Zero crossings of the differential scalar polarizability of Ba${}^{+}$ clock transition

The differential scalar polarizability $\mathrm{\Delta }{\alpha }_0(\omega )$ of the Ba${}^{+}$ S${}_{1\mathrm{/}2}$-to-D${}_{5\mathrm{/}2}$ clock transition has a zero crossing near 481nm, which is measured to be 623.603,13(17),THz. From this measurement, we infer a ratio of reduced matrix elements $⟨P_{3\mathrm{/}2}\mathrm{\parallel }r\mathrm{\parallel }{S}_{1\mathrm{/}2}⟩\mathrm{/}⟨P_{1\mathrm{/}2}\mathrm{\parallel }r\mathrm{\parallel }{S}_{1\mathrm{/}2}⟩=1.41181(13)$, 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 $\mathrm{\Delta }{\alpha }_0(\omega )$, valid for frequencies up to 450,THz, with only one reduced matrix element, $⟨P_{1\mathrm{/}2}\mathrm{\parallel }r\mathrm{\parallel }{S}_{1\mathrm{/}2}⟩$, appearing in the model’s parameterization. We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation (BBR) shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.