Geodesic
Dynamics of cylindrical electron beams injecting in a half-space with perfectly-conducting boundary
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 28 (2020) no. 5, pp. 491-504.

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The purpose of this work is to study the dynamics of electron beams and bunches in a system with an ideally conducting plane and a uniform magnetic field in the presence of delayed electromagnetic waves when a virtual cathode is formed in the system and with parameters close to this state. Methods. The dynamics of a cylindrical electron beam flying through an ideally conducting plane into a strong longitudinal magnetic field is studied by numerical simulation. Particles are large. The study of the beam dynamics is carried out by solving the equations of motion by the 4th-order Runge–Kutta method. The space charge field is calculated using the Lienard–Wiechert relations taking into account the delay effect. The influence of the plane is taken into account by the method of mirror images. The time and place of virtual cathode formation is estimated by one-dimensional model, which determines the time step of the numerical method. Results. It is shown that during the formation of a virtual cathode in a cylindrical beam, particles passing through the virtual cathode region are accelerated by the space charge concentrated near the conducting boundary. The electron beam energy is redistributed between the electrons – the electrons in the tail of the beam are accelerated, the electrons in the vicinity of the entry plane are decelerated. Similar processes also take place in the absence of a virtual cathode when a short electron pulse is injected into a half-space with a magnetic field. Conclusion. Increasing space charge density of the beam in the system, the electrons entering the interaction space first are accelerated more strongly, subsequent electrons slow down more strongly. In this case, the distance traveled by the beam also increases. Especially strong acceleration is observed after the virtual cathode region.
Keywords: electron beam, Lienard–Wiechert potentials, high-current electronics, virtual cathode, O-type devices, large particle method. 
@article{IVP_2020_28_5_a3,
     author = {A. G. Shein and S. A. Alikov},
     title = {Dynamics of cylindrical electron beams injecting in a half-space with perfectly-conducting boundary},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {491--504},
     publisher = {mathdoc},
     volume = {28},
     number = {5},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2020_28_5_a3/}
}
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A. G. Shein; S. A. Alikov. Dynamics of cylindrical electron beams injecting in a half-space with perfectly-conducting boundary. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 28 (2020) no. 5, pp. 491-504. http://geodesic.mathdoc.fr/item/IVP_2020_28_5_a3/