Exploring the statistical properties of the neutron-deficient $^{109}$In isotope with the Oslo method

This study presents the first extraction of the nuclear level density and $\gamma$-ray strength function for the neutron-deficient $^{109}$In isotope using the Oslo method, revealing significant deviations from current model predictions and providing refined constraints for astrophysical $p$-process simulations.

The Gist

The Big Picture: Listening to the Atomic Heartbeat

Imagine an atom's nucleus not as a static ball of dust, but as a bustling city. Inside this city, protons and neutrons are the citizens, constantly moving, vibrating, and interacting. When you poke this city (by hitting it with a particle), it doesn't just sit there; it hums, rings, and emits energy like a bell.

Scientists want to understand two specific things about this "city":

1. How crowded is it? (This is the Nuclear Level Density or NLD). Think of this as counting how many different "floors" or energy states the citizens can occupy.
2. How loud is the ring? (This is the Gamma-ray Strength Function or GSF). This measures how easily the nucleus can emit light (gamma rays) when it gets excited.

The paper focuses on a specific, rare atom called Indium-109 ($^{109}$In). This atom is "neutron-deficient," meaning it has fewer neutrons than its stable cousins. It's like a slightly unbalanced version of a standard atom, living on the edge of stability.

The Experiment: The "Oslo Method" as a Detective Tool

To figure out the properties of this unstable atom, the scientists couldn't just look at it; they had to make it. They used a particle accelerator (a giant atomic slingshot) to fire alpha particles at a Cadmium target. This reaction created Indium-109 and knocked out a proton.

As the new Indium-109 atom settled down, it emitted a cascade of gamma rays (light). The scientists used a massive array of detectors (called OSCAR, which sounds like a fancy award, but is actually a sphere of crystal detectors) to catch these light particles.

They used a clever statistical trick called the Oslo Method.

• The Analogy: Imagine you hear a complex song played on a piano, but you can only hear the final notes and the rhythm, not the sheet music. The Oslo Method is like a super-smart algorithm that listens to the "echoes" of the decay and works backward to reconstruct the entire sheet music (the energy levels and the strength of the transitions).

The Big Surprise: The Missing "Pygmy"

For decades, physicists have been hunting for something called the Pygmy Dipole Resonance (PDR).

• The Analogy: Think of a heavy nucleus as a giant, solid drum. Usually, when you hit it, the whole thing vibrates (the Giant Resonance). But in neutron-rich atoms, scientists thought there was a "skin" of extra neutrons that would wobble independently, like a small, weak drumhead attached to the big one. This wobble creates a specific "hum" or extra strength in the light emission around a certain energy level (about 8 MeV).

The Discovery:
When the scientists looked at Indium-109, they found no such hum.

• In its neighbors (like Tin and Cadmium), this "Pygmy" wobble was visible.
• In Indium-109, it was completely silent.

This is a huge deal because it challenges our understanding of how atomic nuclei behave. It suggests that when you have a nucleus with a low "neutron-to-proton" ratio (like Indium-109), the neutrons don't form a loose, wobbly skin. Instead, the protons seem to take the lead in the vibrations. It's like the "skin" of the city is made of a different material that doesn't wobble the same way.

Why Does This Matter? (The Cosmic Connection)

You might ask, "Why do we care about a specific atom in a lab?" The answer lies in the stars.

There is a cosmic process called the p-process (proton-capture process) that happens in supernovae and exploding stars. This is how the universe creates heavy, rare elements (like gold, platinum, and even the Indium in your electronics).

To simulate these explosions on a computer, astrophysicists need to know exactly how these atoms capture protons or neutrons.

• The Problem: Current computer models are like driving a car with a foggy windshield. They use "best guesses" for how crowded the nucleus is and how it emits light. Because of this fog, the models predict the creation of elements very differently than what actually happens.
• The Solution: This paper provides a clear, high-definition map for Indium-109. By feeding these new, precise numbers into the computer models, scientists can clear up the fog.

