Energies and lifetimes of the 9p and 10p excited states in atomic francium

This paper reports the first experimental measurements of the absolute wavenumbers and radiative lifetimes for the 9p and 10p excited states in francium using Collinear Resonance Ionization Spectroscopy, providing a precise test of relativistic coupled-cluster theory that shows good agreement for lifetimes and relative energies despite a small offset in absolute energies.

The Gist

Imagine the atom as a tiny, multi-story apartment building. The ground floor is where the electron usually lives, but if you give it a little energy, it can jump up to higher floors. In the world of physics, these floors are called "energy levels" or "excited states."

This paper is about a very special, very heavy apartment building called Francium. Francium is the heaviest member of the "Alkali" family (which includes common things like sodium and potassium). Because it's so heavy, its electrons move incredibly fast, and the laws of physics get a bit weird and "relativistic."

Here is the story of what the scientists did, explained simply:

1. The Mystery of the Missing Floors

For a long time, scientists had a map of Francium's apartment building, but it was incomplete. They knew about the lower floors (like the 7th and 8th floors), but the 9th and 10th floors were a mystery. No one had ever measured them directly.

Why does this matter?

• The "Goldilocks" Test: Scientists use Francium to test the "Standard Model" of physics (the rulebook of the universe). They are looking for tiny cracks in the rules, like if the electron has a tiny electric "lopsidedness" (dipole moment).
• The Problem: To find these cracks, they need to know exactly how the building is built. If their theoretical map of the 9th and 10th floors is wrong, they can't tell if a measurement is a new discovery or just a mistake in the map.

2. The High-Speed Elevator (The Experiment)

To measure these high floors, the team used a machine called CRIS (Collinear Resonance Ionization Spectroscopy) at CERN. Think of this as a high-tech, high-speed elevator system.

• The Elevator: They took Francium atoms, ionized them (gave them a charge), and shot them down a long tube at nearly the speed of a bullet.
• The Laser "Flashlight": They used lasers as flashlights to check which floor the electron was on.
  • First, they hit the atom with a laser to boost the electron from the ground floor to the 9th or 10th floor.
  • Then, they hit it with a second laser to knock the electron out completely (ionization) so they could count it.
• The Result: By tuning the laser frequency, they found the exact "key" needed to open the doors to the 9th and 10th floors. They measured the exact energy (wavenumber) and how long the electron stayed there (lifetime) before falling back down.

3. The "Stopwatch" Challenge (Measuring Lifetimes)

Measuring how long an electron stays on a high floor is like trying to time how long a firework stays in the air before it explodes.

• The electron doesn't just fall straight down; it often takes a "scenic route," hopping to intermediate floors (like the 7th or 8th) before hitting the ground.
• The scientists had to be very clever. They used a "stopwatch" (a fast laser pulse) to start the timer and another to stop it. They had to account for all the "scenic route" hops to get the true time.
• The Finding: They found that the electrons on the 9th and 10th floors stay there for a few hundred nanoseconds (billionths of a second).

4. The Theory vs. Reality Check

The team also had a team of "architects" (theoretical physicists) who had built a computer model of the Francium building using complex math (Relativistic Coupled-Cluster theory).

• The Comparison: They compared their new measurements (the real building) with the computer model (the blueprint).
• The Verdict:
  • The Blueprint was mostly right! The model predicted the relative distances between the floors perfectly.
  • The "Global Shift": However, the whole blueprint was shifted up or down by a tiny bit compared to reality. It's like if the architect drew the whole building correctly, but forgot to account for the height of the foundation.
  • The Lifetimes: The model predicted exactly how long the electrons would stay on the floors. This is huge news because it proves the computer model understands the "physics of the walls" (how electrons interact) very well.

Why Should You Care?

Think of this like checking the blueprint of a skyscraper before you build a new one.

1. Trust the Map: Now that we have measured the 9th and 10th floors, we know our computer models are reliable. We can trust them to predict things we can't measure yet.
2. Finding New Physics: Because the models are now so accurate, if we do find a tiny difference between the model and a future experiment, we can be sure it's not a mistake in the math. It might be a new law of physics waiting to be discovered.

In a nutshell: Scientists finally measured the "attic" and "roof" of the heaviest atom, Francium. They found that our best computer models are incredibly accurate, giving us a solid foundation to hunt for the secrets of the universe.

Technical Summary

1. Problem Statement

Francium (Fr, $Z=87$) is the heaviest alkali metal and a premier candidate for precision tests of fundamental symmetries, such as Parity Non-Conservation (PNC) and the search for the electron electric dipole moment (eEDM). These experiments rely heavily on accurate theoretical calculations of atomic structure, specifically Electric Dipole (E1) matrix elements.

However, a significant gap existed in the experimental data for francium:

• Unexplored Spectrum: No experimental data existed for francium $p$-states with principal quantum numbers $n > 8$.
• Theoretical Uncertainty: While relativistic coupled-cluster (RCC) theories exist, they lacked rigorous experimental benchmarks for highly excited states (Rydberg states). Without these benchmarks, it was difficult to validate the accuracy of theoretical models needed to extract fundamental symmetry-violating parameters.
• Existing Data Limitations: Previous values for 9p and 10p states in the NIST database were semi-empirical (based on Rayleigh-Ritz formulas) and lacked isotope shift considerations.

2. Methodology

The study employed Collinear Resonance Ionization Spectroscopy (CRIS) at the ISOLDE facility (CERN) using a beam of the radioactive isotope $^{221}\text{Fr}$.

