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Calculation of the energy levels and hyperfine structure for Xe~II, Rn~II, and Og~II ions

This paper calculates energy levels, Landé g-factors, and hyperfine-structure constants for singly ionized xenon, radon, and oganesson using a configuration-interaction method validated against experimental data, with a specific focus on demonstrating how Breit and quantum-electrodynamic corrections are enhanced by configuration mixing in heavy ions to provide essential electronic factors for future nuclear property studies.

Original authors: T. H. Dinh, V. A. Dzuba, V. V. Flambaum

Published 2026-07-07

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CC BY 4.0

Original authors: T. H. Dinh, V. A. Dzuba, V. V. Flambaum

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the atom as a tiny, bustling solar system. At the center is the sun (the nucleus), and zooming around it are planets (electrons). Usually, we think of these planets as following neat, predictable orbits. But in the world of super-heavy elements—like Oganesson (Og), the heaviest element on the periodic table—things get chaotic. The "planets" are moving so fast that they distort space and time, and they interact with each other in complex, messy ways.

This paper is like a high-tech weather forecast for these chaotic atomic solar systems. Specifically, the authors are trying to predict the "weather patterns" (energy levels and magnetic interactions) for three specific ions: Xenon (Xe), Radon (Rn), and Oganesson (Og), all stripped of one electron.

Here is a breakdown of what they did, using simple analogies:

1. The Goal: Mapping the Invisible

The scientists wanted to create a detailed map of the energy levels for these ions. Think of energy levels as the "floors" of a skyscraper where the electrons live.

Why do this? For Xenon, we have a map because people have measured it in labs. For Radon and especially Oganesson, the map is blank. Oganesson is so heavy and unstable that it's incredibly hard to study in real life.
The Strategy: They used Xenon as a "training ground." They built their computer model, tested it against the known Xenon data, and once it got the Xenon map right (within about 1% error), they trusted it to predict the maps for Radon and Oganesson.

2. The Method: The "Big Picture" vs. The "Fine Print"

Calculating how seven electrons dance around a nucleus is like trying to predict the movement of seven people in a crowded room while they are all holding hands and shouting. It's too complex to calculate every single interaction perfectly.

The Trick (CIPT): The authors used a method called "Configuration Interaction with Perturbation Theory."
- The Analogy: Imagine you are watching a dance floor. You focus intensely on the main dancers (the low-energy states) to see exactly what they are doing. For the people in the back of the room (high-energy states), you don't track their every step. Instead, you just estimate how their general presence might slightly nudge the main dancers. This saves a massive amount of computing power while keeping the result accurate.
The "Relativistic" Factor: Because these atoms are so heavy, the electrons move near the speed of light. This requires using Einstein's rules of relativity, not just Newton's. It's like the difference between driving a car on a flat road versus driving a spaceship near a black hole; the rules change completely.

3. The "Hyperfine" Secret: The Atomic Compass

The paper focuses heavily on "hyperfine structure."

The Analogy: Imagine the nucleus isn't just a solid ball, but a tiny, spinning magnet (like a compass needle). The electrons orbiting it also have their own tiny magnetic fields.
The Interaction: As the electrons zoom by, they feel the tug of the nucleus's magnetic needle. This tug causes the energy "floors" to split into tiny sub-floors. Measuring this split tells us about the shape and magnetism of the nucleus itself.
The Challenge: The authors calculated exactly how strong this magnetic tug would be for Radon and Oganesson. This is crucial because, since we can't easily measure these atoms in a lab yet, these calculations provide the "electronic factor" needed to interpret future measurements. If scientists eventually catch an Oganesson ion, they will use these numbers to figure out what the nucleus looks like.

4. The Surprise: When Small Changes Cause Big Chaos

One of the most interesting findings involves Oganesson.

The Situation: In Oganesson, two different electron configurations (two different "dance routines") are sitting very close to each other in energy. They are almost at the same height.
The Effect: Because they are so close, they start to mix, like two colors of paint blending together.
The Twist: The authors found that tiny, subtle corrections (called Breit and QED corrections—think of these as tiny ripples in the fabric of space-time caused by the electrons' speed and quantum weirdness) usually have a small effect. However, because the two dance routines were mixing so strongly, these tiny ripples caused a massive shift in the magnetic "tug" (hyperfine structure).
The Lesson: It's like a tightrope walker. A tiny breeze might not affect a person standing on the ground, but if they are balancing on a tightrope, that same breeze could knock them over. In Oganesson, the "tightrope" is the mixing of electron states, making the atom incredibly sensitive to these tiny quantum corrections.

5. The Results: A Guide for Future Explorers

The paper concludes with a list of predicted "floors" (energy levels) and "magnetic tugs" (hyperfine constants) for Radon and Oganesson.

For Radon: They identified specific transitions (jumps between floors) that experimentalists could look for to measure the atom's properties.
For Oganesson: They highlighted a specific jump between two high-energy floors that is perfect for study because the magnetic effects are huge and easy to spot.

In Summary:
The authors built a sophisticated computer simulation to predict the behavior of the heaviest atoms in the universe. They proved their method works by matching it to known data (Xenon) and then used it to predict the "magnetic fingerprints" of Radon and Oganesson. They discovered that in the heaviest atoms, tiny quantum effects can be amplified by electron mixing, creating huge changes in how the atom interacts with its own nucleus. These predictions serve as a roadmap for future experiments, helping scientists understand the structure of the "island of stability" in the super-heavy elements.

