On the validity of trajectory methods for calculating the transport of very low energy (<1 keV) electrons in liquids and amorphous media

Abstract It is easily demonstrated that a trajectory picture of low energy electron transport in condensed matter is not compatible with the Heisenberg uncertainty principle. The uncertainty in the position of a low energy electron is large and may in fact be larger than an entire simulated trajectory. This might be interpreted to mean that trajectory methods are not applicable. However, this conclusion is not correct. In the present paper, the evidence for the validity of low energy electron trajectory simulation is discussed, as well as the wave aspects and quantum nature of low energy electron transport in liquids and amorphous solids. It is pointed out that the validity of a trajectory approach to low energy electron transport in a liquid or amorphous solid partly is due to its ability to reproduce the average results of coherent elastic multiple wave scattering in a randomlike medium, and moreover that this ability may be further enhanced by the presence of inelastic scattering. The resulting validity of the trajectory method may be referred to as circumstancial validity, which is of a nature different from the explicit validity of trajectory methods which are compatible with the uncertainty principle. A previous systematic analysis of the limits of circumstancial validity is revisited and discussed for the basic case of multiple elastic scattering of a particle in a random medium of point scatterers. The detailed limits of circumstancial validity are graphically demonstrated in terms of particle wavelength, average distance between scatterers and elastic mean free path. Their immediate applicability to neutron transport is noted. The approximate nature of the point scatterer model as regards electron transport is adressed. In order to obtain an extrapolation of the result of the point scatterer model, it is observed that an increasing error of the trajectory method appears together with an increased amplitude of the multiple wave scattering taking place within the medium. On the basis of this observation, an extrapolation is proposed which provides a rough estimate of the relative error of the trajectory method when applied to multiple elastic scattering of low energy electrons in real liquids or amorphous solids.

[1]  D. Liljequist.  Coherent elastic scattering trajectory simulation , 2012 .

[2]  S. Incerti,et al.  A free-parameter theoretical model for describing the electron elastic scattering in water in the Geant4 toolkit , 2009 .

[3]  John B. Pendry,et al.  Low-Energy Electron Diffraction , 1990 .

[4]  P O'Neill,et al.  Computational Approach for Determining the Spectrum of DNA Damage Induced by Ionizing Radiation , 2001, Radiation research.

[5]  S. Tonzani,et al.  Low-energy electron scattering from DNA including structural water and base-pair irregularities , 2009 .

[6]  Francis A Cucinotta,et al.  A Complete Dielectric Response Model for Liquid Water: A Solution of the Bethe Ridge Problem , 2005, Radiation research.

[7]  T. Liamsuwan,et al.  Elastic scattering cross section models used for Monte Carlo simulation of electron tracks in media of biological and medical interest , 2012, International journal of radiation biology.

[8]  D. Liljequist.  A study of errors in trajectory simulation with relevance for 0.2 - 50 eV electrons in liquid water , 2008 .

[9]  J. Rehr,et al.  Parallel calculation of electron multiple scattering using Lanczos algorithms , 2002 .

[10]  D. Liljequist.  Discussion of coherent and incoherent contributions to the spatial distribution of very low energy electrons elastically scattered in liquid water , 2011 .

[11]  Francis A. Cucinotta,et al.  Track-structure codes in radiation research , 2006 .

[12]  L. Sanche,et al.  Low-energy electron diffraction and resonances in DNA and other helical macromolecules. , 2003, Physical review letters.

[13]  A. Breskin,et al.  Low-energy electron transport in alkali halides , 1994 .

[14]  Hooshang Nikjoo,et al.  A model of the cell nucleus for DNA damage calculations , 2012, International journal of radiation biology.

[15]  D. Emfietzoglou,et al.  Inelastic Cross Sections for Low-Energy Electrons in Liquid Water: Exchange and Correlation Effects , 2013, Radiation research.

[16]  Leonard S. Rodberg,et al.  Introduction to the Quantum Theory of Scattering , 1968 .

[17]  D. Liljequist.  Trajectory calculation as an approximation to the exact quantum treatment of multiple elastic scattering , 2006 .

[18]  S. Uehara,et al.  Interaction of Radiation with Matter , 2016 .

[19]  Iwan Kawrakow,et al.  On the Monte Carlo simulation of electron transport in the sub-1 keV energy range. , 2011, Medical physics.

[20]  D. Liljequist.  Scattering and absorption of particles emitted by a point source in a cluster of point scatterers , 2012 .

[21]  D. Liljequist.  Contribution from inelastic scattering to the validity of trajectory methods , 2013 .

[22]  Lennart Lindborg,et al.  RBE of low energy electrons and photons , 2010, Physics in medicine and biology.

[23]  D. Liljequist.  Limits of validity of trajectory simulation: Correlation of the error with density of scatterers and particle wavelength , 2009 .

[24]  J.-Ch. Kuhr,et al.  Monte Carlo simulation of electron emission from solids , 1999 .

[25]  L. Foldy,et al.  The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers , 1945 .

[26]  Akira Ishimaru,et al.  Wave propagation and scattering in random media , 1997 .