Comparing invariant-mass spectroscopy of 8B with ab initio predictions

This study extends the level scheme of 8B up to 10 MeV by analyzing invariant-mass data from 9C and 13O beam experiments, identifying new levels and decay pathways, and demonstrating strong correspondence between these experimental findings and ab initio symmetry-adapted no-core shell model predictions for positive parity states below 8.4 MeV.

The Gist

Imagine the atomic nucleus not as a solid marble, but as a chaotic, high-energy dance floor where tiny particles called protons and neutrons are constantly spinning, jumping, and occasionally flying apart.

This paper is a report on a scientific investigation into a very specific, unstable dancer on that floor: a nucleus called Boron-8 (or ⁸B). Because this nucleus is so unstable, it doesn't just sit there; it immediately falls apart. The scientists wanted to map out exactly how it falls apart and compare their observations with a super-advanced computer simulation to see if our understanding of the "rules of the dance" is correct.

Here is the breakdown of their experiment and findings, explained with some everyday analogies.

1. The Setup: The "Pinball Machine"

To study Boron-8, the scientists couldn't just wait for it to appear naturally. They had to create it.

• The Beam: They fired a beam of heavy, unstable atoms (Carbon-9 and Oxygen-13) at a target made of Beryllium. Think of this like firing a cannonball at a wall of bricks.
• The Knockout: When the heavy cannonball hits the wall, it knocks pieces off. In this case, the collision knocked a single proton out of the Carbon-9, leaving behind a Boron-8 nucleus.
• The Trap: Since Boron-8 is so unstable, it immediately breaks apart again. The scientists used a giant, high-tech camera array (called HiRA) surrounding the target to catch all the flying debris. They didn't just take a picture; they measured the speed and direction of every single piece flying out.

2. The Detective Work: "Reconstructing the Crime Scene"

The Boron-8 nucleus breaks into different combinations of pieces, like a puzzle falling apart in different ways:

• Scenario A: Two protons and a Lithium-6 nucleus.
• Scenario B: A proton, a Helium-3 nucleus, and an Alpha particle (Helium-4).
• Scenario C: A proton and a Beryllium-7 nucleus (sometimes with a flash of light, or gamma ray).

The scientists used a technique called Invariant-Mass Spectroscopy. Imagine you see a car crash from a distance. You can't see the car anymore, but you see the scattered parts (wheels, doors, glass) flying in different directions. By measuring how fast and where those parts are going, you can mathematically "rewind" the crash to figure out exactly what the car looked like before it hit.

The scientists did this math to figure out the "excitation energy" of the Boron-8 nucleus—essentially, how much energy it had stored up before it exploded.

3. The New Discoveries: Finding Hidden Rooms

Before this study, scientists had a rough map of the Boron-8 "house," but some rooms were missing.

• The New Rooms: By analyzing the debris, they found new energy levels (new "rooms" in the house) that had never been seen before.
• The "Double Proton" Mystery: One of the most exciting finds was a state where the nucleus spits out two protons at the exact same time (a "prompt" decay). Usually, nuclei spit out one particle, wait a moment, and then spit out another. Finding a nucleus that shoots two protons simultaneously is like a firework that explodes two sparks at the exact same microsecond. This is rare and hard to catch.

4. The Computer Simulation: The "Crystal Ball"

On the theoretical side, the team used a supercomputer model called SA-NCSM (Symmetry-Adapted No-Core Shell Model).

• The Analogy: Imagine trying to predict how a complex machine will vibrate. You could try to guess, or you could build a perfect digital twin of the machine and run a simulation.
• The Theory: This computer model uses the fundamental laws of physics (Quantum Chromodynamics) to predict exactly what energy levels the Boron-8 nucleus should have and how it should break apart.
• The Match: The scientists compared their "crime scene reconstruction" (the experiment) with the "digital twin" (the computer).
  • The Good News: For almost every new "room" they found in the experiment, the computer had predicted a matching room. The energies and the way the nucleus broke apart matched up very well.
  • The Twist: There were a few discrepancies where the computer predicted the nucleus was slightly higher in energy than the experiment showed. This suggests that while our "rules of the dance" are mostly right, there are still tiny nuances in how the protons and neutrons interact that we need to refine.

