Higher $eta_c(nS)$ and $eta_b (nS)$ mesons

The hyperfine splittings in heavy quarkonia are studied in a model-independent way using the experimental data on di-electron widths. Relativistic correlations are taken into account together with the smearing of the spin-spin interaction. The radius of smearing is fixed by the known $J/psi-eta_c(1S)$ and $psi(2S)-eta_c(2S)$ splittings and appears to be small, $r_{ss} cong 0.06$ fm. Nevertheless, even with such a small radius an essential suppression of the hyperfine splittings ($sim 50%)$ is observed in bottomonium. For the $nS~ bbar b$ states $(n=1,2,...,6)$ we predict the values (in MeV) 28, 12, 10, 6, 6, and 3, respectively. For the $3S$ and $4S$ charmonium states the splittings 16(2) MeV and 12(4) MeV are obtained.