Decomposition of angular momentum projected nuclear wave function

This paper derives a new identity that decomposes conventional angular momentum projected nuclear wave functions into coupled neutron-proton projected bases, revealing that nucleons in even-even ground states are not fully paired and demonstrating that this decomposition enables further improvements to variation after projection shell model wave functions.

The Gist

Imagine the atomic nucleus not as a solid marble, but as a bustling, chaotic dance floor filled with two types of dancers: Protons (the positively charged ones) and Neutrons (the neutral ones).

Physicists have been trying to write the "perfect choreography" for this dance floor for decades. The goal is to predict exactly how these dancers move, spin, and pair up to form stable nuclei.

The Old Way: The "Group Hug"

For a long time, the standard method to describe this dance was to treat the entire nucleus as one giant, inseparable blob.

• The Analogy: Imagine trying to describe a dance by saying, "The whole crowd spun together."
• The Problem: This method assumes the protons and neutrons are glued together, moving in perfect lockstep. It ignores the fact that sometimes the protons might spin one way while the neutrons spin another, or that they might have their own little sub-groups. It's like trying to describe a complex ballet by only looking at the audience as a single mass.

The New Discovery: The "Coupled Dance"

The authors of this paper, led by Wen Chen and Zao-Chun Gao, have found a new mathematical "lens" to look at the nucleus. They realized that instead of treating the nucleus as one big blob, we can break the dance down into two separate groups: the Proton Dance Team and the Neutron Dance Team.

They developed a new formula (an "identity") that allows them to take a complex, messy nuclear wave function (the description of the dance) and decompose it.

• The Analogy: Instead of saying "The crowd spun," they can now say, "The Proton Team spun with 2 units of energy, while the Neutron Team spun with 2 units, and they combined to make the whole nucleus spin with 4 units."

This is a big deal because it reveals the hidden structure of the nucleus. It lets physicists see exactly how much the protons are doing versus how much the neutrons are doing.

The Big Surprise: "Not Everyone is Paired Up"

In nuclear physics, there's a strong belief that in stable, even-numbered nuclei (like a perfect ballroom dance), every dancer should be paired up with a partner of the same type (proton with proton, neutron with neutron) to cancel out their spins. This is called "pairing."

• The Old Expectation: Everyone is in a perfect couple. The Proton Team has zero net spin, and the Neutron Team has zero net spin.
• The Paper's Finding: When the authors looked at the data, they found that this isn't always true. Even in the most stable ground states, the dancers aren't perfectly paired.
  • Sometimes, the Proton Team is spinning (say, with a spin of 2), and the Neutron Team is also spinning (with a spin of 2), but they are spinning in opposite directions so that the total spin of the nucleus is still zero.
  • Why? It's because the protons and neutrons are talking to each other. The "neutron-proton interaction" is like a conversation between the two dance teams that messes up the perfect pairing, causing them to move in more complex, unpaired ways.

The "Scissors" and the Heavy Deformed Nuclei

The paper also mentions a famous phenomenon called the "Scissors Mode."

• The Analogy: Imagine the Proton Team and the Neutron Team are two blades of a pair of scissors. Sometimes, they open and close relative to each other.
• The authors show that their new method is perfect for studying this. They tested it on heavy, deformed nuclei (nuclei shaped like rugby balls rather than spheres, like Dysprosium-166). They found that the new method could describe these heavy nuclei much better than the old "group hug" method.

The Upgrade: A Better Algorithm

Finally, the authors didn't just stop at looking; they used this new insight to improve the calculation.

• The Old Method: They calculated the dance using the "group hug" assumption.
• The New Method: They rebuilt the calculation using the "coupled dance" (Proton Team + Neutron Team) approach.
• The Result: For simple, even-numbered nuclei, the improvement was small (because the old method was already pretty good there). But for odd-numbered nuclei (where there's an extra dancer left over), the new method made a huge difference, predicting the energy levels much more accurately.

Summary

Think of this paper as upgrading the software used to simulate a nuclear dance floor.

