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Oblate-prolate shape mixing and E0 transition in 28Si

Using antisymmetrized molecular dynamics combined with the generator coordinate method, this study constrains the oblate-prolate shape mixing amplitudes in $^{28}$ Si by reproducing experimental observables, revealing a dominant oblate component in the ground state and establishing an upper limit for the inter-band E0 transition strength.

Original authors: Yasutaka Taniguchi, Masaaki Kimura

Published 2026-06-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yasutaka Taniguchi, Masaaki Kimura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Shape-Shifting Atom

Imagine the nucleus of a silicon-28 atom ( $^{28}\text{Si}$ ) not as a hard, rigid ball, but as a blob of soft, squishy playdough. In nuclear physics, this "playdough" can change its shape. It can be flattened like a pancake (this is called oblate) or stretched out like a rugby ball or a cigar (this is called prolate).

For decades, scientists have known that this specific silicon nucleus is weird. It seems to exist in a state where it’s mostly a pancake, but it also has a hidden tendency to want to be a rugby ball. The big question is: How much does it actually mix these two shapes? Is it 99% pancake and 1% rugby ball? Or is it a 50/50 mix?

This paper tries to answer that question by looking at the "footprints" the nucleus leaves behind—specifically, how it interacts with electricity and how big it is.

The Problem: The Theory Doesn’t Match the Reality

The researchers used a sophisticated computer simulation (called AMD+GCM) to model the nucleus. Think of this simulation as a high-tech recipe for making the nucleus.

When they ran the recipe, they got two distinct results:

A "Pancake State" (Oblate)
A "Rugby Ball State" (Prolate)

However, the simulation showed that these two states barely talked to each other. In the real world, experiments show that the nucleus does switch between these shapes (we know this because of how it emits energy, called E2 transitions). The simulation was too "stiff"—it didn’t allow enough mixing.

So, the authors decided to stop trying to force the computer to predict the mixing perfectly. Instead, they said, "Let’s treat the mixing amount as a mystery variable, and we’ll use real-world data to solve for it."

The Method: The Detective Work

The researchers treated the "Pancake" and "Rugby Ball" states as basic building blocks. They then created a "mixed" state by blending them together in different proportions. Imagine mixing red paint (Pancake) and blue paint (Rugby Ball) to get purple. How much red and how much blue do you need to get the exact shade of purple that nature actually uses?

To find the right mix, they looked at four specific clues from experiments:

The Size (Charge Radius): How big is the nucleus?
The Squashiness (Quadrupole Moment): How flattened or stretched is it?
The Internal Dance (In-band E2): How easily does it spin within its main shape?
The Shape Switch (Inter-band E2): How easily does it jump from the "Pancake" shape to the "Rugby Ball" shape?

They also tweaked a setting in their computer model (called $c_3$ ) to make sure the calculated size matched the real size, because the original model tended to make the nucleus too big.

The Results: Mostly Pancake, Maybe a Little Rugby

After crunching the numbers and seeing which mix of "Pancake" and "Rugby Ball" fit all four experimental clues, they found:

The Ground State (The most stable state): The nucleus is dominated by the Pancake shape. The "Rugby Ball" component is small—less than 20%. So, if you look at a silicon-28 nucleus sitting quietly, it’s mostly flat.
The Excited State ( $2^+_1$ ): When the nucleus is slightly energized, it becomes even more pancake-like. The Rugby Ball component drops to less than 7%.

The Mystery of the "E0" Transition

There is a specific type of energy shift called an E0 transition. This is like a "shape echo." If the nucleus changes its shape without spinning, it creates this signal. The strength of this signal tells us exactly how different the two shapes are and how much they mix.

The paper calculates what this signal should be based on their mixing limits. They found that the signal is likely weak, with an upper limit of about 0.206 (a dimensionless number called $\rho^2$ ).

However, the authors admit this isn't a precise prediction. It’s more like saying, "The signal is definitely not huge, but we can't pin down the exact number yet."

The Conclusion

The paper concludes that the low-lying states of silicon-28 are indeed a mix of pancake and rugby ball shapes, but the pancake wins by a landslide.

The authors suggest that if experimentalists could measure that elusive E0 transition directly, it would be the "smoking gun" that finally tells us the exact percentage of mixing. Until then, we know it’s mostly pancake, with a small hint of rugby ball.

