Formation of gaseous, doubly charged cerium monofluoride CeF$^{2+}$ and its sensitivity to new physics

This paper proposes and experimentally demonstrates the formation of gaseous, doubly charged cerium monofluoride (CeF$^{2+}$) as a stable, valence-isoelectronic surrogate for tricationic protactinium monofluoride ($^{229}$PaF$^{3+}$) to facilitate future searches for physics beyond the Standard Model, supported by quantum chemical calculations that confirm structural parallels and estimate sensitivity to $\mathcal{P,T}$-odd properties.

The Gist

The Big Picture: A "Stand-In" for a Rare Gem

Imagine scientists want to study a very rare, unstable gemstone (a radioactive atom called Protactinium-229) to see if it reveals secrets about the universe that our current "rulebook" (the Standard Model of physics) is missing. Specifically, they want to see if this gemstone has a tiny, hidden "tilt" (an electric dipole moment) that breaks the laws of symmetry.

However, this gemstone is dangerous, rare, and hard to handle. It's like trying to build a delicate clockwork mechanism using a ticking time bomb.

The Solution: The scientists decided to use a "stand-in" or a "body double." They found a stable, safe, and common cousin of the gemstone called Cerium. They built a molecule using this safe cousin (Cerium Fluoride) that looks and acts almost exactly like the dangerous one they actually want to study. This paper is the report on how they successfully built this "stand-in" molecule in a lab and proved it's ready for the job.

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Part 1: Building the Molecule (The Kitchen Analogy)

To study these atoms, the scientists need to turn them into molecules and keep them floating in a gas, not stuck to a wall.

• The Ingredients: They started with a beam of Cerium ions (charged atoms) and injected them into a special trap filled with Helium gas.
• The Secret Sauce: To make the Cerium grab a Fluorine atom, they added a tiny drop of Sulfur Hexafluoride (SF6) gas. Think of SF6 as a delivery truck carrying Fluorine packages.
• The Reaction: Inside the trap, the Cerium ions crashed into the SF6 trucks. They grabbed a Fluorine package and formed a new molecule: Cerium Monofluoride with a double positive charge ($CeF^{2+}$).
• The Proof: They used a super-precise scale (a mass spectrometer) to weigh the new molecules. It's like having a scale so sensitive it can tell the difference between a feather and a feather with a single grain of sand on it. They confirmed they had successfully created the specific molecule they wanted.

The Challenge: They tried to make a version with three positive charges (like the real radioactive target), but it was too unstable and fell apart. However, the two-charged version they made is perfect because it is a "valence-isoelectronic" twin to the radioactive one. This means they have the same number of electrons in their outer shells, so they behave almost identically in experiments.

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Part 2: The Blueprint (The Computer Simulation)

Before they could use this molecule for experiments, they needed to know its internal structure. They ran complex computer simulations (like a high-tech architectural blueprint) to map out the molecule's energy levels.

• The "Staircase" of Energy: They found that the molecule has a set of energy levels (like steps on a staircase) that are very close together and run parallel to each other.
• Why This Matters: In physics, to control a molecule with lasers (like steering a car with a remote), you need these steps to be predictable. The computer showed that the Cerium molecule has a very "clean" set of steps, making it a great candidate for being controlled by lasers.
• The "Dark" Secret: The simulations also showed that this molecule is very sensitive to specific types of physics violations (Parity and Time-reversal violations). It's like a microphone that is tuned to hear a very specific, faint whisper that other microphones miss.

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Part 3: Why This Matters (The Detective Work)

The ultimate goal is to find "New Physics."

• The Current Rulebook: Our current understanding of the universe (the Standard Model) is great, but it doesn't explain everything (like why there is more matter than antimatter).
• The Missing Clue: Scientists are looking for "symmetry violations." Imagine a world where a clock runs backward, or a mirror image behaves differently than the original. The Cerium molecule they built is a highly sensitive detector for these weird behaviors.
• The Strategy: Because the radioactive version (Protactinium) is so hard to get, they are using the stable Cerium version to:
  1. Test the Equipment: Prove that their lab setup can handle these tricky, highly charged molecules.
  2. Refine the Technique: Learn how to cool the molecules down and control them with lasers.
  3. Prepare for the Real Deal: Once they master the Cerium "stand-in," they will be ready to apply the exact same techniques to the real, radioactive Protactinium molecule once they can get a beam of it.

