Single-particle strength toward N = 32: Spectroscopy of 51 Ca via the 50 Ca(d, p) reaction

This paper reports on the spectroscopy of neutron-rich 51Ca via the 50Ca(d,p) reaction, identifying single-particle states and a candidate 9/2+ excitation to provide new constraints on shell evolution in calcium isotopes through comparison with shell-model and ab initio VS-IMSRG calculations.

The Gist

Imagine the atom's nucleus not as a solid ball, but as a bustling apartment building. Inside this building, neutrons and protons live in specific "floors" or orbitals. In a stable neighborhood (like the elements we find on Earth), these floors are arranged in a very predictable pattern. Physicists call these patterns "shells," and when a shell is completely full, the building is extra stable. These full shells are called magic numbers.

For a long time, scientists thought these magic numbers never changed, no matter where you looked in the universe. But recently, they discovered that in the extreme, neutron-rich neighborhoods (like the heavy calcium isotopes studied here), the building's architecture changes. New "magic numbers" appear, and old floors shift up or down.

This paper is about a team of scientists trying to map out the floor plan of a very specific, unstable apartment building: Calcium-51 ($^{51}$Ca). This atom has 20 protons and 31 neutrons. It's right on the edge of a new "magic" zone (near neutron number 32), making it a perfect test case to see how the rules of nuclear physics change when you add too many neutrons.

Here is how they did it, explained simply:

1. The Experiment: A High-Speed Pinball Game

To understand the floor plan of Calcium-51, the scientists couldn't just look at it; they had to build it and see how it reacted.

• The Setup: They took a beam of Calcium-50 (which has 30 neutrons) and fired it like a cannonball at a target made of heavy water (Deuterium).
• The Reaction: When a Calcium-50 nucleus hit a Deuterium nucleus (which is just a proton and a neutron stuck together), the Deuterium would "drop off" its extra neutron, which the Calcium-50 would catch.
• The Result: This turned Calcium-50 into Calcium-51. It's like catching a ball thrown at you; the way you catch it tells you about the ball's speed and spin.

2. The Detective Work: Missing Mass and Proton Tracks

The scientists didn't just watch the Calcium-51 appear; they watched what happened after.

• The Proton: When the neutron was transferred, a proton was kicked out. The scientists used a giant, high-tech "camera" (a silicon detector array called TiNA2) to catch these flying protons.
• The Missing Mass: By measuring exactly how much energy the proton had and where it went, they could calculate the "missing mass." Think of it like a detective solving a crime: "We know the total energy before the crash. We know the energy of the flying debris (the proton). The difference tells us exactly how much energy the new Calcium-51 building has."
• The Spectrum: This calculation revealed a "spectrum" (a list of energy levels). It showed that the new Calcium-51 atom didn't just land on the ground floor; it landed in specific excited rooms (energy states) at 1.7, 2.4, 3.5, and 4.2 MeV.

3. Decoding the Floor Plan: Spin and Shape

Now that they knew where the atom was, they needed to know what it looked like.

• The Angle: They looked at the angle at which the protons flew out. If a proton flies out at a shallow angle, it means the neutron landed in a "wide" room. If it flies out at a sharp angle, it landed in a "tall" or "twisted" room.
• The Verdict: By comparing their observations to computer simulations (like a video game physics engine), they confirmed the "spin" and "parity" (the orientation and shape) of these rooms.
  • They confirmed the ground floor is 3/2-.
  • They found a room at 1.7 MeV is 1/2-.
  • They found rooms at 2.4 and 3.5 MeV are 5/2-.
  • Most excitingly, they found a room at 4.15 MeV that looks like a 9/2+ state. This is a "high-rise" room where a neutron has jumped all the way up to a very high orbital (the $0g_{9/2}$ orbital).

4. Why This Matters: The "Quenching" Mystery

The scientists measured how "strong" the connection was between the neutron and the new room. They call this the spectroscopic factor.

