field, requiring only minimal renormalization.
Quantification of Sensitivity: The study rigorously quantifies the correlation between the resonance energy and the branching ratio, showing that the branching ratio is extremely sensitive to resonance positions within a narrow energy window (0 to 281 keV).
Interaction Independence: By testing multiple Skyrme forces, the authors demonstrate that once the resonance energy is fixed, the calculated branching ratio is largely independent of the specific choice of nuclear interaction, suggesting the result is a robust structural feature rather than a model artifact.

Results

Resonance Sensitivity: The branching ratio exhibits rapid variation with small shifts in resonance energy.
- For a resonance at 171 keV, the branching ratio is approximately $1.3 \times 10^{-5}$ .
- For a resonance at 182 keV (weighted average of experimental data), the branching ratio is calculated as $8.98 \times 10^{-6}$ . This value aligns with the experimental measurement of $8.6 \times 10^{-6}$ from Ref. [9].
- For a resonance at 211 keV, the branching ratio drops to approximately $2.9 \times 10^{-6}$ .
- A resonance at 217 keV yields a branching ratio of $2.2 \times 10^{-6}$ , consistent with the upper limits reported in Refs. [8, 16].
Width Dependence: The study investigates the effect of the resonance width ( $\Gamma_p$ ). Calculations varying the width from 4.7 keV to 12.8 keV show that the branching ratio changes only slightly (from $1.29 \times 10^{-5}$ to $1.22 \times 10^{-5}$ ). This confirms that the branching ratio is considerably more sensitive to the resonance energy than to moderate variations in the resonance width.
Spectroscopic Factor Impact: Using a higher spectroscopic factor ( $S_F = 0.71$ ) increases the branching ratio to the order of $10^{-5}$ , while the adopted value ( $S_F = 0.51$ ) yields the $8.98 \times 10^{-6}$ result.

Significance and Claims
The paper claims to strengthen the link between low-energy proton elastic scattering (a strong interaction process) and beta-delayed proton emission (a weak interaction process). The primary significance lies in demonstrating that small shifts in resonance energy (tens of keV) can induce order-of-magnitude variations in weak-decay observables. The authors posit that their results provide a quantitative benchmark for future measurements, offering a direct criterion to assess the consistency between elastic scattering data and beta-decay observables.

The paper concludes that the "puzzle" of the branching ratio's magnitude can be resolved by precisely determining the resonance energy. Specifically, an experimental confirmation that the resonance lies below 200 keV is crucial to explaining the higher branching ratios, while a location above 200 keV supports the lower upper limits. The work emphasizes that the strong sensitivity of the branching ratio to the resonance position reflects a structural mechanism associated with near-threshold single-particle states.

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