Trajectory calculation as an approximation to the exact quantum treatment of multiple elastic scattering

The relation between trajectory calculation and the exact quantum treatment of purely elastic, multiple scattering is studied for the scattering of a particle incident on a cluster of N point-size scattering centers, randomly distributed in a volume V inside a well-defined boundary. Each scatterer has the same scattering amplitude with s-wave phase shift π/2, corresponding to the cross section σ1. The differential cross section of such a cluster and the distribution of scattering event density inside the cluster are calculated. In the single scattering (kinematic) approximation, the average differential cross section, obtained by sampling over a number of such clusters, is a superposition of a diffraction pattern exactly such as obtained by coherent scattering in a homogeneous continuum of density n = N/V inside the boundary of the cluster, and an incoherent part which is simply the sum N · σ1 of the point scatterer cross sections and thus gives a contribution that is independent of scattering angle. Under certain conditions – tentatively, one condition is that the incident particle wavelength is not appreciably larger than the average distance dnn ≈ n−1/3 between nearest neighbour scattering centers – the exact (dynamic diffraction) differential cross section is found to be a similar superposition of a coherent and an incoherent part. The incoherent part, which now varies with the scattering angle, is approximately reproduced by conventional trajectory simulation, which may thus be regarded as a valid partial approximation. The trajectory simulation also reproduces approximately the scattering event density distribution inside the cluster. The coherent part, which is a diffraction pattern closely similar to that obtained in the kinematic case, is approximately reproduced by a model which assumes it to be due to single coherent scattering of a damped incident wave in the cluster. Under different conditions, however, it is found that trajectory calculation fails to reproduce any part of the exact differential cluster cross section, and also fails to reproduce the scattering event density distribution. This failure seems to occur for example when the conventional mean free path (nσ1)−1 is appreciably smaller than dnn. Some additional aspects of the limits of validity of trajectory calculations due to the external shape of the cluster are discussed.