Comparison of methods for interpolation and extrapolation of boundary trajectories of short-focus electron beams using root-polynomial functions
DOI:
https://doi.org/10.20535/SRIT.2308-8893.2024.3.05Keywords:
interpolation, extrapolation, root-polynomial function, ravine function, average error, electron beam, boundary trajectoryAbstract
The article considers and discusses the comparison of interpolation and extrapolation methods of estimation of the boundary trajectory of electron beams propagated in ionized gas. All estimations have been computed using root-polynomial functions to numerically solve a differential-algebraic system of equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that in the case of solving a self-connected interpolation-extrapolation task, the average error of the beam radius estimation is generally smaller. This approach was especially effective in estimating the focal beam radius. An algorithm for solving self-connected interpolation-extrapolation tasks is given, and its efficiency is explained. Corresponding graphic dependencies are also given and analyzed.
References
J.D. Lawson, The Physics of Charged-Particle Beams. Oxford: Clarendon Press, 1977, 446 p. Available: https://www.semanticscholar.org/paper/The-Physics-of-Charged-Particle-Beams-Stringer/80b5ee5289d5efd8f480b516ec4bade0aa529ea6
M. Reiser, Theory and Design of Charged Particle Beams. John Wiley & Sons, 2008, 634 p. Available: https://www.wiley.com/en-us/Theory+and+Design+of+Charged+Particle+Beams-p-9783527617630
M. Szilagyi, Electron and Ion Optics. Springer Science & Business Media, 2012, 539 p. Available: https://www.amazon.com/Electron-Optics-Microdevices-Miklos-Szilagyi/dp/1461282470
S.J.R. Humphries, Charged Particle Beams. Courier Corporation, 2013, 834 p. Available: https://library.uoh.edu.iq/admin/ebooks/76728-charged-particle-beams---s.-humphries.pdf
S. Schiller, U. Heisig, and S. Panzer, Electron Beam Technology. John Wiley & Sons Inc, 1995, 508 p. Available: https://books.google.com.ua/books/about/Electron_Beam_Technology.html?id=QRJTAAAAMAAJ&redir_esc=y
R.A. Bakish, Introduction to Electron Beam Technology. Wiley, 1962, 452 p. Available: https://books.google.com.ua/books?id=GghTAAAAMAAJ&hl=uk&source=gbs_similarbooks
R.C., Davidson, H. Qin, Physics of Intense Charged Particle Beams in High Energy Accelerators. World Scientific, Singapore, 2001, 604 p. Available: https://books.google.com.ua/books/about/Physics_Of_Intense_Charged_Particle_Beam.html?id=5M02DwAAQBAJ&redir_esc=y
H. Schultz, Electron Beam Welding. Woodhead Publishing, 1993, 240 p. Available: https://books.google.com.ua/books?id=I0xMo28DwcIC&hl=uk&source=gbs_book_similarbooks
G. Brewer, Electron-Beam Technology in Microelectronic Fabrication. Elsevier, 2012, 376 p. Available: https://books.google.com.ua/books?idsnU5sOQD6noC&hl=uk&source=gbs_similarbooks
I. Brodie, J.J. Muray, The Physics of Microfabrication. Springer Science & Business Media, 2013, 504 p. Available: https://books.google.com.ua/books?id=GQYHCAAAQBAJ&hl=uk&source=gbs_similarbooks
A.A. Druzhinin, I.P. Ostrovskii, Y.N. Khoverko, N.S.Liakh-Kaguy, and A.M. Vuytsyk, “Low temperature characteristics of germanium whiskers,” Functional materials 21, no. 2, pp. 130–136, 2014. Available: http://dspace.nbuv.gov.ua/bitstream/handle/123456789/120404/02-Druzhinin.pdf?sequence=1
A.A. Druzhinin, I.A. Bolshakova, I.P. Ostrovskii, Y.N. Khoverko, and N.S. Liakh-Kaguy, “Low temperature magnetoresistance of InSb whiskers,” Materials Science in Semiconductor Processing, vol. 40, pp. 550–555, 2015. Available: https://academic-accelerator.com/search?Journal=Druzhinin
A. Zakharov, S. Rozenko, S. Litvintsev, and M. Ilchenko, “Trisection Bandpass Filter with Mixed Cross-Coupling and Different Paths for Signal Propagation,” IEEE Microwave Wireless Component Letters, vol. 30, no. 1, pp. 12–15, Jan. 2020.