Summary

This paper is a "proof of concept." The scientists said, "We can't easily study the rare, radioactive atom we really want. So, let's build a safe, stable twin instead." They successfully built the twin in a gas trap, weighed it to prove it exists, and used computers to confirm it has the right properties to be used as a high-precision detector for new physics. They have now paved the road for the future experiment with the real radioactive atom.

Technical Summary

1. Problem Statement

The search for physics beyond the Standard Model (BSM) often relies on measuring Electric Dipole Moments (EDMs) and other Parity ($P$) and Time-reversal ($T$) violating effects. A promising candidate for such precision measurements is the triply charged protactinium monofluoride ion ($^{229}\text{PaF}^{3+}$). The $^{229}\text{Pa}$ nucleus possesses an octupole deformation that significantly enhances the nuclear Schiff moment, a key observable for $P,T$-odd physics.

However, realizing experiments with $^{229}\text{PaF}^{3+}$ faces three major hurdles:

1. Radioactivity: $^{229}\text{Pa}$ has a short half-life (1.5 days), requiring rare isotope facilities.
2. Refractory Nature: Protactinium is difficult to deliver to ion sources using conventional techniques.
3. Molecular Stability: Creating and controlling highly charged molecular ions (like $3+$) is prone to Coulomb explosion and complex chemistry.

To overcome these challenges, the authors propose using a stable, valence-isoelectronic surrogate: doubly charged cerium monofluoride ($\text{CeF}^{2+}$). $\text{Ce}^{2+}$ ($[\text{Xe}]4f^2$) is isoelectronic to $\text{Pa}^{3+}$ ($[\text{Rn}]5f^2$). The paper aims to experimentally demonstrate the formation of $\text{CeF}^{2+}$ in the gas phase, characterize its electronic structure, and evaluate its potential as a platform for quantum control and new physics searches.

2. Methodology

Experimental Approach

The experiments were conducted at TRIUMF (Canada) using the Off-Line Ion Source (OLIS) and the TITAN (TRIUMF's Ion Trap for Atomic and Nuclear science) facility.

• Ion Production: Stable $^{140}\text{Ce}$ cations were generated using a Multi-Charge Ion Source (MCIS) via electron cyclotron resonance (ECR) sputtering. Both singly ($\text{Ce}^+$) and triply ($\text{Ce}^{3+}$) charged beams were produced.
• Molecular Formation: The ion beams were injected into a Radio-Frequency Quadrupole ion beam Cooler and Buncher (RFQcb) filled with a helium buffer gas mixed with trace amounts of sulfur hexafluoride ($\text{SF}_6$).
  • $\text{Ce}^+$ beams were used to demonstrate the basic formation of $\text{CeF}^+$ and $\text{CeF}_2^+$.
  • $\text{Ce}^{3+}$ beams were specifically targeted to form the doubly charged $\text{CeF}^{2+}$ (and attempt $\text{CeF}^{3+}$).
• Identification: Trapped ions were transferred to TITAN's Multiple-Reflection Time-of-Flight (MR-TOF) mass spectrometer. This system provided high mass resolving power ($R > 10^5$) to unambiguously identify the molecular ions against background species.

Theoretical Approach

• Electronic Structure: Ab initio calculations were performed using Unrestricted Coupled-Cluster with Single, Double, and perturbative Triple excitations [UCCSD(T)] and Two-Component Complex Generalized Hartree–Fock (2c-ZORA-cGHF) methods. These included relativistic effects essential for heavy elements.
• Properties Calculated: The team computed potential energy surfaces, equilibrium bond lengths, vibrational frequencies, transition dipole moments, Franck-Condon factors, and molecular enhancement factors for various $P,T$-odd interactions (Schiff moment, electron EDM, etc.).

