Excitation function measurement of $^{144}$Sm($α$,n) reaction at sub-Coulomb energies and detailed covariance analysis

This study presents the measurement of the $^{144}$Sm($\alpha$,n)$^{147}$Gd reaction cross-section at sub-Coulomb energies (14–21 MeV) using the stacked foil activation technique, accompanied by a detailed covariance analysis of uncertainties and a comparison with both literature data and Hauser-Feshbach theoretical predictions.

The Gist

Imagine you are a chef trying to bake the perfect cake, but instead of flour and sugar, you are mixing atomic particles. This paper is a detailed recipe report from a team of scientists who tried to bake a very specific "atomic cake" to understand two big mysteries: how the universe creates heavy elements, and how we can make better medicines for cancer patients.

Here is the story of their experiment, broken down into simple, everyday concepts.

1. The Goal: Baking a Rare "Atomic Cake"

The scientists wanted to study a specific reaction: smashing Alpha particles (tiny helium nuclei) into Samarium-144 (a rare earth metal).

• The Reaction: When they smash these together, the Samarium eats the Alpha particle and spits out a neutron. This turns it into Gadolinium-147.
• Why do this?
  1. Cosmic Cooking: In the universe, there are rare elements called "p-nuclei" that are hard to explain. Samarium is one of them. By understanding how it reacts, scientists can better understand how stars cook up the elements we are made of.
  2. Medical Magic: The result, Gadolinium-147, is a "superhero" for medical imaging. It has a perfect lifespan (about 38 hours) and emits a specific type of light (gamma ray) that doctors can use to take pictures inside the human body to find tumors.

2. The Challenge: The "Sub-Coulomb" Barrier

Imagine trying to push two strong magnets together. They repel each other fiercely. In the atomic world, positive charges repel just as strongly. This repulsion is called the Coulomb Barrier.

• Usually, to smash atoms together, you need to hit them with the speed of a bullet to overcome this repulsion.
• The Twist: These scientists wanted to hit the atoms slower than usual (below the barrier), between 14 and 21 MeV. It's like trying to push two magnets together when they are barely touching. It's incredibly difficult, and the reaction happens very rarely. They had to be extremely precise.

3. The Experiment: The "Sandwich" Technique

To catch these rare reactions, they used a clever method called Stacked Foil Activation.

• The Setup: Imagine a stack of pancakes. They took thin sheets of Samarium oxide (the "filling") and sandwiched them between layers of aluminum (the "bread").
• The Beam: They fired a beam of Alpha particles at this stack.
• The Trick: As the beam passed through the first layer of aluminum, it slowed down a little. By the time it hit the second layer, it was even slower. By the time it hit the last layer, it was moving very slowly.
• The Result: In one single experiment, they effectively tested five different "speeds" (energies) of the beam, creating a map of how the reaction works at different speeds.

4. The Simulation: The "Virtual Reality" Check

Because the beam slowed down as it passed through the metal layers, the scientists didn't know the exact speed of the beam hitting each specific layer.

• The Solution: They used a supercomputer program called GEANT4. Think of this as a high-tech flight simulator. They built a virtual version of their experiment and simulated millions of alpha particles flying through the stack.
• The Outcome: This told them exactly how much the beam slowed down and how "fuzzy" the speed was at each layer. This was crucial because if you don't know the speed, you can't know the recipe.

5. The Measurement: The "Ghost Hunter"

After the bombardment, the Samarium turned into Gadolinium-147. This new isotope is radioactive; it's like a tiny, glowing firefly that emits gamma rays as it decays.

• The Detection: They put the foils next to a super-sensitive camera (a Germanium detector) that can "see" these gamma rays.
• The Count: They counted how many "glows" they saw. More glows meant more reactions happened.
• The Correction: Because the foils were thick and close to the camera, some gamma rays would hit the detector at the same time and get "counted" as one big ray instead of two small ones. The scientists used math to fix this "double-counting" error, ensuring their numbers were accurate.

6. The "Covariance" Puzzle: The Web of Uncertainty

This is the most unique part of the paper. Usually, scientists just say, "Our result is X, with an error of Y."