A. Zakharov, S. Litvintsev, and M. Ilchenko, “Trisection Bandpass Filters with All Mixed Couplings,” IEEE Microwave Wireless Components Letter, vol. 29, no. 9, pp. 592–594, 2019. Available: https://ieeexplore.ieee.org/abstract/document/8782802
A. Zakharov, S. Rozenko, and M. Ilchenko, “Varactor-tuned microstrip bandpass filter with loop hairpin and combline resonators,” IEEE Transactions on Circuits Systems. II. Experimental Briefs, vol. 66, no. 6, pp. 953–957, 2019. Available: https://ieeexplore.ieee.org/document/8477112
T. Kemmotsu, T. Nagai, and M. Maeda, “Removal Rate of Phosphorous form Melting Silicon,” High Temperature Materials and Processes, vol. 30, issue 1-2, pp. 17–22, 2011. Available: https://www.degruyter.com/journal/key/htmp/30/1-2/html
J.C.S. Pires, A.F.B. Barga, and P.R. May, “The purification of metallurgically grade silicon by electron beam melting,” Journal of Materials Processing Technology, vol. 169, no. 1, pp. 347–355, 2005. Available: https://www.academia.edu/9442020/The_purification_of_metallurgical_grade_silicon_by_electron_beam_melting
D. Luo, N. Liu, Y. Lu, G.Zhang, and T. Li, “Removal of impurities from metallurgically grade silicon by electron beam melting,” Journal of Semiconductors, vol. 32, issue 3, article ID 033003, 2011. Available: http://www.jos.ac.cn/en/article/doi/10.1088/1674-4926/32/3/033003
D. Jiang, Y. Tan, S. Shi, W. Dong, Z. Gu, and R. Zou, “Removal of phosphorous in molten silicon by electron beam candle melting,” Materials Letters, vol. 78, pp. 4–7, 2012.
A.F. Tseluyko, V.T. Lazurik, D.L. Ryabchikov, V.I. Maslov, and I.N. Sereda, “Experimental study of radiation in the wavelength range 12.2-15.8 nm from a pulsed high-current plasma diode,” Plasma Physics Reports, 34(11), pp. 963–968, 2008. Available: https://link.springer.com/article/10.1134/S1063780X0811010X
V.G. Rudychev, V.T. Lazurik, and Y.V. Rudychev, “Influence of the electron beams incidence angles on the depth-dose distribution of the irradiated object,” Radiation Physics and Chemistry, 186, 109527, 2021. Available: https://www.sciencedirect.com/science/article/abs/pii/S0969806X21001778
V.M. Lazurik, V.T. Lazurik, G. Popov, and Z. Zimek, “Two-parametric model of electron beam in computational dosimetry for radiation processing, Radiation Physics and Chemistry, 124, pp. 230–234, 2016. Available: https://www.sciencedirect.com/science/article/abs/pii/S0969806X1530133X
I. Melnyk, S. Tuhai, and A. Pochynok, “Universal Complex Model for Estimation the Beam Current Density of High Voltage Glow Discharge Electron Guns,” Lecture Notes in Networks and Systems; Eds: M. Ilchenko, L. Uryvsky, L. Globa, vol. 152, pp. 319–341, 2021. Available: https://www.springer.com/gp/book/9783030583583
S.V. Denbnovetsky, V.G. Melnyk, and I.V. Melnyk, “High voltage glow discharge electron sources and possibilities of its application in industry for realizing different technological operations,” IEEE Transactions on Plasma Science, vol. 31, issue 5, pp. 987–993, October, 2003. doi: 10.1109/TPS.2003.818444
S. Denbnovetskiy et al., “Principles of operation of high voltage glow discharge electron guns and particularities of its technological application,” Proceedings of SPIE, The International Society of Optical Engineering, pp. 10445–10455, 2017. Available: https://spie.org/Publications/Proceedings/Paper/10.1117/12.2280736
I.V. Melnyk, “Numerical simulation of distribution of electric field and particle trajectories in electron sources based on high-voltage glow discharge,” Radioelectronic and Communication Systems, vol. 48, no. 6, pp. 61–71, 2005. doi: https://doi.org/10.3103/S0735272705060087
S.V. Denbnovetsky, J. Felba, V.I. Melnik, and I.V. Melnik, “Model of Beam Formation in a Glow Discharge Electron Gun with a Cold Cathode,” Applied Surface Science, 111, pp. 288–294, 1997. Available: https://www.sciencedirect.com/science/article/pii/S0169433296007611?via%3Dihub
J.I. Etcheverry, N. Mingolo, J.J. Rocca, and O.E. Martınez, “A Simple Model of a Glow Discharge Electron Beam for Materials Processing,” IEEE Transactions on Plasma Science, vol. 25, no. 3, pp. 427–432, June, 1997.