3. Key Contributions and Results

A. Experimental Formation of $\text{CeF}^{2+}$

• Success: The team successfully formed and identified gaseous $\text{CeF}^{2+}$ using $\text{Ce}^{3+}$ ions and $\text{SF}_6$.
• Mass Spectrometry: High-resolution MR-TOF spectra confirmed the presence of $\text{CeF}^{2+}$ (mass-to-charge ratio $m/q \approx 79.5$). The signal was distinct from contaminants and calibrants (e.g., $\text{Rb}^+$).
• Charge State Limitations: While $\text{CeF}^{2+}$ was formed efficiently, attempts to create $\text{CeF}^{3+}$ (using $\text{Ce}^{3+}$) failed. Theoretical analysis suggested that $\text{CeF}^{3+}$ is metastable and prone to Coulomb explosion or that the reaction pathway is endothermic. Instead, fragmentation products like $\text{SF}^+$ were observed.
• Efficiency: The production rate was approximately 350 $\text{CeF}^{2+}$ ions per second after mass separation, corresponding to an overall efficiency of $\sim 10^{-6}$ relative to the injected $\text{Ce}^{3+}$ beam.

B. Electronic Structure and Quantum Control

• Quasi-Parallel States: Calculations revealed a compressed, quasi-parallel manifold of the seven lowest electronic states (all within $\sim 4000 \text{ cm}^{-1}$). This structure is remarkably similar to the predicted structure of $\text{PaF}^{3+}$.
• Orbital Character: The singly occupied molecular orbital (SOMO) for these low-lying states is dominated by $f$-spinors from the Cerium atom, with some admixture of $p$-orbitals from Fluorine.
• Laser Cooling Potential: The transitions between these low-lying states exhibit diagonal Franck-Condon factors (close to unity), which is a prerequisite for laser cooling and optical cycling. However, the transition dipole moments for some of these transitions are relatively low.
• State Manipulation: The paper outlines a potential optical pumping scheme using the $(1)_{5/2}$ excited state to prepare the ground state $(X)_{5/2}$, noting the need for vibrational repump lasers.

C. Sensitivity to New Physics

• Enhancement Factors: The calculated enhancement factors ($W$) for $P,T$-odd properties were compared to known systems like $\text{HfF}^+$ and $\text{ThO}$.
  • Electron Spin-Dependent ($W_d, W_s$): These are suppressed in $\text{CeF}^{2+}$ (and $\text{PaF}^{3+}$) compared to lighter molecules because the SOMO is $f$-character, which has low electron density at the nucleus compared to $s/p$ orbitals.
  • Electron Spin-Independent ($W_T, W_S$): These factors, particularly the Nuclear Schiff Moment ($W_S$), are significantly enhanced. This makes $\text{CeF}^{2+}$ (and by proxy $\text{PaF}^{3+}$) an ideal probe for quark-sector $P,T$-odd physics and nuclear structure effects.
• Global Analysis: The authors argue that adding $\text{CeF}^{2+}$ data to global analyses of $P,T$-odd parameters will help break degeneracies in the parameter space, complementing existing heavy-molecule experiments.

4. Significance and Future Outlook

• Proof of Concept: The successful formation of $\text{CeF}^{2+}$ validates the experimental methodology (RFQcb with $\text{SF}_6$ injection) for creating highly charged molecular ions. This is a critical stepping stone toward the ultimate goal of forming and trapping radioactive $^{229}\text{PaF}^{3+}$.
• Surrogate System: $\text{CeF}^{2+}$ serves as a stable, abundant proxy to develop and refine quantum control techniques (laser cooling, state preparation, detection) that will be directly transferable to the short-lived $^{229}\text{Pa}$ system.
• New Physics Reach: The paper establishes that medium-heavy systems with $f$-electron character offer unique sensitivity to nuclear Schiff moments, filling a gap in the current landscape of EDM searches which are dominated by $s/p$-electron systems.
• Path Forward: The authors propose specific upgrades for future $\text{PaF}^{3+}$ experiments, such as dedicated gas exchange cells to prevent charge exchange of $\text{Pa}^{4+}$ with helium, or using electron-impact ionization (EBIT) to create $\text{PaF}^{3+}$ from lower charge states.

In conclusion, this work bridges the gap between theoretical proposals for exotic molecular ions and experimental reality, demonstrating that the infrastructure required to probe the most extreme nuclear deformations for new physics is within reach.