Calculation of the hyperfine interaction using an effective-operator form of many-body theory

The effective-operator form of many-body theory is reviewed and applied to the calculation of the hyperfine structure. Numerical results are given for the $2p$, $3p$, and $4p$ excited states of Li and the $3p$ state of Na. This is the first complete calculation of the hyperfine structure using an effective-operator form of perturbation theory. As in the Brueckner-Goldstone form of many-body theory, the various terms in the perturbation expansion are represented by Feynman diagrams which correspond to basic physical processes. The angular part of the perturbation diagrams are evaluated by taking advantage of the formal analogy between the Feynman diagrams and the angular-momentum diagrams, introduced by Jucys et al. The radial part of the diagrams is calculated by solving one- and two-particle equations for the particular linear combination of excited states that contribute to the Feynman diagrams. In this way all second- and third-order effects are accurately evaluated without explicitly constructing the excited orbitals. For the $2p$ state of Li our results are in agreement with the calculations of Nesbet and of Hameed and Foley. However, our quadrupole calculation disagrees with the work of Das and co-workers. The many-body results for Li and Na are compared with semiempirical methods for evaluating the quadrupole moment from the hyperfine interaction and a new quadrupole moment of ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ is given.