S.V. Denbnovetsky, V.I. Melnyk, I.V. Melnyk, and B.A. Tugay, “Model of control of glow discharge electron gun current for microelectronics production applications,” Proceedings of SPIE. Sixth International Conference on “Material Science and Material Properties for Infrared Optoelectronics”, vol. 5065, pp. 64–76, 2003. doi: https://doi.org/10.1117/12.502174
I.V. Melnyk, “Estimating of current rise time of glow discharge in triode electrode system in case of control pulsing,” Radioelectronic and Communication Systems, vol. 56, no. 12, pp. 51–61, 2017.
A. Mitchell, T. Wang, “Electron beam melting technology review,” Proceedings of the Conference “Electron Beam Melting and Refining State of the Art 2000, Reno, NV, USA, 2000, ed. R. Bakish, pp. 2–13.
D.V. Kovalchuk, N.P. Kondraty, “Electron-beam remelting of titanium – problems and development prospects,” Titan 2009, no. 1(23), pp. 29–38.
V.A. Savenko, N.I. Grechanyuk, and O.V. Churakov, “Electron beam refining in production of platinum and platinum-base alloys. Information 1. Electron beam refining of platinum,” Advances in Electrometallurgy, no. 1, pp. 14–16, 2008.
T.O. Prikhna et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrALY(Si) Alloys I. Fundamentals of the Production Technology, Structure, and Phase Composition of Cast NiCrAlY Alloys,” Powder Metallurgy and Metal Ceramics, vol. 61, issue 1-2, pp. 70–76, 2022. Available: https://link.springer.com/article/10.1007/s11106-022-00296-8
T.O. Prikhna et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrAlY(Si) Alloys Produced by Electron-Beam Melting II. Structure and Chemical and Phase Composition of Cast CoCrAlY Alloys,” Powder Metallurgy and Metal Ceramicsthis, vol. 61, issue 3-4, pp. 230–237, 2022.
I.M. Grechanyuk et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrAlY(Si) Alloys Produced by Electron Beam Melting IV. Chemical and Phase Composition and Structure of Cocralysi Powder Alloys and Their Use,” Powder Metallurgy and Metal Ceramics, vol. 61, issue 7-8, pp. 459–464, 2022.
M.I. Grechanyuk et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni (Co)CrAlY (Si) Alloys Produced by Electron Beam Melting III. Formation, Structure, and Chemical and Phase Composition of Thermal-Barrier Ni(Co)CrAlY/ZrO2–Y2O3 Coatings Produced by Physical Vapor Deposition in One Process Cycle,” Powder Metallurgy and Metal Ceramics, vol. 61, issue 5-6, pp. 328–336, 2022.
J. Zhang et al., “Fine equiaxed β grains and superior tensile property in Ti–6Al–4V alloy deposited by coaxial electron beam wire feeding additive manufacturing,” Acta Metallurgica Sinica (English Letters), 33(10), pp. 1311–1320, 2020.
D. Kovalchuk, O. Ivasishin, “Profile electron beam 3D metal printing,” in Additive Manufacturing for the Aerospace Industry. Elsevier Inc., 2019, pp. 213–233.
M. Wang et al., “Microstructure and mechanical properties of Ti-6Al-4V cruciform structure fabricated by coaxial electron beam wire-feed additive manufacturing,” Journal of Alloys and Compounds, vol. 960. article 170943. doi: https://doi.org/10.1016/j.jallcom.2023.170943
I. Melnyk, S. Tuhai, M. Surzhykov, I. Shved, V. Melnyk, and D. Kovalchuk, “Analytical Estimation of the Deep of Seam Penetration for the Electron-Beam Welding Technologies with Application of Glow Discharge Electron Guns,” 2022 IEEE 41-st International Conference on Electronics and Nanotechnology (ELNANO), 2022, pp. 1–5.
