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Book Cover
E-book
Author Danilov, V. I. (Vladimir Ivanovich), 1943-

Title Mathematical modeling of emission in small-size cathode / Vladimir Danilov, Roman Gaydukov, Vadim Kretov
Published Singapore : Springer, ©2020

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Description 1 online resource (205 pages)
Series Heat and Mass Transfer Ser
Heat and mass transfer.
Contents Intro; Preface; Preface to the English Edition; Acknowledgements; Contents; 1 Introduction; 1.1 Brief History of the Electron Emission Discovery; 1.2 Types of Electron Emission; 1.3 Statement of the Problem; 1.4 Mathematical Statement of the Problem. Heat Transfer Model; References; 2 Physical Basis for Field Emission; 2.1 Energy-Band Theory and Fermi Level; 2.2 Conductivity of Semiconductors; 2.2.1 Electron and Hole Concentration; 2.2.2 Effective Mass; 2.2.3 Electron and Hole Mobilities; 2.2.4 Temperature-Dependence of Conductivity in Silicon; 2.3 Thermoelectricity
2.4 Emission Current Density and Nottingham Effect2.4.1 Support Function in Metals; 2.4.2 Electron Tunneling Through the Potential Barrier; 2.4.3 Formula for the Barrier Transparency Factor in the Case of Field Emission Cathode; 2.4.4 Emission Current Density in Metals; 2.4.5 Specific Characteristics of Field Emission from a Semiconductor Cathode; 2.4.6 Approximation of the Formula for the Emission Current Density; 2.4.7 Nottingham Effect; 2.4.8 Optimal Values of Approximation Parameters; 2.4.9 Dependence of the Inversion Temperature on the External Electric Field Strength; References
3 Mathematical Model3.1 Phase Field System and its Use in Heat Transfer Modeling; 3.2 Phase Field System as Regularization of Limiting Problems with Free Boundary; 3.3 Asymptotic Solution of the Phase Field System and Modified Stefan Problem; 3.3.1 Construction of an Asymptotic Solution; 3.3.2 Examples; 3.4 Weak Solution of the Phase Field System and the Melting Zone Model; 3.4.1 Weak Solutions and Rankine-Hugoniot-Type Conditions; 3.4.2 Solutions of ̀̀Wave Train'' Type and the Corresponding Limiting Problem
3.5 Derivation of the Solution of the Limit Stefan-Gibbs-Thomson Problem from a Numerical Solution of the Phase Field System3.6 Generation and Coalescence of Dissipative Waves; References; 4 Numerical Simulation and its Results; 4.1 Nanocathode Model; 4.2 Calculation of the Current Density Inside the Cathode; 4.3 Calculation of the Emission Current Density and Modeling of the Nottingham Effect; 4.4 Difference Scheme; 4.4.1 Difference Scheme for the Equation for the Potential; 4.4.2 Difference Scheme for the Equation for the Order Function; 4.4.3 Difference Scheme for the Heat Equation
4.4.4 Stability of the Difference Scheme4.4.5 One More Version of the Difference Scheme; 4.4.6 Choice of Steps of the Difference Scheme; 4.5 Algorithm for Solving the Difference Equations and Possible Versions of its Parallelization; 4.6 Some Remarks About the Calculation of the Electric Potential; 4.7 Results of Numerical Experiments; 4.7.1 Nonmonotone Behavior of Free Boundaries; 4.7.2 Results of Modeling with Physical Parameters Corresponding to Experimental Data; 4.8 Formation of Melting and Crystallization Nuclei in the Model; 4.9 Conclusion; References
Summary This book deals with mathematical modeling, namely, it describes the mathematical model of heat transfer in a silicon cathode of small (nano) dimensions with the possibility of partial melting taken into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type free boundary problems. The approach used is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the mathematical model including its parallel implementation. The results of numerical simulation concludes the book. The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates
Bibliography Includes bibliographical references
Notes Print version record
Subject Electrons -- Emission -- Mathematical models
Cathodes -- Mathematical models
Form Electronic book
Author Gaydukov, Roman
Kretov, Vadim
ISBN 9811501955
9789811501951