Determination of nuclear quadrupole moments for $^{25}$Mg, $^{87}$Sr, and $^{135,137}$Ba via configuration-interaction combined with a coupled-cluster approach

This paper employs a configuration-interaction plus coupled-cluster approach to calculate electric-field gradients and magnetic dipole hyperfine-structure constants for low-lying states of Mg, Sr, and Ba, enabling the accurate determination of nuclear quadrupole moments for $^{25}$Mg, $^{87}$Sr, and $^{135,137}$Ba that reveal significant discrepancies with previously adopted values for strontium and barium.

The Gist

Imagine the nucleus of an atom not as a perfect, round ball, but as a slightly squashed or stretched balloon. This shape isn't random; it's a specific "signature" of how the protons and neutrons are arranged inside. Scientists call this shape the nuclear quadrupole moment. Knowing the exact shape of this "balloon" is crucial for understanding the fundamental rules of physics, from how atoms stick together to how materials behave.

However, measuring this squishiness directly is incredibly hard. It's like trying to guess the exact shape of a balloon inside a sealed, dark box just by listening to the sound it makes when you tap it.

The Experiment: Listening to the Atomic "Hum"

In this paper, the author, Yong-Bo Tang, acts as a master audio engineer. He focuses on three specific "families" of atoms: Magnesium (Mg), Strontium (Sr), and Barium (Ba).

When these atoms are excited (like a guitar string being plucked), they emit a very specific "hum" or vibration called hyperfine structure. This hum is caused by the interaction between the spinning electrons on the outside and the squashed nucleus on the inside.

• The Measured Part: Scientists have already measured the pitch of this hum very precisely in a lab.
• The Missing Link: To figure out the shape of the nucleus (the quadrupole moment) from that pitch, you need to know exactly how the electrons are arranged around the nucleus. This arrangement creates an "electric field gradient" (think of it as the slope of a hill that the electrons are rolling down).

The Problem: The Hill is Too Steep to Calculate

Calculating the shape of that "hill" (the electric field gradient) is a nightmare for computers. Electrons don't just sit still; they dance around each other, pushing and pulling in complex ways called electron correlation.

• If you ignore these dances, your calculation of the hill is wrong.
• If you try to calculate every single dance, your computer crashes.

Previous attempts to calculate this were like trying to map a mountain range using a blurry satellite photo. The results were inconsistent. For Strontium and Barium, different studies gave different answers, with some results differing by up to 10%.

The Solution: A Hybrid "Swiss Army Knife" Approach

To solve this, Tang developed a new computational method that combines two powerful techniques:

1. Configuration Interaction (CI): This is like looking at every possible way the electrons could arrange themselves, one by one. It's thorough but slow.
2. Coupled-Cluster (CC): This is like using a sophisticated shortcut to predict how the electrons influence each other in groups. It's fast but sometimes misses the tiny details.

Tang's method, CI+CC, is the best of both worlds. It uses the "shortcut" to handle the big, heavy interactions between the core electrons, and then uses the "thorough" method to fine-tune the details of the outer electrons. It's like using a drone to map the general shape of a forest, then sending a team of hikers to measure the exact height of every specific tree.

The Results: Clearing Up the Confusion

Using this high-precision "Swiss Army knife," Tang calculated the electric field gradients for several low-energy states of Mg, Sr, and Ba. He then combined his calculations with the known experimental "hums" to determine the nuclear shapes.

Here is what he found:

• Magnesium (25Mg): The result was a perfect match with previous experiments. It's like tuning a radio and finding the station crystal clear. The calculated shape agrees with what was found using "muonic X-ray" experiments (a different, high-tech way of measuring).
• Strontium (87Sr): Here, the plot thickens. Tang's result suggests the nucleus is about 10% more squashed than the currently accepted value in textbooks. The old value came from looking at a Strontium ion (an atom that lost an electron), while Tang looked at the neutral atom. The difference suggests the old calculation might have missed some subtle electron dances.
• Barium (135,137Ba): Similar to Strontium, Tang's results for Barium are about 4% different from the currently accepted values derived from Barium ions.

The Takeaway

The paper concludes that while the method works beautifully for Magnesium, there is a significant discrepancy for Strontium and Barium when compared to the "gold standard" values currently used by scientists.

Tang suggests that the difference might be because the current "gold standard" calculations missed a specific type of electron interaction called triple excitation (where three electrons interact simultaneously). Just as a choir sounds different if three singers harmonize in a way no one predicted, these triple interactions might be shifting the "pitch" of the atom's shape.

In summary: The author built a better computer model to measure the shape of atomic nuclei. For Magnesium, the model confirmed what we already knew. For Strontium and Barium, the model suggests the current "official" measurements might be slightly off, hinting that we need to look closer at how three electrons interact to get the true shape of these atomic nuclei.

Technical Summary

Technical Summary: Determination of Nuclear Quadrupole Moments for 25Mg, 87Sr, and 135,137Ba

Problem Statement
The nuclear electric quadrupole moment ($Q$) is a fundamental parameter describing nuclear structure symmetry, with significant implications for atomic, nuclear, and condensed matter physics. While experimental hyperfine-structure constants ($B$) can be measured with high precision, extracting accurate values of $Q$ requires equally precise theoretical calculations of the electric-field gradients ($q$). Existing literature reveals notable discrepancies in the recommended values for the nuclear quadrupole moments of $^{87}\text{Sr}$ and $^{137}\text{Ba}$, particularly when comparing values derived from neutral atoms versus ions. For instance, values derived from the neutral Sr atom are approximately 8% larger than those recommended from ion data, and similar inconsistencies exist for Ba. Furthermore, standard theoretical approaches often struggle to account for the complex electron correlation effects necessary for high-precision gradient calculations in heavy alkaline-earth systems.

