Wigner time delay of a particle elastically scattered by a cluster of zero-range potentials

TL;DR

This paper investigates the Wigner time delay in elastic scattering of slow particles by complex targets made of zero-range potentials, deriving formulas and analyzing specific multi-center models.

Contribution

It introduces general formulas linking phase shifts of individual zero-range potentials to the overall scattering phase, and applies these to multi-center targets.

Findings

Derived formulas connecting individual and cluster phase shifts.

Analyzed time delays for two-, three-, and four-center targets.

Applied results to electron and meson scattering scenarios.

Abstract

The Wigner time delay of slow particles in the process of their elastic scattering by complex targets formed by several zero-range potentials is investigated. It is shown that at asymptotically large distances from the target, the Huygens-Fresnel interference pattern formed by spherical waves emitted by each of the potentials is transformed into a system of spherical waves generated by the geometric center of the target. These functions determine flows of particles in and out through the surface of the sphere surrounding the target. The energy derivatives of phase shifts of these functions are the partial Wigner time delay. General formulas that connect the s-phase shifts of particle scattering by each of the zero-range potentials with the phases of particle scattering by the potential cluster forming the target are obtained. Model targets consisting of two-, three- and 4-centers are considered. It is assumed that these targets are built from identical delta-potentials with equal distances between their centers. The partial Wigner time delay of slow particles by considered targets are obtained. We apply the derived general formulas to consideration of electron scattering by atomic clusters that trap electron near the target, and by calculating the times delay of mesons scattered by few-nucleons systems.