Global isotopic analysis of hyperfine-resolved rotational spectroscopic data for barium monofluoride, BaF

This paper reports new high-precision Fourier-transform microwave spectroscopic measurements and a global isotopic analysis of five BaF isotopologues, which significantly improve hyperfine parameters and reveal distinctive Born-Oppenheimer breakdown structures driven by nuclear field shifts, thereby supporting future searches for physics beyond the Standard Model such as the electron electric dipole moment.

The Gist

The Big Picture: Why Are We Listening to Barium Fluoride?

Imagine the universe as a giant, quiet room. Physicists are trying to hear a very faint whisper that would prove a new rule of nature exists—a rule that breaks the "Standard Model" (the current rulebook of physics).

To hear this whisper, they need a microphone that is incredibly sensitive. In this case, the "microphone" is a molecule called Barium Monofluoride (BaF). It's a heavy molecule made of one Barium atom and one Fluorine atom. Because it's heavy and has a specific shape, it acts like a super-sensitive detector for two mysterious things:

1. The Electron's Electric Dipole Moment (eEDM): A tiny, hidden "lopsidedness" in the electron that shouldn't exist according to current theories.
2. The Nuclear Anapole Moment: A weird magnetic twist inside the nucleus of the atom.

To find these whispers, the scientists need to know exactly how the BaF molecule behaves. They can't just guess; they need a perfect map of its energy levels. This paper is about drawing that map with extreme precision.

---

The Experiment: The "Molecular Jet" and the "Echo Chamber"

The Setup:
The scientists didn't just sit a bottle of BaF on a table. That would be too messy and hot. Instead, they used a technique called laser ablation.

• The Analogy: Imagine a tiny piece of Barium metal is the target. They hit it with a super-fast laser pulse (like a high-tech paintball gun), blasting tiny bits of Barium into the air.
• The Mix: They mix this with a gas (mostly Neon) that has a little bit of Fluorine in it. The Barium and Fluorine snap together instantly to form BaF molecules.
• The Cooling: This mixture shoots out of a nozzle like a supersonic jet. As it expands, it cools down instantly to about -271°C (just a few degrees above absolute zero).
• Why? At this temperature, the molecules stop spinning wildly and settle into their calmest, lowest-energy states. It's like calming a chaotic crowd of people so you can hear them whisper.

The Measurement:
They put this cold molecular jet inside a special microwave cavity (an echo chamber). They blast it with microwaves.

• The Analogy: Think of a guitar string. If you pluck it at the exact right frequency, it vibrates loudly. The scientists are "plucking" the BaF molecules with microwaves. When the frequency matches the molecule's natural spin, the molecule absorbs the energy.
• By measuring exactly which frequencies make the molecules "sing," they can calculate the molecule's internal structure.

---

The Challenge: The "Odd" and "Even" Families

The researchers didn't just study one type of BaF. They studied five different versions (isotopologues) of the molecule, based on the different types of Barium atoms available in nature.

• The "Even" Family (138, 136, 134): These Barium atoms have a "quiet" nucleus. They don't spin on their own. They are like a calm, still pond.
• The "Odd" Family (137, 135): These Barium atoms have a "noisy" nucleus. They spin like a top. This spin creates a magnetic field that interacts with the rest of the molecule, making the "song" much more complex.

The Problem:
In the past, scientists had studied the "Even" family and the "Odd" family separately. They had good maps for the "Even" ones, but the "Odd" ones were a bit fuzzy, and they had never mapped the rare "Odd" ones (135 and 134) with high precision before.

The Solution:
This paper combines all five families into one giant "Global Fit."

• The Analogy: Imagine you are trying to tune a piano. You have 5 different pianos (the 5 isotopes). Previously, people tuned Piano A and Piano B separately. This team tuned all 5 at the same time, using the fact that they are all made of the same materials to cross-check their work. This allowed them to correct tiny errors in the tuning of the rare pianos that no one had ever seen clearly before.

---

The Discovery: The "Broken Rule" (Born-Oppenheimer Breakdown)

Here is the most interesting part of the paper.

The Old Rule (Born-Oppenheimer Approximation):
In physics, there's a handy rule that says: "If you change the weight of the atom (the nucleus), the molecule's behavior changes in a perfectly predictable, smooth way." It's like saying if you make a car heavier, it will drive slower in a perfectly straight line.

The Reality:
When the scientists looked at their new, ultra-precise data, they found the "smooth line" was actually wobbly.

