Constraining cross sections for unstable $^{153,159}$Gd$(n,γ)$ and their astrophysical implications

This study proposes a Bayesian-optimized approach to constrain $\gamma$-ray strength functions and nuclear level densities, successfully inferring $(n,\gamma)$ cross sections for unstable $^{153,159}$Gd isotopes with significantly reduced uncertainty and revealing that the resulting enhanced reaction rates substantially increase $^{160}$Gd abundance in $s$-process nucleosynthesis simulations.

The Gist

Imagine the universe as a giant, cosmic kitchen where stars are the chefs. To make the heavy elements that make up our world (like the gold in your jewelry or the gadolinium in your MRI machine), these chefs use a special recipe called neutron capture. They slowly add neutrons to atoms, one by one, building heavier and heavier elements.

However, there's a problem. Sometimes, the recipe gets tricky. At certain steps, the atom is unstable and wants to fall apart (decay) before the chef can add the next ingredient. This is called a "branching point." To know exactly how much of the final dish gets made, we need to know the speed at which the chef adds the next neutron. This speed is called the "cross-section."

The problem is, for some unstable ingredients (like specific types of Gadolinium), we can't go into the kitchen and measure this speed directly. The ingredients are too radioactive and short-lived. So, scientists have to guess using computer models. But until now, these guesses have been like trying to bake a cake without a recipe—sometimes the result is a cake, sometimes a brick. The uncertainty was huge (about 167% off!).

The Solution: A New Way to Guess

The team in this paper, led by Zhang and Li, came up with a clever way to make a much better guess. Instead of just guessing the final speed, they decided to fix the fundamental rules of the kitchen first.

Think of the atom like a complex machine with two main dials:

1. The Gamma-Ray Strength Function (The "Flashlight"): When the atom catches a neutron, it gets excited and needs to calm down by flashing light (gamma rays). This dial controls how bright and how many flashes it emits.
2. The Nuclear Level Density (The "Staircase"): Imagine the atom has a staircase of energy levels. This dial controls how many steps are on the staircase and how crowded they are.

The Analogy:
Imagine you are trying to predict how fast a car will drive down a hill, but you can't test the car because it's broken.

• Old Method: You just guess the speed based on the hill's shape. You might be way off.
• This Paper's Method: You first measure the car's engine power (the "Flashlight") and the friction of its tires (the "Staircase") using data from similar, working cars (stable Gadolinium). Once you know exactly how the engine and tires work, you can calculate the speed of the broken car with much higher confidence.

What They Did

1. Calibrated the Dials: They took data from stable Gadolinium isotopes (the "working cars") to fine-tune the "Flashlight" and "Staircase" settings.
2. Applied the Rules: They used these fine-tuned settings to calculate the speed for the unstable Gadolinium isotopes (153 and 159) that we can't measure directly.
3. The Result: They reduced the uncertainty from a wild 167% down to a manageable 30%. That's like going from guessing a cake will weigh between 1 and 5 pounds to knowing it weighs between 2.8 and 3.2 pounds.

Why Does This Matter? (The Cosmic Impact)

Why do we care about Gadolinium? Because it affects the "branching points" in the cosmic recipe.

• The 159Gd Problem: One specific unstable atom, 159Gd, is a fork in the road. It can either catch a neutron and become heavier (160Gd) or decay into something else.
• The Discovery: The team found that the "Flashlight" settings make the neutron-catch happen almost 3 times faster than previous databases thought.
• The Consequence: Because the catch is faster, more atoms successfully make the turn to become 160Gd. Their simulations show that the universe might actually produce twice as much 160Gd as we previously thought.

The Bottom Line

This paper is like upgrading the GPS for cosmic chefs. By fixing the underlying physics rules (the engine and tires), they provided a much more accurate map for how heavy elements are forged in stars.

• For Astronomers: It means our models of how the universe creates elements are now more accurate.
• For Engineers: Gadolinium is used in nuclear reactors and medical therapies. Having better data on how it interacts with neutrons helps design safer reactors and more effective cancer treatments.

In short, they turned a blurry, shaky guess into a sharp, reliable prediction, helping us understand the recipe of the universe a little better.

Technical Summary

1. Problem Statement

Neutron capture $(n, \gamma)$ cross sections for Gadolinium (Gd) isotopes are critical for nuclear reactor design, medical applications (e.g., Gadolinium Neutron Capture Therapy), and astrophysical $s$-process nucleosynthesis models.

• The Gap: While data for stable Gd isotopes ($^{155}$Gd, $^{157}$Gd) are available, experimental data for the crucial unstable isotopes $^{153}$Gd and $^{159}$Gd are scarce. Direct measurement is hindered by the high radioactivity of targets and the lack of free-neutron targets for inverse kinematics experiments.
• The Challenge: In the absence of data, stellar models rely on theoretical predictions (e.g., TALYS). However, these predictions suffer from massive uncertainties (up to several orders of magnitude) due to poorly constrained $\gamma$-ray strength functions ($\gamma$SF) and nuclear level densities (NLD). These uncertainties propagate directly into astrophysical reaction rates, limiting the reliability of nucleosynthesis simulations, particularly for "branching points" where neutron capture competes with $\beta$-decay.

