Inelastic collision characteristics of electrons in liquid water

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Abstract

Calculations of inelastic mean-free-paths and collision stopping-powers for low energy electrons (<10 keV) in liquid water are presented and compared with the existing values in the literature. A semi-empirical inelastic model was developed which makes use of (a) the dielectric formalism for the valence shells responsible for condensed-phase effects, and (b) the binary-encounter-approximation for the K-shell. The energy and momentum dependence dielectric-response function of liquid water was constructed based on a modified Drude model, optical data, and appropriate dispersion algorithms. Binding effects for the K-shell were based on the initial (unperturbed) molecular state. Calculations were performed under the first Born approximation with the inclusion of empirical correction functions at very low energies (<300 eV).

Introduction

The interaction of electrons in water in the liquid phase is of particular importance to biomedical applications due to the abundance of this substance in biological tissues. The development of Monte-Carlo (MC) codes for full electron slowing-down simulation necessitates that accurate and computationally simple scattering models are available over a wide range of impact energies down to about a few eVs.

Treating the inelastic scattering process as a two-body collision between the projectile and a quasi-free electron should generally suffice for ionization events producing fast secondaries. Inner-shell ionization events (much above threshold) associated with small impact parameters fall under this scheme. However, the majority of inelastic collisions involve small energy losses to both discrete and continuum states which cannot be properly described by binary models. This is illustrated by Bethe's asymptotic cross-section where the soft collision spectrum is determined by the optical properties of the target [1]. Small energy losses are dissipated to the diffused outer shells that exhibit condensed-phase properties.

The present work is part of an effort to extend the already developed MC code [2], [3] to the transport of electrons in liquid water. As a first step, an inelastic model for the liquid suitable for full MC simulation is developed. The electron subsystem of liquid water was divided into a valence band, representing the smearing of the four outer shells of H2O, and a core shell representing the oxygen K-shell. The large difference in binding energy between valence and core shells provides justification for this approximation. The dielectric formalism, which accounts for condensed-phase effects, was used for the valence band, while the binary-encounter-approximation (BEA) for the core shell. Important transport quantities are then calculated in the first Born approximation (FBA) supplemented by correction functions at very low energies.

Section snippets

Valence band

The inelastic interaction with the valence electrons of condensed targets (e.g. liquids, solids) is best described in terms of the energy and momentum dependence dielectric-response function (DRF) of the material ε(E,K), E and ℏK being the energy and momentum transfer, respectively (assuming an isotropic and homogeneous medium). For sufficiently energetic projectiles the FBA applies, and the doubly differential inverse mean-free-path (MFP) becomes (non-relativistic limit) [4]d2ΣdEdK=1πα0TKIm−1

Results

Based on the parametrization of the optical spectrum of the liquid described in the text (see also Table 1), Fig. 1 depicts the predicted real and imaginary parts of the DRF (optical limit) along with the experimental data. Fig. 2 depicts the ELF (optical limit) obtained by Eq. (12) along with the experimental data and the model predictions of Ritchie et al. [13] and the more recent calculations of Dingfelder et al. [8]. Evidently, an improved representation of the data was attained by the

Acknowledgements

Research sponsored by the US Department of Energy under contract DE-AC05-960R22464 (NN-20 Program).

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