Three-Body Effective Interaction in Deuterium

Abstract

The reduction of a simple, nonrelativistic three-body problem to an equivalent two-body problem is discussed. The theory is applied to deuterium, which is treated as a spinless three-body system with only two pair interactions present. The neutron-electron interaction is ignored and the neutron-proton interaction is assumed to be a separable $s$-state Yamaguchi potential. The electron-proton interaction is taken as a static Coulomb interaction. With these assumptions, an exact expression for an effective electron-deuteron potential, which depends only on the electron-deuteron relative coordinates and the total three-body energy, is derived. It is shown how this effective interaction may be compared using first-order perturbation theory with the corresponding effective potential of a model problem in which the structure of the deuteron is ignored. The three-body theory is used to estimate the contribution to the Lamb shift in deuterium arising from the finite size and polarizability of the deuteron. The effects of polarizability are found to be insignificant, and hence the usual two-body description of deuterium is justified.