Microscopic theory of the $\gamma$ decay of giant resonances in superfluid nuclei

This paper presents a microscopic Skyrme quasiparticle vibration (QPVC) model to calculate $\gamma$-decay widths in superfluid nuclei, successfully applying it to predict the decay from the giant dipole resonance to the $2_{1}^{+}$ state in $^{140}$Ce with results consistent with recent experimental data.

The Gist

The Big Picture: Listening to the Nucleus Sing

Imagine an atomic nucleus not as a static ball of clay, but as a giant, vibrating drum. When you hit this drum hard (by adding energy), it doesn't just vibrate in one simple way; it creates a chaotic, loud roar called a Giant Resonance. This is like the whole drum skin shaking violently all at once.

For a long time, scientists have known how to measure the "loudness" of this roar. But recently, experiments have become so sensitive that we can now hear the whispers that happen after the roar. Specifically, we can see how the nucleus calms down by emitting a tiny flash of light (a gamma ray) to settle into a lower, calmer vibration.

This paper is about building a microscopic blueprint to predict exactly how loud those whispers are, especially in nuclei that act like superfluids (a special state where particles flow without friction, like a super-cooled liquid).

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The Problem: The Missing Puzzle Piece

Think of the nucleus as a complex dance floor.

1. The Dancers (Quasiparticles): These are the individual protons and neutrons moving around.
2. The Music (Phonons): These are the collective vibrations of the whole group.

In the past, scientists could calculate the music (the vibrations) or the individual dancers, but they struggled to calculate what happens when the dancers and the music interact while the nucleus is trying to calm down. It's like trying to predict the exact path of a dancer who is tripping over their own feet while the floor is shaking beneath them.

Previous models were like "macro" models—they looked at the nucleus from far away and guessed the shape. But for superfluid nuclei (like the one studied here, Cerium-140), we needed a "micro" model that accounts for the fact that the dancers are paired up (like partners holding hands) and that the floor is constantly reshaping itself.

The Solution: The "Dressed" Dancer Model

The authors developed a new mathematical tool called the Skyrme Quasiparticle Vibration Coupling (QPVC) model. Here is how it works, using an analogy:

Imagine a celebrity (the nucleus) walking down a red carpet.

• The Simple View: You just see the celebrity walking.
• The Real View: The celebrity is surrounded by a swirling crowd of fans (the "cloud" of particles). When the celebrity tries to move or change direction (emit a gamma ray), the fans push and pull them. This changes how they move.

The authors' model calculates this "cloud effect" (which they call polarization). They realized that when the nucleus emits a gamma ray, it doesn't just happen in a vacuum. The emission triggers a ripple in the surrounding "cloud" of particles, which in turn pushes back on the nucleus, either helping or hindering the emission.

They used a method called Nuclear Field Theory, which is essentially a way of drawing Feynman diagrams (like comic strips of particle interactions). They didn't just draw one interaction; they drew 24 different scenarios (second-order diagrams) to make sure they didn't miss any subtle ways the particles could interact.

The Experiment: The "Ceiling" Test

To test their new blueprint, they looked at a specific nucleus: Cerium-140 ($^{140}\text{Ce}$).

• The Setup: Scientists at a high-intensity light source (HI$\gamma$S) hit the Cerium nucleus with high-energy photons, making it scream (Giant Dipole Resonance).
• The Goal: They wanted to see how much of that energy was released as a gamma ray dropping the nucleus down to a specific low-energy state (the $2^+$ state).
• The Result: The authors ran their model using four different "rulesets" (Skyrme functionals) to describe how the particles interact.

The Findings:

1. The Numbers: They predicted that the "whisper" (gamma decay width) would be between 200 and 420 electron-volts. This is incredibly small, but their model matched the recent experimental data very well.
2. The Branching Ratio: They found that only about 0.75% to 1.2% of the energy goes into this specific "whisper" path. The rest goes elsewhere.
3. The Polarization Effect: They confirmed that the "cloud of fans" (polarization) actually changes the outcome. Interestingly, their microscopic calculation agreed with a famous old formula (Bohr-Mottelson) that was derived from a macroscopic view, proving that both the "big picture" and the "tiny picture" tell the same story.