I. Melnyk, S. Tuhai, and A. Pochynok, “Calculation of Focal Paramters of Electron Beam Formed in Soft Vacuum at the Plane which Sloped to Beam Axis,” The Forth IEEE International Conference on Information-Communication Technologies and Radioelectronics UkrMiCo’2019. Collections of Proceedings of the Scientific and Technical Conference, Odesa, Ukraine, September 9-13, 2019. doi: 10.1109/UkrMiCo47782.2019.9165328
I. Melnyk, S. Tuhai, and A. Pochynok, “Interpolation of the Boundary Trajectories of Electron Beams by the Roots from Polynomic Functions of Corresponded Order,” 2020 IEEE 40th International Conference on Electronics and Nanotechnology (ELNANO), pp. 28–33. doi: 10.1109/ELNANO50318.2020.9088786
I. Melnik, S. Tugay, and A. Pochynok, “Interpolation Functions for Describing the Boundary Trajectories of Electron Beams Propagated in Ionised Gas,” 15-th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET – 2020), pp. 79–83. doi: 10.1109/TCSET49122.2020.235395
I.V. Melnyk, A.V. Pochynok, “Study of a Class of Algebraic Functions for Interpolation of Boundary Trajectories of Short-Focus Electron Beams,” System Researches and Information Technologies, no. 3, pp. 23–39, 2020. doi: https://doi.org/10.20535/SRIT.2308-8893.2020.3.02
I. Melnyk, S. Tuhai, M. Skrypka, A. Pochynok, and D. Kovalchuk, “Approximation of the Boundary Trajectory of a Short-Focus Electron Beam using Third-Order Root-Polynomial Functions and Recurrent Matrixes Approach,” 2023 International Conference on Information and Digital Technologies (IDT), Zilina, Slovakia, 2023, pp. 133–138, doi: 10.1109/IDT59031.2023.10194399
I. Melnyk, A. Luntovskyy, “Estimation of Energy Efficiency and Quality of Service in Cloud Realizations of Parallel Computing Algorithms for IBN,” in Klymash, M., Beshley, M., Luntovskyy, A. (eds) Future Intent-Based Networking. Lecture Notes in Electrical Engineering, vol. 831, Springer, Cham, pp. 339–379. doi: https://doi.org/10.1007/978-3-030-92435-5_20
G.M. Phillips, Interpolation and Approximation by Polynomials. Springer, 2023, 312 p. Available: http://bayanbox.ir/view/2518803974255898294/George-M.-Phillips-Interpolation-and-Approximation-by-Polynomials-Springer-2003.pdf
N. Draper, H. Smith, Applied Regression Analysis; 3 Edition. Wiley Series, 1998, 706 p. Available: https://www.wiley.com/en-us/Applied+Regression+Analysis,+3rd+Edition-p-9780471170822
C. Mohan, K. Deep, Optimization Techniques. New Age Science, 2009, 628 p. Available: https://www.amazon.com/Optimization-Techniques-C-Mohan/dp/1906574219
M.K. Jain, S.R.K. Iengar, and R.K. Jain, Numerical Methods for Scientific & Engineering Computation. New Age International Pvt. Ltd., 2010, 733 p. Available: https://www.researchgate.net/profile/Abiodun-Opanuga-2/post/how_can_solve_a_non_linear_PDE_using_numerical_method/attachment/59d61f7279197b807797de30/AS%3A284742038638596%401444899200343/download/Numerical+Methods.pdf
S.C. Chapra, R.P. Canale, Numerical Methods for Engineers; 7th Edition. McGraw Hill, 2014, 992 p. Available: https://www.amazon.com/Numerical-Methods-Engineers-Steven-Chapra/dp/007339792X
E. Wentzel, L. Ovcharov, Applied Problems of Probability Theory. Mir, 1998, 432 p. Available: https://mirtitles.org/2022/06/03/applied-problems-in-probability-theory-wentzel-ovcharov/
J.A. Gubner, Probability and Random Processes for Electrical and Computer Engineers. UK, Cambridge: Cambridge University Press, 2006. Available: http://www.amazon.com/Probability-Processes-Electrical-Computer-Engineers/dp/0521864704
M. Lutz, Learning Python; 5th Edition. O’Reilly, 2013, 1643 p.
W. McKinney, Python for Data Analysis: Data Wrangling with Pandas, NumPy, and Jupyter; 3rd Edition. O’Reilly Media, 2023, 579 p.
