Nearly elastic scattering and the trajectory approximation

We consider the scattering of an incident particle by bosons. We assume the coupling is linear in the boson coordinates, but can be quite general in the incident particle coordinate. We show that when the final distribution is close to elastic, the independent-boson model applies to this problem. It yields a loss distribution which is simply the exponentiation of the first-order distorted-wave Born approximation (DWBA). Furthermore, we demonstrate that in this regime, for short wavelengths of the scattered particle, this exponentiated Born approximation reduces to the trajectory approximation (TA). Thus it includes both the DWBA and the TA as special cases. We also give a simple recipe for estimating the error this approximation makes. We illustrate these results on a simple one-dimensional model involving a single oscillator, and discuss their relation to previous studies of overlapping regimes. This approximation and accompanying error estimate should prove very useful in analyzing multiphonon effects in atom-surface scattering.