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  • 1.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    A method to measure the validity of trajectory simulation by removing the boundary-dependent coherent scattering contribution2008In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, ISSN 0168-583X, E-ISSN 1872-9584, Vol. 266, no 2, p. 211-223Article in journal (Refereed)
    Abstract [en]

    A method is shown for measuring the validity of trajectory simulation, as compared to exact quantum calculation. The object is to calculate the differential cluster cross section obtained by multiple elastic scattering of a particle in a cluster of N randomly distributed point scatterers. After discussing the application of the analytic Foldy theory to the calculation of the coherent part of the cluster cross section, i.e. the diffraction pattern that is associated with the finite size of the cluster, it is shown that, for N up to 103, this coherent part can be removed by a simple, exact procedure, allowing the trajectory simulation to be compared with the remaining incoherent part. A scalar measure of the relative error in the trajectory differential cluster cross section is introduced. In the examples shown, this relative error is negligible at a low density of scatterers, but increases substantially with increasing density. The relevance for low-energy electron scattering is discussed.

  • 2.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    A model calculation of coherence effects in the elastic backscattering of very low energy electrons (1 - 20 eV) from amorphous ice2012In: International Journal of Radiation Biology, ISSN 0955-3002, E-ISSN 1362-3095, Vol. 88, no 1-2, p. 50-53Article in journal (Refereed)
    Abstract [en]

    Purpose: Backscattering of very low energy electrons in thin layers of amorphous ice is known to provide experimental data for the elastic and inelastic cross sections and indicates values to be expected in liquid water. The extraction of cross sections was based on a transport analysis consistent with Monte Carlo simulation of electron trajectories. However, at electron energies below 20 eV, quantum coherence effects may be important and trajectory-based methods may be in significant error. This possibility is here investigated by calculating quantum multiple elastic scattering of electrons in a simple model of a very small, thin foil of amorphous ice.

    Method: The average quantum multiple elastic scattering of electrons is calculated for a large number of simulated foils, using a point-scatterer model for the water molecule and taking inelastic absorption into account. The calculation is compared with a corresponding trajectory simulation.

    Results: The difference between average quantum scattering and trajectory simulation at energies below about 20 eV is large, in particular in the forward scattering direction, and is found to be almost entirely due to coherence effects associated with the short-range order in the amorphous ice. For electrons backscattered at the experimental detection angle (45° relative to the surface normal) the difference is however small except at electron energies below about 10 eV.

    Conclusion: Although coherence effects are in general found to be strong, the mean free path values derived by trajectory-based analysis may actually be in fair agreement with the result of an analysis based on quantum scattering, at least for electron energies

  • 3.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    A study of errors in trajectory simulation with relevance for 0.2 - 50 eV electrons in liquid water2008In: Radiation Physics and Chemistry, ISSN 0969-806X, E-ISSN 1879-0895, Vol. 77, no 7, p. 835-853Article in journal (Refereed)
    Abstract [en]

    A highly simplified model of the elastic scattering of electrons in a nm-size volume of liquid water, where the water molecules are regarded as point scatterers and inelastic scattering is neglected, is studied for electron energies 0.2–50 eV. This model allows an exact quantum mechanical solution of the multiple elastic scattering problem. The exact solutions are compared with the corresponding trajectory simulations. Two different data sets for the elastic scattering mean free path are discussed and used, one based on cross sections for water molecules in the gas phase and the other on cross sections derived from scattering in amorphous ice. In each case, the comparison provides a detailed insight into the character and magnitude of the error in the trajectory simulation, and gives a preliminary indication of an upper bound to the limits of validity of trajectory simulation in the real case of electron scattering in liquid water. Main results: with a fully random distribution of scatterers, the trajectory simulation is found to be a surprisingly good approximation down to quite low electron energies (). Below a few eV, the error increases rapidly with decreasing electron energy. A substantial increase of the error in trajectory simulation results when short-range order (minimum distance between scatterers) is introduced; the observed effect may partly be understood from the theory of kinematic diffraction in liquids. At very low electron energies (a few eV) one may note a quantum effect in the form of a standing wave pattern in the average distribution of scattering events.

