Elastic particle transport in non-crystalline matter - a comparison of trajectories and diffraction

Due to doubts which may arise in the case of low energy (long wavelength) particles, the validity of Monte Carlo trajectory simulation or the equivalent transport theory is considered. After a preliminary discussion, the present analysis is focused on elastic plural or multiple scattering in non-crystalline matter. The effect of kinematic diffraction from molecules in a gas is demonstrated, but argumentation based on the assumption of random phases in kinematic diffraction is shown to be inadequate for scattering in condensed matter. Dynamic diffraction represents a full quantum treatment of multiple elastic scattering, which may be used to test and explore the limits of validity of the trajectory formalism. Dynamic s-wave diffraction of particles from clusters of identical point-size scatterers allows an exact solution of the quantum problem; for practical reasons, the cluster size is however limited. The average dynamical diffraction cluster cross section 〈σNdyn〉 is calculated for a large number of simulated clusters, each with an equal number N (≈10-1000) of point-size scatterers, randomly distributed inside a volume V of prescribed shape. This is compared with the trajectory cluster cross section σNtraj calculated by the trajectory formalism for particles elastically scattered by a homogeneous density n = N/V of scatterers inside the same volume. Besides the trivial agreement which is seen when multiple scattering is negligible, non-trivial, approximate agreement 〈σNdyn〉≈σNtraj is found for thin layers of high density. Approximate agreement has also been found with trajectory simulated transmission and backscattering in such foils; otherwise, however, agreement is not found. This indicates, though not conclusively, that trajectory simulation may, under certain, restrictive conditions, be an approximation of the average effect of dynamic diffraction in samples with randomly varying internal structure. It is hoped that the method of comparison may contribute to the understanding of the basis and limits of validity of trajectory calculations of multiple scattering. For example, present results indicate that in a trajectory simulation, a mean free path which is orders of magnitude smaller than the average distance between the scatterers may be consistent with the full quantum treatment.