1. Old Software: Treated the nucleus as one big, blurry blob.
2. New Software: Separates the Protons and Neutrons, allowing us to see their individual moves and how they interact.
3. Key Insight: Even in the most stable nuclei, the protons and neutrons aren't always perfectly paired up; they are constantly interacting and influencing each other's spins.
4. Benefit: This new view helps us understand the "scissors" motion of nuclei and gives us much more accurate predictions for complex, heavy atoms.

It's a move from seeing the nucleus as a single object to seeing it as a dynamic partnership between two distinct teams.

Technical Summary

1. Problem Statement

• Context: Constructing high-quality nuclear wave functions with good angular momentum ($J$) is a fundamental challenge in nuclear many-body physics. Traditional methods, such as the Variation After Projection Shell Model (VAPSM), typically perform angular momentum projection (AMP) directly on a reference state (e.g., a Slater determinant) representing the entire nuclear system as a single entity.
• Limitation: This traditional approach implicitly assumes that neutrons and protons move coherently as a whole, neglecting relative collective motion between the two subsystems. However, experimental evidence (e.g., the "scissors mode") and theoretical studies suggest that relative motion between neutron and proton fluids is a significant degree of freedom, particularly in deformed nuclei.
• Goal: The authors aim to derive a mathematical framework to decompose conventional AMP wave functions into components based on the individual angular momenta of neutrons ($J_\nu$) and protons ($J_\pi$). This decomposition allows for a microscopic analysis of the internal structure of nuclear states and offers a pathway to improve existing shell model approximations by explicitly including coupled neutron-proton bases.

2. Methodology

The paper employs a rigorous algebraic derivation followed by numerical applications using the VAPSM and full Shell Model (SM) calculations.

• Derivation of a General Identity:

  • The authors derive a new identity for the angular momentum projection operator $\hat{P}^J_{MK}$ acting on a product state $|\Phi\rangle = |\Phi_\pi\rangle |\Phi_\nu\rangle$.
  • They demonstrate that the projection of the total system can be expressed as a sum over products of individual neutron and proton projection operators, coupled via Clebsch-Gordan coefficients:
    $$\hat{P}^J_{MK} = \sum_{J_\pi J_\nu K_\pi K_\nu} \langle J_\pi K_\pi J_\nu K_\nu | J K \rangle \hat{P}^{J_\pi J_\nu}_{JM}(K_\pi K_\nu)$$
    where $\hat{P}^{J_\pi J_\nu}_{JM}(K_\pi K_\nu)$ is a "coupled projected basis" operator projecting four quantum numbers: $J_\pi, J_\nu, J, M$.
  • This identity holds for arbitrary reference states, not just axial ones.

• Wave Function Decomposition:

  • Using the derived identity, any projected nuclear wave function $|\Psi^{JM}\rangle$ is expanded into a sum of coupled projected bases:
    $$|\Psi^{JM}\rangle = \sum_{J_\pi J_\nu} \sum_{K_\pi K_\nu i} f_{Ki} \langle J_\pi K_\pi J_\nu K_\nu | J K \rangle \hat{P}^{J_\pi J_\nu}_{JM}(K_\pi K_\nu) |\Phi_i\rangle$$
  • A metric, $C(J_\pi, J_\nu)$, is defined to quantify the contribution (weight) of each $(J_\pi, J_\nu)$ component to the total wave function.

• Numerical Implementation:

  • Systems Studied: Calculations were performed for $sd$-shell nuclei (even-even, odd-mass, and odd-odd) and a heavy deformed nucleus ($^{166}\text{Dy}$).
  • Models:
    • VAPSM: Used with the USDB interaction for $sd$-shell and jj56pnb for heavy nuclei.
    • Full Shell Model (SM): Used as a benchmark (via the NuShellX code) to validate the decomposition formula.
  • Improvement Strategy: The authors propose a new VAPSM variant where the wave function is constructed directly from the coupled projected bases (Eq. 35) rather than the traditional single-system projection. They diagonalize the Hamiltonian in this new basis space to test for energy improvements.

