Radiative decay and electromagnetic moments in $^{229}$Th determined within nuclear DFT

This paper utilizes nuclear density functional theory with symmetry breaking and restoration to calculate the radiative decay properties and electromagnetic moments of the ground and isomeric states in $^{229}$Th, finding good agreement with experimental data while highlighting the need for improved octupole descriptions in future Skyrme functional parametrizations.

The Gist

Imagine the atomic nucleus not as a boring, solid marble, but as a chaotic, spinning dance floor filled with thousands of tiny dancers (protons and neutrons). Usually, this dance floor is perfectly round or slightly oval. But in the case of a specific atom called Thorium-229, the dance floor is doing something weird: it's shaped like a pear.

This pear shape is the key to a scientific mystery that has puzzled physicists for 50 years. This paper is like a high-tech detective story where the authors use a super-computer to figure out exactly how this pear-shaped nucleus behaves, specifically how it "wiggles" and how it interacts with magnetic fields.

Here is the breakdown of the story in simple terms:

1. The Mystery: The "Sleeping Giant"

Thorium-229 has a special "sleeping" state (an isomer) that is incredibly close to its "awake" state (the ground state). The energy difference between them is so tiny that it's like comparing the height of a single grain of sand to a mountain.

• Why it matters: Because this gap is so small, scientists hope to use this atom as the world's most precise clock (a "nuclear clock"), which would be billions of times more accurate than the atomic clocks in your GPS.
• The Problem: To build this clock, we need to know exactly how the nucleus behaves. But because the energy is so low, the electrons surrounding the nucleus get in the way, making it hard to measure the nucleus's true properties directly.

2. The Tool: The "Virtual Laboratory"

Since we can't easily measure these tiny wiggles in a real lab without interference, the authors built a Virtual Laboratory using a method called Nuclear Density Functional Theory (DFT).

• The Analogy: Think of this as a video game physics engine. Instead of building a real nuclear clock, they simulated the entire nucleus on a computer. They didn't just guess; they used a set of mathematical rules (called "Skyrme functionals") that describe how protons and neutrons interact.
• The Challenge: They tried seven different sets of rules (like trying seven different video game physics engines) to see which one gave the most realistic result.

3. The Three Big Secrets They Uncovered

The paper focuses on three specific things that make this nucleus special. The authors realized that previous computer models were missing three crucial ingredients:

• The Pear Shape (Octupole Deformation):
  • The Metaphor: Imagine a spinning top. If it's perfectly round, it spins smoothly. If it's pear-shaped, it wobbles. The authors found that ignoring this "wobble" (the pear shape) made their predictions completely wrong. You can't understand Thorium-229 without acknowledging its pear shape.
• The "Time-Odd" Effect (Core Polarization):
  • The Metaphor: Imagine a crowd of people (the nucleus) watching a solo dancer (the odd neutron). When the dancer spins, the crowd doesn't just stand still; they lean and sway in response, creating a magnetic field. Previous models ignored this "crowd sway." The authors found that including this effect was essential to get the right answer.
• Mixing the Dancers (Configuration Mixing):
  • The Metaphor: The nucleus isn't just one static pose; it's a mix of different dance moves. The authors allowed their computer to mix different possible states together, rather than picking just one. They found that while this mixing is important, the "pear shape" and the "crowd sway" were the real heavy lifters.

4. The Results: A "Good Enough" Prediction

The authors ran their simulations and compared them to the few real measurements we have.

• The Good News: Without tweaking their numbers to force a match (a "parameter-free" approach), their predictions were surprisingly close to reality. They successfully predicted how strong the magnetic "wobble" (transition strength) is and how the nucleus reacts to magnetic fields.
• The Bad News: The different sets of rules (the seven functionals) gave slightly different answers. It's like asking seven different architects to design a bridge; they all built a bridge, but they are all slightly different sizes.
• The Fix: The authors realized their "rules" (the functionals) weren't detailed enough to perfectly describe the "pear shape." They need to update their mathematical rules in the future to better capture this specific type of deformation.

