High-resolution spectroscopy of barium monofluoride: Odd isotopologues, hyperfine structure and isotope shifts

TL;DR

This work delivers a comprehensive experimental and theoretical study of BaF spectra across both even and odd isotopologues, with a focus on 135BaF and 137BaF. By combining high-resolution fluorescence and in-cell absorption spectroscopy with state-of-the-art Dirac-Coulomb four-component ab initio calculations, the authors benchmark excited-state hyperfine structure and extract precise molecular constants. The results enable a robust King-plot analysis of isotope shifts, revealing odd-even staggering in the Ba nuclear charge radius and offering nuclear-structure insights from molecular spectroscopy. The findings also provide essential spectroscopic input for laser cooling and future precision measurements of nuclear moments and fundamental-symmetry tests using BaF, including short-lived isotopes. Overall, the work tightly links detailed molecular spectroscopy to nuclear structure and beyond-Standard-Model physics, while advancing practical capabilities for cooling and detecting rare BaF isotopologues.

Abstract

Barium monofluoride (BaF) is a promising molecular species for precision tests of fundamental symmetries and interactions. We present a combined theoretical and experimental study of BaF spectra and isotope shifts, focusing in particular on the poorly understood odd isotopologues 137BaF and 135BaF. By comparing state-of-the-art ab initio calculations with high-resolution fluorescence and absorption spectroscopy data, we provide a benchmark for electronic structure theory and disentangle the hyperfine and rovibrational spectra of the five most abundant isotopologues, from 138BaF to 134BaF. The comprehensive knowledge gained enables a King plot analysis of the isotope shifts that reveals the odd-even staggering of the barium nuclear charge radii. It also paths the way for improved laser cooling of rare BaF isotopologues and crucially supports future measurements of nuclear anapole and Schiff moments.

Figures (9)

