Proton radioactivity in deformed nuclei with microscopic optical potential: A novel angular-dependent emission mechanism in the nanosecond-lived $^{149}$Lu

This paper presents a novel theoretical framework combining deformed microscopic optical potentials with angular-dependent emission mechanisms to accurately predict the half-lives of oblate deformed proton emitters like $^{149}$Lu, revealing unprecedented angular emission phenomena and validating the model's predictive power for exotic nuclear decays.

The Gist

Imagine an atom as a tiny, crowded dance floor. Usually, the dancers (protons and neutrons) hold hands tightly, staying in a stable circle. But sometimes, in very strange, "overcrowded" atoms, a proton gets pushed so hard that it tries to escape the dance floor entirely. This escape is called proton radioactivity.

This paper is about a specific, very short-lived dancer named Lutetium-149 (149Lu). Scientists have known this atom exists for a while, but they couldn't quite explain how it escapes or how long it lasts before vanishing. The authors of this paper built a new, more accurate map to solve the mystery.

Here is the breakdown of their discovery, using simple analogies:

1. The Old Map vs. The New GPS

Previously, scientists tried to predict how long 149Lu would last using "old maps." These maps were based on rough guesses and simplified rules that worked well for normal, round atoms but failed for weird, squashed ones.

The authors created a new GPS system called a "microscopic optical potential."

• The Analogy: Imagine trying to walk through a forest. The old maps just said, "The trees are thick here." The new GPS actually counts every single tree, measures the distance between them, and calculates exactly how hard it is to push through the branches.
• The Result: This new map is built from the fundamental rules of how particles interact (the "real" physics), rather than just guessing based on how other atoms behave.

2. The Squashed Ball and the "Dead Zones"

Most atoms are like perfect spheres (like a basketball). But 149Lu is oblate, meaning it's squashed flat like a pancake or a hamburger bun.

The authors discovered something completely new because of this shape: The "Dead Zones."

• The Analogy: Imagine a round trampoline (a normal atom). If you jump off, you can launch in any direction. But now, imagine a trampoline that is squashed flat. If you try to jump off the very top or bottom (the "poles" of the pancake), the surface is so steep and the barrier so high that you literally cannot get off. You are stuck.
• The Discovery: For 149Lu, the authors found that if the proton tries to escape at a steep angle (near the "top" or "bottom" of the squashed nucleus), the path is completely blocked. The proton cannot escape in those directions. It can only escape from the "sides" (the equator).
• Why it matters: Previous theories missed this. They thought the proton could escape anywhere. The authors showed that the shape of the atom actually shuts down escape routes at small angles.

3. The "Bounce" and the Escape Time

To figure out how long the atom lasts (its "half-life"), you need to know two things:

1. How hard is the wall? (The barrier the proton must tunnel through).
2. How often does the proton hit the wall? (The "assault frequency").

The authors used a clever trick to figure out the second part.

• The Analogy: Imagine a ball bouncing inside a bowl. If the bowl is deep and narrow, the ball bounces very fast. If it's wide and shallow, it bounces slowly. The authors looked at the shape of the energy "bowl" holding the proton and used a new method (inspired by a simple spring) to calculate exactly how fast the proton was bouncing against the wall before it escaped.

4. The Perfect Match

When they ran the numbers with their new "GPS" and "Bounce Calculator":

• The Prediction: They calculated that 149Lu should last about 467 nanoseconds (a billionth of a second).
• The Reality: Experiments had measured it at about 450 nanoseconds.
• The Verdict: This is an incredible match. Their new method worked perfectly, whereas the old "rough guess" methods were way off.

5. What They Did Next

Because their new method worked so well for 149Lu, they used it to check its neighbors:

• 150Lu and 151Lu: They predicted how long these atoms last, and the numbers matched the experiments perfectly.
• 148Lu: They predicted a brand-new atom (148Lu) that hasn't been measured yet. They think it will be even shorter-lived (about 4.4 nanoseconds), making it the fastest-decaying proton emitter ever known.

Summary

The paper claims that by using a highly detailed, fundamental physics map (the microscopic optical potential) and accounting for the fact that this atom is squashed like a pancake, they discovered a new rule: Protons in squashed atoms cannot escape from the poles.

This new understanding allows them to predict exactly how long these exotic atoms will live, solving a puzzle that had stumped scientists for years. They didn't just guess; they built a model that explains the "why" and "how" of the atom's escape, proving that the shape of the nucleus is the key to unlocking its secrets.

