Correlation trends in the hyperfine structure for Rb, Cs, and Fr, and high-accuracy predictions for hyperfine constants

We perform high-precision calculations of the hyperfine structure for nS(1/2) and nP(1/2) states of the alkali-metal atoms Rb, Cs, and Fr across the principal quantum number n and study the trend in the size of the correlations. Our calculations are performed in the all-orders correlation potential method. We demonstrate that the relative correlation corrections fall off quickly with n and tend towards constant and nonzero values for highly excited states. This trend is supported by experiments, and we utilize the smooth dependence on n to make high-accuracy predictions of the hyperfine constants, with uncertainties to within 0.1% for most considered S-1/2 states of Rb and Cs.