• Figure 1: Level structure of the $\mathrm{X}^2\Sigma^+$$(\nu)\rightarrow$$\mathrm{A}^2\Pi_{1/2}$$(\nu')$ transition in the odd isotopologues of BaF. Here, $N$ denotes the rotational, $J$ the total angular momentum without nuclear spins, and $G$, $F_1$ and $F$ additional intermediate and hyperfine angular momentum quantum numbers described in the text. Solid arrows indicate transitions probed by fluorescence spectroscopy, which resolves the full hyperfine structure, as shown in Fig. \ref{['fig:fluorescencespectroscopy']}. Dashed arrows indicate transitions investigated via absorption spectroscopy, as shown in Fig. \ref{['fig:absorptionspectroscopy']}, with the dash-dotted arrows summarizing any higher transitions probed. Absorption spectroscopy is performed for transitions between different vibrational quantum numbers $\nu$ and $\nu'$, which are of important interest as vibrational repumpers in laser cooling. Dots indicate substructures not shown, energy spacings are not to scale, and colors are used for visual clarity. A version of this figure containing all energy spacings extracted in this work can be found in the appendix.
• Figure 2: Fluorescence spectroscopy. Experimental (gray points) and modeled (black curve) spectra spanning all measured transitions arising from different $N$ and $G$ levels in the ${}^{137}$BaF and ${}^{135}$BaF $\mathrm{X}^2\Sigma^+$ ground states, and $J^P$ levels in the $\mathrm{A}^2\Pi_{1/2}$ states. Each experimental data point in a spectrum scan is typically averaged over $10$ repetitions, each taken at a different spots on the ablation target to minimize drifts in molecular signal. With $\approx\! 100~\mu$W of incident laser light, signal strengths of $100-1400\,$counts/pulse are observed, on a background of about $280$/pulse from scattered laser light. Backgrounds from scattered laser light are determined from signals outside the time window of the molecular pulse and subtracted. The insets show zoomed in example scans of the $X(N=1,G=2)\rightarrow A(J^p=1/2^+)$ and $X(N=0,G=2)\rightarrow A(J^p=3/2^-)$ transitions, the former being one of the transitions relevant for laser cooling Kogel2025lasercooling137. These plots also show two types of simulated spectra. The first is a "stick plot", showing narrow lines with heights set by the square of the associated transition dipole matrix element. The second is a complete model of the expected spectrum that uses the optical Bloch equations to determine the height and power-broadened width of each line (see appendix for details). In the simulated spectra, relative line heights are scaled by the natural abundance of the respective isotopologue, and line positions and dipole matrix elements are set by the final parameters determined in this work. Hyperfine Hamiltonian parameters were determined, for each isotope separately, from splittings between pairs of lines within each continuous spectrum. Separations between the hyperfine centers-of-gravity for the two isotopes within each continuous spectrum, as well as separations between the disconnected black traces, were varied to minimize the overall goodness of fit. The resulting separations were consistent, within uncertainties, with expectations based on non-hyperfine parameters extracted from the absorption spectroscopy reported in Fig. \ref{['fig:absorptionspectroscopy']}.
• Figure 3: Example absorption spectroscopy of BaF. The lines shown belong to the $X(\nu=0)\rightarrow A(\nu'=0)$ transition, which is the main cooling transition for laser cooling. Experimental data is shown in the top half of the plot, a prediction based the constants determined in this work in the bottom half. Labels denote the respective ground state angular momenta $G$ and $N$ involved in the transitions. Example transitions for $N=1,2,3,\ldots$ and $G=1,2$ are indicated in Fig. \ref{['fig:levelstructure']}. Individual predicted lines for ${}^{137}$BaF through ${}^{134}$BaF are shown in blue, orange, green and red, respectively. The envelope shown as a solid black line takes into account the Doppler broadening at $3.5\,$K, using the molecular constants determined in this work. The dashed line is based on the previous best set of constants Steimle2011, which predict significantly shifted transitions and even transitions that were not observed. This discrepancy makes accurate line assignments challenging, underscoring the importance of our fluorescence spectroscopy and theoretical results for reliable assignments. For clarity, the amplitude of the data in every x-axis sub-segment has been individually normalized. Residual differences in amplitude between experiment and prediction are due to collisional and optical pumping effects inside the buffer gas source, which will be discussed in future work.
• Figure 4: King plot analysis and odd-even staggering of the barium nuclear charge radius. (a) Isotope shifts of BaF's (from top to bottom) $T_{01}$ (green), $T_{11}$ (orange) and $T_{10}$ (blue) constants relative to the $T_{00}$ constant. (b) Isotope shifts of $T_{00}$ (red), as well as the transitions above, compared to the $553.6\,$nm transition in atomic barium FrickeDatabase. Lines in (a,b) have been vertically offset for visual clarity, with their offset values indicated. (c) Relative change in nuclear charge radius $\delta\langle r^2\rangle$, as determined by combining our data with previously known mass and field shift values Athanasakis2023. Data points are averaged over all molecular transitions studied in (a,b), and are in excellent agreement with atomic reference data (solid line). Error bars of the atomic data (shaded area) are derived from uncertainties in the atomic isotope, mass and field shifts FrickeDatabase, and the inset highlights the agreement at the 1–2% level.
• Figure 5: Experimental (gray dots) and modeled (black curve) spectra for the $X\ket{N=0, G=2, F_1,F} \to A\ket{J'^{ P} = 1/2^-,F_1',F'}$ transitions in $^{135}$BaF and $^{137}$BaF (with individual modeled spectra and quantum numbers $\ket{F_1,F} \to \ket{F_1',F'}$ labeled in green and blue, respectively). Also shown is the $X\ket{N=1, G=1,F_1,F} \to A\ket{J'^{ P} = 1/2^+,F_1',F'}$ transition in $^{137}$BaF (with quantum numbers in blue), which overlaps partially with this frequency range. Among the transitions labeled in the diagram, only the ones in bold were used in the quantitative analysis. A stick plot depicting all transitions, with strength proportional to the square of the transition dipole matrix elements, is overlaid on the full modeled spectrum generated using simulations based on the optical Bloch equations. Note that for the simulated spectra shown here (and in all figures), the center of mass of each isotopologue and rotation state was adjusted to maximize overlap with the data. The resulting adjustments were well within the range of uncertainties for the rotational and isotope splittings determined via absorption spectroscopy.