#### For citation:

Shein A. G., Alikov S. A. Dynamics of cylindrical electron beams injecting in a half-space with perfectly-conducting boundary. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2020, vol. 28, iss. 5, pp. 491-504. DOI: 10.18500/0869-6632-2020-28-5-491-504

# Dynamics of cylindrical electron beams injecting in a half-space with perfectly-conducting boundary

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.

- Child C.D. Discharge from hot CaO // Phys. Rev. (Series I). 1911. Vol. 32, no. 5. P. 492.
- Langmuir I. The effect of space charge and residual gases on thermionic currents in high vacuum // Phys. Rev. (Second Series). 1913. Vol. 2, no. 6. P. 450–486.
- Langmuir I. The effect of space charge and initial velocities on the potential distribution and thermionic current between parallel plane electrodes // Phys. Rev. 1923. Vol. 21, no. 4. P. 419–435.
- Langmuir I., Blodgett K.B. Currents limited by space charge between coaxial cylinders // Phys. Rev. 1923. Vol. 22, no. 4. P. 347–356.
- Bursian V.R. and Pavlov V.I. Journal of the Russian Physical and Chemical Society, 1923, vol. 55, pp.71–80 (in Russian).
- Boguslavskii S.A. Proceedings of the State Experimental Electro-Technical Institute, 1924, no. 3, pp. 18–27 (in Russian).
- Morozov M.Yu. and Hramov A.E. Effect of the external magnetic field on the critical current for the onset of a virtual cathode in an electron beam. Plasma Physics Reports, 2007, vol. 33, no. 7, pp. 553–561.
- Filatov R.A. and Hramov A.E. Simulation of oscillatory processes in a beam-plasma system with a virtual cathode in gas-filled interaction space. Plasma Physics Reports, 2011, vol. 37, no. 5, pp. 395–408.
- Kellin N.S. and Tolmachev A.I. Effect of space charge and the initial electron velocity on the potential distribution in a plane diode. Technical Physics. The Russian Journal of Applied Physics, 2012, vol. 57, no. 4, pp. 512–515.
- Magda I.I., Melezhik O.G., Pashchenko A.V., Romanov S.S., Shapoval I.M., Novikov V.E. Modification of the Child–Langmuir–Boguslavsky law for the diode gap in the system with virtual cathode. Problems of Atomic Science and Technology (PAST), 2012, vol. 80, no. 4, pp. 133–137.
- Kurkin S.A., Koronovskii A.A., Hramov A.E. Specific features of virtual cathode formation and dynamics with allowance for the magnetic self-field of a relativistic electron beam. Plasma Physics Reports, 2013, vol. 39, № 4, pp. 296–306.
- Rukhadze A.A., Bogdankevich L.S., Rosinskii S.E., Rukhlin V.G. Physics of high-current relativistic electron beams. Moscow, Atom-izdat, 1980, 200 p. (in Russian).
- Shein A.G., Bakulin V.M., Mutovkin A.N. Computing the space-charge fields of M-type tubes // Journal of Communications Technology and Electronics. 2000. Vol. 45, no. 10. P. 1146–1149.
- Alikov S.A., Shein A.G. Peculiar properties of the electron beam dynamics simulation by particleparticle methods taking into account delay effects // ITM Web of Conferences. 29th International Crimean Conference «Microwave & Telecommunication Technology» (CriMiCo’2019) (Sevastopol, Russia, September 8–14, 2019). 2019. Vol. 30. DOI: https://doi.org/10.1051/itmconf/20193009005.
- Tarakanov V.P. Theoretical and numerical analysis of nonlinear problems in plasma physics using the KARAT code: Doctor of Physical and Mathematical Sciences thesis. Moscow, 2011, 264 p. (in Russian).
- Ginzburg S.L., D’yachenko V.F., Paleichik V.P., Khodataev K.V. Calculation of the characteristics of the radiation from a generator with a virtual cathode. Technical Physics, 1999, vol. 44, no. 2. pp. 212–217. DOI: 10.1134/1.1259286.

- 901 reads