• The Analogy: Imagine you are walking down a hill. You expect the slope to be steady. But as you walk, you notice the ground bumps up and down in a specific pattern: Up, Down, Up, Down.
• The Cause: This "wobble" happens because the Barium nuclei aren't just different weights; they are different sizes and shapes. The "Odd" Barium nuclei are slightly squashed or stretched compared to the "Even" ones. This changes how the electrons feel the nucleus.

The Breakthrough:
The team had to invent a new way to describe this wobble, called Born-Oppenheimer Breakdown (BOB).

• They realized they had to account for two things:
  1. Mass Shift: The simple weight difference.
  2. Field Shift: The difference in the size of the nucleus (the "squashiness").

By separating these two effects, they found that the "size" of the Barium nucleus plays a huge role. They even found that their new measurements matched a separate "King Plot" (a different type of graph used by nuclear physicists) perfectly. This confirms that their map of the molecule is accurate and that they understand the tiny differences in the Barium nucleus.

---

Why Does This Matter?

1. Better Microscopes: By understanding the molecule so perfectly, scientists can use it to hunt for the "whispers" of new physics (like the electron's dipole moment). If the map of the molecule is wrong, you might think you found a new law of physics when you actually just miscalculated the molecule's spin.
2. Nuclear Secrets: This study gives us a new way to measure the size and shape of Barium nuclei. It's like using a molecule as a ruler to measure the inside of an atom.
3. Future Tech: The techniques used here (cooling molecules, precise microwave measurements) are the same ones needed to build quantum computers and ultra-precise atomic clocks.

Summary

This paper is about listening to a molecule very, very carefully. The scientists took five different versions of a Barium-Fluoride molecule, cooled them down, and measured their spins with sub-kilohertz precision. By comparing all five versions at once, they discovered that the "rules" of how atoms behave are slightly wobbly due to the weird shapes of the Barium nuclei. This creates a much clearer picture of the molecule, which is essential for future experiments trying to discover the deepest secrets of the universe.

Technical Summary

1. Problem Statement

Barium monofluoride (BaF) is a critical candidate molecule for searching for physics beyond the Standard Model, specifically the measurement of the electron electric dipole moment (eEDM) and the nuclear anapole moment. These searches require extreme precision in understanding the molecule's internal structure and energy levels.

However, a comprehensive understanding of BaF's spectroscopic parameters has been hindered by:

• Fragmented Data: Previous studies focused on specific isotopologues (mostly $^{138}\text{Ba}^{19}\text{F}$ and $^{137}\text{Ba}^{19}\text{F}$) or specific vibrational states, lacking a unified global analysis.
• Missing Isotopologue Data: High-precision data for the less abundant isotopologues ($^{135}\text{Ba}^{19}\text{F}$ and $^{134}\text{Ba}^{19}\text{F}$) were previously unavailable in the literature.
• Complex Hyperfine Structure: The presence of nuclear spins in odd-mass barium isotopes ($I=3/2$) creates complex hyperfine splitting and electric quadrupole interactions that differ significantly from the even-mass isotopes (which rely solely on fluorine spin, $I=1/2$).
• Born-Oppenheimer Breakdown (BOB): Standard mass-scaling models are insufficient for heavy polar molecules like BaF at sub-kHz precision. The interplay between nuclear size effects (field shifts) and mass-dependent effects requires a more sophisticated analysis to resolve discrepancies between isotopologues.

2. Methodology

The authors employed a combination of new high-precision experiments and a global theoretical fit:

• Experimental Technique:

  • Instrument: Fourier-transform microwave (FTMW) spectroscopy using the COBRA (Coaxially Oriented Beam-Resonator Arrangement) spectrometer at Leibniz University Hannover.
  • Sample Generation: BaF molecules were generated via laser ablation of a barium target using a Nd:YAG laser (532 nm) in a supersonic expansion of neon buffer gas containing $\sim$2% $\text{CF}_4$. This cooled the rotational temperature to $\sim$2 K.
  • Measurements: Pure rotational transitions in the ground vibronic state ($X^2\Sigma^+, v=0$) were measured for the five most abundant isotopologues: $^{138}\text{Ba}$, $^{137}\text{Ba}$, $^{136}\text{Ba}$, $^{135}\text{Ba}$, and $^{134}\text{Ba}$.
  • Precision: Achieved sub-kHz accuracy (0.5 kHz or better) for unblended lines.