3. Key Contributions

1. Mathematical Identity: The derivation of Eq. (15) and Eq. (17), which provides an explicit expansion of the total angular momentum projection operator in terms of separate neutron and proton projection operators. This bridges the gap between traditional AMP and the "coupled projected bases" used in Projected Shell Model (PSM) studies of the scissors mode.
2. Decomposition Framework: The introduction of the $C(J_\pi, J_\nu)$ metric allows for the direct extraction of neutron and proton angular momentum distributions from complex many-body wave functions.
3. Validation of VAPSM: The study confirms that VAPSM wave functions (even with a single Slater determinant) yield $C(J_\pi, J_\nu)$ distributions nearly identical to exact Shell Model results, validating the VAPSM as a robust approximation.
4. Structural Insight: The method reveals that even in the ground states of even-even nuclei, nucleons are not fully paired (i.e., $J_\pi = J_\nu = 0$ is not the sole component). Significant contributions from $J_\pi = J_\nu \neq 0$ components (e.g., $2, 4$) are observed, driven by the neutron-proton interaction.
5. Improved Basis Set: The proposal and preliminary testing of a "coupled projected basis" VAPSM, which treats the coefficients of the coupled bases as variational parameters.

4. Results

• Decomposition of Ground States:
  • For even-even nuclei (e.g., $^{20}\text{Ne}, ^{24}\text{Mg}$), the $C(J_\pi=0, J_\nu=0)$ component is less than 1. In some cases (like $^{20}\text{Ne}$ and $^{24}\text{Mg}$), the $C(J_\pi=2, J_\nu=2)$ component is even larger than the $0+0$ component.
  • This indicates that like nucleons are not fully paired in the ground state due to the mixing induced by the neutron-proton interaction.
  • As the neutron number deviates from $N=Z$, the $C(0,0)$ component increases, suggesting the neutron-proton interaction weakens.
• Odd-Mass and Odd-Odd Nuclei:
  • In $^{25}\text{Mg}$ (odd-mass) and $^{26}\text{Al}$ (odd-odd), the wave functions are dominated by specific $(J_\pi, J_\nu)$ pairs (e.g., $J_\pi=5/2, J_\nu=5/2$ in $^{26}\text{Al}$), reflecting parallel spin alignment.
  • Components with odd $J_\pi$ or $J_\nu$ are generally suppressed in even-even nuclei due to time-reversal symmetry, though small non-zero values indicate slight symmetry breaking.
• Heavy Deformed Nuclei ($^{166}\text{Dy}$):
  • In the $J=0$ ground state, the dominant components are $J_\pi = J_\nu = 2, 4, 6$ (anti-parallel), while $J_\pi=J_\nu=0$ is only ~3.2%.
  • For higher spin states ($J=2, 10$), the components where $J_\pi + J_\nu = J$ become predominant, indicating that the proton and neutron ellipsoids rotate in the same direction.
• Energy Improvements:
  • Re-optimizing the wave function using the coupled projected basis (Eq. 35) showed negligible improvement for even-even nuclei (confirming that relative scissors motion is less critical for their ground states).
  • However, for odd-mass and odd-odd nuclei, the new basis significantly lowered the calculated energies, bringing them closer to exact Shell Model results.

5. Significance

• Microscopic Understanding: The decomposition provides a direct window into the microscopic structure of nuclear states, quantifying the degree of "neutron-proton decoupling" or relative motion.
• Methodological Advancement: It validates the use of coupled projected bases as a superior basis set for describing nuclei where neutron-proton correlations are strong (e.g., $N \approx Z$ or odd-mass systems).
• Future Applications: The framework paves the way for a full "Variation After Projection" calculation using coupled bases (varying both the Slater determinants and the coupling coefficients simultaneously). This is expected to be particularly powerful for studying the scissors mode in heavy deformed nuclei, a region where standard shell model calculations are computationally intractable.
• Theoretical Consistency: The work unifies the traditional AMP approach with the Projected Shell Model (PSM) approach, showing they are mathematically connected through the derived identity.