2. Methodology

The authors propose a constrained approach to infer cross sections for unstable isotopes by calibrating nuclear model inputs using available data from neighboring stable isotopes.

• $\gamma$-Ray Strength Function ($\gamma$SF) Extraction:
  • $E1$ Component: The authors fitted experimental photo-absorption data (from the IAEA database) for stable isotopes $^{154,156,158,160}$Gd using a Standard Lorentzian (SLO) model. They determined peak parameters (centroid energy, width, peak cross-section) for two Lorentzian components to account for nuclear deformation.
  • $M1$ Component: Three phenomenological models were considered: (i) Standard SLO (RIPL-3 systematics), (ii) Renormalization based on $E1$ strength, and (iii) A model including Spin-flip and Scissors-mode contributions (crucial for deformed nuclei).
• Nuclear Level Density (NLD) Normalization:
  • Since experimental average s-wave neutron resonance spacings ($D_0$) are unavailable for $^{160}$Gd (due to the instability of $^{159}$Gd), the authors estimated $D_0$ via a linear fit of $\ln(D_0)$ vs. mass number $A$ for neighboring nuclei (Eu, Gd, Tb), accounting for odd-even effects.
  • They utilized HFB+c (Hartree-Fock-Bogoliubov + collective) microscopic level densities.
  • Bayesian Optimization (BO): The microscopic NLDs were renormalized using a Bayesian optimization method. The parameters were adjusted to simultaneously satisfy two constraints:
    1. Matching known discrete low-energy levels ($1.05 \le E_x \le 2.05$ MeV).
    2. Matching the total level density at the neutron separation energy ($S_n$) derived from the estimated $D_0$ and spin-cutoff parameters.
• Cross Section Calculation:
  • The constrained $\gamma$SF and NLD parameters were implemented in the nuclear reaction code TALYS-2.0.
  • The approach was first validated against experimental data for stable $^{155}$Gd and $^{157}$Gd.
  • Finally, the validated model was used to infer cross sections for unstable $^{153}$Gd and $^{159}$Gd in the energy range of 0.01–5.0 MeV.

3. Key Contributions

• Novel Constraint Strategy: Successfully demonstrated a method to infer cross sections for unstable isotopes by constraining fundamental nuclear properties ($\gamma$SF and NLD) using data from stable neighbors, bypassing the need for direct radioactive target measurements.
• Uncertainty Reduction: Quantified the reduction in model uncertainty. Standard TALYS predictions (using various model combinations) yielded an Average Relative Uncertainty (ARU) of ~167%. The constrained approach reduced this to ~30% (a reduction factor of 5.5).
• Astrophysical Rate Revision: Calculated new astrophysical reaction rates for $^{153,159}$Gd and compared them with the standard JINA REACLIB database.

4. Key Results

• Validation: The constrained model showed excellent agreement with experimental data for stable $^{155}$Gd and $^{157}$Gd across the entire energy range, validating the reliability of the method.
• Unstable Isotopes ($^{153,159}$Gd):
  • $^{153}$Gd: The calculated rates are consistent with the upper limit of the JINA REACLIB recommendation.
  • $^{159}$Gd: The calculated $(n, \gamma)$ rates are significantly higher than JINA REACLIB recommendations, by a factor of ~2.9 at the mean value.
• Branching Ratios:
  • For $^{153}$Gd, neutron capture dominates ($f_{n,\gamma} \approx 70\%$ at $T=0.2$ GK), and the branching ratio remains similar to JINA predictions.
  • For $^{159}$Gd, $\beta$-decay usually dominates due to its short half-life (18.5 h). However, the enhanced $(n, \gamma)$ rate increases the branching ratio ($f_{n,\gamma}$) by a factor of 2–3 compared to JINA REACLIB.
• Nucleosynthesis Impact ($^{160}$Gd):
  • $s$-process simulations using the new rates show that the production of $^{160}$Gd is enhanced by a factor of ~2 compared to simulations using JINA REACLIB rates.
  • This suggests that previous models relying on JINA rates may have significantly underestimated the abundance of $^{160}$Gd in $s$-process environments.

5. Significance

• Astrophysics: The study resolves a critical uncertainty in the $s$-process nucleosynthesis path. The enhanced flow through the $^{159}$Gd branching point alters the predicted abundance of stable $^{160}$Gd, which is an "s-only" isotope shielded from the $r$-process. This has implications for understanding stellar evolution and the chemical composition of the universe.
• Nuclear Data: The work provides a robust framework for generating reliable nuclear data for unstable isotopes where direct measurement is impossible. This is essential for improving reaction networks in both astrophysics and nuclear engineering.
• Future Applications: The authors plan to extend this methodology to other isotopic chains (Sm, Er) relevant to cosmic-ray exposure and $r$-process freeze-out. They emphasize the need for continued experimental measurements of stable neighbors at facilities like CSNS, n_TOF, and LANSCE to further constrain these models.