Chapter 6

Ratio of hyperfine splittings The spin and magnetic moment of the proton are \(\left(1 / 2,2.79 \mu_{\mathrm{N}}\right)\), of the deuteron \(\left(1,0.857 \mu_{\mathrm{N}}\right)\) and of \({ }^{3} \mathrm{He}\left(1 / 2,-2.13 \mu_{\mathrm{N}}\right)\). Calculate the ratio of the ground-state hyperfine splittings of (a) atomic hydrogen and deuterium and (b) atomic hydrogen and the hydrogen-like ion \({ }^{3} \mathrm{He}^{+}\).

Interval for hyperfine structure The \(3 \mathrm{~d}^{5} 444 \mathrm{p}^{6} \mathrm{P}_{7 / 2}\) level of \({ }^{55} \mathrm{Mn}\) is split by hyperfine interaction into six levels that have separations \(2599,2146,1696,1258\) and \(838 \mathrm{MHz}\). Deduce the nuclear spin of \({ }^{55} \mathrm{Mn}\) and show that the separations confirm your value.

Hyperfine structure When studied by means of high-resolution spectroscopy, the resonance line \(4 \mathrm{~s}^{2} \mathrm{~S}_{1 / 2}-4 \mathrm{p}^{2} \mathrm{P}_{1 / 2}\) of naturally-occurring potassium consists of four components with spacings and intensity ratios as shown in the following diagram.Natural potassium is a mixture of \({ }^{39} \mathrm{~K}\) and \({ }^{41} \mathrm{~K}\) in the ratio \(14: 1\). Explain the origin of the structure, and deduce the nuclear spins and the ratio of the magnetic moments of the two isotopes.