The differential scalar polarizability $\mathrm{\Delta }{\alpha }_0\left(\omega \right)$ of the ${\mathrm{Ba}}^{+}$ $S_{1/2}\text{−}{D}_{5/2}$ clock transition has a zero crossing near 481 nm, which is measured to be 623.603 13(17) THz. From this measurement, we infer a ratio of reduced matrix elements $\langle P_{3/2}\parallel r\parallel S_{1/2}\rangle /\langle P_{1/2}\parallel r\parallel S_{1/2}\rangle =1.41181\left(13\right)$, which provides a stringent test of atomic structure calculations and experimental determination of matrix elements. Additionally, it enables the construction of an accurate approximation to $\mathrm{\Delta }{\alpha }_0\left(\omega \right)$, valid for frequencies up to 450 THz, with only one reduced matrix element, $\langle P_{1/2}\parallel r\parallel S_{1/2}\rangle$, appearing in the model's parametrization. We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.

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• Atomic & molecular structure
• Atomic, optical & lattice clocks