Multiple shape coexistence near Sn118: First 03+ lifetime measurement

This paper reports the first measurement of the $0^+_3$ state lifetime in $^{118}$Sn using thermal-neutron capture, where the observed enhanced $E0$ transition strength provides compelling evidence for multiple shape coexistence, a finding corroborated by theoretical calculations predicting three distinct nuclear shapes in the Sn isotopes.

The Gist

Imagine the nucleus of an atom as a tiny, bustling dance floor. Usually, in certain "semi-magic" atoms like Tin (Sn), the dancers (protons and neutrons) prefer to stand still in a perfect, round circle. This is their comfortable, spherical shape.

However, this new research reveals that in the Tin isotopes around mass 118, the dance floor is actually hosting a wild party where three different dance styles are happening at the exact same time. This phenomenon is called "multiple shape coexistence."

Here is a simple breakdown of what the scientists found:

1. The Mystery of the "Intruder" Dancers

For a long time, physicists knew that Tin atoms could sometimes switch from a round shape to a stretched-out, football-like shape (called "prolate"). They thought this was just a simple switch between two styles: Round vs. Football.

But there was a missing piece of the puzzle. In the Tin-118 atom, there is a specific excited state (a high-energy dance move) called the $0^+_3$ state. Scientists knew it existed, but they didn't know how long it lasted before it changed. Without knowing its "lifetime," they couldn't tell if it was a distinct third dance style or just a messy mix of the other two.

2. The Stopwatch Experiment

To solve this, the team went to a research reactor in France (the Institut Laue-Langevin). They acted like high-speed photographers.

• The Setup: They bombarded a target of Tin-117 with slow-moving neutrons. When a neutron was caught, it turned the atom into Tin-118 in a super-excited state.
• The Race: As the atom calmed down, it emitted gamma rays (flashes of light). The scientists used special detectors (LaBr3 crystals) that act like incredibly fast stopwatches.
• The Measurement: They timed exactly how long the mysterious $0^+_3$ state existed before it decayed. They found it lasted about 74 picoseconds (that's 0.000000000074 seconds).

3. The "Shape-Shifter" Evidence

Knowing the lifetime allowed them to calculate how much the atom's shape changed during its transitions.

• The Analogy: Imagine two dancers. If they are very similar, they can switch places easily without much effort. If they are very different (one is a ballerina, the other is a breakdancer), switching between them is a huge, dramatic leap.
• The Result: The scientists measured a massive "jump" (called an E0 transition strength) between the second excited state and the third excited state. This huge jump proved that these two states are fundamentally different shapes.

4. The Three-Way Coexistence

By combining their new stopwatch data with advanced computer simulations (which act like a virtual physics lab), they concluded that Tin-118 (and its neighbors Tin-116 and Tin-120) isn't just switching between two shapes. It is hosting three distinct shapes simultaneously:

1. The Spherical Ball: The normal, round ground state.
2. The Prolate Football: A stretched-out shape (like a rugby ball).
3. The Oblate Pancake: A flattened shape (like a pancake or a frisbee).

The paper suggests that the "intruder" states (the excited ones) are actually these different shapes fighting for dominance. The $0^+_2$ state is mostly the "Football," while the newly measured $0^+_3$ state is likely the "Pancake."

Why This Matters

This is a rare discovery. Usually, we only see two shapes coexisting in a nucleus. Finding three in the same atom is like finding a single room where a round table, a long dining table, and a flat coffee table are all occupying the same space at once.

The researchers used a sophisticated computer model (called the Generator Coordinate Method) to confirm this. The model showed that these three shapes naturally emerge from the way protons and neutrons interact, without needing to tweak the math to make it fit.

In short: The scientists finally timed a fleeting atomic state, and that timing proved that Tin atoms are not just round or stretched; they are complex shape-shifters capable of holding three distinct forms at the same time.

Technical Summary

Technical Summary: Multiple Shape Coexistence near $^{118}$Sn: First $0^+_3$ Lifetime Measurement

Problem Statement
The semi-magic Tin (Sn) isotopes ($Z=50$) are a benchmark system for nuclear structure studies, particularly regarding shape coexistence. In the neutron mid-shell region ($^{112-118}$Sn), excited $0^+$ states around 2 MeV are interpreted as arising from proton two-particle-two-hole (2p-2h) excitations across the $Z=50$ shell gap, forming intruder bands with prolate deformation distinct from the spherical ground state. While enhanced electric monopole (E0) transitions between $0^+$ band heads serve as spectroscopic signatures of shape mixing, a complete understanding of the coexistence in $^{118}$Sn has been hindered by incomplete lifetime data. Specifically, the lifetime of the third excited $0^+$ state ($0^+_3$) in $^{118}$Sn was previously unknown, with only broad limits ($60 \text{ ps} < \tau < 290 \text{ ps}$) available. This lack of precise lifetime information prevented the calculation of the E0 transition strength $\rho^2(E0; 0^+_3 \to 0^+_2)$, leaving the nature of the mixing between the $0^+_2$ and $0^+_3$ states and the potential existence of multiple distinct shapes unresolved.