• The Problem: In this experiment, the errors were connected. If the beam current was slightly off, it affected all the measurements. If the target thickness was slightly wrong, it affected all of them. They weren't independent mistakes; they were a web of linked uncertainties.
• The Solution: The team created a Covariance Matrix. Imagine a giant spreadsheet where every cell shows how much one measurement's error is linked to another's.
  • Analogy: If you are guessing the weight of five apples, and you use a slightly broken scale, your error for Apple #1 is linked to Apple #2. This matrix maps out those links. It's the first time this has been done for this specific reaction, making their data much more trustworthy for other scientists to use.

7. The Verdict: Theory vs. Reality

Finally, they compared their real-world data with computer predictions (using a code called TALYS).

• The Finding: The computer models are like different chefs guessing the recipe. Some chefs (models) guessed the reaction rate was too high; others guessed it was too low.
• The Winner: They found that the "flavor" of the reaction depended mostly on how they modeled the force between the particles (the Alpha Optical Model). Some models matched their data well, while others missed the mark, especially at the slower speeds.

Summary

In short, this paper is a masterclass in precision. The scientists:

1. Built a "speed trap" for atomic particles using stacked foils.
2. Used a virtual reality simulator to know exactly how fast the particles were moving.
3. Counted the radioactive "glows" with a high-tech camera.
4. Created a complex "error map" (covariance matrix) to show exactly how their measurements were linked.

Their work helps astronomers understand how the universe makes heavy elements and helps doctors improve the tools they use to find cancer. It's a perfect blend of cosmic curiosity and medical practicality.

Technical Summary

1. Problem Statement and Motivation

The study addresses two primary scientific needs:

• Nuclear Astrophysics: The $^{144}\text{Sm}(\alpha,n)^{147}\text{Gd}$ reaction is a critical channel in the astrophysical p-process (photodisintegration process), which is responsible for the synthesis of neutron-deficient nuclei (p-nuclei) heavier than iron. While $^{144}\text{Sm}$ is a "neutron magic" p-nucleus with high natural abundance, its reaction rates at sub-Coulomb energies (below the Coulomb barrier of $\approx 21.8$ MeV) are not well-constrained by existing experimental data. Accurate cross-sections are essential for modeling stellar nucleosynthesis.
• Medical Applications: The reaction product, $^{147}\text{Gd}$, is a promising radioisotope for Single-Photon Emission Tomography (SPET) due to its suitable half-life ($38.06$ hours) and a strong gamma-ray emission at $229.32$ keV ($60.7\%$ intensity).
• Methodological Gap: Previous measurements of this reaction lacked a rigorous treatment of uncertainties. Specifically, the correlations between different data points (covariance) arising from shared systematic errors (e.g., target thickness, beam flux, detector efficiency) were often ignored, leading to potential over- or under-estimation of reaction rates.

2. Methodology

The experiment was conducted at the Variable Energy Cyclotron Centre (VECC), Kolkata, India, utilizing the stacked foil activation technique followed by offline gamma-ray spectroscopy.

A. Target Preparation and Irradiation

• Target Material: Targets were prepared using molecular deposition of isotopically enriched ($67\%$) $^{144}\text{Sm}_2\text{O}_3$ powder onto high-purity aluminum backings.
• Thickness: Five targets were prepared with areal densities between $280$ and $350 \, \mu\text{g/cm}^2$. Thickness was verified via weighing and alpha-source transmission ($^{229}\text{Th}$), with an uncertainty of $15\text{--}25\%$.
• Beam Configuration: Two stacks of targets were irradiated with a $28$ MeV $\alpha$-beam. Aluminum degrader foils ($135 \, \mu\text{m}$ and $210 \, \mu\text{m}$) were used to reduce the beam energy, creating effective incident energies ranging from $14$ to $21$ MeV (sub-Coulomb barrier).
• Beam Current: Irradiation lasted $17\text{--}22$ hours with an average current of $\sim 210\text{--}270$ nA.

B. Energy Determination (GEANT4 Simulation)

To address the uncertainty in the effective beam energy caused by energy straggling in the degraders and targets:

• A detailed GEANT4 Monte Carlo simulation was performed.
• The simulation modeled the beam transport through the specific geometry of the stacks, including the initial cyclotron energy spread ($200$ keV FWHM).
• This provided precise mean energies and $1\sigma$ energy straggling values for each target position (ranging from $\pm 0.18$ to $\pm 0.30$ MeV).