Why This Matters

Think of nuclear physics like trying to understand a symphony.

• Old way: We knew the orchestra was loud (Giant Resonance).
• New way: We can now hear the specific notes the violins play as the music fades out.

This paper is important because:

1. It's a New Tool: It provides the first fully self-consistent way to calculate these "whispers" in superfluid nuclei.
2. It Validates Theory: It shows that our complex mathematical models can predict real-world experiments with high precision.
3. It Helps the Stars: Understanding how nuclei emit energy is crucial for nuclear astrophysics. It helps us understand how stars explode (supernovae) and how heavy elements are forged in the universe.

In a Nutshell

The authors built a super-detailed, microscopic simulation of a vibrating atomic nucleus. They showed that when a nucleus tries to calm down from a violent vibration, the surrounding particles push and pull on it, changing how much light it emits. Their calculations match new, high-precision experiments perfectly, giving us a clearer picture of the quantum dance floor inside the atom.

Technical Summary

1. Problem Statement

Nuclear electromagnetic transitions, particularly the $\gamma$-decay from high-energy excited states (such as Giant Resonances and Pygmy Resonances) to low-lying states, serve as a critical probe for nuclear structure. While recent experimental advances (e.g., at the High Intensity $\gamma$-ray Source, HI$\gamma$S) have enabled the measurement of these rare decay branches, a microscopic theoretical description for superfluid nuclei has been lacking.

Existing models often rely on phenomenological inputs or fail to consistently include:

• Pairing correlations: Essential for superfluid nuclei but previously omitted in Quasiparticle-Phonon Coupling (PVC) studies of $\gamma$-decay.
• Dynamic correlations: The interplay between collective nuclear motion and individual quasiparticles.
• Self-consistency: Many approaches use different interactions for the ground state and the transition vertices.

The specific challenge addressed is calculating the $\gamma$-decay width from the Giant Dipole Resonance (GDR) to the first excited $2^+_1$ state in superfluid nuclei, a process where branching ratios are extremely small ($\sim 1\%$) and sensitive to many-body dynamics.

2. Methodology

The authors developed a fully self-consistent Skyrme Quasiparticle Vibration Coupling (QPVC) model within the framework of Nuclear Field Theory (NFT).

• Theoretical Framework:

  • Hamiltonian: The system is described by interacting fermions (quasiparticles) and bosons (QRPA phonons). The interaction $V$ includes both the particle-hole (ph) channel (Skyrme force) and the particle-particle (pp) channel (pairing interaction).
  • Wave Function Expansion: The initial ($|N_i\rangle$) and final ($|N_f\rangle$) vibrational states are treated as QRPA phonons dressed by the interaction. The wave functions are expanded perturbatively up to the second order in $V$, incorporating configurations such as $2qp$ (two-quasiparticle) and $2qp \otimes \text{phonon}$.
  • Transition Operator: The electric multipole operator $Q_{\lambda\mu}$ is used, including effective charges to account for nuclear recoil.

• Diagrammatic Approach:

  • The calculation involves summing 24 second-order Feynman diagrams representing the $\gamma$-decay process.
  • These diagrams account for:
    1. Direct transitions between QRPA states.
    2. Scattering of the initial phonon into $2qp \otimes \text{phonon}$ configurations.
    3. Polarization Effects: A crucial component where the transition operator induces nuclear vibrations (a cloud of quanta) that modify the transition moment. This is treated via a factorization approach, inserting the wave function expansion into the polarization term.

• Self-Consistency:

  • The same Skyrme Energy Density Functional (EDF) is used for the ground state (HFB), the construction of QRPA phonons, and the interaction vertices in the diagrams.
  • Volume pairing is consistently applied across all stages.