Methodology
To address this gap, the authors performed a direct lifetime measurement of the $0^+_3$ state in $^{118}$Sn using the fast-timing technique following thermal-neutron capture.

• Experiment: The experiment was conducted at the Institut Laue-Langevin (ILL) using the FIssion Product Prompt gamma-ray Spectrometer (FIPPS). A powder target enriched to 89.2% $^{117}$Sn was irradiated with thermal neutrons.
• Detection: The setup utilized 15 $LaBr_3$ fast-timing detectors (positioned at $\approx 45^\circ$ and $135^\circ$) for timing measurements and 8 Compton-suppressed High-Purity Germanium (HPGe) clover detectors (at $90^\circ$) for energy resolution.
• Data Analysis: States were populated via the $(n, \gamma)$ reaction. The lifetime was extracted using the Generalized Centroid Difference (GCD) method. The analysis focused on triple-gamma coincidences ($\gamma\gamma\gamma$) to isolate the decay cascade from the capture state ($S_n = 9326$ keV) down to the $0^+_3$ state and its subsequent decay. By gating on the primary $\gamma$ ray (4765 keV) and specific feeder/decay transitions (2504 keV and 827 keV), the authors minimized background. The centroid difference ($\Delta C$) between "delayed" and "anti-delayed" time distributions was measured and corrected for the prompt response difference (PRD) to extract the lifetime.
• Theoretical Modeling: To interpret the experimental results, the authors employed the Multi-Reference Covariant Density Functional Theory (MR-CDFT) combined with the Generator Coordinate Method (GCM). This approach uses a relativistic energy density functional without adjusting free parameters to experimental excitation energies. It constructs collective wave functions as superpositions of symmetry-conserving mean-field states with varying quadrupole deformations ($\beta_2$) to describe shape mixing and coexistence.

Key Results

• Lifetime Measurement: The lifetime of the $0^+_3$ state in $^{118}$Sn was determined to be $\tau(0^+_3) = 74(13)$ ps. This value was derived from a weighted average of measurements involving three different cascades (primary $\gamma$ rays at 4765, 4407, and 4972 keV).
• Transition Strengths: Using the measured lifetime, the reduced transition probability $B(E2; 0^+_3 \to 2^+_1)$ was calculated as $0.80(14)$ W.u. Combined with existing $X(E0/E2)$ data, the E0 transition strength was determined:
  • $\rho^2(E0; 0^+_3 \to 0^+_1) = 2.0(5) \times 10^{-3}$
  • $\rho^2(E0; 0^+_3 \to 0^+_2) = 150(30) \times 10^{-3}$ (150(30) milliunits).
• Theoretical Agreement: The MR-CDFT calculations successfully reproduced the low-lying energy spectra and transition strengths for $^{116,118,120}$Sn. The calculations revealed that the $0^+_1$ states are nearly spherical ($\pi 1p-1h$), the $0^+_2$ states are dominated by prolate shapes ($\pi 4p-4h$ in $^{116,118}$Sn), and the $0^+_3$ states correspond to oblate shapes (evolving from $\pi 2p-2h$ to $\pi 3p-3h$). The theoretical results confirmed a large shape difference ($\Delta \beta_2 > 0.24$) between the $0^+_2$ and $0^+_3$ states.

Significance and Claims
The primary significance of this work is the first experimental confirmation of multiple shape coexistence in the semi-magic $^{118}$Sn nucleus.

• Evidence for Three Shapes: The large observed $\rho^2(E0; 0^+_3 \to 0^+_2)$ value of 150(30) milliunits provides compelling evidence for strong mixing between states of significantly different shapes. Combined with the known spherical ground state and the prolate intruder band, the authors propose the coexistence of three distinct shapes: spherical, prolate, and oblate.
• Systematic Trend: This finding extends the observation of multiple shape coexistence to the neighboring $^{116}$Sn and $^{120}$Sn isotopes, suggesting a systematic phenomenon across the neutron mid-shell region of Sn isotopes.
• Model Validation: The study highlights the necessity of including proton excitations across the $Z=50$ closed shell and shape fluctuations in theoretical models. The success of the MR-CDFT approach in reproducing these complex transition strengths contrasts with shell-model studies using a $Z=50$ core, which fail to capture these features.
• Broader Context: The authors suggest that multiple shape coexistence (involving three or more shapes) may be more common than previously recognized, drawing parallels with similar phenomena observed in Ni, Pb, and Cd isotopes.

The paper concludes that while the MR-CDFT reproduces transition strengths well, the overestimation of excitation energies suggests a need for future refinements, such as including time-odd components or explicit non-collective quasi-particle configurations. The authors note that further experimental work, such as fusion-evaporation reactions to search for rotational bands built on the $0^+_3$ states and Coulomb excitation studies to map the $(\beta_2, \gamma)$ plane, is required to firmly establish the multiple shape coexistence scenario.