  For twenty years research into the anomalies in the HF spectra was going in a
wrong direction by fighting the related Bohr-Weisskopf effect. As a way out,
the model-independent way is proposed of estimating the nuclear radii from the
hyper-fine splitting. The way is based on analogy of HFS to internal conversion
coefficients, and the Bohr-Weisskopf effect - to the anomalies in the internal
conversion coefficients. This makes transparent It is shown that the parameters
which can be extracted from the data are the even nuclear moments of the
magnetization distribution. The radii $R_2$ and (for the first time) $R_4$ are
thus obtained by analysis of the experimental HFS for the H- and Li-like ions
of $^{209}$Bi. The critical prediction of the HFS for the $2p_{1/2}$ state is
discussed. The moments may be determined in this way only if the higher QED
effects are properly taken into account. Therefore, this set of the parameters
form a basis of a strict QED test. Experimental prospects are discussed, aimed
at retrieving data on the HFS values for a set of a few-electron configurations
of the atom.