  • 4.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics. Stockholm University.
    A study of the effect of inelastic absorption on the validity of trajectory simulation of elastic scattering2010In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, ISSN 0168-583X, E-ISSN 1872-9584, Vol. 268, no 24, p. 3546-3553Article in journal (Refereed)
  • 5.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Coherent elastic scattering trajectory simulation2012In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, ISSN 0168-583X, E-ISSN 1872-9584, Vol. 275, p. 79-85Article in journal (Refereed)
    Abstract [en]

    A method for trajectory simulation is proposed which takes coherent scattering as well as the structure (granularity) of the scattering medium into account. The simulation generates a sequence of inelastic events localized to single scatterers; this sequence is the trajectory. The location of an inelastic event in the sequence is sampled by means of probabilities calculated from a wavefield created by coherent elastic scattering of the wave emitted from the previous event. The method is applied to the case of a particle scattered in clusters of point scatterers with randomly varying positions, simulating scattering in amorphous matter. Effects of coherence and structure on the trajectory simulated spatial distributions of inelastic events are demonstrated. Agreement with conventional trajectory simulation is observed when short-range order is absent and the wavelength is about equal to or smaller than the average distance between nearest neighbour scatterers.

  • 6.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Contribution from inelastic scattering to the validity of trajectory methods2013In: Journal of Electron Spectroscopy and Related Phenomena, ISSN 0368-2048, E-ISSN 1873-2526, Vol. 189, p. 5-11Article in journal (Refereed)
    Abstract [en]

    Previous calculations of elastic particle scattering in clusters of randomly distributed point scatterers have shown that Monte Carlo trajectory simulation may be valid as an approximation of quantum multiple elastic scattering, provided that the particle wavelength is about equal to or smaller than the average distance between nearest neighbour scatterers. For the biologically important case of electrons in liquid water, this suggests that trajectory simulation of multiple elastic scattering may be valid (less than 5% error) for electron energies down to about 15 eV. Short range order increases the error somewhat, and produces, as well known, manifest diffraction effects under single scattering conditions. Present similar calculations show that trajectory simulation may be a reasonably good approximation also for particles with wavelengths substantially larger than d(nn), if plural or multiple inelastic scattering is introduced. This suggests that for electrons in liquid water trajectory simulation might, depending on the nature and frequency of inelastic scattering, be valid as an approximation at electron energies as low as a few eV.

  • 7.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Discussion of coherent and incoherent contributions to the spatial distribution of very low energy electrons elastically scattered in liquid water2011In: Radiation Physics and Chemistry, ISSN 0969-806X, E-ISSN 1879-0895, Vol. 80, no 3, p. 291-303Article in journal (Refereed)
    Abstract [en]

    The occurrence of diffraction effects versus the validity of trajectory simulation of the elastic scattering of very low energy electrons 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.

  • 8.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Limits of validity of trajectory simulation: correlation of the error with density of scatterers and particle wavelength2009In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, ISSN 0168-583X, E-ISSN 1872-9584, Vol. 267, p. 3409-3419Article in journal (Refereed)
    Abstract [en]

    To a first approximation, the elastic scattering of long wavelength particles in amorphous matter may be modelled as scattering in a volume filled with a density n of N point scatterers in random positions. For not too large N (up to about 2×103), the error in trajectory simulation (classical transport theory) due to the neglect of interference effects can then be determined in detail by means of a comparison with an exact quantum calculation of the plural or multiple scattering process. A relative error RE is defined and calculated for the scattering in different directions as well as for the distribution of scattering events inside the volume. A very strong correlation is found between the relative error and the ratio λ/dnn, where λ is the wavelength of the incident particle and dnn=n-1/3 is an average distance between nearest neighbour scatterers. For scattering in a volume of dimensions large compared to the particle wavelength, present calculations suggest that the correlation can be described as REa·(λ/dnn)b, where the parameters a<0.05 and b2 depend on the s-wave phaseshift δ0 in the scattering process. The condition for validity of trajectory simulation, defined in terms of a limit of validity L (maximum acceptable relative error), may thus be written λ/dnn<ξ, where ξ=(L/a)1/b1. For λ/dnn<1, the relative error is generally less than 5%, and trajectory simulation may be regarded as valid with at least 95% accuracy. In the exact quantum calculation, two features of pronounced quantum character are observed in the distribution of scattering events: oscillations due to quantum interference in finite volumes, and, for small negative δ0, randomly localized peaks due to proximity resonance.

