Multireference covariant density functional theory for shape coexistence and isomerism in $^{43}$S

This study extends multireference covariant density functional theory to successfully reproduce the low-energy structure of the odd-mass nucleus $^{43}$S, revealing a ground state dominated by an intruder prolate configuration, a high-$K$ isomer built on a prolate 1qp state, and a second excited state characterized by shape coexistence and $K$-mixing.

The Gist

Imagine an atomic nucleus not as a rigid, solid ball, but as a drop of liquid that can stretch, squish, and change shape. Sometimes, this drop can be round like a basketball, other times it can stretch into a rugby ball (prolate), or flatten like a pancake (oblate).

In the world of nuclear physics, there's a special rulebook called the "Shell Model" that predicts how protons and neutrons arrange themselves. Usually, when a nucleus has a "magic number" of neutrons (like 28), it's supposed to be very stable and spherical, like a perfectly round marble.

However, in the nucleus of Sulfur-43 (an atom with 16 protons and 27 neutrons), things get messy. The "magic" rule starts to break down. This nucleus is like a dancer who can't decide on a single pose; it wants to be a rugby ball, a pancake, and a sphere all at the same time. This phenomenon is called Shape Coexistence.

Here is a simple breakdown of what the scientists in this paper did and what they found:

1. The Problem: A Confused Dancer

Sulfur-43 is an "odd-mass" nucleus, meaning it has an odd number of neutrons. This leaves one "lonely" neutron wandering around the core. Because of this, the nucleus is very sensitive to how that lonely neutron moves.

Scientists knew that Sulfur-43 had some strange behaviors:

• It had a "ground state" (its resting position) that looked like a rugby ball.
• It had an "isomer" (a long-lived excited state) that acted like a high-energy dancer, but nobody was sure exactly what shape it was or why it stayed excited for so long.
• Different computer models gave different answers, like a group of people trying to describe an elephant while blindfolded.

2. The Solution: A New Super-Computer Recipe

The authors used a sophisticated method called Multireference Covariant Density Functional Theory (MR-CDFT).

Think of this method like a high-end photo editing software for nuclei.

• Old methods would take a single photo of the nucleus in one shape and say, "This is it."
• This new method takes thousands of photos of the nucleus in every possible shape (stretched, flattened, twisted) and every possible spin (how fast it's rotating).
• Then, it uses a mathematical "mixing bowl" to blend all these photos together. It asks: "If we mix 30% of this rugby shape with 20% of that pancake shape, does it match the real experiment?"

They also added a special ingredient called K-mixing. Imagine the nucleus spinning like a top. "K" is a measure of how tilted that spin is. Sometimes, the nucleus doesn't just spin straight up; it wobbles. This new recipe accounts for that wobble.

3. The Discovery: Solving the Mystery of Sulfur-43

By running this complex simulation, the scientists finally got a clear picture of what's happening inside Sulfur-43:

• The Ground State (The Resting Pose): The lowest energy state is indeed a rugby ball (prolate). The lonely neutron is sitting in a specific spot that makes the whole nucleus stretch out.
• The Isomer (The High-Energy Dancer): They identified the mysterious long-lived state (the 7/2- state) as a High-K Isomer.
  • Analogy: Imagine a figure skater spinning. If they spin with their arms straight up, they are stable. If they try to spin with their arms tilted at a weird angle, it takes a lot of effort to change their spin direction.
  • In Sulfur-43, the lonely neutron is spinning in a way that is "tilted" (High-K). To get back to the ground state, it has to make a huge, difficult jump. This "forbidden" jump is why the state lasts so long (it's an isomer).
• The Confused Third State: There was a third state (3/2-2) that looked like a pancake (oblate). The simulation showed it's a mix of a rugby ball and a pancake, but mostly a pancake.

4. Why It Matters

This paper is a big deal because it proves that their new "photo-mixing" recipe works.

• It successfully predicted the energy levels, the shapes, and even the magnetic properties of the nucleus.
• It confirmed that the "magic number" 28 is indeed crumbling in these heavy, neutron-rich atoms, allowing these wild shape changes to happen.
• It showed that to understand these tiny particles, you can't just look at them in one shape; you have to consider all the shapes they could be and how they mix.

In a nutshell:
The scientists used a super-advanced mathematical blender to mix different shapes and spins of a Sulfur-43 nucleus. They found that the nucleus is a shape-shifter, and they finally figured out why one of its excited states is so stubborn and long-lived: it's stuck in a "tilted spin" that makes it very hard to relax back to normal. This helps us understand how the fundamental rules of the universe change when atoms get very heavy and neutron-rich.

Technical Summary

1. Problem Statement

The paper addresses the complex low-energy structure of the odd-mass nucleus $^{43}$S (Sulfur-43), located near the neutron magic number $N=28$.

• Context: The $N=28$ shell gap, traditionally a magic number, weakens in neutron-rich isotones (like $^{42}$Si, $^{40}$Mg, and Sulfur isotopes) due to the erosion of the shell gap and the crossing of the $\nu f_{7/2}$ and $\nu p_{3/2}$ orbitals.
• Specific Challenge: $^{43}$S exhibits shape coexistence (simultaneous existence of prolate and oblate deformations) and isomerism. Previous studies (Shell Model, AMD+GCM, and standard CDFT) have offered conflicting interpretations regarding the dominant configurations of specific states, particularly the nature of the $7/2^-_1$ isomer and the $3/2^-_2$ excited state.
• Gap: Existing theoretical frameworks often struggle to simultaneously account for shape mixing (superposition of different deformations) and $K$-mixing (mixing of different angular momentum projections on the symmetry axis) in odd-mass nuclei within a relativistic framework.

