⚛️ nuclear theory

Relativistic dynamical effects in proton emission: the Wentzel-Kramers-Brillouin method for 1+1 dimensional Dirac equation

This paper employs the WKB approximation on the 1+1 dimensional Dirac equation to derive a corrected relativistic penetration probability using an effective potential that systematically increases predicted proton emission half-lives, with effects reaching up to 84% for high orbital angular momentum cases like $^{144}\mathrm{Tm}$ .

Original authors: Guangping Chen, Wenmin Deng, Ganlong Ding, Sibo Wang, Jing Peng, Haozhao Liang

Published 2026-08-14

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CC BY 4.0

Original authors: Guangping Chen, Wenmin Deng, Ganlong Ding, Sibo Wang, Jing Peng, Haozhao Liang

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, crowded dance floor where protons and neutrons are constantly jostling. Sometimes, a proton gets too excited or finds itself on the edge of the crowd and decides to make a run for it, escaping the nucleus entirely. This dramatic exit is called proton radioactivity. It's a rare event, happening only in very specific, unstable atoms that have too many protons to hold together comfortably.

To understand how a proton escapes, physicists treat it like a ghost trying to walk through a wall. In the quantum world, particles don't just bounce off barriers; they can sometimes "tunnel" right through them, a phenomenon known as quantum tunneling. The speed at which this happens depends on two main things: how often the proton bumps against the wall (the "assault frequency") and how likely it is to actually pass through (the "penetration probability"). For decades, scientists have used a set of mathematical rules called the WKB approximation to calculate these odds. However, there's been a lingering debate about exactly which "map" to use when drawing the wall the proton is trying to cross. Should we use a simple, straightforward map, or a more complex one that accounts for the weird, high-speed rules of relativity?

This paper, titled "Relativistic dynamical effects in proton emission," dives into that debate by taking a fresh look at the math. The authors, a team of physicists from China and Japan, start with the Dirac equation, which is the gold standard for describing particles moving at speeds where Einstein's theory of relativity matters. They apply the WKB method to this equation to derive a new, more accurate formula for the "wall" the proton faces.

Here is the big discovery: The paper argues that the old, simple way of calculating the barrier—just adding up two types of forces (scalar and vector potentials) like you would add apples and oranges—is actually incorrect for this specific high-speed scenario. Instead, the authors show that the correct "effective potential" is a more complicated recipe involving squares and fractions of those forces. They call this new map $U_{eff}$ .

When the team ran the numbers using this new, more complex map, they found some surprising results. First, the "wall" the proton has to tunnel through becomes effectively higher and wider in the middle. This makes it harder for the proton to escape, which means the penetration probability drops. Second, the rate at which the proton hits the wall (the assault frequency) also changes, often slowing down.

The most exciting part is how much this matters. The authors found that for most protons, using the new map increases the predicted time it takes for the atom to decay (its half-life). In some cases, the difference is massive. For a specific isotope called $^{144}\text{Tm}$ , the new calculation suggests the atom lives about 84% longer than the old method predicted. This effect gets even stronger if the escaping proton is spinning fast (has high orbital angular momentum).

The paper doesn't just suggest this is a small tweak; it demonstrates that the old, simple method systematically underestimates how long these atoms should last. By correcting the math to be consistent with relativity, the authors provide a more reliable way to predict how these unstable atoms behave. This isn't just about fixing a number; it's about ensuring our understanding of the quantum world is built on a solid, self-consistent foundation, especially when particles are moving fast enough that the rules of the universe start to get a little strange.

Technical Summary: Relativistic Dynamical Effects in Proton Emission via the WKB Method for the 1+1 Dimensional Dirac Equation

Problem Statement
Proton radioactivity, the spontaneous emission of protons from nuclei beyond the proton drip line, is a critical process for validating nuclear models and investigating nuclear structure. In practical half-life calculations, the penetration probability through the potential barrier is a highly sensitive quantity, typically evaluated using the Wentzel–Kramers–Brillouin (WKB) approximation. While relativistic mean-field (RMF) theories describe nucleon dynamics using the Dirac equation with scalar ( $S$ ) and vector ( $V$ ) potentials, previous practical calculations often approximated the nuclear barrier simply as the sum $S(r) + V(r)$ . This choice, while intuitively linked to the large component of the Dirac spinor, lacks theoretical consistency when applied to the WKB formalism, which is originally derived from the Schrödinger equation. The paper addresses the ambiguity regarding which potential form should consistently enter the semiclassical exponent in a relativistic WKB treatment of proton emission.