  • 9.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    On the sampling of step length in Monte Carlo simulation of trajectories with very small mean free path2012In: Radiation Physics and Chemistry, ISSN 0969-806X, E-ISSN 1879-0895, Vol. 81, no 11, p. 1703-1709Article in journal (Refereed)
    Abstract [en]

    In an event-by-event simulation of the trajectory of a particle moving in matter it is usually assumed that the probability for the particle to travel a distance s without interaction is exp(-s/lambda), where lambda = (n . sigma)(-1) is the total mean free path, n the number of scatterers per unit volume and sigma the total cross section per scatterer. The step length s between scattering events is then generated by means of a sampling formula s = -lambda ln(1-R), where R a random number in the interval 0 < R < 1. It is here argued that this conventional sampling method, which basically assumes that the scattering medium may be regarded as a homogeneous continuum, may be erroneous unless lambda is much larger than the average distance d(nn) between nearest neighbour scatterers, estimated by d(nn) = n(-1/3). An alternative sampling method (M sampling) is proposed with a fixed step length D = d(nn) and a finite probability I = 1-exp(-D/lambda) of a single elastic or inelastic scattering event at the end of each step. According to this method, conventional sampling may exaggerate the number of events per unit path length; the corrected mean free path between events is found to be lambda(c) = D/(1-exp(-D/lambda)). The correction is substantial when lambda is comparable to or smaller than D, in practice for very low energy particles in liquids and solids. Consequently, quantities like stopping power may then be overestimated, while transport mean free path may be underestimated. In the opposite limit lambda >> D, conventional and M sampling produce the same result. Present results further indicate that conventional sampling using the corrected total mean free path lambda(c) is a good approximation to M sampling.

  • 10.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Scattering and absorption of particles emitted by a point source in a cluster of point scatterers2012In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, ISSN 0168-583X, E-ISSN 1872-9584, Vol. 275, p. 69-78Article in journal (Refereed)
    Abstract [en]

    A theory for the scattering and absorption of particles isotropically emitted by a point source in a cluster of point scatterers is described and related to the theory for the scattering of an incident particle beam. The quantum mechanical probability of escape from the cluster in different directions is calculated, as well as the spatial distribution of absorption events within the cluster. A source strength renormalization procedure is required. The average quantum scattering in clusters with randomly shifting scatterer positions is compared to trajectory simulation with the aim of studying the validity of the trajectory method. Differences between the results of the quantum arid trajectory methods are found primarily for wavelengths larger than the average distance between nearest neighbour scatterers. The average quantum results include, for example, a local minimum in the number of absorption events at the location of the point source and interference patterns in the angle-dependent escape probability as well as in the distribution of absorption events. The relative error of the trajectory method is in general, though not generally, of similar magnitude as that obtained for beam scattering.

  • 11.
    Liljequist, David
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Elfving, Britt
    Skavberg Roaldsen, Kirsti
    Intraclass correlation - A discussion and demonstration of basic features2019In: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 14, no 7, article id e0219854Article in journal (Refereed)
    Abstract [en]

    A re-analysis of intraclass correlation (ICC) theory is presented together with Monte Carlo simulations of ICC probability distributions. A partly revised and simplified theory of the single-score ICC is obtained, together with an alternative and simple recipe for its use in reliability studies. Our main, practical conclusion is that in the analysis of a reliability study it is neither necessary nor convenient to start from an initial choice of a specified statistical model. Rather, one may impartially use all three single-score ICC formulas. A near equality of the three ICC values indicates the absence of bias (systematic error), in which case the classical (one-way random) ICC may be used. A consistency ICC larger than absolute agreement ICC indicates the presence of non-negligible bias; if so, classical ICC is invalid and misleading. An F-test may be used to confirm whether biases are present. From the resulting model (without or with bias) variances and confidence intervals may then be calculated. In presence of bias, both absolute agreement ICC and consistency ICC should be reported, since they give different and complementary information about the reliability of the method. A clinical example with data from the literature is given.

