Modified Effective-Range Theory

Developments in effective-range theory as applied to low-energy atomic scattering processes are reviewed. The focus is on the seminal contributions to this theory made by Larry Spruch, who showed how its original form could be extended to account for the presence of long-range forces arising from Coulomb interactions. The influence of this body of work on the very recent developments stimulated by advances in experimental techniques is emphasized. Some time ago, Wigner [1], in the context of the theory of nuclear reactions, formulated a law governing the energy dependence of transition amplitudes for energies near the threshold for the opening of a new reaction channel. The significance of the law lies in its generality; the threshold law is independent of the detailed dynamics of the reaction other than the short-range nature of the nuclear interaction. It was then seen [2] that with the threshold behavior accounted for, the amplitude for nucleon-nucleon scattering could be represented as a linear function of the scattering energy, in terms of two parameters, referred to as the scattering length and effective range. Here again, the significance of this “effective-range” theory lies in its generality; one requires only that the nuclear potential be of short range. In particular, it became apparent that an observation of low-energy neutron-proton scattering could provide information on, say, the depth and range of the potential but not on any of its finer details. A typical application may involve the extrapolation of scattering data based on measurements performed at a few different energies. Bound-state properties may be probed as well. Suitably extended, the theory allows for the inclusion of a Coulomb interaction [3].