Methodology
The authors derive the relativistic penetration probability by applying the WKB approximation directly to the 1+1 dimensional (one spatial, one temporal) Dirac equation.

Derivation of the Effective Potential: Starting with the 1+1 Dirac equation containing scalar $S(x)$ and vector $V(x)$ potentials, the authors reduce the coupled first-order equations to a second-order differential equation for the upper spinor component. By eliminating the first-order derivative term via a specific transformation, they obtain a Schrödinger-like equation.
WKB Expansion: Applying the standard WKB ansatz to this equation, they derive the zeroth-order phase function. This process reveals that the classical momentum $p(x)$ is determined not by $S+V$ , but by a Schrödinger-equivalent potential:
$U_{\text{eff}}(r) = S(r) + \frac{E}{m}V(r) + \frac{S^2(r) - V^2(r)}{2m}$
where $E$ is the single-particle energy and $m$ is the proton mass.
Connection Formulas: The authors establish connection formulas to match wave functions across classically allowed and forbidden regions, deriving the transmission probability $T$ . They also discuss the assault frequency $\nu$ , utilizing a quasi-classical normalization condition that avoids the inapplicability of bound-state quantization conditions for decay energies.
Numerical Application: The framework is extended to the radial coordinate $r$ to calculate proton emission properties. The authors compare results using the conventional potential $V_N = S(r) + V(r)$ against the derived $U_{\text{eff}}(r)$ . The scalar and vector potentials are parameterized using Woods-Saxon forms based on established nuclear parameters. Calculations are performed for a range of proton emitters, with a focus on $^{144}\text{Tm}$ , $^{155}\text{Ta}$ , and $^{170}\text{Au}$ to illustrate varying magnitudes of relativistic effects.

Key Contributions

Theoretical Consistency: The paper rigorously demonstrates that the semiclassical momentum in a relativistic WKB treatment must be governed by $U_{\text{eff}}(r)$ rather than the simple sum $S(r) + V(r)$ .
Quantification of Dynamical Effects: The study quantifies the impact of this potential choice on three key observables: penetration probability ( $P$ ), assault frequency ( $\nu$ ), and half-life ( $T_{1/2}$ ).
Dependence on Angular Momentum: The analysis reveals that relativistic dynamical effects are not uniform but depend significantly on the orbital angular momentum ( $l$ ) of the emitted proton.

Results

Penetration Probability: Using $U_{\text{eff}}(r)$ systematically reduces the penetration probability compared to using $S(r) + V(r)$ . This reduction occurs because $U_{\text{eff}}$ modifies the inner barrier, shifting the second classical turning point ( $r_2$ ) inward, thereby increasing the WKB action integral. The reduction is generally weakly dependent on decay energy ( $Q_p$ ) but becomes more pronounced with higher orbital angular momentum. For $l=6$ in $^{144}\text{Tm}$ , the ratio $P(U_{\text{eff}})/P(S+V)$ drops to approximately 0.794.
Assault Frequency: The assault frequency also exhibits relativistic dynamical effects. In most cases, $U_{\text{eff}}$ leads to a reduction in $\nu$ . However, the behavior is non-monotonic with respect to $l$ ; for certain nuclei (e.g., $^{144}\text{Tm}$ at $l=4$ ), the frequency is enhanced. This oscillatory behavior is attributed to the detailed phase structure of the wave function in the inner region, specifically the cosine factor in the normalization condition, which cannot be approximated by its average value.
Half-Life: The combined effect of reduced penetration probability and generally reduced assault frequency results in a systematic increase in predicted half-lives when $U_{\text{eff}}(r)$ is used. The magnitude of this effect varies with mass number and $l$ . For ground-state emitters, the half-life ratio $T_{1/2}(U_{\text{eff}})/T_{1/2}(S+V)$ ranges from 1.2 to 1.4 for low $l$ , reaching up to 1.84 for $^{144}\text{Tm}$ ( $l=5$ ), corresponding to an 84% increase. The effect is most pronounced for lighter nuclei and higher orbital angular momenta.

Significance
The paper establishes a benchmark for improving the self-consistency of proton radioactivity descriptions within relativistic WKB frameworks. By identifying $U_{\text{eff}}(r)$ as the correct effective potential, the work corrects a common approximation in previous studies. The authors note that while the 1+1 dimensional framework captures the core relativistic effects, it serves as a necessary starting point for extending such analyses to the full 3+1 dimensional Dirac equation. The findings suggest that relativistic dynamical corrections are essential for accurate predictions, particularly for high- $l$ proton emission processes.

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