Breit corrections to moderately charged ions in all-orders calculations

This paper presents all-orders calculations incorporating the Breit interaction for moderately charged Cs and Fr ions, revealing that while these corrections significantly improve fine-structure intervals and are substantial for f states, they do not fully resolve existing discrepancies with experimental energy levels.

The Gist

The Big Picture: Why Are We Doing This?

Imagine you are trying to build the most accurate clock in the universe. Scientists are currently trying to build a nuclear clock using a Thorium atom. This clock would be so precise it could detect changes in the fabric of space-time or dark matter.

To build this clock, we need to understand exactly how the electrons inside heavy atoms (like Thorium, Francium, or Cesium) move and interact. It's like trying to predict the exact path of a single dancer in a crowded ballroom. If our math is off by even a tiny bit, the clock won't work, and we might miss out on discovering "new physics" beyond our current understanding of the universe.

The Problem: The "Heavy" Dancers

In these heavy atoms, the electrons are moving incredibly fast—close to the speed of light. Because they are moving so fast, the rules of simple physics (Newton) don't work; we have to use Relativity (Einstein).

The authors of this paper found a specific problem:

• When they calculated the energy levels of certain electrons (specifically the f-orbitals, which are like the "outer rings" of the atom), their predictions were way off compared to real-world experiments.
• It was like predicting a dancer would jump 1 meter high, but they actually jumped 1.5 meters. The gap was huge.

The Culprit: The "Breit Interaction"

In the world of atoms, electrons usually repel each other because they have the same electric charge (like two magnets pushing apart). This is the Coulomb interaction.

However, because these electrons are moving so fast, they also create magnetic fields and interact in more complex ways. This extra interaction is called the Breit interaction.

Think of it this way:

• The Coulomb Interaction: Two people pushing each other away in a hallway.
• The Breit Interaction: The same two people, but they are also running past each other on skateboards. Because they are moving, they create wind currents and magnetic fields that push or pull them in ways they wouldn't if they were standing still.

The authors suspected that this "skateboard effect" (Breit) was the reason their calculations were wrong for the heavy atoms.

The Solution: The "All-Orders" Method

Usually, scientists calculate these effects by adding them up one by one, like stacking bricks.

1. First, calculate the basic push.
2. Then, add the first layer of the "skateboard effect."
3. Then, add a second layer, and so on.

The problem is that for heavy atoms, the "skateboard effect" is so strong that you can't just add a few layers; you need to account for the effect happening simultaneously with the basic push, over and over again.

The authors used a technique called "All-Orders Calculations."

• Analogy: Imagine trying to predict the traffic flow in a city.
  • Old Method: You count the cars, then add the effect of a traffic light, then add the effect of a construction zone, one by one.
  • All-Orders Method: You simulate the entire city's traffic flow at once, where every car reacts to every other car and every traffic light instantly.

They modified their mathematical "traffic simulation" (specifically the Green's function, which is a tool that tracks how an electron moves) to include the Breit interaction right from the start.

The Results: A Mixed Bag

Here is what they found, broken down simply:

1. The Good News: Fine Structure Intervals
When they looked at the difference in energy between two very similar electron states (called "fine structure"), their new method worked perfectly.

• Analogy: If the old method said the gap between two steps was 10 inches, and the real gap was 10.1 inches, the new method said 10.1 inches. Perfect match.
• This proves that their new "All-Orders" method is mathematically sound and handles the complex interactions correctly.

2. The Bad News: Total Energy Levels
However, when they looked at the total energy of the electron (how high the step is from the ground), the big gap between their math and the experiment did not go away.

• Even though they included the massive "skateboard effect" (Breit) in their super-advanced simulation, the numbers were still off by a large amount.
• They also tried adding a more complex version of the Breit effect (one that changes depending on the "frequency" or rhythm of the interaction), but that turned out to be a tiny, negligible change.

The Conclusion: What Does This Mean?

