Spectroscopy of $^{11}$Be from the $^{10}$Be($d,p$) reaction measured in inverse kinematics by the AT-TPC in SOLARIS

This study utilizes the AT-TPC within the SOLARIS solenoid to perform the first high-luminosity inverse kinematics $^{10}$Be($d,p$) transfer reaction measurement, determining spectroscopic factors for $^{11}$Be states up to 3.40 MeV and providing evidence through comparison with NCCI calculations that the 3.40 MeV state has positive parity, consistent with it being the second excited state of the ground-state rotational band.

The Gist

The Big Picture: Building a Better Map of the Atomic World

Imagine the nucleus of an atom as a tiny, bustling city. Most of the time, the citizens (protons and neutrons) live in very predictable neighborhoods, following strict rules of order. But sometimes, in rare and exotic cities (called "rare isotopes"), the rules get flipped upside down. The citizens rearrange themselves in ways that shouldn't happen according to the old rulebooks.

This paper is about a team of scientists who went to investigate one of these "rule-breaking" cities: Beryllium-11 (11Be).

The Problem: The "Ghost" in the Machine

In the city of 11Be, there is a specific building (a quantum state) at a height of 3.40 MeV (think of this as a specific floor in a skyscraper). For decades, scientists have argued about what kind of building this is.

• Team A says it's a "Negative Parity" building (like a house with a basement).
• Team B says it's a "Positive Parity" building (like a house with an attic).

Knowing which one it is is crucial because it tells us if the whole city is built on a rotating, spinning foundation (a "rotational band") or if it's just a static pile of bricks. If it's the "Positive Parity" version, it fits perfectly into a beautiful, spinning dance pattern predicted by modern math. If it's the other, the whole dance breaks.

The Challenge: Catching a Ghost with a Net

To figure this out, you need to bump into these atoms and see how they bounce. But 11Be is a "rare isotope." It's like trying to study a specific, rare bird that only flies by once every few hours.

• Old Method: Scientists usually use a "passive target" (like a net waiting for the bird). But because the beam of 11Be atoms is so weak (only 1,000 particles per second!), a passive net would catch almost nothing. It's like trying to catch a single raindrop with a bucket in a drought.
• The New Solution: The team built a Super-Active Net. They used a device called the AT-TPC (Active Target Time Projection Chamber) filled with pure deuterium gas. Instead of waiting for the bird to hit a wall, the bird flies through the gas, and the gas itself acts as the target. Every time the bird bumps into a gas atom, the gas lights up like a neon sign, recording the exact path of the collision.

They also put this giant net inside a SOLARIS machine, which is essentially a giant, super-strong magnet (a solenoid). This magnet acts like a curved slide, bending the paths of the particles so the scientists can measure exactly how hard they bounced.

The Experiment: The "Slow-Motion" Collision

The scientists fired a beam of 10Be atoms (the parent of 11Be) at this gas net.

1. The Collision: When a 10Be atom hit a deuterium atom in the gas, it stole a neutron and became 11Be, while kicking out a proton (a hydrogen nucleus).
2. The Tracking: The AT-TPC recorded the 3D path of every single particle. It's like having a high-speed camera that doesn't just take a photo, but reconstructs the entire movie of the collision in 3D space.
3. The Result: They successfully mapped out the "neighborhood" of 11Be up to the mysterious 3.40 MeV floor.

The Detective Work: Reading the Angles

Once they had the data, they had to play detective. They looked at the angles at which the protons flew away.

• Think of it like throwing a ball at a wall. If you throw it straight on, it bounces back. If you throw it at a shallow angle, it skims off.
• The angle at which the particle bounces off tells you about the "spin" and "shape" of the nucleus it hit.

They used a complex math tool called DWBA (Distorted-Wave Born Approximation) to compare their real-world angles with theoretical predictions.

• The 1.78 MeV State: They were very sure about this one. It was clearly a "Positive Parity" building.
• The 3.40 MeV State: This was the tricky one. The data didn't give a perfect angle to make a 100% definitive call. However, when they looked at the "spectroscopic factors" (a measure of how much the new neutron "fits" into the existing structure), the numbers leaned heavily toward the Positive Parity theory.

The Theory Check: The "Ab Initio" Crystal Ball

To be sure, they compared their experimental results with the most advanced computer simulations available, called Ab Initio calculations (which means "from the beginning," using only the fundamental laws of physics without guessing).

They ran these simulations using different "interaction recipes" (mathematical rules for how particles talk to each other).

• One recipe, called Daejeon16, was like a perfectly tuned engine. It predicted that the 3.40 MeV state should be at exactly 3.40 MeV and have Positive Parity.
• This matched their experimental data almost perfectly.

The Conclusion: A Spinning City Confirmed

The paper concludes that the mysterious 3.40 MeV state is almost certainly Positive Parity.

What does this mean?
It confirms that 11Be isn't just a random pile of bricks. It is a spinning, deformed city.

• The ground state (the bottom floor) is a "one-neutron halo" (a fuzzy cloud of a neutron orbiting a core).
• This core is spinning, creating a "rotational band" (like a dance troupe).
• The 1.78 MeV state, the 3.40 MeV state, and others are all members of this same dance troupe, spinning together.

Why This Matters

This experiment was a "proof of concept." It showed that you can study extremely rare, weak beams of atoms by combining a super-sensitive gas detector with a giant magnet. This opens the door to studying even stranger, more exotic atoms in the future, helping us understand how the universe builds its heaviest elements.

In short: They built a better net, caught a rare bird, figured out its dance moves, and confirmed that the whole atomic city spins in a beautiful, predictable pattern.

