Precision Tests of Isospin Symmetry through Coulomb excitation of A = 62 Nuclei

This study presents the most accurate test to date of isospin symmetry in the $A=62$ mass system by utilizing precision Coulomb excitation measurements of $^{62}$Zn, $^{62}$Ga, and $^{62}$Ge at RIKEN to confirm a linear relationship between their proton matrix elements, a finding supported by large-scale shell-model calculations.

The Gist

Imagine the atomic nucleus as a bustling, tiny city made of two types of citizens: Protons (who carry a positive electric charge) and Neutrons (who are electrically neutral).

For a long time, physicists have believed that inside this city, the "Strong Force" (the glue holding the city together) treats both citizens exactly the same, regardless of their charge. This idea is called Isospin Symmetry. It's like saying that if you swapped a proton for a neutron, the city's layout, its buildings, and its rules would remain perfectly unchanged.

However, protons repel each other because they are all positively charged (like magnets with the same pole facing each other), while neutrons don't. This electrical repulsion is a small "glitch" that might break the perfect symmetry. The big question for scientists is: Does this electrical glitch ruin the symmetry, or is the city still perfectly balanced?

The Experiment: A High-Stakes "Twin" Test

To answer this, the researchers in this paper decided to look at a specific family of nuclear "triplets" (three related nuclei) with a total weight of 62 units:

1. Zinc-62 (Lots of protons, fewer neutrons)
2. Gallium-62 (A mix)
3. Germanium-62 (Fewer protons, lots of neutrons)

Think of these three as identical triplets born into different families. They have the same "DNA" (nuclear structure), but their "family names" (number of protons) are different.

The Old Way vs. The New Way:
In the past, scientists studied these triplets one by one, using different machines, different settings, and different methods for each. It was like trying to compare the height of three people by measuring one with a ruler, the second with a tape measure, and the third by guessing. The differences in the tools made it hard to tell if the people were actually different heights or if it was just the tools lying.

This paper's breakthrough: The team used the RIKEN facility in Japan to shoot all three nuclei at targets at the exact same time, with the exact same machine, and under the exact same conditions.

It's like putting all three triplets on the same scale, in the same room, at the same moment. This cancels out all the "measurement errors" and lets them see the true differences (or lack thereof) between the nuclei.

The Test: The "Shape" of the Nucleus

The scientists wanted to see how these nuclei "wobble" or vibrate when hit. Specifically, they looked at how the protons inside move when the nucleus gets excited. They measured something called the E2 Matrix Element (a fancy number that tells us how "squishy" or deformable the nucleus is).

According to the rules of Isospin Symmetry, if you plot these numbers for the three triplets, they should form a perfectly straight line.

• If the line is straight, symmetry is perfect.
• If the line bends or breaks, symmetry is broken.

The Results: A Perfect Line

The results were stunning. When they plotted the data for Zinc, Gallium, and Germanium, the points fell on a perfectly straight line.

• The Analogy: Imagine drawing a line through three dots on a piece of paper. Usually, you might miss by a tiny bit. But here, the dots were so perfectly aligned that the line didn't wobble at all.
• The Conclusion: In this specific region of the nuclear chart (around mass 62), the "glitch" caused by electric charge is so small that the symmetry remains intact. The protons and neutrons are still playing by the same rules.

Why This Matters

The paper also looked at heavier nuclei (around mass 70). In those heavier families, the line did break. Why? Because those heavier nuclei are "shape-shifters." They are so big and floppy that they can change their shape easily (like a balloon vs. a marble). The electrical repulsion in those big, floppy nuclei causes them to deform differently, breaking the symmetry.

But the A=62 nuclei studied here are more like marbles—they are stiff and spherical. In these "stiff" nuclei, the symmetry holds strong.

The Takeaway

This experiment is like the most precise ruler ever invented. By using a unified method to measure three nuclear "siblings" simultaneously, the team proved that Isospin Symmetry is a very robust rule, at least for nuclei that aren't too big or too floppy.

They didn't just guess; they provided the most accurate proof to date that nature treats protons and neutrons as near-perfect twins, unless the nucleus gets so large and wobbly that the electric charge starts to pull the twins apart. This helps physicists build better models of how the universe's building blocks stick together.

Technical Summary

1. Problem Statement

The study addresses the fundamental question of isospin symmetry in atomic nuclei. While the strong nuclear force is assumed to be symmetric under isospin rotation (treating protons and neutrons equivalently), this symmetry is broken by the electromagnetic interaction (Coulomb force) and potentially by isospin-breaking components of the nuclear force.

• The Core Issue: In an isobaric multiplet (nuclei with the same mass number $A$ and total isospin $T$, but different isospin projections $T_z$), the reduced proton matrix elements ($M_p$) for electromagnetic transitions are predicted to scale linearly with $T_z$ if isospin symmetry is conserved.
• Previous Limitations: Previous tests of this linearity often utilized different experimental techniques for the three members of a triplet (e.g., lifetime measurements for one, Coulomb excitation for another). This introduced significant systematic uncertainties that obscured the precision of the test. Furthermore, deviations from linearity had been observed in heavier systems (e.g., $A=70$), potentially linked to nuclear deformation, but a high-precision baseline in the $A=62$ region was lacking.

