Measurement of the hyperfine coupling constant for $n{S}_{1\mathrm{/}2}$ Rydberg states of ${}^{85}$Rb

We present measurements of the hyperfine structure splittings of $n{S}_{1\mathrm{/}2}$ Rydberg states of ${}^{85}$Rb for $n$= 43, 44, 45 and 46. From the splittings, the hyperfine coupling constant, $A_{\mathrm{H}\mathrm{F}\mathrm{S}}$, is determined to be 15.372(80) GHz. This result is an order-of-magnitude improvement from previous measurements. We study and account for systematic uncertainty sources, such as unwanted electric and magnetic fields, dipolar Rydberg-Rydberg interactions, and AC shifts. Initial evidence for hyperfine-mixed Rydberg pair states is found.