  • 12.
    Liljequist, David
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Liamsuwan, Thiansin
    Nikjoo, Hooshang
    Elastic scattering cross section models used for Monte Carlo simulation of electron tracks in media of biological and medical interest2012In: International Journal of Radiation Biology, ISSN 0955-3002, E-ISSN 1362-3095, Vol. 88, no 1-2, p. 29-37Article in journal (Refereed)
    Abstract [en]

    Purpose: Elastic scattering is important for the spatial distribution of electrons penetrating matter, and thus for the distribution of deposited energy and DNA damage. Scattering media of interest are in particular liquid and gaseous water and gaseous nitrogen. The former are used as surrogates for tissue and cell environments (since more than 70% of the cell consists of water), while cross section data for nitrogen have been scaled and used as input in Monte Carlo (MC) codes simulating scattering in biologically relevant media. A short review is given of electron elastic scattering cross section models used in a biological and medical context and their experimental and theoretical background.

    Conclusions: Adequate theories and models exist for calculating elastic electron scattering in gaseous nitrogen and gaseous water (i.e., by free molecules) down to electron energies well below 100 eV. However, elastic electron scattering in liquid water at such low energies is apparently uncertain and not well understood. Further studies in the case of liquid water are thus motivated due to its biological importance.

  • 13.
    Liljequist, David
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Nikjoo, H.
    On the validity of trajectory methods for calculating the transport of very low energy ( < 1 keV) electrons in liquids and amorphous media2014In: Radiation Physics and Chemistry, ISSN 0969-806X, E-ISSN 1879-0895, Vol. 99, p. 45-52Article in journal (Refereed)
    Abstract [en]

    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.

  • 14. Nikjoo, H.
    et al.
    Emfietzoglou, D.
    Liamsuwan, T.
    Taleei, R.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Uehara, S.
    Radiation track, DNA damage and response-a review2016In: Reports on progress in physics (Print), ISSN 0034-4885, E-ISSN 1361-6633, Vol. 79, no 11, article id 116601Article, review/survey (Refereed)
    Abstract [en]

    The purpose of this paper has been to review the current status and progress of the field of radiation biophysics, and draw attention to the fact that physics, in general, and radiation physics in particular, with the aid of mathematical modeling, can help elucidate biological mechanisms and cancer therapies. We hypothesize that concepts of condensed-matter physics along with the new genomic knowledge and technologies and mechanistic mathematical modeling in conjunction with advances in experimental DNA (Deoxyrinonucleic acid molecule) repair and cell signaling have now provided us with unprecedented opportunities in radiation biophysics to address problems in targeted cancer therapy, and genetic risk estimation in humans. Obviously, one is not dealing with 'low-hanging fruit', but it will be a major scientific achievement if it becomes possible to state, in another decade or so, that we can link mechanistically the stages between the initial radiation-induced DNA damage; in particular, at doses of radiation less than 2 Gy and with structural changes in genomic DNA as a precursor to cell inactivation and/or mutations leading to genetic diseases. The paper presents recent development in the physics of radiation track structure contained in the computer code system KURBUC, in particular for low-energy electrons in the condensed phase of water for which we provide a comprehensive discussion of the dielectric response function approach. The state-of-the-art in the simulation of proton and carbon ion tracks in the Bragg peak region is also presented. The paper presents a critical discussion of the models used for elastic scattering, and the validity of the trajectory approach in low-electron transport. Brief discussions of mechanistic and quantitative aspects of microdosimetry, DNA damage and DNA repair are also included as developed by the authors' work.

  • 15. Nikjoo, H.
    et al.
    Taleei, R.
    Liamsuwan, T.
    Liljequist, David
    Stockholm University, Faculty of Science, Department of Physics.
    Emfietzoglou, D.
    Perspectives in radiation biophysics: From radiation track structure simulation to mechanistic models of DNA damage and repair2016In: Radiation Physics and Chemistry, ISSN 0969-806X, E-ISSN 1879-0895, Vol. 128, p. 3-10Article in journal (Refereed)
    Abstract [en]

    In radiation targeted therapy and genetic risk estimation of low dose radiation protection there is a crucial need for full description of DNA damage response and repair (DDR) leading to cell death and cell mutation. We propose such a description can be arrived through realistic track-structure simulations together with mechanistic mathematical formulation of DDR and the availability of experimental data for testing the proof of principle. In this paper we review briefly first the state of the art in DNA damage and repair, and then the recent advances in the physics of track structure which represents an essential tool in radiation biophysics.

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