The authors have done a very important job:

1. They proved the method works: They showed that including the Breit interaction in their "All-Orders" simulation is necessary and improves the accuracy of fine details.
2. They identified a mystery: They confirmed that the Breit interaction is huge for these heavy atoms, but it is not the whole story. There is still something missing in our understanding of these heavy ions that causes the big error in total energy.

The Takeaway:
Think of the atom as a complex machine. The authors fixed the gears related to the "skateboard effect" (Breit), and the machine runs smoother than before. But the machine is still making a weird noise (the energy discrepancy). Now, scientists know that the "skateboard effect" isn't the cause of the noise; they have to look for a different broken part to fix the nuclear clock project.

It's a classic scientific victory: They didn't solve the whole mystery, but they successfully ruled out the biggest suspect, which is a huge step forward.

Technical Summary

1. Problem Statement

The accurate theoretical prediction of atomic properties for heavy, moderately charged ions (specifically in the Cs and Fr isoelectronic sequences) is critical for precision tests of fundamental physics, including parity violation studies and the development of nuclear and atomic clocks.

• The Discrepancy: While standard Dirac-Fock (DF) calculations combined with correlation potentials generally agree well with experiment for many states, significant deviations remain for specific $f$-states (e.g., $4f$ in La$^{2+}$, $5f$ in Th$^{3+}$). The deviation is often on the order of hundreds of wavenumbers ($cm^{-1}$), comparable to the magnitude of Breit corrections.
• The Limitation of Current Methods: Previous calculations typically included the Breit interaction only as a second-order perturbation or included it in the DF potential but treated correlations separately. It was unclear whether the large discrepancies in $f$-state energies were due to the omission of Breit effects in higher-order correlation diagrams (all-orders) or if the frequency-dependent nature of the Breit interaction was significant.
• Specific Motivation: Resolving these discrepancies is vital for the nuclear clock project (using the $^{229}$Th nucleus), which relies on precise knowledge of the hyperfine structure of the Th$^{3+}$ ion (ground state $5f_{5/2}$).

2. Methodology

The authors employed an all-orders correlation potential method based on Feynman diagram techniques to sum dominant series of perturbation diagrams exactly.

• Integration of Breit into the Green's Function:

  • The core innovation was modifying the electron Green's function ($G$) to include the one-body part of the Breit interaction ($V_{Br}$) directly within the Dirac-Fock (DF) potential.
  • Instead of treating Breit solely as a perturbation added after solving the DF equations, they solved the Dyson equation using a Breit-Dirac-Fock (BDF) Green's function:
    $$\bar{G} = [1 - G_0(V_x + V_{Br})]^{-1}G_0$$
    where $G_0$ is the local potential Green's function, $V_x$ is the exchange potential, and $V_{Br}$ is the effective one-body Breit operator.
  • This approach sums the one-body Breit interaction to all orders in the residual Coulomb interaction, capturing non-linear screening effects and relaxation of the core wavefunctions that occur when Breit is included in the mean field.

• Technical Implementation:

  • The authors constructed the coordinate representation of the non-local Breit operator $V_{Br}$ using a complete set of hydrogenic basis states (up to $l=6$).
  • They utilized the full spinor structure of the Green's function (mixing large-large, large-small, and small-small components), noting that the one-body Breit operator significantly mixes relativistic and non-relativistic spinor components, unlike the Coulomb interaction.
  • Frequency-Dependent Breit: They also incorporated the frequency-dependent Breit interaction (Møller interaction) into the DF procedure to test if higher-order relativistic corrections (order $(v/c)^3$) could resolve the energy discrepancies.

• Scope of Calculation:

  • Calculations were performed for ions in the Cs and Fr isoelectronic sequences, including Fr, Ra$^+$, La$^{2+}$, Ce$^{3+}$, Ac$^{2+}$, and Th$^{3+}$.
  • Contributions calculated included: All-orders correlations ($\Sigma^{(\infty)}$), Breit corrections at second-order ($\Sigma^{(2)}$), additional Breit corrections from the all-orders Green's function ($\Sigma^{(\infty)}$), and QED radiative corrections.

