Theoretical calculations on half-lives of spontaneous one-proton radioactivity

Technical Summary: Theoretical Calculations on Half-Lives of Spontaneous One-Proton Radioactivity

Problem Statement
Research into unstable nuclei beyond the nucleon drip lines is essential for understanding nuclear interactions and structure in extreme systems. While the neutron drip line remains largely unexplored, the proton drip line has been the subject of extensive experimental and theoretical investigation. Proton radioactivity, a spontaneous decay process occurring in nuclei with negative one-proton separation energies, serves as a probe for open quantum systems, shell evolution, and pairing correlations. Despite the availability of experimental data for various proton emitters (ranging from $^{53}\text{Co}^m$ to $^{185}\text{Bi}$), a comprehensive theoretical framework that systematically incorporates nuclear deformation and evaluates the sensitivity of half-life calculations to specific nuclear inputs (decay energy and spectroscopic factors) is required. Furthermore, there is a need for theoretical predictions for lighter proton emitters in the $fpg$ shell ($30 < Z < 50$) to guide future experimental searches.

Methodology
The authors employ a quantum tunneling model based on the microscopic Gamow state theory to calculate proton emission half-lives. The core of the methodology involves constructing a two-body potential between the emitted proton and the daughter nucleus, explicitly accounting for nuclear deformation.

1. Potential Construction: The interaction potential combines a deformed Woods-Saxon nuclear potential, a Thomas-form spin-orbit potential, and an expanded Coulomb potential.

   • Nuclear Potential: The Woods-Saxon and spin-orbit terms utilize radius parameters that depend on the polar angle $\theta$ via a multipole expansion involving quadrupole ($\beta_2$) and hexadecapole ($\beta_4$) deformation parameters. An isospin-dependent radius formula is used for the Woods-Saxon potential to account for the strong isospin asymmetry of drip-line nuclei.
   • Coulomb Potential: For deformed nuclei, the Coulomb potential is calculated using a multipole expansion method rather than a simple spherical approximation.
   • Gamow States: The emitted proton is treated as a Gamow state satisfying the time-independent Schrödinger equation with outgoing Coulomb wavefunction boundary conditions at large distances. The potential depth ($V_0$) is adjusted to reproduce the experimental decay energy ($Q$-value).

2. Half-Life Calculation:

   • The decay width $\Gamma(\theta)$ is calculated using the amplitude of the Gamow-state wavefunction in the exterior region, offering a more microscopic description than semiclassical WKB approximations.
   • The total decay width is obtained by averaging the partial widths over all azimuthal angles.
   • The half-life ($T_{1/2}$) is derived from the relation $T_{1/2} = \hbar \ln 2 / (S_p \Gamma)$, where $S_p$ is the spectroscopic factor.

3. Data Sources:

   • Decay Energies: Calculations utilize both experimental one-proton separation energies (from NNDC and AME 2020) and theoretical values from the Finite-Range Droplet Model (FRDM).
   • Spectroscopic Factors: Two sets of factors are employed: those derived from Relativistic Mean Field (RMF) theory and those obtained from deformation-dependent fitting formulas.
   • Deformation Parameters: Quadrupole and hexadecapole parameters are sourced from FRDM calculations.

Key Contributions

• Systematic Deformed Model: The study extends the quantum tunneling model to include quadrupole and hexadecapole deformations for both parent and daughter nuclei, providing a microscopic calculation of proton emission half-lives under deformation.
• Comprehensive Dataset: The authors compile a unified dataset of half-lives for experimentally confirmed proton emitters ($50 < Z < 84$) and theoretically predicted emitters ($30 < Z < 50$).
• Sensitivity Analysis: The work quantitatively analyzes the dependence of calculated half-lives on the choice of decay energy (experimental vs. FRDM) and spectroscopic factors (RMF vs. fitted), demonstrating the critical role of experimental decay energies in achieving accuracy.
• Predictions for Light Nuclei: The paper provides theoretical predictions for proton emitters in the $fpg$ shell ($Z=30$ to $50$), including a specific prediction for $^{89}\text{Rh}$, a nucleus already experimentally identified but with unmeasured half-life details in this context.

Results

• Validation: The model was validated using $^{151}\text{Lu}$ as a representative case. By incorporating deformation ($\beta_2 = -0.167, \beta_4 = -0.035$) and an RMF spectroscopic factor, the calculated half-life (70.2 ms) showed reasonable agreement with the experimental value (127 ms).
• Accuracy and RMS Deviations:
  • When using experimental decay energies ($Q_{\text{exp}}$) and RMF spectroscopic factors ($S_p^{\text{RMF}}$), the root-mean-square (RMS) deviation between calculated and experimental logarithmic half-lives is $\sigma = 0.46$.
  • Using FRDM decay energies significantly increases the deviation ($\sigma \approx 4.90$), highlighting the sensitivity of the model to decay energy inputs.
  • The model's performance is comparable to other theoretical approaches, such as an analytical semi-classical method and an extended Geiger-Nuttall law.
• Lighter Proton Emitters ($Z < 50$):
  • For $^{89}\text{Rh}$, the model predicts half-lives of 274 ns and 972 ns using RMF and fitted spectroscopic factors, respectively.
  • Predictions for other nuclei in the $fpg$ shell (e.g., $^{60}\text{Ga}$, $^{63}\text{As}$, $^{72}\text{Rb}$) indicate generally shorter half-lives (ranging from nanoseconds to microseconds) compared to heavier emitters.
  • The results confirm that for a fixed angular momentum, the half-life decreases as decay energy increases, consistent with the Geiger-Nuttall law ($\log_{10} T_{1/2} = a Q^{-1/2} + b$).

Significance
The paper claims that its primary significance lies in providing a "useful reference for future experimental investigations related to the proton drip line." By compiling a comprehensive dataset of calculated and predicted half-lives for nuclei from $^{60}\text{Ga}$ to $^{185}\text{Bi}$, the authors aim to assist experimentalists in identifying and characterizing proton-rich nuclei. The study underscores that while theoretical models can achieve precise predictions when using experimental decay energies, the extension of these methods to lighter, unobserved nuclei offers critical guidance for experiments utilizing advanced techniques (such as in-flight decay measurements) capable of probing nanosecond to picosecond timescales. The dataset is made openly available to facilitate further research in this domain.