Effects of shape coexistence and configuration mixing on low-lying states in tellurium isotopes

This paper utilizes the interacting boson model with configuration mixing, informed by microscopic mean-field calculations, to demonstrate that strong mixing between intruder prolate and normal oblate configurations drives the parabolic shape-coexistence behavior observed in the low-lying states of even-even tellurium isotopes near the middle of the neutron major shell.

The Gist

Imagine a nucleus not as a static, hard marble, but as a squishy, living blob of dough. Sometimes this dough is perfectly round (spherical), sometimes it's squashed flat like a pancake (oblate), and sometimes it's stretched out like a rugby ball (prolate).

For decades, physicists thought that nuclei near a certain "magic number" of protons (50) were like perfect, vibrating marbles. They thought they just wiggled in place. But recent experiments showed that these nuclei are actually much more dramatic: they can change their shape entirely, existing in a state of shape coexistence. It's like a chameleon that can be a round ball and a stretched-out sausage at the same time, or a person who can be both a calm librarian and a wild rock star simultaneously.

This paper by Kosuke Nomura investigates Tellurium (Te) isotopes—specifically the ones in the middle of the neutron range—to see how this "shape-shifting" works.

Here is the breakdown of the research using simple analogies:

1. The Two "Personalities" of the Nucleus

The researchers found that Tellurium nuclei have two distinct "personalities" or configurations:

• The Normal Personality (Oblate): This is the default state, like a slightly flattened pancake. It's the "ground state" or the most comfortable position for the nucleus.
• The Intruder Personality (Prolate): This is an excited state where the nucleus stretches out into a rugby ball shape. It's called an "intruder" because it comes from a different energy level, like a guest crashing a party.

In the middle of the neutron range (around Tellurium-116 to 120), these two personalities are fighting for dominance. They are so close in energy that the nucleus can't decide which shape to be, so it mixes them.

2. The Computer Simulation: The "Map Maker"

To understand this, the author didn't just guess; he used a sophisticated computer simulation. Think of it like a GPS map maker.

• Step 1 (The Terrain): First, he used a high-tech model (Relativistic Hartree-Bogoliubov) to map out the "terrain" of the nucleus. This map showed two valleys: a deep valley for the pancake shape and a slightly higher, nearby valley for the rugby ball shape.
• Step 2 (The Translation): He then translated this complex quantum map into a simpler language called the Interacting Boson Model (IBM). Imagine taking a complex 3D topographical map and turning it into a simple board game where you move pieces (bosons) around to represent the nucleus's energy.
• Step 3 (The Mixing): Crucially, he didn't just look at one valley. He allowed the game pieces to jump between the "pancake valley" and the "rugby ball valley." This is the Configuration Mixing.

3. The Results: Why Mixing Matters

When the researchers ran the simulation without mixing (only looking at the pancake shape), the results didn't match reality. The predicted energy levels were wrong, and the nucleus seemed too rigid.

However, when they turned on the mixing (allowing the nucleus to be both shapes at once), the simulation suddenly matched the experimental data perfectly.

• The "Parabolic" Curve: The energy levels of the excited states formed a U-shape (parabola). This is the "smoking gun" signature of shape coexistence. It's like seeing a ball roll down into a valley and then back up; the lowest point of that curve tells us exactly where the two shapes are mixing the most.
• The "Intruder" is Key: They found that the "intruder" rugby-ball shape isn't just a rare visitor; it's a major player. In many of these nuclei, the excited states are actually made of a huge chunk of this intruder shape mixed with the normal shape.

4. The Electromagnetic Clues

How do we know this mixing is real? The researchers looked at how the nucleus emits energy (light/radio waves) when it changes states.

• The "Flash" Test: When the nucleus jumps from the mixed state to a lower state, it flashes a specific amount of energy (an electric quadrupole transition).
• The simulation showed that because the nucleus is a mix of two shapes, these flashes are much brighter and more frequent than if the nucleus were just one shape. This "brightness" confirmed that the two shapes are indeed dancing together.

5. The Limitations: The Map Isn't Perfect

The paper also admits that the map isn't perfect.

• The EDF Problem: The "terrain" (the energy map) depends on the specific mathematical rules (Energy Density Functional) used to build it. The author tried two different rulebooks (DD-PC1 and SLy6). While both showed shape coexistence, they disagreed on exactly which shape was the "winner" (the ground state) for heavier Tellurium isotopes.
• The "Anomaly": There was one specific nucleus (Tellurium-114) where the experiment showed a weird drop in energy emission that the computer model couldn't explain. It's like the GPS saying "Turn Left," but the car actually turned Right. This suggests that while the model is great, it might be missing a tiny, subtle ingredient (perhaps a specific type of wobble or triaxiality) in its recipe.