2. Methodology

The authors conducted a unified experiment at the RIKEN Radioactive Isotope Beam Factory (RIBF) to study the $A=62$, $T=1$ triplet: $^{62}\text{Ge}$ ($T_z = +1$), $^{62}\text{Ga}$ ($T_z = 0$), and $^{62}\text{Zn}$ ($T_z = -1$).

• Experimental Setup:

  • Beam Production: A stable $^{78}\text{Kr}$ primary beam (300 pnA, 345 AMeV) was fragmented on a Be target. The resulting secondary beams were purified and selected using the BigRIPS separator and ZeroDegree spectrometer.
  • Unified Conditions: Crucially, all three isotopes were studied under identical experimental conditions (beam energy $\approx$ 150 AMeV, same targets, same detection setup). This allows for the cancellation of nearly all systematic uncertainties in relative comparisons.
  • Targets: Two targets were used to disentangle nuclear and Coulomb effects: a heavy $^{197}\text{Au}$ target (dominated by Coulomb excitation) and a light $^{12}\text{C}$ target (mixed Coulomb and nuclear).
  • Detection: Gamma rays from the inelastic scattering ($2^+_1 \to 0^+_1$ transition) were detected using the DALI2+ array (226 NaI(Tl) crystals). Doppler correction was applied using ion velocity and scattering angle measurements.

• Analysis Framework:

  • Cross Sections: Inelastic scattering cross sections ($\sigma$) were extracted by fitting $\gamma$-ray yields with Geant4-simulated response functions.
  • Extraction of Matrix Elements: Using the distorted-wave coupled-channels code Fresco, the measured cross sections were analyzed to extract:
    1. Nuclear deformation lengths ($\delta_N$).
    2. Proton E2 matrix elements ($M_p$).
  • Theoretical Interpretation: Large-scale shell-model calculations were performed using the KB3GR interaction in the $pf$ model space (using the ANTOINE code) and compared with EXVAM (beyond-mean-field) calculations.

3. Key Contributions

• Unprecedented Precision: By measuring all three triplet members with the same setup, the authors achieved a relative precision of 1.2% in testing the linearity of $M_p$. This represents a fourfold improvement over previous tests on lower-mass triplets.
• Systematic Uncertainty Cancellation: The study demonstrates that using identical conditions for the entire triplet effectively eliminates reaction modeling uncertainties and detection efficiency variations that plagued previous comparative studies.
• Direct Probe of Wave Functions: Unlike energy-based tests (e.g., Isobaric Multiplet Mass Equation), this method probes the transition matrix elements directly, offering a sensitive test of the wave function composition and isospin purity.

4. Key Results

• Linearity of $M_p$: The extracted proton matrix elements for the $0^+_1 \to 2^+_1$ transition are:
  • $^{62}\text{Ge}$: $38.1(24)$ efm$^2$
  • $^{62}\text{Ga}$: $37.5(31)$ efm$^2$
  • $^{62}\text{Zn}$: $37.4(23)$ efm$^2$
  • Finding: These values lie on a straight line within experimental uncertainties, confirming the linear relationship $M_p(T_z) = \frac{1}{2}(M_0 - M_1 T_z)$.
• Isoscalar and Isovector Components: The data yields an isoscalar component $M_0 \approx 75.4$ efm$^2$ and an isovector component $M_1 \approx +0.6$ efm$^2$, consistent with systematics.
• Quadratic Deviation: A quadratic fit ($M_p = a + bT_z + cT_z^2$) yielded a coefficient $c$ consistent with zero, providing the most stringent test to date that no significant isospin-breaking quadratic terms exist in this mass region.
• Comparison with Theory:
  • Large-scale shell-model calculations (KB3GR) with microscopically derived effective charges successfully reproduced the experimental linearity and magnitude.
  • The results contrast with the $A=70$ triplet, where a $3\sigma$ deviation from linearity was observed (attributed to shape coexistence and deformation in $^{70}\text{Kr}$). The $A=62$ nuclei are described as weakly collective and well-modeled by the spherical shell model.

5. Significance

• Validation of Isospin Symmetry: The study provides the most accurate confirmation to date that isospin symmetry is preserved in weakly collective nuclei, with deviations limited to the percent level.
• Benchmark for Theory: The high-precision data serves as a critical benchmark for nuclear structure models, specifically validating the KB3GR interaction and effective charges in the $pf$ shell.
• Deformation vs. Symmetry: The results suggest that the deviations from isospin symmetry observed in heavier nuclei (like $A=70$) are driven by nuclear deformation effects (shape changes/coexistence) rather than intrinsic isospin-breaking forces in the wave function.
• Future Directions: The methodology establishes a framework for future high-precision tests in intermediate mass regions (e.g., $A=66$) and highly collective systems (e.g., $A=74$) to further map the interplay between isospin symmetry and nuclear deformation.

In conclusion, this paper establishes a new standard for precision in nuclear spectroscopy, proving that isospin symmetry holds robustly in the $A=62$ region when systematic errors are rigorously controlled, and highlighting deformation as the primary driver for symmetry breaking in more complex nuclei.