Extrapolation of Cross-Section Data to Zero Energy for Long-Range Effective Potentials

Abstract

The effective one-body potential representing a particle-compound-system interaction usually has a very complicated short-range behavior, but a relatively simple, and more or less known, long-range tail $U$ which can be nonlocal and energy-dependent. With $U$ present, the usual effective-range-theory (ERT) expansion for a single-channel scattering phase shift $\ensuremath{\eta}$ is inapplicable. Recent extensions of ERT include the effect of long-range potentials, but since $\ensuremath{\eta}$ then becomes very energy-dependent, one would need many terms to obtain the desired accuracy, so the results are of restricted use. We show that one can first solve, numerically, a one-body problem which includes only $U$, and then use a modified ERT expansion for the difference $\ensuremath{\delta}$ between $\ensuremath{\eta}$ and the phase shift due to $U$ alone. Since the rapid energy dependence due to $U$ is thus extracted out (exactly), terms in the expansion which depend upon the short-range potential in the presence of $U$ are more slowly varying in energy; they involve a number of adjustable parameters to be determined experimentally, but only a few are usually needed. The procedure thus makes possible the extrapolation of low-energy scattering data down to zero energy, even when effective long-range potentials are present.