Technical Summary

1. Problem Statement

The study addresses two primary challenges in nuclear physics:

• Spectroscopy of Rare Isotopes: Investigating the structure of nuclei far from stability, specifically $^{11}$Be, which resides in the $N=8$ "island of inversion." $^{11}$Be is famous for its parity-inverted ground state ($1/2^+$), which contradicts standard shell model predictions (expecting $1/2^-$). While the ground state is well understood as a rotational band head on a deformed core, the nature of the excited state at 3.40 MeV remains debated. Previous experiments have assigned it either negative parity ($3/2^-$) or positive parity ($3/2^+$), a distinction crucial for confirming the existence of a $K^\pi = 1/2^+$ rotational band.
• Experimental Limitations in Inverse Kinematics: Traditional passive target techniques struggle with low-intensity radioactive beams (often $<10^3$ particles/second) required for rare isotopes. High-luminosity transfer reactions in inverse kinematics have historically been difficult to perform without sacrificing energy or angular resolution.

2. Methodology

The authors performed a commissioning experiment to demonstrate a novel coupling of experimental tools:

• Reaction: The neutron-adding transfer reaction $^{10}$Be(d, p)$^{11}$Be was measured in inverse kinematics.
• Beam: A $^{10}$Be beam at 9.6 MeV/u with a very low intensity of 1,000 particles per second.
• Detector System:
  • AT-TPC (Active Target Time Projection Chamber): Filled with 250 liters of pure deuterium gas (acting as both target and detector). This allows for high luminosity even with low beam intensities.
  • SOLARIS Solenoid: The AT-TPC was placed inside a 3 Tesla superconducting solenoid. This magnetic field bends charged particle trajectories, enabling precise momentum reconstruction and particle identification (PID) via magnetic rigidity ($B\rho$) and energy loss ($dE/dx$).
• Data Analysis:
  • Event-by-event reconstruction using the spyral analysis package to extract kinematic parameters (angles, energy, vertex).
  • Distorted-Wave Born Approximation (DWBA) calculations (using the code ptolemy) were performed to analyze angular distributions and extract spectroscopic factors.
  • Theoretical Comparisons: Results were compared against phenomenological shell model interactions (WBP, WBT, YSOX, FSU) and ab initio No-Core Configuration Interaction (NCCI) calculations using NNLOopt and Daejeon16 nucleon-nucleon interactions.

3. Key Contributions

• First High-Luminosity Inverse Kinematics Measurement: This is the first successful coupling of the AT-TPC with the SOLARIS solenoid to perform a transfer reaction measurement. It proved that high-resolution spectroscopy is possible with extremely low beam intensities (orders of magnitude lower than typical requirements).
• Resolution of the 3.40 MeV State: The experiment provided new constraints on the 3.40 MeV state in $^{11}$Be, a state whose parity has been controversial.
• Validation of Ab Initio Theory: The study provided a critical benchmark for ab initio NCCI calculations, specifically testing the convergence and predictive power of the Daejeon16 interaction for light nuclei with deformation.

4. Results

• Excitation Spectrum: States in $^{11}$Be were populated up to 3.40 MeV with an energy resolution of approximately 350 keV (FWHM). The ground state ($1/2^+$), 0.32 MeV state ($1/2^-$), 1.78 MeV state ($5/2^+$), 2.65 MeV state ($3/2^-$), and the 3.40 MeV state were identified.
• Angular Distributions & Spectroscopic Factors:
  • The 1.78 MeV state ($5/2^+$) showed a clear $\ell=2$ transfer with a spectroscopic factor of 0.72 ± 0.03, consistent with a $0d_{5/2}$ orbital.
  • The 3.40 MeV state had limited angular coverage due to kinematic constraints, making a direct parity assignment from angular distributions alone difficult.
  • However, the extracted spectroscopic factors for the 3.40 MeV state were significantly more consistent with a positive parity assignment ($\ell=2$, $0d_{3/2}$) than a negative parity assignment ($\ell=1$, $0p_{3/2}$). The value for the $3/2^+$ hypothesis (0.08 ± 0.02) aligned better with previous neutron-width derivations and shell model predictions than the $3/2^-$ hypothesis.
• Theoretical Agreement:
  • Shell Model: Various multi-shell interactions showed significant scatter in predicting the 3.40 MeV energy, failing to simultaneously reproduce the 1.78 MeV and 3.40 MeV levels within 0.4 MeV.
  • Ab Initio (NCCI): Calculations using the Daejeon16 interaction showed excellent convergence. Extrapolated excitation energies for the positive parity band members were ~1.7–1.8 MeV (for $5/2^+$) and ~3.2–3.4 MeV (for $3/2^+$), matching the experimental data remarkably well. The NNLOopt interaction showed poor convergence.

5. Significance

• Confirmation of Rotational Band Structure: The results strongly support the assignment of $J^\pi = 3/2^+$ to the 3.40 MeV state. This confirms that $^{11}$Be possesses a $K^\pi = 1/2^+$ rotational band built on the parity-inverted ground state, consisting of the $1/2^+$, $5/2^+$, and $3/2^+$ states.
• Deformation and Clustering: The findings reinforce the picture of $^{11}$Be as a system where a single neutron orbits a highly deformed, clustered core (resembling an $\alpha+\alpha$ dimer plus neutrons), rather than a simple spherical shell model system.
• Technological Breakthrough: The experiment demonstrates that the AT-TPC + SOLARIS combination is a viable, high-performance spectrometer for rare isotope research. It enables detailed spectroscopic studies of nuclei that can only be produced at intensities of a few hundred particles per second, opening new avenues for exploring nuclear structure far from stability.
• Future Directions: The authors suggest that higher beam energies would be required to access the next predicted member of the rotational band (the $9/2^+$ state at ~6 MeV) and to fully resolve the angular distributions for definitive parity assignments in future campaigns.