The Result:

• The team calculated how likely Indium-109 is to capture a neutron or a proton.
• Their new data matched real-world measurements for proton capture perfectly.
• However, it showed that the standard "library" of data used by astronomers (JINA REACLIB) was wrong for neutron capture. The old models were off by a significant margin.

The Takeaway

This paper is a story of correcting the map.

1. We found a new atom's "fingerprint": We measured exactly how Indium-109 vibrates and emits light.
2. We found a missing piece: We discovered that the "Pygmy" wobble (the extra strength seen in other atoms) is missing here, which tells us something profound about the internal structure of neutron-poor atoms.
3. We fixed the cosmic calculator: By using these new measurements, we can now simulate how stars create heavy elements much more accurately.

In short, the scientists listened to a tiny, unstable atom, realized it was singing a different song than its neighbors, and used that new song to help us understand how the universe builds the heavy elements that make up our world.

Technical Summary

1. Problem Statement

The paper addresses critical gaps in nuclear physics and astrophysics regarding the neutron-deficient region of the nuclear chart, specifically around the Sn (Tin) mass region.

• Scientific Context: While the evolution of low-lying electric dipole (E1) strength, often associated with the Pygmy Dipole Resonance (PDR), has been extensively studied in neutron-rich nuclei (linked to neutron skins), the behavior in neutron-deficient nuclei remains poorly understood.
• Specific Gap: There is a lack of experimental data on odd-Z, neutron-deficient nuclei like $^{109}$In. Previous studies on even-Z neighbors (Cd, Sn, Pd) showed mixed results regarding PDR enhancement, but the behavior of odd-Z isotopes near the $Z=50$ shell closure was largely unexplored.
• Astrophysical Impact: Accurate Nuclear Level Densities (NLD) and $\gamma$-ray Strength Functions (GSF) are essential inputs for Hauser-Feshbach statistical models used to calculate reaction rates for the astrophysical p-process (proton-rich nucleosynthesis). Current models often rely on extrapolations with large uncertainties, potentially compromising the accuracy of p-process simulations.

2. Methodology

The study utilized a combination of experimental techniques and theoretical analysis:

• Experimental Setup:

  • Reaction: Performed at the Oslo Cyclotron Laboratory (OCL) using the $^{106}$Cd($\alpha, p\gamma$)$^{109}$In reaction with a 23-MeV $\alpha$-beam.
  • Detectors: Used the OSCAR array (30 LaBr$_3$(Ce) crystals) for $\gamma$-ray detection and the SiRi telescope array for particle identification.
  • Data Collection: Recorded $\approx 1.3 \times 10^9$ $p-\gamma$ coincidence events.

• Analysis Techniques:

  • Oslo Method: Extracted the NLD and GSF from the primary $\gamma$-ray distributions derived from the $p-\gamma$ coincidence matrix. This method relies on the proportionality between the first-generation $\gamma$-ray distribution and the product of NLD and GSF.
  • Shape Method: Used to constrain the slope of the GSF (and consequently the NLD) by analyzing direct decays to low-lying discrete states (specifically the ground state and a cluster of excited states). This was crucial because standard neutron resonance data (needed for absolute normalization) are unavailable for unstable $^{109}$In.
  • Normalization Strategy:
    • Since direct neutron resonance data for $^{109}$In is missing, the authors used systematics from neighboring even-odd Cd and Sn isotopes to estimate the average total radiative width ($\langle \Gamma_\gamma \rangle$).
    • They employed a $\chi^2$ minimization to find a consistent spin-cutoff parameter ($\sigma(S_n)$) that aligns the Oslo method results with the Shape method results and stays within the 95% confidence interval of the systematics.

• Theoretical Modeling:

  • Microscopic Calculations: Performed Relativistic Equation of Motion (REOM2/RQTBA) calculations to interpret the microscopic origin of the E1 strength, comparing $^{109}$In with $^{110,112}$Sn.
  • Reaction Rates: Used the TALYS code to calculate radiative neutron- and proton-capture cross sections ($^{108}$In(n,$\gamma$)$^{109}$In and $^{108}$Cd(p,$\gamma$)$^{109}$In) using the newly extracted experimental NLD and GSF as inputs.