5. Why Should You Care?

This isn't just about abstract physics.

• Better Clocks: Understanding this nucleus helps us build the next generation of clocks. These clocks could detect dark matter, test if the laws of physics change over time, or even help us navigate space with pinpoint accuracy.
• The "Pear" is Key: The paper proves that to understand the future of nuclear technology, we have to stop treating nuclei as simple spheres and start treating them as complex, pear-shaped, wobbling objects.

In a nutshell: The authors used a super-computer to simulate a pear-shaped atomic nucleus. They discovered that to get the physics right, you have to account for the pear shape and how the nucleus "leans" when it spins. Their computer model worked well, proving that we are getting closer to building a nuclear clock that could revolutionize how we measure time and space.

Technical Summary

1. Problem Statement

The paper addresses the theoretical characterization of the unique nuclear isomer in Thorium-229 ($^{229}$Th), which possesses the lowest known excited state energy in the nuclear chart ($\sim 8.3$ eV). This system is a prime candidate for a nuclear clock and tests of fundamental physics. However, its theoretical description is challenging due to:

• Low Energy Transition: The transition between the ground state ($5/2^+$) and the isomeric state ($3/2^+$) is dominated by Magnetic Dipole (M1) multipolarity, with the competing Electric Quadrupole (E2) component suppressed by $\sim 10^{13}$.
• Structural Complexity: The states involve strong core polarization effects, octupole deformations (pear-shape), and configuration mixing that standard models often fail to capture without empirical adjustments.
• Theoretical Gaps: Previous self-consistent approaches often neglected parity breaking (octupole deformation), time-reversal symmetry breaking, or required adjustable parameters (effective charges/g-factors) to reproduce experimental data.

The authors aim to provide the first parameter-free analysis of the radiative decay rate ($B(M1)$) and electromagnetic moments of $^{229}$Th using a fully self-consistent framework that incorporates all relevant symmetry breakings and restorations.

2. Methodology

The study employs Multi-Reference Nuclear Density Functional Theory (MR-DFT) using the HFODD code (v3.33k) with seven different Skyrme energy density functionals (SkXc, SkM*, UNEDF1, SIII, SkO', SLy4, UNEDF0).

Key Technical Steps:

1. Symmetry Breaking and Configuration Selection:
   • The authors blocked specific odd-neutron axial quasiparticle configurations in $^{229}$Th near the Fermi surface.
   • They relaxed parity constraints to allow for intrinsic octupole deformation ($Q_3^0 \neq 0$).
   • Three configurations were selected for each angular momentum projection ($\Omega = 3/2$ and $\Omega = 5/2$) based on Nilsson labels (e.g., $(631)_{3/2}$, $(633)_{5/2}$).
2. Symmetry Restoration:
   • The symmetry-broken mean-field wave functions were projected onto good angular momentum ($I$) and positive parity ($\pi=+$).
   • Particle-number restoration was omitted to save computational cost, as previous tests showed negligible impact on magnetic moments.
3. Configuration Interaction (CI):
   • A $3 \times 3$ matrix of the Hamiltonian and overlap was constructed for the projected band-heads.
   • The Hill-Wheeler equations were solved to obtain mixing coefficients and the final mixed wave functions for the $5/2^+$ and $3/2^+$ states.
4. Time-Even vs. Time-Odd (TE vs. TO) Terms:
   • Calculations were performed in two variants: (i) Time-Even (TE) terms only, and (ii) both TE and Time-Odd (TO) terms.
   • TO terms are crucial as they drive the angular-momentum core polarization, which significantly affects magnetic moments and M1 transition rates.
5. Regression Analysis:
   • Since the Skyrme functionals used were not fitted to octupole data, the authors performed a linear regression.
   • They correlated the calculated intrinsic octupole moments ($Q_3^0$) and transition strengths ($B(M1)$) with experimental $Q_3^0$ values of neighboring even-even nuclei ($^{226}$Ra and $^{230}$Th) to extrapolate results for $^{229}$Th.