5. Why Does This Matter?

You might ask, "Why do we care about a tiny, unstable atom that doesn't exist in nature?"

• Testing the Rules: Boron-8 and its "mirror twin" (Lithium-8) are the perfect test subjects. They are simple enough for our supercomputers to handle, but complex enough to be tricky. If our computer models work here, we can trust them to predict the behavior of much heavier, more complex elements.
• The Stars: Boron-8 plays a role in how stars burn and how they produce energy. Understanding its structure helps us understand the life cycles of stars and the creation of elements in the universe.
• The "Mirror" Effect: By comparing Boron-8 with Lithium-8 (which has the same number of particles but swapped protons and neutrons), scientists can study how the universe treats matter slightly differently based on its charge (isospin breaking).

The Bottom Line

This paper is a success story of experiment meeting theory. The scientists built a "time machine" using particle collisions to see how Boron-8 falls apart, found some new ways it breaks, and confirmed that our most advanced computer models of the atomic world are getting it right. They found that the nucleus is a bit more deformed (squashed or stretched) than a perfect sphere, and that our understanding of the forces holding it together is solid, even if there are still a few tiny details to polish.

Technical Summary

1. Problem Statement

The nucleus $^8$B (and its mirror $^8$Li) serves as a critical testing ground for ab initio nuclear theories, particularly those utilizing chiral effective field theory (EFT) potentials. While low-lying states are well-studied, the excited spectrum above the proton separation energy ($S_p \approx 136$ keV) remains complex. All excited states in $^8$B are particle-unbound resonances, making their study challenging.
The specific challenges addressed in this work include:

• Completing the Level Scheme: Extending the known energy level scheme of $^8$B up to $E^* \approx 10$ MeV.
• Decay Mechanisms: Determining whether decays are prompt (simultaneous emission) or sequential (via intermediate resonances) and identifying the specific decay pathways (e.g., $2p$, $p+\alpha$, $p+^7$Be).
• Theoretical Validation: Testing the predictive power of the Symmetry-Adapted No-Core Shell Model (SA-NCSM) in the continuum, specifically regarding its ability to predict excitation energies, spins, parities, and decay widths for positive-parity states populated via proton knockout.

2. Methodology

Experimental Approach

• Reactions: The study utilized proton knockout reactions and projectile fragmentation using fast beams ($E/A \approx 69$ MeV) of $^9$C and $^{13}$O incident on a 1-mm-thick Beryllium (Be) target.
  • $^9$C Beam: Used for single-proton knockout to populate $^8$B states. Two datasets were analyzed: one optimized for heavy element detection ($9C_{1st}$) and an upgraded set ($9C_{2nd}$) with improved electronics.
  • $^{13}$O Beam: Used for projectile fragmentation (loss of 5 nucleons) to verify resonances observed in knockout reactions.
• Detection: Experiments were conducted using the HiRA (High Resolution Array) detector system, consisting of Si and CsI(Tl) telescopes arranged to cover polar angles of $2^\circ$–$12^\circ$.
  • Coincidence Measurements: The CAESAR $\gamma$-ray detector array was employed in the $9C_{2nd}$ experiment to detect coincident $\gamma$-rays, crucial for distinguishing ground-state decays from decays to excited states of fragments (e.g., $^6$Li, $^7$Be).
• Analysis Technique:
  • Invariant-Mass Spectroscopy: Excitation energies ($E^*$) were reconstructed from the momenta of decay fragments.
  • Correlation Analysis: Momentum correlations between decay fragments were analyzed to deduce decay pathways (sequential vs. prompt) and intermediate states.
  • Background Subtraction: Weighted event mixing techniques were used to model and subtract non-resonant backgrounds.

Theoretical Approach

• Model: Symmetry-Adapted No-Core Shell Model (SA-NCSM). This approach uses an $SU(3)$ symmetry-adapted basis to handle large model spaces and nuclear deformation efficiently.
• Interaction: The NNLOopt chiral potential was used.
• Continuum Treatment: The interior wavefunctions were calculated using SA-NCSM, while the asymptotic behavior was handled via a microscopic R-matrix technique to couple to continuum degrees of freedom.
• Calculations: Performed with harmonic oscillator frequencies ($\hbar\Omega$) of 15, 20, and 25 MeV and model spaces up to $N_{max}=12$, with extrapolations to $N_{max} \to \infty$.
• Observables: Calculated excitation energies, spectroscopic factors, and partial decay widths for states with $J^\pi = 0^+, 1^+, 2^+, 3^+$.