3. Key Contributions

1. Formalism: Successfully integrated the one-body Breit interaction into the all-orders correlation potential method by modifying the Feynman Green's function. This allows for the exact summation of Breit effects alongside dominant correlation diagrams.
2. Differentiation of Approaches: Demonstrated that including Breit in the DF potential (BDF) yields corrections of different magnitudes and sometimes signs compared to treating Breit as a simple second-order perturbation on top of standard DF orbitals.
3. Frequency Dependence Analysis: Systematically evaluated the impact of the frequency-dependent Breit interaction within the DF procedure, finding it to be a negligible correction compared to the static Breit term.

4. Results

A. Energy Levels

• Large Breit Corrections: The study confirmed that Breit corrections to $f$-states in moderately charged ions are exceptionally large (hundreds of $cm^{-1}$). For example, in La$^{2+}$, the Breit correction at the $\Sigma^{(2)}$ level for the $4f$ state is $\sim 748$ $cm^{-1}$.
• All-Orders Effect: Including Breit in the all-orders correlation potential ($\Sigma^{(\infty)}$) provides an additional correction that partially cancels the second-order term. For La$^{2+}$ $4f$, the total Breit correction drops from $748$ $cm^{-1}$ to $\sim 719$ $cm^{-1}$.
• Persistent Discrepancy: Despite the large magnitude of Breit corrections, including them does not resolve the discrepancy between theory and experiment for the absolute energy levels of $f$-states. The theoretical values still deviate from experimental values by $\sim 0.5\% - 0.7\%$ (hundreds of $cm^{-1}$).
• Level Ordering: The inclusion of Breit was crucial for correctly predicting the ordering of energy levels in Radium (Ra), specifically correcting the inversion between the $7d_{5/2}$ and $5f_{5/2}$ levels.

B. Fine Structure Intervals

• Significant Improvement: While absolute energies remained discrepant, the inclusion of Breit into the all-orders correlation potential led to near-perfect agreement with experimental fine-structure intervals.
• Example: For the $4f$ interval in La$^{2+}$, the theoretical value with all corrections matched the experiment within $0.2\%$. Similar improvements were observed for $4f$ in Ce$^{3+}$ and $5f$ in Th$^{3+}$.
• Mechanism: Since fine structure is a highly relativistic effect, the proper treatment of Breit in the correlation potential is essential for capturing the splitting accurately.

C. Frequency-Dependent Breit

• The inclusion of the frequency-dependent Breit interaction ($B(\omega)$) into the DF procedure resulted in very small corrections (typically $< 1\%$ of the total Breit correction).
• While the difference between static and frequency-dependent Breit is larger in the DF procedure than in first-order perturbation, it is insufficient to explain the large remaining discrepancy in absolute energy levels.

5. Significance and Conclusion

• Validation of Method: The work validates the necessity of treating Breit and all-orders correlations on the same footing. The method successfully reproduces fine-structure intervals, proving the theoretical framework is robust for relativistic effects.
• Unresolved Mystery: The persistence of large errors in absolute $f$-state energies suggests that the Breit interaction, even when treated to all orders in the correlation potential, is not the sole source of the discrepancy. The authors conclude that other physical mechanisms (potentially higher-order QED effects, nuclear size/shape effects, or missing correlation diagrams like ladder diagrams for specific states) must be investigated to resolve the energy level deviations.
• Impact on Nuclear Clocks: The results highlight the complexity of modeling Th$^{3+}$ and similar ions. While the fine-structure intervals are now well-understood, the large uncertainty in absolute energy levels remains a challenge for the precise hyperfine structure studies required for the nuclear clock initiative.

In summary, the paper establishes that while Breit corrections are massive and essential for fine structure, their inclusion in all-orders calculations does not fully solve the energy level discrepancy for $f$-states in heavy ions, pointing to the need for further theoretical investigation beyond the current Breit and correlation models.