Emerging collectivity and phase transition in mass A$\approx$150 region. New information for Nd isotopes

This study investigates low and medium spin excitations in $^{146,148,150,152}$Nd isotopes using Gammasphere data to identify 159 new levels and 305 new transitions, providing insights into emerging quadrupole collectivity and the role of the 11/2$^-$[505] neutron extruder in the A$\approx$150 phase transition region.

The Gist

Imagine the nucleus of an atom not as a static ball of clay, but as a bustling, energetic dance floor. Inside this tiny space, protons and neutrons are constantly moving, pairing up, and changing their formation. Sometimes, the whole nucleus wobbles, stretches, or spins in a coordinated way. This paper is like a high-resolution video recording of that dance floor, specifically focusing on a group of dancers called Neodymium (Nd) isotopes.

The researchers used a giant, ultra-sensitive camera array called Gammasphere to watch these atomic nuclei after they were "excited" (shaken up) by two methods: letting them decay naturally or smashing them apart in a nuclear fission experiment.

Here is the breakdown of what they found, translated into everyday concepts:

1. The "Extruder" and the Shape-Shifting

The central character in this story is a specific neutron (a particle in the nucleus) that acts like a mischief-maker or a "catalyst." The scientists call this the 11/2-[505] neutron extruder.

• The Analogy: Imagine a dance floor where the dancers usually move in a circle (spherical shape). Suddenly, one specific dancer (the extruder) decides to push the group into a different formation.
• What Happened: The researchers found that this "extruder" neutron is responsible for creating new, low-energy states in the nucleus. It's like this neutron grabs a pair of its partners and moves them to a different spot on the dance floor, causing the whole group to change shape from round to oval (prolate) or flat (oblate).
• The Discovery: They mapped out how this "extruder" helps the nucleus transition from a spherical shape to a deformed, rotating shape. This transition is what physicists call a Quantum Phase Transition—a sudden shift in the fundamental nature of the nucleus, similar to how water suddenly turns to ice.

2. The "New Dancers" (New Levels and Transitions)

Before this study, we only knew the choreography for the main dancers. This paper added 159 new levels (new positions the nucleus can be in), 305 new transitions (new ways the nucleus jumps between positions), and 83 new spin-parity assignments (figuring out exactly how the nucleus is spinning and oriented).

• The Analogy: It's like discovering 159 new dance moves and 305 new steps that the dancers can take.
• The Isomers: They found two new "sleeping" states called isomers. Think of these as dancers who get stuck in a pose and hold it for a while before relaxing.
  • In Neodymium-150, they found a dancer holding a pose for about 41 nanoseconds (billionths of a second).
  • In Neodymium-152, they found another holding a pose for about 42 nanoseconds.
  • They determined that these "sleeping" states are caused by two neutrons pairing up in a specific, high-energy configuration involving that "extruder" neutron.

3. The "Parabola" Mystery Solved

The researchers noticed something strange in the data from previous studies. The energy levels of certain excited states seemed to follow two wide, U-shaped curves (parabolas) that didn't quite make sense together.

• The Analogy: Imagine looking at a graph of dance heights and seeing two wide, blurry arches. You might think there are two different types of dancers.
• The Solution: The paper argues these "wide arches" are actually an optical illusion. When you look closer, you see that the arches are made of different types of dancers mixed together. Some are "spherical" (round), some are "oblate" (flat), and some are "prolate" (stretched). By separating them, the confusing wide arches break down into clear, distinct patterns.

4. The "Proton" Cousin

While the neutron "extruder" was the main star, the paper also suggests there might be a proton "extruder" (a specific proton acting similarly) that helps drive these shape changes, especially in heavier elements in this region. It's like having a second catalyst on the dance floor that helps the protons (the other half of the team) change their formation too.

5. Why This Matters (According to the Paper)

The paper concludes that the current ways we try to describe these atomic nuclei are missing the mark.