Technical Summary

Technical Summary: Proton Radioactivity in Deformed Nuclei with Microscopic Optical Potential

Problem Statement
Proton radioactivity serves as a critical probe for nuclear structure and dynamics beyond the drip line. However, predictive models for decay properties in deformed nuclei face significant challenges due to the interplay of nuclear deformation and continuum coupling. Specifically, the nucleus $^{149}$Lu, identified as a ground-state proton emitter with a record-high decay energy ($Q_p = 1920$ keV) and a nanosecond-scale half-life ($T_{1/2} \approx 450$ ns), presents a theoretical puzzle. Previous studies utilizing nuclear energy density functionals (EDF) and phenomenological particle-rotor models have yielded conflicting results regarding its deformation (oblate vs. prolate) and decay mechanisms. A critical limitation in prior theoretical treatments is the reliance on approximations that omit the quadratic contribution inherent in the Schrödinger equivalent potential ($U_{SEP}$), artificially deepening the potential barrier and obscuring deformation-sensitive emission dynamics. Furthermore, existing models often lack a description based on realistic nucleon-nucleon (NN) interactions, which is essential for accurately describing such exotic systems.

Methodology
The authors present a theoretical framework that integrates three primary components to describe proton emission in deformed nuclei:

1. Deformed Microscopic Optical Potential (RBOM): Instead of phenomenological interactions, the nuclear potential ($V_N$) is derived from the real part of a microscopic nucleon-nucleus optical potential constructed via the Relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space. This potential is adapted for deformed nuclei using an improved Local Density Approximation (LDA) in two-dimensional space, where the Schrödinger equivalent potential ($U_{SEP}$) is explicitly angular-dependent. The nuclear densities are calculated using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the PC-PK1 density functional.
2. WKB Penetration Probability: The tunneling probability ($P$) is calculated using the Wentzel-Kramers-Brillouin (WKB) approximation. Crucially, the calculation accounts for the angular dependence of the potential barrier $V(r, \theta)$, integrating over orientations to obtain the total penetration probability.
3. Harmonic-Oscillator-Inspired Assault Frequency: A new scheme is proposed to estimate the assault frequency ($\nu_0$) of the emitted proton. By approximating the potential barrier near its minimum as a harmonic oscillator, the frequency $\omega(\theta)$ is extracted, and $\nu_0(\theta)$ is defined as $\omega(\theta)/(2\pi)$.

Key Contributions and Results

• Novel Angular-Dependent Mechanism: The study predicts a phenomenon unprecedented in spherical proton emitters: the complete suppression of classically allowed regions at small polar angles ($\theta \le 21^\circ$) for oblate deformed nuclei. In these regions, the potential barrier minimum exceeds the decay energy $Q_p$, causing the inner and outer turning points to vanish and resulting in zero penetration probability ($P_\theta = 0$). This contrasts with spherical nuclei where the barrier is rotationally invariant.
• Accurate Half-Life Prediction for $^{149}$Lu: The framework yields a predicted half-life of $T_{1/2} = 467^{+143}_{-108}$ ns for $^{149}$Lu. This is in excellent agreement with the experimental value of $450^{+170}_{-100}$ ns. The calculation utilizes $Q_p = 1920$ keV and the Bonn A NN interaction.
• Deformation Constraints: Rigorous deformation analysis indicates that configurations with $|\beta_2| \ge 0.32$ are excluded by the experimental half-life data. The ground-state deformation of $\beta_2 = -0.168$ (oblate) is consistent with the data, matching the central experimental value within 4%.
• Comparison with Previous Models: The paper demonstrates that using the full Schrödinger equivalent potential ($U_{SEP}$) is superior to the linear approximation ($U_S + U_0$) often used in EDF studies. The linear approximation artificially deepens the barrier by approximately 20 MeV, failing to reproduce the angular-dependent suppression and yielding incorrect half-life trends.
• Systematic Validation: The model is extended to $^{150}$Lu, $^{151}$Lu, and their isomers, achieving excellent agreement with experimental half-lives across four orders of magnitude. Additionally, the framework predicts $^{148}$Lu as a highly oblate ($\beta_2 = -0.166$) proton emitter with a half-life of $4.42$ ns, which would be the shortest-lived proton emitter known if confirmed.

Significance
The paper claims to validate deformed microscopic optical potentials derived from realistic NN interactions as a robust predictive tool for drip-line proton emitters. By moving beyond phenomenological interactions and incorporating the full relativistic structure of the potential (including quadratic terms), the work provides quantitative evidence for the role of deformation in exotic decays. The identification of the angular-dependent emission mechanism offers a new physical perspective on proton radioactivity, suggesting that future experimental analyses, such as those involving angular anisotropy in polarized nuclei, could provide critical insights into these underlying dynamics. The work underscores the necessity of realistic microscopic approaches to resolve discrepancies in the theoretical description of the shortest directly measured proton half-lives.