For twenty years research into the anomalies in the HF spectra was going in a wrong direction in fighting the Bohr-Weisskopf effect. As way out, we propose the model-independent way, which enables the nuclear radii and their moments to be obtained 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. 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 obtained in this way by analysis of the experimental HFS for the H-and Li-like ions of 209 Bi. The critical prediction is made concerning the HFS for the 2p 1 2 state. 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 recommendations are given, aimed at retrieving data on the HFS values for a set of a few-electron configurations of various atoms.