• Numerical Implementation:

  • Calculations were performed for $^{140}\text{Ce}$ (a superfluid nucleus).
  • Four Skyrme functionals were tested: SIII, SGII, SkM, and LNS*.
  • The model space was large enough to exhaust the isovector energy-weighted sum rule.
  • Imaginary parts ($\eta_1, \eta_2$) were introduced in energy denominators to simulate the damping of phonons and coupling to more complex configurations not explicitly included.

3. Key Contributions

1. Extension to Superfluid Systems: This is the first fully self-consistent microscopic calculation of $\gamma$-decay from Giant Resonances to low-lying states that explicitly includes pairing correlations via the QPVC model.
2. Comprehensive Diagrammatic Treatment: The authors systematically included all second-order diagrams (24 in total) and consistently treated the polarization effect, which had been previously handled only macroscopically or phenomenologically.
3. Microscopic Polarization Extraction: The study extracts the polarization factor microscopically, allowing for a direct comparison with the macroscopic Bohr-Mottelson formula.
4. Benchmarking against Experiment: The model was applied to the specific case of the GDR $\to$ $2^+_1$ decay in $^{140}\text{Ce}$, a system recently measured at HI$\gamma$S, providing a direct test of the theory.

4. Results

• Ground State and Resonance Properties:

  • The four Skyrme functionals provided a reasonable description of the $2^+_1$ state excitation energy and $B(E2)$ strength, as well as the GDR centroid energy in $^{140}\text{Ce}$.
  • SIII slightly overestimated energies but reproduced $B(E2)$ well; SGII, SkM*, and LNS gave better energy descriptions but overestimated $B(E2)$.

• $\gamma$-Decay Widths and Branching Ratios:

  • Total Decay Width ($\sum \Gamma_\gamma$): The calculated total width for the collective dipole states in the GDR region decaying to the $2^+_1$ state ranges from 200 eV to 420 eV, depending on the functional used (LNS: ~216 eV, SkM*: ~419 eV).
  • Branching Ratio: The corresponding branching ratio ($\Gamma_\gamma / \Gamma_{\text{total}}$) ranges from 0.75% to 1.20%.
  • Energy Dependence: The partial decay widths exhibit an energy dependence similar to the $E1$ strength distribution, showing enhancement around the main IVGDR peak.

• Polarization Effect:

  • The microscopic polarization factor $|1+\chi|^2$ was calculated as a function of transition energy $\Delta E$.
  • Trend: The results agree in trend with the macroscopic Bohr-Mottelson formula.
  • Suppression vs. Enhancement: For transition energies well below the GDR, the polarization factor is $<1$ (suppression). As $\Delta E$ approaches the GDR energy, the factor exceeds 1 (enhancement), peaking when the virtual excitation of the collective mode is most efficient.
  • Deviations: Small deviations from the macroscopic formula were observed for individual states, attributed to the microscopic model's inclusion of fragmented dipole phonon states rather than a single collective phonon.

• Stability: The results were found to be stable with respect to the imaginary damping parameters $\eta_1$ and $\eta_2$ when set to values corresponding to half the experimental GDR width (~2.2 MeV).

5. Significance

• Validation of Microscopic Theory: The work demonstrates that fully self-consistent microscopic models can successfully predict rare $\gamma$-decay branches in superfluid nuclei, bridging the gap between macroscopic phenomenological models and complex many-body dynamics.
• Impact on Nuclear Astrophysics: Accurate theoretical predictions of $\gamma$-decay widths and branching ratios are essential for refining inputs to nuclear astrophysics models (e.g., r-process nucleosynthesis), where statistical models often rely on uncertain photon strength functions.
• Interpretation of New Data: The study provides a theoretical framework to interpret recent and future high-precision measurements at facilities like HI$\gamma$S, LNL, and SSRF, particularly regarding the structure of the Giant Dipole Resonance and the nature of low-lying states in superfluid nuclei.
• Methodological Advancement: By establishing a consistent QPVC framework that includes pairing and polarization, this work sets a new standard for calculating electromagnetic transitions between excited states in open-shell nuclei.