C. Gamma-Ray Spectroscopy and Efficiency

• Detection: Irradiated targets were measured using a high-purity Germanium (HPGe) detector ($40\%$ relative efficiency, $\sim 1.8$ keV resolution at $1.33$ MeV).
• Calibration: Efficiency was calibrated using a standard $^{152}\text{Eu}$ source.
• Corrections:
  • Coincidence Summing: Corrected using the EFFTRAN code (Monte Carlo) for close geometry ($12.5$ mm) measurements.
  • Geometry: Corrections were applied to account for the finite size of the irradiated area ($\sim 8$ mm diameter) compared to the point-source calibration.
• Cross-Section Calculation: The cross-section ($\sigma$) was derived from the $229.3$ keV gamma peak of $^{147}\text{Gd}$ using the standard activation equation, incorporating beam flux, target atoms, efficiency, and decay factors.

D. Covariance Analysis (Key Innovation)

Unlike previous studies, this work performed a comprehensive covariance analysis:

• A covariance matrix was constructed to quantify how uncertainties in input parameters (beam current, target thickness, efficiency, gamma intensity, etc.) propagate and correlate across different energy points.
• The Jacobian matrix method was used to relate input parameter uncertainties to the final cross-section uncertainties.
• This allowed the authors to generate a correlation matrix, revealing that systematic errors create a mutual correlation of $\approx 7\text{--}8\%$ between all measured data points.

3. Key Results

• Excitation Function: The study successfully measured the cross-section for $^{144}\text{Sm}(\alpha,n)^{147}\text{Gd}$ at five sub-Coulomb energies:
  • $20.90 \pm 0.18$ MeV: $594.61 \pm 154.38$ mb
  • $19.34 \pm 0.20$ MeV: $431.44 \pm 112.03$ mb
  • $17.68 \pm 0.22$ MeV: $124.53 \pm 32.45$ mb
  • $16.03 \pm 0.27$ MeV: $20.52 \pm 5.35$ mb
  • $14.09 \pm 0.30$ MeV: $0.79 \pm 0.20$ mb
• Total Uncertainty: The combined uncertainty for the measurements was approximately $25\%$, dominated by target thickness uncertainty ($15\text{--}25\%$).
• Theoretical Comparison:
  • Data was compared with TALYS-2.0 statistical model calculations using $432$ different parameter combinations (varying $\alpha$-optical potentials, level densities, and gamma-strength functions).
  • Sensitivity: The cross-sections were found to be most sensitive to the choice of the $\alpha$-optical model potential (AOMP).
  • Best Fits: The AOMP3 (Demetriou & Goriely) and AOMP6 (Avrigeanu et al.) potentials provided the best reproduction of the experimental data, particularly at lower energies, though some models slightly underestimated the highest energy points.

4. Significance and Contributions

1. First Covariance Analysis: This is the first study to provide a full covariance and correlation matrix for the $^{144}\text{Sm}(\alpha,n)$ reaction. This is crucial for astrophysical network calculations, as it allows researchers to correctly propagate uncertainties without double-counting systematic errors.
2. Sub-Coulomb Data: It fills a gap in the experimental database for energies below the Coulomb barrier, which is the relevant regime for p-process nucleosynthesis in stellar environments.
3. Methodological Rigor: The combination of GEANT4 for energy degradation analysis and EFFTRAN for efficiency/summing corrections sets a high standard for activation experiments involving stacked foils and low-energy beams.
4. Medical Relevance: The data supports the production optimization of $^{147}\text{Gd}$ for SPET imaging applications.

5. Conclusion

The paper presents a robust measurement of the $^{144}\text{Sm}(\alpha,n)^{147}\text{Gd}$ excitation function at sub-Coulomb energies. By integrating advanced simulation (GEANT4) with a rigorous statistical treatment of uncertainties (covariance analysis), the authors have provided high-quality data that significantly improves the reliability of astrophysical models involving p-nuclei and offers valuable benchmarks for nuclear reaction codes like TALYS. The finding that optical model potentials are the dominant factor in theoretical discrepancies highlights the need for further refinement of $\alpha$-nucleus interaction models.