3. Key Contributions

• First Parameter-Free MR-DFT Analysis: This is the first study to calculate the $B(M1)$ transition strength and electromagnetic moments of $^{229}$Th without adjusting effective charges or g-factors, relying solely on the underlying nuclear energy density functional.
• Inclusion of Time-Odd Core Polarization: The work explicitly demonstrates the critical role of time-odd mean-field terms. Calculations including TO terms yielded results consistent with experiment, whereas TE-only calculations failed.
• Octupole Correlation Necessity: The study proves that neglecting octupole deformation (parity breaking) leads to significant discrepancies with experimental $B(M1)$ values, validating the necessity of including octupole degrees of freedom.
• Systematic Uncertainty Quantification: By using a set of seven Skyrme functionals and regression against $^{226}$Ra and $^{230}$Th, the authors quantified the theoretical uncertainty arising from the choice of functional, particularly regarding octupole polarizability.

4. Results

The authors compared their regression-aligned results with the latest experimental data:

• Magnetic Dipole Transition ($B(M1)$):
  • Experimental Value: $0.0388(12) \, \mu_N^2$.
  • Theoretical Result (TE+TO, Regressed): $0.03(2) \, \mu_N^2$ (aligned with $^{230}$Th) and $0.04(3) \, \mu_N^2$ (aligned with $^{226}$Ra).
  • Conclusion: The results agree favorably with experiment within uncertainties. TE-only results were inconsistent and far from experimental values.
• Magnetic Dipole Moments ($\mu$):
  • $5/2^+$ Ground State: Calculated $\approx 0.3(3) \, \mu_N$ vs. Exp. $0.366(6) \, \mu_N$. (Well reproduced).
  • $3/2^+$ Isomer: Calculated $\approx -0.18(7) \, \mu_N$ vs. Exp. $-0.378(8) \, \mu_N$. (Underestimated by $\sim 50\%$).
  • Note: Despite the underestimation of the isomer's moment, the transition strength $B(M1)$ remains accurate due to the specific cancellation of matrix elements.
• Electric Quadrupole Moments ($Q$):
  • Both states were reproduced with high accuracy (within a few percent of experimental values: $Q(5/2^+) \approx 2.98$ b vs. 3.11 b; $Q(3/2^+) \approx 1.73$ b vs. 1.77 b).
• Magnetic Octupole Moments ($\Omega$):
  • Predicted for the first time theoretically: $\Omega(5/2^+) \approx -0.054(10) \, \mu_N \text{ b}$ and $\Omega(3/2^+) \approx 0.129(6) \, \mu_N \text{ b}$. These are currently unmeasured experimentally.

5. Significance and Future Outlook

• Validation of Theory: The success of the parameter-free approach validates the use of MR-DFT with symmetry restoration for complex, low-energy nuclear structures.
• Technological Implications: Accurate theoretical values for $B(M1)$ and moments are essential for interpreting nuclear clock experiments and designing laser excitation schemes for $^{229}$Th in solid-state hosts.
• Limitations and Recommendations:
  • The current Skyrme functionals struggle with self-interaction and self-pairing errors, leading to singularities in some matrix elements.
  • The primary source of theoretical uncertainty remains the octupole polarizability of the functionals.
  • Future Work: The authors emphasize the need to systematically adjust the octupole degrees of freedom in future global functional parametrizations to reduce theoretical uncertainties and improve predictive power for actinide nuclei.

In summary, this paper establishes a robust, ab initio framework for describing the electromagnetic properties of the $^{229}$Th isomer, confirming that time-odd core polarization and octupole correlations are the dominant physical mechanisms governing its unique low-energy structure.