Technical Summary: Calculation of Energy Levels and Hyperfine Structure for Xe II, Rn II, and Og II Ions

Problem Statement
Theoretical investigations into the electronic structure of superheavy elements (SHEs) are critical for understanding the interplay between strong relativistic and correlation effects, particularly in the search for the predicted "island of stability." However, direct experimental spectroscopic data for superheavy elements with atomic numbers $Z > 103$ are currently unavailable due to short half-lives and low production rates. While neutral oganesson (Og, $Z=118$ ) has been studied theoretically, the spectroscopic properties of singly ionized species, specifically Radon II (Rn II) and Oganesson II (Og II), have not been previously investigated. Furthermore, singly ionized systems offer a favorable compromise for theoretical accuracy: they possess a single hole in the outermost shell (and typically two in low-lying excited states) while retaining seven valence electrons (including the $ns^2$ subshell). This configuration presents a challenging many-body problem for standard configuration interaction (CI) methods but is amenable to advanced extensions.

Methodology
The authors employ the Configuration Interaction with Perturbation Theory (CIPT) method, specifically designed for systems with a large number of valence electrons. The calculations treat the $ns^2np^5$ outer shell configurations ( $n=5, 6, 7$ for Xe II, Rn II, and Og II, respectively) with the $ns$ electrons included in the valence space, resulting in seven valence electrons.

Hamiltonian and Basis: The CI Hamiltonian includes the relativistic Hartree-Fock (RHF) potential, the Breit interaction, and Quantum Electrodynamic (QED) corrections. Single-electron basis states are generated using B-splines ( $l_{max}=4, n_{max}=20$ ) within a cavity of radius $R_{max} = 40a_B$ .
Hyperfine Structure (HFS): HFS constants are calculated using the Time-Dependent Hartree-Fock (TDHF) method, equivalent to the Random-Phase Approximation (RPA), to account for core polarization effects.
Relativistic and QED Corrections:
- Breit Interaction: Treated on equal footing with the Coulomb interaction throughout the RHF, basis generation, and CI stages.
- QED Corrections: Implemented via a radiative potential (vacuum polarization and self-energy) added to the RHF potential to influence configuration mixing. Additionally, specific corrections to the hyperfine interaction matrix elements for $s_{1/2}$ and $p_{1/2}$ states are applied using scaling factors derived from single-electron ion calculations, approximated as functions of nuclear charge $Z$ .
Validation: The methodology is first applied to Xe II, where experimental data is abundant, to benchmark the accuracy of the approach before applying it to Rn II and Og II.

Key Results

Xe II Benchmark: The calculations for Xe II show agreement with experimental data at the level of approximately 1% for energy levels and typically 10% for hyperfine constants. Accuracy improves to ~1% for states with large hyperfine constants, where cancellation effects are minimal.
Rn II and Og II Predictions: The paper presents calculated energy levels, Landé $g$ -factors, and magnetic dipole ( $A$ ) and electric quadrupole ( $B$ ) hyperfine-structure constants for Rn II and Og II. These values are provided as electronic factors, excluding nuclear moments ( $g_I$ and $Q$ ), to facilitate future extraction of nuclear properties from experimental measurements.
Configuration Mixing and Corrections: A significant finding concerns the Og II ion, specifically the odd-parity $J=1/2$ $J = 1/2$ states. The paper demonstrates that strong configuration mixing between the $7s^27p^48p$ $7 s^{2} 7 p^{4} 8 p$ and $7s^27p^5$ $7 s^{2} 7 p^{5}$ configurations leads to a situation where states with very different hyperfine matrix elements are separated by small energy intervals.
- In this regime, Breit and QED corrections, while small in magnitude for energy separations, induce substantial changes in the hyperfine constants.
- The corrections are shown to be non-additive; including Breit and QED effects simultaneously yields results significantly different from the sum of individual corrections. For example, in the second and third $J=1/2$ states of Og II, the inclusion of these corrections alters the hyperfine constants by up to ~50% and ~20%, respectively, due to the sensitivity of the mixing coefficients to energy shifts.
Spectroscopic Transitions: The authors identify specific transitions promising for future spectroscopic studies. For Og II, an electric-dipole (E1) transition between the $7s^27p^48s$ $^2S_{1/2}$ state and the $7s^27p^5$ $^2P^o_{1/2}$ state is highlighted. Both states exhibit large hyperfine structure, making the transition suitable for determining magnetic dipole constants, though the electric quadrupole constant cannot be measured for these $J=1/2$ states.

Significance and Claims
The paper claims to fill a critical gap in the theoretical understanding of singly ionized superheavy elements. By validating the CIPT approach against Xe II data, the authors establish confidence in the reliability of their predictions for Rn II and Og II. The work provides the necessary electronic structure factors to interpret future spectroscopic measurements of radon and oganesson ions, which could be trapped in Paul/Penning traps or storage rings.

Crucially, the study highlights that in heavy open-shell ions with strong configuration mixing, relativistic and QED corrections cannot be treated as simple perturbations to energy levels alone; they fundamentally alter the mixing of states, thereby dramatically affecting hyperfine structure. This insight is vital for the accurate interpretation of future experiments aimed at extracting nuclear magnetic dipole and electric quadrupole moments for superheavy nuclei, contributing to the broader effort to test nuclear models in the superheavy region.

1. The Goal: Mapping the Invisible

2. The Method: The "Big Picture" vs. The "Fine Print"

3. The "Hyperfine" Secret: The Atomic Compass

4. The Surprise: When Small Changes Cause Big Chaos

5. The Results: A Guide for Future Explorers

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