Theory of the Hyperfine Anomalies of Deuterium, Tritium, and Helium-3+

The covariant equation for the three-body deuterium system is discussed, and reduced to a three-dimensional equation for the case of instantaneous two-body interactions. A noncovariant perturbation scheme, based on this three-dimensional equation, was employed to calculate the hyperfine structure (hfs) to order $\frac{\alpha m}{M}$ (hfs) for ${}_1{}^{}{}_{}{}^{}\mathrm{H}_{}^2{}_{}{}^{}$, ${}_1{}^{}{}_{}{}^{}\mathrm{H}_{}^3{}_{}{}^{}$, ${}_2{}^{}{}_{}{}^{}\mathrm{He}_{}^{3+}{}_{}{}^{}$. The results are compared with the experimental values, and shown to be compatible. Final determination of the theoretical values and the adequacy of the theory is shown to depend critically on the determination of a number of experimental quantities, in particular on a more precise measurement of $\alpha$, of the nucleon electric and magnetic form factors and associated nucleon polarization, of the presence and magnitude of a two-nucleon spin-orbit potential, and of the singlet $n-p$ effective range.