• The Expectation: In a perfect, simple world, if a room is empty, a neutron should fill it with 100% strength.
• The Reality: They found the strength was "quenched" (weakened). It was only about 35-50% of what was expected.
• The Analogy: Imagine trying to park a car in a garage. In a simple model, the car fits perfectly. In reality, the garage is crowded with other invisible cars (complex quantum interactions), so your car only takes up half the space. This "crowding" is a sign that the nucleus is a very complex, interacting system, not just a simple stack of blocks.

5. The Big Picture: Testing the Architects

Finally, the team compared their real-world data with two different "architects" (theoretical models):

1. The Shell Model (GXPF1Br): A traditional model that treats the nucleus like a building with specific floors.
2. The Ab Initio Model (VS-IMSRG): A modern, "from-scratch" model that tries to calculate everything from the fundamental forces between particles.

The Result: Both models did a great job predicting the location of the rooms, but they struggled a bit with the strength of the connections. The data confirmed that the "magic number" at N=32 is real, but the transition to the next shell (N=40) is messy and complex.

Summary

This paper is a successful mission to map the interior of a strange, unstable atomic building. By firing neutrons at it and catching the debris, the scientists confirmed the layout of the rooms and proved that even in these extreme conditions, the "magic numbers" of the universe hold up, though the walls are a bit wobblier than we thought. This helps us understand how the heaviest elements in the universe are built and why some atoms are stable while others fall apart.

Technical Summary

1. Problem Statement

The neutron-rich calcium isotopes serve as a critical testing ground for modern nuclear structure theory, particularly regarding shell evolution and the emergence of new magic numbers (specifically $N=32$ and $N=34$) far from the valley of stability. While mass measurements and $\gamma$-ray spectroscopy have confirmed the existence of these shell closures, there remains a significant lack of experimental data concerning the single-particle structure in this region.

Specifically for the isotope $^{51}$Ca ($N=31$), previous studies provided conflicting or incomplete information:

• Early three-neutron transfer reactions failed to confirm low-lying states.
• $\beta$-decay studies and multinucleon transfer reactions suggested candidates for $1/2^-$ and $5/2^-$ states, as well as a positive-parity $9/2^+$ state, but spin-parity assignments were often tentative.
• The nature of the $9/2^+$ state was debated: is it a single-particle neutron excitation into the $0g_{9/2}$ orbital, or a proton-dominated octupole deformation coupled to the core?
• There is a need to benchmark theoretical models (Shell Model and ab initio approaches) with precise spectroscopic factors and angular distributions to understand the evolution of neutron orbitals ($1p_{3/2}$, $1p_{1/2}$, $0f_{5/2}$, $0g_{9/2}$) as the neutron number increases.

2. Methodology

The study employed a direct one-neutron transfer reaction in inverse kinematics to probe the single-particle character of $^{51}$Ca.

• Reaction: $^{50}\text{Ca}(d, p)^{51}\text{Ca}$.
• Facility: RI Beam Factory (RIBF) at RIKEN, utilizing the OEDO (Optimized Energy Degrading Optics) beamline and the SHARAQ magnetic spectrometer.
• Beam: A secondary beam of $^{50}$Ca was produced via projectile fragmentation of a $^{70}$Zn primary beam. The beam was decelerated to approximately 14 AMeV (14.2 AMeV in 2022, 13.1 AMeV in 2024) using a wedge-shaped degrader.
• Target: A CD$_2$ (deuterated polyethylene) target.
• Detection System:
  • TiNA2 Array: A silicon detector array (Double-Sided Silicon Strip Detectors - DSSSDs) positioned at backward angles ($103^\circ$–$170^\circ$) to detect outgoing protons.
  • SHARAQ Spectrometer: Used to detect and identify the heavy recoil nuclei ($^{51}$Ca) via magnetic rigidity ($B\rho$), time-of-flight, and energy loss in an ionization chamber.
• Data Analysis:
  • Missing Mass Spectroscopy: Excitation energies of $^{51}$Ca were reconstructed by combining the kinematics of the detected protons (TiNA2) and the recoiling nuclei (SHARAQ).
  • Angular Distributions: Differential cross-sections were extracted by binning data into angular intervals.
  • Theoretical Modeling: Experimental cross-sections were compared with Adiabatic Distorted Wave Approximation (ADWA) calculations using the FRESCO code to extract spectroscopic factors ($C^2S$).
  • Comparison: Results were benchmarked against Shell-Model calculations (GXPF1Br interaction) and ab initio Valence-Space In-Medium Similarity Renormalization Group (VS-IMSRG) predictions.

