Discussion of coherent and incoherent contributions to the spatial distribution of very low energy electrons elastically scattered in liquid water

The occurrence of diffraction effects versus the validity of trajectory simulation of the elastic scattering of very low energy electrons (<20eV) in liquid water is discussed. A simple model is used where the water molecules are represented by point scatterers, distributed randomly with or without short-range order. It is shown that the average spatial distribution of elastically scattered electrons within such a medium may be unambiguously divided into a coherent and an incoherent part. The calculation is based on the method of self-consistent quantum multiple scattering, and is performed for one wavelength where trajectory simulation is a valid approximation and one wavelength where it is not. The relation of the point scatterer model to advanced methods used for calculating quantum multiple scattering of electrons within clusters of atoms is briefly discussed. The point-scatterer quantum calculations are compared to corresponding trajectory simulations and to solutions of the Helmholtz-Foldy equation. Results indicate that 1) the coherence length for electrons scattered in a medium with random-like variations in scatterer positions is limited by elastic as well as inelastic scattering, and may taken to be equal to the total mean free path; 2) diffraction effects may occur due to short-range order in the medium, or by means of coherent scattering from spatially fixed structures (e.g., boundaries or interfaces) provided that the distance between such objects does not greatly exceed the coherence length; 3) trajectory simulation of the elastic scattering process gives a good approximation of the average quantum scattering in the medium, provided that the wavelength is not larger than the average distance between the scatterers; the effect of coherent scattering on the electron spatial distribution within the medium is then small or absent. The results further show that the Helmholtz-Foldy equation, which otherwise may be used to calculate the coherent part, is not generally a good approximation at long wavelengths in the presence of short-range order.