Experimental Setup:

• Ion Production: $^{221}\text{Fr}^+$ ions were produced via proton bombardment of a uranium carbide target, followed by surface ionization, mass separation (HRS), and cooling/bunching in a radio-frequency quadrupole trap (ISCOOL).
• Neutralization: Ions were neutralized in a charge-exchange cell (CEC) filled with sodium vapor.
• Laser Excitation Scheme: A two-step resonant ionization process was used:
  1. Excitation: Tunable lasers excited atoms from the ground state ($7s\ ^2S_{1/2}$) to the target $np\ ^2P_{1/2,3/2}$ states.
     • 8p: 422 nm (Ti:Sa laser).
     • 9p: 365–368 nm (Ti:Sa laser).
     • 10p: 342–343 nm (Optical Parametric Oscillator, OPO).
  2. Ionization: A 1064 nm Nd:YAG laser ionized the excited atoms.
• Detection: Ionized atoms were deflected onto a single-ion counting detector.

Measurement Techniques:

• Wavenumbers: Spectra were recorded by scanning laser frequencies. Centroids were extracted using a least-squares fit (SATLAS package). Hyperfine structure was unresolved for the upper states due to laser linewidth, so fits relied on the known ground-state hyperfine coefficient ($A_{7s}$).
• Lifetimes: Radiative lifetimes were measured using a pulsed delay method. The time delay between the excitation pulse and the ionization pulse was varied. The decay curve of the ion signal was fitted to a model accounting for:
  • Direct exponential decay of the excited state.
  • Cascade contributions: Population feeding from the excited state to lower-lying states (which are then ionized by the 1064 nm laser).
  • Background noise.

Theoretical Framework:

• RCCSDT: The authors performed new ab initio calculations using the Relativistic Coupled-Cluster with Single, Double, and Triple excitations (RCCSDT) method.
• Comparison Strategy: Since theoretical calculations provide binding energies relative to the ionization limit, the authors adjusted the theoretical ground-state energy to match the experimental 7s$\to$8p transition. This allowed for a direct comparison of relative excitation energies and lifetimes.

3. Key Contributions

1. First Experimental Data: This work presents the first-ever experimental measurements of absolute wavenumbers and radiative lifetimes for the 9p and 10p excited states in francium.
2. High-Precision Benchmarking: It provides a rigorous benchmark for relativistic many-body theories (specifically RCCSDT) in the heavy alkali regime, where electron correlation and relativistic effects are extreme.
3. Validation of E1 Matrix Elements: By comparing measured lifetimes with theory, the study validates the accuracy of calculated E1 matrix elements, which are critical inputs for atomic parity violation (APV) calculations.
4. Improved Theoretical Models: The paper highlights that while absolute energies in RCCSDT have a global offset (due to missing high-order correlations), the relative level spacings and lifetimes are reproduced with high accuracy, confirming the method's reliability for transition properties.

4. Key Results

A. Absolute Wavenumbers (cm$^{-1}$) for $^{221}\text{Fr}$:

• 8p $^2P_{3/2}$: $23,657.5355(11)$ (Benchmarked against previous work).
• 9p $^2P_{1/2}$: $27,111.238(5)$
• 9p $^2P_{3/2}$: $27,360.097(5)$
• 10p $^2P_{1/2}$: $29,058.1(6)$
• 10p $^2P_{3/2}$: $29,192.8(8)$
• Note: These values correct and supersede previous semi-empirical NIST data.

B. Radiative Lifetimes (ns):
The experimental lifetimes showed excellent agreement with RCCSDT predictions (within uncertainties):

• 9p $^2P_{1/2}$: Exp: $329(6)$ ns | Theory: $318(9)$ ns
• 9p $^2P_{3/2}$: Exp: $179(5)$ ns | Theory: $186(6)$ ns
• 10p $^2P_{1/2}$: Exp: $553(21)$ ns | Theory: $574(4)$ ns
• 10p $^2P_{3/2}$: Exp: $362(5)$ ns | Theory: $356(5)$ ns
• Observation: The relative deviations are generally $\le 4\%$. The $P_{3/2}$ states have shorter lifetimes than $P_{1/2}$ due to stronger coupling to the $7s$ ground state.

C. Theoretical Analysis:

• Energy Offset: The RCCSDT calculation showed a systematic global offset of $\sim 154 \text{ cm}^{-1}$ in absolute binding energies compared to experiment. This is attributed to missing higher-order correlation effects (quadruple excitations) and high-angular-momentum orbitals.
• Relative Accuracy: Despite the absolute offset, the relative excitation energies (level spacings) were reproduced with high precision (scatter $< 5 \text{ cm}^{-1}$).
• Lifetime Agreement: The close match in lifetimes confirms that the RCCSDT wavefunctions accurately capture the state-dependent electron correlation physics required for E1 matrix elements.

5. Significance

• Fundamental Physics: The validated theoretical models and E1 matrix elements are now more reliable for interpreting future atomic parity violation experiments in francium, which aim to probe physics beyond the Standard Model.
• Methodological Validation: The study demonstrates that RCCSDT is a robust tool for heavy atoms, provided that relative energies and transition properties are the focus rather than absolute binding energies.
• Future Directions: This work paves the way for measuring higher members of the P series and $d$-states, completing the spectroscopic picture of francium and further refining tests of fundamental symmetries.

In summary, this paper bridges a critical gap between experiment and theory for heavy alkali atoms, providing the essential data needed to trust theoretical predictions for precision tests of fundamental physics in francium.