3. Key Contributions

• First Direct Spectroscopy: This work provides the first direct measurement of single-particle strength in $^{51}$Ca using the $(d,p)$ transfer reaction, a method highly sensitive to orbital angular momentum ($\Delta L$) and spectroscopic strength.
• Spin-Parity Confirmation: The study definitively confirms the spin-parity assignments for several excited states based on angular distribution shapes and ADWA fits.
• Resolution of the $9/2^+$ State: It provides strong evidence supporting the assignment of the state at 4.155 MeV as a $9/2^+$ state with a structure dominated by neutron excitation into the $0g_{9/2}$ orbital, rather than a proton-dominated octupole mode.
• Quantification of Quenching: The paper quantifies the spectroscopic factors, offering data on the "quenching" (reduction) of single-particle strength in neutron-rich nuclei compared to independent-particle model predictions.

4. Key Results

• Excitation Spectrum: Five bound states were identified and populated:
  1. Ground State ($0$ MeV): $3/2^-$ (confirmed).
  2. 1.718 MeV: $1/2^-$.
  3. 2.378 MeV: $5/2^-$ (previously proposed but not strongly populated in earlier transfer reactions; confirmed here as a weakly populated state).
  4. 3.478 MeV: $5/2^-$.
  5. 4.155 MeV: $9/2^+$.
• Spectroscopic Factors ($C^2S$):
  • The ground state ($3/2^-$) and the 1.718 MeV state ($1/2^-$) show strong single-particle character consistent with the $1p_{3/2}$ and $1p_{1/2}$ orbitals, respectively.
  • The 3.478 MeV state ($5/2^-$) corresponds to the $0f_{5/2}$ orbital but shows a reduced spectroscopic factor ($C^2S \approx 0.35$), indicating fragmentation or configuration mixing.
  • The 4.155 MeV state ($9/2^+$) has a spectroscopic factor of $0.15(3)$. The angular distribution ($\Delta L = 4$) strongly favors the $0g_{9/2}$ assignment. A $\Delta L = 1$ assignment (which would imply $1p$ orbitals) is ruled out as it would exhaust the available single-particle strength, contradicting the data.
• Comparison with Theory:
  • Shell Model (GXPF1Br): Shows remarkable agreement with experimental energies and spectroscopic factors. It predicts the $9/2^+$ state as a mix of single-particle and core-coupled components (coupling to the $5^-$ state in $^{50}$Ca), rather than the octupole $3^-$ coupling suggested by some previous inelastic scattering studies.
  • VS-IMSRG: Correctly predicts the spectroscopic strength distribution but significantly overestimates the excitation energy of the $9/2^+$ state (predicting ~7.9 MeV vs. observed 4.155 MeV).

5. Significance

• Constraints on Shell Evolution: The results provide new constraints on the evolution of the $0g_{9/2}$ orbital in neutron-rich calcium isotopes. The data indicates that the $9/2^+$ state in $^{51}$Ca is significantly higher in energy and has much weaker single-particle strength compared to its isotones (e.g., $^{55}$Cr, $^{57}$Fe), suggesting a breakdown of simple single-particle trends near $N=32$.
• Validation of Theoretical Models: The agreement between the experimental spectroscopic factors and the GXPF1Br shell model validates the use of large-scale shell-model calculations with extended model spaces ($sd-fp-sdg$) for describing neutron-rich nuclei.
• Core Excitations: The findings highlight the importance of complex core excitations (coupling to excited states of the $^{50}$Ca core) in shaping the level structure of $^{51}$Ca, particularly for states at higher excitation energies.
• Future Directions: These results serve as a benchmark for ab initio theories (like VS-IMSRG) to improve their predictions for excitation energies in the neutron-rich regime, a crucial step toward understanding the limits of nuclear existence near the neutron drip line.