• The Problem: Some models treat the nucleus like a smooth, collective fluid (like a liquid drop), while others treat it like individual particles (like billiard balls).
• The Reality: The paper suggests the truth is a mix. The nucleus starts as individual particles (single-particle excitations), but as you add more dancers (neutrons), they start to "dress up" in collective costumes and move together.
• The Takeaway: To truly understand these atomic nuclei, we need a new approach that combines the behavior of individual particles with the emerging group dance. The "extruder" neutrons are the key to unlocking how this transition happens.

In summary: This paper is a detailed map of a specific atomic neighborhood. It identifies a key "instigator" neutron that triggers shape changes, discovers new "dance moves" (energy levels) and "pauses" (isomers), and argues that we need a new way of thinking about how these atomic nuclei move to understand the transition from round to stretched shapes.

Technical Summary

Technical Summary: Emerging Collectivity and Phase Transition in the Mass A≈150 Region

Problem and Motivation
The nuclear structure of the mass $A \approx 150$ region is characterized by a complex interplay between spherical and deformed configurations, often manifesting as coexisting nuclear structures. A central challenge in this region is understanding the nature of low-lying $0^+$ excitations, which are frequently interpreted as $\beta$ vibrations but whose existence in well-deformed nuclei remains debated. Recent studies in the $A \approx 100$ region suggest that many low-energy $0^+$ states arise from the excitation of nucleon pairs at crossings of up-sloping and down-sloping Nilsson orbitals, specifically involving "extruder" orbitals like the $\nu 9/2^+[404]$ neutron orbital.

In the $A \approx 150$ region, it has been proposed that the $\nu 11/2^-[505]$ neutron extruder plays a similar catalytic role in generating low-lying $0^+$ excitations. However, experimental data for Neodymium (Nd) isotopes, particularly in the lighter isotopes where collectivity begins to build up, has been incomplete. There is a need to verify and extend experimental information on Nd isotopes to clarify the emergence of quadrupole collectivity, the evolution of various collective modes, and the quantum phase transition occurring in this mass region.

Methodology
The authors investigated low and medium spin excitations in four Nd isotopes: $^{146}\text{Nd}$, $^{148}\text{Nd}$, $^{150}\text{Nd}$, and $^{152}\text{Nd}$. The nuclei were populated via two primary mechanisms:

1. $\beta^-$ decay of the corresponding Pr isotopes ($^{146,148,150,152}\text{Pr}$).
2. Prompt-$\gamma$ fission of $^{252}\text{Cf}$.

Experiments were conducted using the Gammasphere array of Compton-suppressed Ge spectrometers. The analysis relied on:

• Coincidence measurements: Double- and triple-$\gamma$ coincidences were used to construct partial level schemes and identify new transitions.
• Angular correlations: Determined for strong $\gamma\gamma$ cascades to assign spin and parity ($I^\pi$) to excited levels.
• Time-delayed analysis: Techniques involving time-delayed windows were employed to identify isomeric states and measure their half-lives.
• Systematics and Comparisons: Experimental results were compared with phenomenological classifications, existing systematics, and theoretical calculations (including Large-Scale Shell Model, confined $\beta$-soft models, and SD-pair shell models) reported in other works.

Key Results
The study significantly expanded the known level schemes for the four Nd isotopes:

• New Levels and Transitions: A total of 159 new levels, 305 new $\gamma$ transitions, and 83 new spin-parity assignments were added to the four nuclei.
• Isomers: Two new isomeric states were identified:
  • A 2285.4 keV state in $^{150}\text{Nd}$ with $T_{1/2} = 41(14)$ ns and tentative $I^\pi = (7^+)$.
  • A 2389.85 keV state in $^{152}\text{Nd}$ with $T_{1/2} = 42(8)$ ns and tentative $I^\pi = (8^-)$.
• $^{146}\text{Nd}$: The study confirmed $\gamma$ collectivity at $N=86$ by identifying a $\gamma$-band head at 1470.58 keV ($2^+$) and its associated band members. The 307.0 keV decay from the 1777.5 keV ($3^+$) level to the 1470.58 keV level was a key finding.
• $^{148}\text{Nd}$: A rotational band was firmly established on top of the $0^+_2$ level at 916.80 keV. The study revised spin-parity assignments for several levels, including the 1687.85 keV level ($4^+$) and the 1683.37 keV level ($2^+$), the latter being a candidate for a spherical seniority-type excitation.
• $^{150}\text{Nd}$: The ground-state band was extended to spin $(18^+)$. A new isomer at 2285.4 keV was identified. The study proposed tentative spin-parity assignments for several $\gamma$-band candidates.
• $^{152}\text{Nd}$: The band structure above the 2242.70 keV isomer was significantly revised and extended. The 2242.70 keV isomer was assigned $I^\pi = 7^+$, and a new isomer at 2389.85 keV was found. The study identified high-$K$ $\Delta I=1$ M1+E2 cascades.