3. Key Contributions and Results

A. Experimental Observations

The study identified and characterized several new resonances in $^8$B across three exit channels:

1. $2p + ^6$Li Channel:

   • Confirmed the prompt $2p$ decay of the $0^+_2$ isobaric analog state at $E^* \approx 10.6$ MeV (observed as a narrow peak at 7.1 MeV in the $n\gamma$ spectrum due to the $^6$Li $\gamma$-ray).
   • Observed a new broad state at $E^* \approx 8.4$ MeV (assigned as $1^+_4$) undergoing prompt $2p$ decay.
   • Identified a wide state at $E^* \approx 10.1$ MeV (tentatively assigned as $3^+_4$).

2. $p + ^3$He + $\alpha$ Channel:

   • Observed a state at $E^* \approx 6.1$ MeV (assigned as $3^+_2$) decaying sequentially via $^7$Be($7/2^-$).
   • Identified a state at $E^* \approx 5.4$ MeV (tentatively $2^+_3$) and a state at $E^* \approx 8.2$ MeV (tentatively $2^+_4$ or $3^+_3$).
   • Decay Mechanism: Correlation analysis revealed that the 8.2 MeV state decays predominantly (73%) via $^3$He + $^5$Li(g.s.) and secondarily (27%) via $p + ^7$Be($5/2^-$).

3. $p + ^7$Be Channel:

   • Discovered a new broad state at $E^* \approx 5.1$ MeV (assigned as $1^+_3$) decaying to both the ground and first excited ($1/2^-$) states of $^7$Be.
   • Established that the 8.4 MeV state has a significant branching ratio ($\ge 23\%$) to the $2p$ channel, with limits placed on $p+^7$Be branches.

B. Theoretical-Experimental Comparison

• Level Assignment: The SA-NCSM predictions successfully matched all newly observed levels to theoretical counterparts for $E^* \le 8.4$ MeV.
  • Energy Agreement: Generally within 500 keV for low-lying states, though deviations were noted for the $3^+_2$ and $2^+_2$ states (likely due to spin-mixing effects not fully captured).
  • Spin/Parity: Theoretical spectroscopic factors and decay widths strongly supported the experimental spin assignments (e.g., confirming the $3^+_2$ assignment via strong predicted decay to $^7$Be($7/2^-$)).
• Deformation: The study confirmed that low-lying states in $^8$B share a triaxial deformation, while states above 8 MeV exhibit a coexistence of triaxial and oblate deformations.
• Prompt 2p Decay: The SA-NCSM calculations supported the identification of the 8.4 MeV state as a prompt $2p$ emitter, consistent with the companion paper [14].

4. Significance

• Validation of Ab Initio Continuum Models: This work provides a rigorous benchmark for the SA-NCSM framework in the continuum. The successful prediction of excitation energies, spins, and complex decay branching ratios (including sequential vs. prompt modes) demonstrates the model's capability to describe light, unbound nuclei with high fidelity.
• Refinement of Nuclear Structure: The identification of new levels and the mapping of their decay pathways resolve ambiguities in the $^8$B level scheme. It clarifies the nature of the $2p$ emitter at 8.4 MeV and the role of intermediate states like $^5$Li and $^7$Be in multi-particle decays.
• Isospin Symmetry: The comparison between $^8$B and its mirror $^8$Li showed minimal Thomas-Ehrman shifts, suggesting that the double occupancy of the proton $s$-orbital is not a dominant feature in these states, reinforcing the validity of the chiral potentials used.
• Methodological Advancement: The study demonstrates the power of combining invariant-mass spectroscopy with advanced correlation analysis and $\gamma$-ray tagging to disentangle complex decay chains in unbound nuclei.

In conclusion, the paper successfully expands the experimental knowledge of the $^8$B spectrum and validates the predictive power of modern ab initio theories, specifically the SA-NCSM, in describing the structure and decay of light, particle-unbound nuclei.