The Big Picture Takeaway

This paper confirms that Tellurium isotopes are shape-shifters. They aren't just vibrating marbles; they are complex systems where two different shapes (flat and stretched) coexist and mix together.

Why does this matter?
Understanding this mixing helps physicists build a "Periodic Table of Nuclear Shapes." It tells us that as we add or remove neutrons, nuclei don't just get bigger or smaller; they fundamentally change their personality. This research helps us predict how unstable nuclei behave, which is crucial for understanding how stars explode (supernovae) and how heavy elements are forged in the universe.

In short: The nucleus is a chameleon. This paper successfully used a computer to prove that Tellurium is wearing two costumes at once, and that wearing both costumes is actually the key to understanding its behavior.

Technical Summary

Here is a detailed technical summary of the paper "Effects of shape coexistence and configuration mixing on low-lying states in tellurium isotopes" by Kosuke Nomura.

1. Problem Statement

The low-energy structure of even-even nuclei near the proton major shell closure ($Z=50$), specifically the Tellurium (Te) isotopes, has historically been interpreted as exhibiting vibrational multiphonon excitations. However, recent experimental data suggests this interpretation is insufficient.

• The Issue: Many Te isotopes, particularly those in the middle of the neutron major shell ($N=50$ to $N=82$), exhibit low-lying excited $0^+$ states and non-yrast bands that cannot be explained by a pure vibrational model.
• The Hypothesis: These anomalies are likely due to shape coexistence, where different intrinsic nuclear shapes (e.g., oblate vs. prolate) coexist at low energies. This is driven by intruder excitations (proton particle-hole excitations across the $Z=50$ shell gap).
• The Gap: While shape coexistence is well-established in nuclei below $Z=50$ (like Cd and Sn) and near $Z=82$ (Pb and Hg), its systematic presence and the specific role of configuration mixing in Te isotopes ($Z=52$) require further theoretical clarification using a consistent microscopic framework.

2. Methodology

The authors employ a microscopic-macroscopic approach combining the Self-Consistent Mean-Field (SCMF) method with the Interacting Boson Model with Configuration Mixing (IBM2-CM).

• Step 1: Microscopic Input (SCMF):

  • Calculations are performed using the Relativistic Hartree-Bogoliubov (RHB) model.
  • The energy density functional (EDF) used is the density-dependent point-coupling interaction (DD-PC1).
  • A separable pairing force is applied.
  • Constrained calculations are performed on mass quadrupole moments to generate Potential Energy Surfaces (PES) in terms of quadrupole deformation ($\beta$) and triaxiality ($\gamma$).

• Step 2: Mapping to IBM:

  • The RHB-derived PESs are mapped onto the neutron-proton Interacting Boson Model with Configuration Mixing (IBM2-CM).
  • Configuration Space: The model space includes two configurations:
    1. Normal (0p-0h): Corresponding to the ground-state configuration (typically oblate in midshell Te).
    2. Intruder (2p-2h): Corresponding to proton excitations across the $Z=50$ shell (typically prolate).
  • Hamiltonian Construction:
    • The IBM2-CM Hamiltonian is a $2 \times 2$ matrix in the boson coherent state space.
    • Diagonal elements represent the unperturbed normal and intruder Hamiltonians.
    • Off-diagonal elements represent the mixing interaction ($\hat{V}_{mix}$).
    • Key Parameter Adjustment: The energy offset ($\Delta$) between configurations is determined not just by mean-field energy differences, but by accounting for quantum correlation energies to ensure the ground state remains the normal configuration (avoiding the "intruder ground state" problem seen in previous Pb studies).

• Step 3: Spectroscopic Calculations:

  • The Hamiltonian is diagonalized to obtain energy levels and wave functions.
  • Electromagnetic transition rates ($B(E2)$, $B(M1)$, $B(E0)$) and moments ($Q$, $\mu$) are calculated using effective charges and $g$-factors.
  • Sensitivity Analysis: The study also tests the dependence of results on the underlying EDF by comparing DD-PC1 (RHB) with Skyrme SLy6 (HF+BCS) inputs.

