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- Liljequist, David, 1946-
(författare)
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A model calculation of coherence effects in the elastic backscattering of very low energy electrons (1 - 20 eV) from amorphous ice
- 2012
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Ingår i: International Journal of Radiation Biology. - : Informa UK Limited. - 0955-3002 .- 1362-3095. ; 88:1-2, s. 50-53
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Tidskriftsartikel (refereegranskat)abstract
- 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
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| 2. |
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| 3. |
- Liljequist, David, 1946-
(författare)
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Discussion of coherent and incoherent contributions to the spatial distribution of very low energy electrons elastically scattered in liquid water
- 2011
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Ingår i: Radiation Physics and Chemistry. - : Elsevier BV. - 0969-806X .- 1879-0895. ; 80:3, s. 291-303
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Liljequist, David, 1946-
(författare)
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Electron penetration in solids and its application to Mössbauer spectroscopy
- 1979
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The penetration of elecirons in solid matter is of considerable interest in many fields of applied physics. One application is found in Mössbauer spectroscopy, where spectra recorded by means of electrons are used for surface studies. This thesis deals with models for the penetration of medium fast electrons in solids, and with the electron transport and corresponding analysis involved in Mössbauer spectroscopical surface studies. Particular emphasis is given to the case where the electron energy loss is used to increase the analysing capability.
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| 5. |
- Liljequist, David, 1946-
(författare)
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Limits of validity of trajectory simulation: correlation of the error with density of scatterers and particle wavelength
- 2009
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Ingår i: Nuclear Instruments and Methods in Physics Research Section B. - : Elsevier. - 0168-583X .- 1872-9584. ; 267, s. 3409-3419
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Tidskriftsartikel (refereegranskat)abstract
- 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 RE≈a·(λ/dnn)b, where the parameters a<0.05 and b∼2 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/b∼1. 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.
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