Technical Summary: Oblate-prolate shape mixing and E0 transition in $^{28}$ Si

Problem Statement
The $^{28}$ Si nucleus is a canonical example of oblate-prolate shape coexistence. While the positive quadrupole moment of the $2^+_1$ state indicates that the ground band is predominantly oblate, theoretical calculations consistently suggest a prolate deformation for the $K^\pi = 0^+_3$ band built on the $0^+_3$ state. A critical unresolved issue is the degree of quantum mechanical mixing between these oblate and prolate configurations. Although the observed finite inter-band $E2$ transition strength, $B(E2; 0^+_3 \rightarrow 2^+_1)$ , implies shape mixing, previous microscopic calculations (including shell-model and Antisymmetrized Molecular Dynamics (AMD)) have failed to reproduce this value, leaving the mixing amplitudes poorly constrained. Consequently, the inter-band $E0$ transition strength, the most direct probe of shape mixing, remains unmeasured and theoretically uncertain.

Methodology
The authors adopt a phenomenological approach to constrain the mixing amplitudes rather than predicting them solely from an effective interaction. The methodology involves the following steps:

Basis State Generation: Oblate- and prolate-dominant basis states ( $0^+_{ob}, 2^+_{ob}, 0^+_{pr}, 2^+_{pr}$ ) are generated using Antisymmetrized Molecular Dynamics (AMD) combined with the Generator Coordinate Method (GCM). The Gogny D1S interaction is used, with a scaling parameter $c_3$ introduced for the density-dependent term to address systematic overestimations of charge radii in light nuclei.
Mixed State Construction: The physical states ( $0^+_1, 0^+_3, 2^+_1$ $0_{1}^{+}, 0_{3}^{+}, 2_{1}^{+}$ ) are constructed as linear combinations of the basis states, parameterized by mixing angles $\theta_0$ $θ_{0}$ and $\theta_2$ $θ_{2}$ :
- $|0^+_1\rangle = \cos\theta_0 |0^+_{ob}\rangle + \sin\theta_0 |0^+_{pr}\rangle$
- $|0^+_3\rangle = -\sin\theta_0 |0^+_{ob}\rangle + \cos\theta_0 |0^+_{pr}\rangle$
- $|2^+_1\rangle = \cos\theta_2 |2^+_{ob}\rangle + \sin\theta_2 |2^+_{pr}\rangle$
Constraint Procedure: The mixing angles and the scaling factor $c_3$ $c_{3}$ are treated as free parameters. The allowed parameter space is determined by simultaneously reproducing four experimental observables within their uncertainties:
- The ground state charge radius.
- The quadrupole moment of the $2^+_1$ state.
- The in-band $B(E2; 2^+_1 \rightarrow 0^+_1)$ value.
- The inter-band $B(E2; 0^+_3 \rightarrow 2^+_1)$ value.

Key Contributions and Results

Basis State Properties: The direct AMD+GCM calculation reproduces the coexistence of oblate and prolate bands but yields negligible inter-band transition strengths, confirming that the basis states are approximately pure configurations with insufficient intrinsic mixing.
Constraint on Mixing Amplitudes: By analyzing the overlap of constraints imposed by the experimental data, the authors determine the allowed region for the mixing parameters.
- Ground State ( $0^+_1$ ): The analysis yields $\cos^2\theta_0 \gtrsim 0.8$ . This indicates that the oblate component is dominant, and the prolate component in the ground state is limited to less than approximately 20%.
- First Excited State ( $2^+_1$ ): The allowed prolate component is even smaller, less than 7%.
- Scaling Factor: The density-dependent term scaling factor is constrained to the range $0.96 \lesssim c_3 \lesssim 0.98$ .
Estimation of E0 Strength: Using the allowed mixing amplitudes, the authors estimate the dimensionless inter-band $E0$ transition strength, $\rho^2(E0; 0^+_3 \rightarrow 0^+_1)$ . The constraints do not provide a sharp prediction but establish an upper limit of $\rho^2(E0; 0^+_3 \rightarrow 0^+_1) \lesssim 0.206$ .

Significance and Conclusions
The paper concludes that the low-lying $0^+$ states of $^{28}$ Si exhibit substantial oblate-prolate mixing, quantified by the derived limits on the mixing angles. The primary significance of this work lies in providing a quantitative framework to constrain these mixing amplitudes using existing experimental data. The authors emphasize that while the current analysis provides an upper limit for the $E0$ transition strength, it does not tightly constrain it. Therefore, a future experimental measurement of the inter-band $E0$ transition strength would provide a crucial test for the two-state mixing picture and allow for a more precise determination of the mixing amplitude.