Methodology
The authors employ a relativistic hybrid approach combining the Configuration Interaction (CI) method with the Coupled-Cluster (CC) method, referred to as the RCI+all-order method. This framework is designed to comprehensively account for core-core, core-valence, and valence-valence electron correlations, which are decisive for accurate electric-field gradient calculations.

Key methodological components include:

• Hamiltonian and Interaction: The hyperfine interaction is modeled using spherical tensor operators for magnetic dipole ($k=1$) and electric quadrupole ($k=2$) interactions. The nucleus is treated as a uniformly magnetized and charged sphere.
• Hybrid Calculation Scheme: The method utilizes a finite basis set of even-tempered Gaussian-type functions to expand Dirac wave functions. The correlation potentials are derived via five distinct computational strategies to assess uncertainty:
  1. CI + Second-order Many-Body Perturbation Theory (MBPT(2)).
  2. CI + Linearized Coupled-Cluster Singles and Doubles (LCCSD).
  3. CI + Full Coupled-Cluster Singles and Doubles (CCSD).
  4. CI + LCCSD with rescaling parameters (CI+LCCSDs).
  5. CI + CCSD with rescaling parameters (CI+CCSDs).
• Corrections: Transition matrix elements include Random Phase Approximation (RPA), core Brueckner effects, structural radiation, and normalization corrections (collectively termed high-order or HO corrections), as well as two-particle (TP) interactions up to the second order.
• Extraction of Q: The nuclear quadrupole moment is extracted using the relation $Q = B / (234.9648867 \cdot q)$, where $B$ is the experimentally measured hyperfine-structure constant and $q$ is the theoretically calculated electric-field gradient.

Key Contributions and Results
The study systematically calculates electric-field gradients and magnetic dipole hyperfine-structure constants ($A$) for low-lying states of neutral $^{25}\text{Mg}$, $^{87}\text{Sr}$, and $^{135,137}\text{Ba}$. The specific states analyzed include:

• Mg: $3s3p \ ^3P_{1,2}$
• Sr: $5s4d \ ^1D_2$ and $5s5p \ ^3P_{1,2}$
• Ba: $6s5d \ ^3D_{1,2,3}$, $6s5d \ ^1D_2$, and $6s6p \ ^1P_1$

Energy and Hyperfine Constants:
The calculated energies and magnetic dipole constants ($A$) show excellent agreement with experimental data (NIST) and other theoretical results. The CI+LCCSD method is identified as the most reliable, with final recommended values for $A$ matching experimental measurements within their uncertainties (typically within 2% for most states). This validates the hybrid CI+CC approach for alkaline-earth systems.

Nuclear Quadrupole Moments ($Q$):
By combining the calculated gradients with experimental $B$ values, the authors determine the following nuclear quadrupole moments:

• $^{25}\text{Mg}$: $0.203(2)$ b. This result is in perfect agreement with the mesonic X-ray experiment ($0.201(3)$ b) and previous hyperfine-based determinations.
• $^{87}\text{Sr}$: $0.336(4)$ b. This value is approximately 10% larger than the currently adopted value of $0.305(2)$ b (derived from the $4d_{5/2}$ state of Sr$^+$) but aligns well with values derived from the $5p_{3/2}$ state of Sr$^+$ and neutral Sr states ($5s5p \ ^3P_{1,2}$).
• $^{137}\text{Ba}$: $0.246(4)$ b. This is approximately 4% larger than the recommended value of $0.236(3)$ b (derived from the $5d_{3/2,5/2}$ states of Ba$^+$). However, it agrees excellently with values derived from the $6p_{3/2}$ state of Ba$^+$.
• $^{135}\text{Ba}$: $0.161(2)$ b.

Significance and Claims
The paper claims that the configuration-interaction plus coupled-cluster approach provides a robust and accurate framework for determining nuclear quadrupole moments, particularly for unstable or heavy nuclei where direct measurement is difficult.

The authors highlight a significant discrepancy between values derived from neutral atoms and those from ions for Sr and Ba. Specifically, the results for $^{87}\text{Sr}$ and $^{137}\text{Ba}$ derived from neutral atoms in this work differ by ~10% and ~4%, respectively, from the values currently adopted in literature (which rely on ion data). The authors suggest that these discrepancies may stem from the omission of triple-excitation contributions in standard coupled-cluster calculations, noting that recent work on ions indicates triple excitations contribute 1–2% to hyperfine constants.

The study concludes that while their results for $^{25}\text{Mg}$ are consistent with established data, the findings for $^{87}\text{Sr}$ and $^{137}\text{Ba}$ challenge the currently adopted values derived from ion states. The authors assert that their neutral-atom-based extractions are reliable, supported by the high precision of their magnetic dipole constant calculations, but acknowledge that further theoretical work including triple excitations and three-body interactions is necessary to fully resolve the differences between neutral-atom and ion-based determinations.