2. Methodology

The authors extend the Multireference Covariant Density Functional Theory (MR-CDFT) to describe odd-mass nuclei with a novel implementation that includes both $K$-mixing and shape-mixing.

• Theoretical Framework:
  • Mean-Field: Calculations start with a single-reference CDFT using the PC-PK1 relativistic energy density functional (EDF). The false quantum vacuum (FQV) scheme is used to generate one-quasiparticle (1qp) states for the odd-mass system.
  • Symmetry Restoration: The wave functions are projected onto good particle numbers (neutrons and protons) and angular momentum ($J$) using projection operators ($\hat{P}^J_{MK}$, $\hat{P}^N$, $\hat{P}^Z$).
  • Configuration Mixing: The total wave function $|\Psi^{J\pi}_{\alpha}\rangle$ is constructed as a superposition of projected 1qp configurations with different intrinsic shapes (quadrupole deformation $\beta_2$) and $K$ quantum numbers:
    $$|\Psi^{J\pi}_{\alpha}\rangle = \sum_c f^{J\pi\alpha}_c |NZJ\pi; c\rangle$$
    where $c$ represents the collective label $(K, \beta_2)$.
  • Generator Coordinate Method (GCM): The mixing coefficients ($f$) are determined by solving the Hill-Wheeler-Griffin (HWG) equation, which minimizes the energy functional.
  • Observables: Transition strengths ($B(E2)$, $B(M1)$), spectroscopic quadrupole moments ($Q_s$), and magnetic dipole moments ($\mu$) are calculated using the mixed-density prescription.

3. Key Contributions

• Extension of MR-CDFT: This is the first application of MR-CDFT to $^{43}$S that explicitly incorporates $K$-mixing alongside shape-mixing. Previous applications to odd-mass nuclei often treated these effects separately or neglected $K$-mixing.
• Unified Description: The study provides a unified relativistic description of the interplay between shape coexistence, $K$-forbiddenness, and isomerism in the $N=28$ region.
• Resolution of Conflicts: The work clarifies the structural nature of the $3/2^-_2$ and $5/2^-_2$ states, offering a perspective that differs from some previous AMD+GCM predictions.

4. Key Results

The MR-CDFT calculations successfully reproduce the experimental energy spectra and electromagnetic properties of $^{43}$S.

• Ground State ($3/2^-_1$):

  • Identified as predominantly a prolate-deformed configuration ($\beta_2 \approx 0.3$).
  • Dominated by the intruder $\nu 1/2^-[321]$ orbital (originating from the spherical $\nu p_{3/2}$ shell).
  • Forms the base of a rotational band following the weak-coupling limit of the particle-rotor model.

• Isomeric State ($7/2^-_1$):

  • Identified as a high-$K$ isomer.
  • Dominated by the prolate configuration $\nu 7/2^-[303]$ (from the $\nu f_{7/2}$ shell) with $K^\pi = 7/2^-$.
  • The decay to the ground state is strongly hindered due to the large change in $K$ ($\Delta K = 3$), explaining its isomeric nature.
  • The inclusion of $K$-mixing significantly improved the calculated excitation energy, bringing it closer to experimental values.

• Excited States ($3/2^-_2$ and $5/2^-_2$):

  • $3/2^-_2$: Dominated by an oblate configuration ($\beta_2 \approx -0.3$) with $K^\pi = 1/2^-$. This differs from some AMD predictions that suggested a $K=3/2$ dominance.
  • $5/2^-_2$: Found to be dominated by a prolate configuration ($K^\pi = 5/2^-$). This contrasts with AMD+GCM results which predicted an oblate shape for this state, aligning instead with recent shell-model calculations.

• Electromagnetic Properties:

  • The calculated $B(E2)$ and $B(M1)$ transition strengths, as well as spectroscopic quadrupole moments ($Q_s$) and magnetic moments ($\mu$), show good agreement with experimental data (e.g., $Q_s(7/2^-_1) = 20.9$ efm$^2$ vs. exp. $23(3)$ efm$^2$).
  • The model predicts two distinct rotational bands: one built on the $K=1/2$ ground state and another on the $K=7/2$ isomer.

5. Significance

• Validation of MR-CDFT: The study demonstrates that MR-CDFT is a robust tool for describing complex phenomena in odd-mass nuclei, specifically the intricate coupling between single-particle motion and collective deformation.
• Insight into $N=28$ Erosion: The results reinforce the picture of the erosion of the $N=28$ shell gap, where the crossing of $\nu f_{7/2}$ and $\nu p_{3/2}$ orbitals drives shape coexistence and the emergence of high-$K$ isomers.
• Theoretical Discrepancy Resolution: By identifying the prolate nature of the $5/2^-_2$ state and the oblate nature of the $3/2^-_2$ state, the paper highlights the sensitivity of nuclear structure to the treatment of $K$-mixing and shape mixing, suggesting that future experimental lifetime measurements are needed to definitively distinguish between competing theoretical models (MR-CDFT vs. AMD).
• Future Directions: The authors note that the remaining discrepancy in the $7/2^-_1$ excitation energy is likely due to the omission of triaxial deformation. They indicate that extending the framework to include triaxiality in odd-mass nuclei is a priority for future work.