3. Key Contributions

• First Extraction: This is the first time the NLD and GSF of the neutron-deficient isotope $^{109}$In have been extracted experimentally.
• Novel Normalization: Developed a robust procedure to normalize data for unstable nuclei lacking neutron resonance data by combining the Oslo method, Shape method, and isotopic systematics.
• Theoretical Insight: Provided a detailed microscopic interpretation of the E1 strength using REOM2/RQTBA, revealing a qualitative change in the nature of low-lying dipole states in neutron-deficient nuclei compared to neutron-rich ones.

4. Key Results

• Nuclear Level Density (NLD):

  • The experimental NLD follows a Back-Shifted Fermi Gas (BSFG) trend starting at relatively low excitation energies ($\approx 5$ MeV).
  • The extracted NLD is significantly higher than predictions from most standard microscopic models (e.g., Skyrme-HFB, Gogny-HFB) unless these models are adjusted.
  • The values lie between those of neighboring $^{105}$Cd and $^{111}$Cd.

• $\gamma$-ray Strength Function (GSF):

  • No PDR Enhancement: Unlike neighboring Sn, Cd, and Pd isotopes which often show an enhancement of dipole strength near the neutron separation energy (associated with the PDR or "upbend"), $^{109}$In shows no significant enhancement in the 7–8 MeV region.
  • Low-Lying Strength: The integrated low-lying E1 strength above the Isovector Giant Dipole Resonance (IVGDR) tail is estimated to be only $\approx 0.53\%$ of the Thomas-Reiche-Kuhn (TRK) sum rule, which is considerably lower than in neighboring nuclei.
  • Agreement: The GSF shape agrees well with neighboring Sn and Cd isotopes at low energies but is distinct in the 7–8 MeV region.

• Microscopic Interpretation (REOM2/RQTBA):

  • Calculations reveal that in neutron-rich $^{112}$Sn, low-lying dipole states are driven by neutron oscillations (neutron skin).
  • In neutron-deficient $^{109}$In, the lowest dipole states are dominated by proton transitions. The neutron contribution is suppressed, and the destructive interference between proton and neutron transition densities leads to the observed reduction in total strength.

• Astrophysical Reaction Rates:

  • Proton Capture ($^{108}$Cd(p,$\gamma$)): The calculated cross sections using the new experimental inputs show excellent agreement with direct experimental measurements over a wide energy range (3–7 MeV), validating the Oslo method inputs.
  • Neutron Capture ($^{108}$In(n,$\gamma$)): The experimental constraints reveal that the JINA REACLIB library (specifically the ths8 model) significantly overestimates the reaction rate compared to the new experimental data. The BRUSLIB library performs better but still shows deviations.
  • Uncertainty Reduction: The new data significantly reduce the model uncertainties for NLD and GSF in this mass region, particularly showing that standard default models in TALYS often underestimate the NLD and GSF for neutron-deficient nuclei.

5. Significance

• Astrophysical p-Process: The results provide critical constraints for p-process nucleosynthesis simulations. The finding that JINA REACLIB rates for neutron-deficient nuclei may be inaccurate suggests a need to re-evaluate reaction networks involving these isotopes.
• Model Validation: The study highlights that theoretical models predicting a strong PDR or enhanced low-lying strength based on neutron-skin oscillations fail to describe neutron-deficient nuclei. The data necessitate models that account for the dominance of proton transitions in this region.
• Future Research: The work establishes a benchmark for studying other unstable, odd-Z nuclei in the Sn mass region and demonstrates the viability of combining the Oslo and Shape methods to study nuclei where traditional neutron resonance data is unavailable.

In summary, this paper fundamentally alters the understanding of dipole strength in neutron-deficient nuclei, showing a suppression of the PDR-like enhancement and providing high-precision experimental inputs that correct existing astrophysical reaction rate libraries.