Modelling of electron beam absorption in complex geometries

Computational modelling of processes that involve highly energetic electrons like electron beam melting, welding, drilling or electron beam lithography, to name but a few, requires information about the attenuation of the electron beam as it passes through the sample. Depth–dose curves as a function of electron energy, target material as well as local surface obliquity have to be provided in situ during the calculation. The most efficient way to address this issue is by employing mathematical expressions. Therefore, we propose an electron beam model based on a set of semi-empirical equations available from different published literature and on theoretical considerations. Particular stress is thereby put on accuracy and the range of validity of the theoretical approach by comparison with experimental data. Finally, we apply our model to powder-bed based additive manufacturing. The numerical results demonstrate that electron beam absorption and depth of penetration have a strong influence on the quality of the fabricated product.

[1]  K. Kanaya,et al.  The energy dependence of a diffusion model of an electron probe into solid targets , 1978 .

[2]  H. Niedrig,et al.  Electron backscattering from thin films , 1982 .

[3]  R. K. Bull,et al.  Radiation dosimetry: Electron beams with energies between 1 and 50 MeV: ICRU Report 35; 157 pp.; 85 figures; US $23.00. , 1986 .

[4]  H. Seliger.  TRANSMISSION OF POSITRONS AND ELECTRONS , 1955 .

[5]  K. Weber.  Eine einfache reichweite-energie-beziehung für elektronen im energiebereich von 3 keV bis 3 MeV , 1963 .

[6]  Carolin Körner,et al.  Lattice Boltzmann model for thermal free surface flows with liquid-solid phase transition , 2011 .

[7]  S. Okayama,et al.  Penetration and energy-loss theory of electrons in solid targets , 1972 .

[8]  P. Staub.  Bulk target backscattering coefficient and energy distribution of 0.5-100 keV electrons : an empirical and synthetic study , 1994 .

[9]  U. Rüde,et al.  Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming , 2005 .

[10]  Carolin Körner,et al.  Lattice Boltzmann method for dynamic wetting problems. , 2009, Journal of colloid and interface science.

[11]  C. Körner,et al.  Mesoscopic simulation of selective beam melting processes , 2011 .

[12]  E. H. Darlington,et al.  Backscattering of 10-100 keV electrons from thick targets , 1975 .

[13]  T. E. Everhart,et al.  Determination of Kilovolt Electron Energy Dissipation vs Penetration Distance in Solid Materials , 1971 .

[14]  H. Fitting,et al.  Transmission, energy distribution, and SE excitation of fast electrons in thin solid films , 1974 .

[15]  R. K. Yadav,et al.  Backscattering of 8–28 keV electrons from a thick Al, Ti, Ag and Pt targets , 2006 .

[16]  C. Ronning,et al.  Ion beam irradiation of nanostructures: sputtering, dopant incorporation, and dynamic annealing , 2015 .

[17]  Alain Dubus,et al.  Evaluation of theoretical models for elastic electron backscattering from surfaces , 2000 .

[18]  S. Rogaschewski,et al.  Backscattering coefficient measurements of 15 to 60 keV electrons for solids at various angles of incidence , 1980 .

[19]  Erik René Kieft,et al.  Refinement of Monte Carlo simulations of electron–specimen interaction in low-voltage SEM , 2008 .

[20]  H. August,et al.  Analytical Expressions for the Electron Backscattering Coefficient , 1989 .

[21]  Michael Schmidt,et al.  Numerical Simulation of Drilling with Pulsed Beams , 2012 .

[22]  V. Cosslett,et al.  Multiple scattering of 5 - 30 keV electrons in evaporated metal films III: Backscattering and absorption , 1965 .

[23]  Peter Petrov,et al.  Experimental investigation of weld pool formation in electron beam welding , 1998 .

[24]  R N Thomas,et al.  Multiple scattering of 5-30 keV electrons in evaporated metal films II: Range-energy relations , 1964 .

[25]  P. Andreo,et al.  Fractional energies of backscattered electrons and photon yields by electrons , 1999 .

[26]  H. Hunger,et al.  Measurements of the Electron Backscattering Coefficient for Quantitative EPMA in the Energy Range of 4 to 40 keV , 1979, November 16.

[27]  Robert F. Singer,et al.  Cellular Titanium by Selective Electron Beam Melting , 2007 .

[28]  Effect of electron beam obliquity on lateral buildup ratio: a Monte Carlo dosimetry evaluation. , 2007, Physics in medicine and biology.

[29]  Robert Katz,et al.  Energy Deposition by Electron Beams and δ Rays , 1968 .

[30]  R. Singer,et al.  Cellular Ti-6Al-4V structures with interconnected macro porosity for bone implants fabricated by selective electron beam melting. , 2008, Acta biomaterialia.

[31]  H. Fitting,et al.  Electron Backscattering at Various Angles of Incidence , 1983, April 16.

[32]  G. D. Archard,et al.  Back Scattering of Electrons , 1961 .

[34]  V. Cosslett,et al.  Multiple scattering of 5-30 keV electrons in evaporated metal films: I. Total transmission and angular distribution , 1964 .

[35]  Reflection Ratios of Electrons and Photons from Solids , 1992 .

[36]  Grant J. Lockwood,et al.  Calorimetric measurement of electron energy deposition in extended media. Theory vs experiment , 1980 .

[37]  G Neubert,et al.  Measurements of the back-scattering and absorption of 15-60 keV electrons for transparent solid films at various angles of incidence , 1984 .

[38]  C. Klein.  Further remarks on electron beam pumping of laser materials. , 1966, Applied optics.

[39]  A. E. Grün,et al.  Lumineszenz-photometrische Messungen der Energieabsorption im Strahlungsfeld von Elektronenquellen Eindimensionaler Fall in Luft , 1957 .