3. Key Contributions

• Systematic Microscopic Mapping: This work provides a systematic, parameter-free (except for effective charges) description of Te isotopes ($^{108-126}\text{Te}$) using EDF-mapped IBM2-CM, avoiding phenomenological fitting of Hamiltonian parameters to experimental spectra.
• Resolution of the $\Delta$ Parameter Issue: The authors refine the method for determining the energy offset ($\Delta$) between normal and intruder configurations. They demonstrate that using mean-field energy differences alone leads to incorrect ground states; instead, they adjust $\Delta$ to reproduce the experimental $0^+_2$ excitation energy, ensuring the correct hierarchy of states.
• Identification of Shape Coexistence: The study confirms that shape coexistence is a dominant feature in midshell Te isotopes ($N \approx 66$), driven by the mixing of oblate (normal) and prolate (intruder) configurations.

4. Key Results

A. Potential Energy Surfaces (PES)

• The RHB-SCMF calculations predict a transition from prolate to oblate shapes as neutron number increases.
• Coexistence: In transitional nuclei ($^{114-120}\text{Te}$), the PES exhibits a pronounced oblate global minimum ($\beta \approx 0.15-0.2$) and a prolate local minimum ($\beta \approx 0.3$).
• The mapped IBM2-CM PESs successfully reproduce the topology of the microscopic PESs, including the $\gamma$-softness observed in $^{114}\text{Te}$.

B. Low-Energy Spectra

• Parabolic Behavior: The calculated excitation energies for non-yrast states ($0^+_2, 2^+_2, 0^+_3, 2^+_3$) exhibit a characteristic **parabolic behavior**, reaching a minimum at the neutron midshell ($N=66$,$^{118}\text{Te}$). This is a signature of shape coexistence.
• Configuration Mixing:
  • In $^{114-120}\text{Te}$, the intruder (prolate) configuration contributes significantly (often >50%) to the wave functions of low-lying states, including the first excited $0^+_2$ state.
  • The IBM2-CM successfully reproduces the ordering and energy spacing of these non-yrast states, whereas the standard single-configuration IBM-2 fails to predict the low-lying $0^+_2$and$0^+_3$ states correctly.
• Discrepancies: While the energy spectra are well-reproduced, the model struggles with the $B(E2)$ anomaly in $^{114}\text{Te}$, where experimental data shows a reduction in $B(E2; 4^+_1 \to 2^+_1)$ that neither the standard IBM nor the mixing model can fully explain without higher-order terms or triaxiality effects.

C. Electromagnetic Properties

• $B(E2)$ Transitions:
  • Strong $B(E2; 0^+_2 \to 2^+_1)$ transitions are predicted for $^{114}\text{Te}$ and $^{116}\text{Te}$, confirming strong mixing between oblate and prolate shapes.
  • The model predicts enhanced transitions for non-yrast bands (e.g., $2^+_2 \to 0^+_2$), suggesting the existence of excited rotational bands built on the$0^+_2$ state.
• Quadrupole Moments: The calculated spectroscopic quadrupole moments ($Q(2^+_1)$) show a sign change from positive to negative around $N=62-64$, reflecting the shape transition. However, for heavier isotopes ($N \ge 70$), the model predicts oblate shapes ($Q < 0$) which contradict some experimental interpretations, suggesting limitations in the current mapping for very neutron-rich nuclei.

D. EDF Dependence

• Comparisons between DD-PC1 (RHB) and SLy6 (Skyrme) inputs show that while the general trend of shape coexistence is robust, the degree of mixing and specific energy splittings are sensitive to the EDF.
• The Skyrme SLy6 input produces a softer $\gamma$ potential, leading to more compressed spectra and stronger configuration mixing compared to the DD-PC1 results.

5. Significance and Conclusion

• Validation of Shape Coexistence: The study firmly establishes that shape coexistence involving oblate and prolate configurations is the driving mechanism for the low-energy structure of midshell Te isotopes.
• Role of Intruder States: It quantifies the critical role of proton intruder configurations (2p-2h excitations) in determining the spectroscopic properties of these nuclei, extending the understanding of this phenomenon from the $Z<50$ and $Z>82$ regions to the $Z>50$ region.
• Methodological Advancement: The paper demonstrates the viability of the EDF-mapped IBM2-CM as a predictive tool for nuclear structure. It successfully bridges the gap between microscopic mean-field theory and phenomenological boson models without arbitrary parameter fitting.
• Future Directions: The inability to fully reproduce the $B(E2)$ anomaly in $^{114}\text{Te}$ and the discrepancies in quadrupole moments for heavy isotopes suggest that future improvements are needed, potentially involving higher-order boson terms, explicit triaxiality, or refined EDFs to better capture quantum correlations and shape dynamics.

In summary, this work provides a comprehensive theoretical framework that explains the complex low-energy spectra of Tellurium isotopes as a result of the interplay between normal and intruder configurations, offering a unified picture of shape coexistence in the $Z \approx 50$ mass region.