Discussion and Interpretation
The authors analyzed the results through the lens of emerging collectivity and the role of specific single-particle orbitals:

• $0^+$ Excitations and the Extruder: The study proposes that the $\nu 11/2^-[505]$ neutron extruder is central to the formation of low-lying $0^+$ states. The mechanism involves a pair of neutrons being transferred from the extruder orbital to deformation-driving low-$\Omega$ (prolate) or high-$\Omega$ (oblate) orbitals. This explains the "wide parabolas" observed in $0^+_2$ energies as artifacts of mixing different structural origins (spherical 2-quasiparticle, oblate, and prolate collective configurations).
• $S_{2n}$ and Transfer Reactions: The authors introduced a local excess of two-neutron separation energy, $\Delta S_{2n}(N,Z)$, to highlight deviations from a linear trend. A sudden increase in $\Delta S_{2n}$ above $N=88$ correlates with the onset of deformation. This analysis, combined with $(p,t)$ and $(t,p)$ transfer reaction data, supports the proposed configuration exchange between $0^+_1$ and $0^+_2$ levels around $N=89$.
• Proton Extruder: Evidence suggests an analogous role for the $\pi 9/2^+[404]$ proton extruder, particularly in explaining anomalies in $0^+_2$ deformation in $^{150}\text{Nd}$ and $^{152}\text{Nd}$ compared to other isotones.
• $\gamma$ Collectivity: The study identifies two types of $\gamma$ bands in the region, differing in proton structure. Below $N=90$, $\gamma$ bands exhibit staggering characteristic of $\gamma$-soft structures, evolving into triaxial rotational bands at $N=90$.
• Two-Quasiparticle (2-qp) Isomers: The newly identified isomers in $^{150}\text{Nd}$ and $^{152}\text{Nd}$ are interpreted as 2-qp configurations involving the $\nu 11/2^-[505]$ extruder, specifically $\nu(11/2^-[505] \otimes 3/2^-[521])$ and $\nu(11/2^-[505] \otimes 5/2^+[642])$.

Significance and Claims
The paper claims to provide essential new experimental data to test nuclear models in the $A \approx 150$ region. Its primary contributions are:

1. Experimental Verification: It confirms the presence of the $\nu 11/2^-[505]$ extruder at the Fermi surface and its vital role in creating excited configurations, including the specific oblate structure of the 915.4 keV $0^+_2$ level in $^{146}\text{Nd}$.
2. Structural Classification: It proposes a classification of low-energy $0^+$ and $2^+$ excitations based on nucleon-pair excitations and the "catalytic" action of extruder orbitals, offering a consistent explanation for abrupt changes in transfer cross-sections around $N=90$.
3. Model Limitations: The authors note that current collective models tend to overestimate the collective nature of excitations in transitional nuclei, while Large-Scale Shell Model (LSSM) calculations struggle to properly describe $0^+$ excitations due to basis truncation.
4. Future Direction: The work suggests that a new theoretical approach is needed for transitional nuclei, one that combines nucleon-pair excitations at orbital crossings with emerging collectivity, rather than relying solely on collective modes or standard shell models.

The authors remain modest regarding definitive conclusions on specific spin-parity assignments for some tentative levels and emphasize the need for further dedicated $\beta$-decay measurements and transfer reaction studies to verify the proposed configurations and the role of the proton extruder.