• Data Analysis:

  • Global Fit: The new data were combined with previously published datasets (Ryzlewicz et al., Ernst et al.) covering a wide range of rotational ($N=1-22$) and vibrational ($v=0-4$) states.
  • Software: The SPFIT program (from Pickett's CALPGM suite) was used to fit the data to an effective Hamiltonian.
  • Hamiltonian Model:
    • Even Isotopologues: Modeled using Hund's case (b$\beta_J$) with fluorine hyperfine terms.
    • Odd Isotopologues: Modeled using a sequential Hund's case (b$\beta_S$) coupling scheme to account for the strong magnetic hyperfine interaction of the barium nucleus.
  • BOB Analysis: A Born-Oppenheimer breakdown analysis was implemented on the primary rotational constant ($Y_{01}$). This included:
    • Mass-dependent terms: Watson-type corrections ($\Delta_{01}$).
    • Field shift terms: Corrections based on the change in mean square nuclear charge radii ($\delta\langle r^2 \rangle$) between isotopes.

3. Key Contributions

1. First High-Precision Data for Rare Isotopologues: The paper reports the first sub-kHz precision microwave spectroscopic data for $^{135}\text{Ba}^{19}\text{F}$ and $^{134}\text{Ba}^{19}\text{F}$.
2. Comprehensive Global Fit: A unified fit of 95 spectral lines across five isotopologues, spanning rotational transitions from $N=1-0$ to $N=22-21$ and vibrational levels up to $v=4$.
3. Refinement of Hyperfine Parameters:
   • Determined nuclear spin-rotation coupling constants ($C_I$) for both barium and fluorine for the first time.
   • Significantly improved the precision of the Fermi-contact ($b_F$) and dipolar ($c$) hyperfine parameters for both Ba and F.
   • Refined the nuclear electric quadrupole coupling constants ($eQq_0$) for the odd isotopes, validating them against known nuclear quadrupole moments.
4. BOB Decomposition: Successfully disentangled the nuclear field shift (due to nuclear size variations) from mass-dependent BOB effects in the rotational constant.

4. Key Results

• Spectroscopic Parameters: The global fit yielded a root-mean-square (RMS) deviation of 1.083, indicating excellent agreement with experimental uncertainties.
  • The primary rotational constant ($B_0$) and spin-rotation constant ($\gamma$) were determined with significantly reduced uncertainties compared to previous literature.
  • New values for the nuclear spin-rotation constants ($C_I$) were derived: $C_I(\text{F}) \approx 26.13$ MHz, while $C_I(\text{Ba})$ for odd isotopes were found to be much smaller ($\sim$1.5–1.7 MHz).
• Hyperfine Structure:
  • The $^{137}\text{Ba}$ and $^{135}\text{Ba}$ electric quadrupole coupling constants ($eQq_0$) were determined independently. The ratio of these values ($0.6501$) agreed excellently with the ratio of nuclear quadrupole moments ($Q$) from IUPAC and Pyykkö compilations.
• Born-Oppenheimer Breakdown (BOB) Analysis:
  • The analysis revealed a distinct odd-even staggering in the nuclear charge radii of barium isotopes.
  • The field shift effect (nuclear size) was found to dominate the BOB corrections, accounting for roughly 2/3 of the total deviation, while the mass-dependent effect accounted for the remaining 1/3.
  • The field shift parameters ($V_{Ba}$) derived from the fit ($\sim 10.6 \times 10^{-7} \text{ fm}^{-2}$) showed strong agreement with independent nuclear physics data (Landolt-Börnstein) and recent molecular King plot analyses.
• Validation: The derived BOB parameters and field shifts provided a consistency check that supports the "odd-even staggering" model of barium nuclear radii, distinguishing it from smoother trends seen in other heavy elements like Lead (Pb).

5. Significance

• Foundation for eEDM and Anapole Moment Searches: By providing a rigorous, globally consistent set of spectroscopic parameters, this work removes systematic uncertainties that could obscure the tiny signals of parity non-conservation (PNC) in future experiments.
• Nuclear Physics Insights: The ability to separate mass-dependent and field-shift effects in a molecular system offers a unique probe of nuclear structure, specifically the distribution of charge radii in barium isotopes. The results corroborate the "odd-even staggering" of nuclear sizes.
• Benchmark for Theory: The precise hyperfine parameters serve as a stringent test for ab initio quantum chemical calculations, particularly those predicting effective electric fields for eEDM experiments.
• Methodological Advance: The study demonstrates that combining high-precision FTMW data with a global BOB analysis is essential for heavy diatomic molecules, setting a precedent for future studies of other heavy systems (e.g., RaF, YbF, PbF).

In summary, this paper establishes a new standard of precision for BaF spectroscopy, resolving long-standing discrepancies in hyperfine parameters and providing a critical link between molecular spectroscopy and fundamental nuclear physics.