Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3220310.22363/2658-4670-2022-30-3-217-230Research ArticleNumerical simulation of cold emission in coaxial diode with magnetic isolationBelovAlexandr A.<p>Candidate of Physical and Mathematical Sciences, Researcher of Faculty of Physics, M. V. Lomonosov Moscow State University; Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia</p>aa.belov@physics.msu.ruhttps://orcid.org/0000-0002-0918-9263LozaOleg T.<p>Doctor of Physical and Mathematical Sciences, Professor of Institute of Physical Research and Technology</p>loza-ot@rudn.ruhttps://orcid.org/0000-0003-4676-6303LovetskiyKonstantin P.<p>Candidate of Physical and Mathematical Sciences, Associate professor of Department of Applied Probability and Informatics</p>lovetskiy-kp@rudn.ruhttps://orcid.org/0000-0002-3645-1060KarnilovichSergey P.<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Institute of Physical Research and Technology</p>karnilovich-sp@rudn.ruhttps://orcid.org/0000-0001-5696-1546SevastianovLeonid A.<p>Doctor of Physical and Mathematical Sciences, Professor of Department of Applied Probability and Informatics</p>sevastianov-la@rudn.ruhttps://orcid.org/0000-0002-1856-4643Lomonosov Moscow State UniversityPeoples’ Friendship University of Russia (RUDN University)0510202230321723005102022Copyright © 2022, Belov A.A., Loza O.T., Lovetskiy K.P., Karnilovich S.P., Sevastianov L.A.2022<p style="text-align: justify;">Due to the emergence and active development of new areas of application of powerful and super-powerful microwave vacuum devices, interest in studying the behavior of ensembles of charged particles moving in the interaction space has increased. An example is an electron beam formed in a coaxial diode with magnetic isolation. Numerical simulation of emission in such a diode is traditionally carried out using particle-in-cell methods. They are based on the simultaneous calculation of the equations of motion of particles and the Maxwell’s equations for the electromagnetic field. In the present work, a new computational approach called the point macroparticle method is proposed. In it, the motion of particles is described by the equations of relativistic mechanics, and explicit expressions are written out for fields in a quasi-static approximation. Calculations of the formation of a relativistic electron beam in a coaxial diode with magnetic isolation are performed and a comparison is made with the known theoretical relations for the electron velocity in the beam and for the beam current. Excellent agreement of calculation results with theoretical formulas is obtained.</p>coaxial diode with magnetic isolationcold emissionpoint macroparticlesкоаксиальный диод с магнитной изоляциейхолодная эмиссияточечные макрочастицы1. Introduction Relativistic electron beams. The existing plasma relativistic microwave generators and amplifiers (plasma masers) are based on the interaction of tubular plasma with tubular high-current relativistic electron beam (REB) [1]. The explosive-emission cathode [2] forms a tubular REB with an internal radius of ∼2 cm and a thickness of ∼0.15 cm, which propagates in a magnetic field of 1 T created by a solenoid. The electron energy in such a beam is ∼106 eV, the electron current density is 103-104 A/cm2. The power of the REB, as a rule, exceeds 109 W, the current pulse lasts from several nanoseconds to several microseconds [3]. High-current relativistic electron beams are formed directly in the diode, which is supplied with a voltage pulse from the primary energy storage. Electrons receive energy only in the diode, no additional means of particle acceleration (similar to sections of linear inductive or resonant accelerators) are used. Installations for generating high-current REB are also, in some works, called direct-acting accelerators [4]. The creation of controlled beams (streams) of charged particles is carried out using a variety of devices, the main element of which is a source of charged particles. A fairly common element of such a system that provides an intense, well-focused electron beam is an electron gun. The most commonly used are thermionic guns, in which the primary element is a vacuum diode [5]. Calculation methods. To calculate the dynamics of electron beams, a gasdynamic approximation is used (see, for example, [6]). As is known, the system of equations of gas dynamics is valid for thermodynamically equilibrium continuous media. Various types of equilibrium violation are taken into account using additional model assumptions. The success of this approach depends on how well the nonequilibrium model is chosen. Models that have proven themselves well in some applications (for example, nonequilibrium electronic processes of solid-state electronics) may not be applicable in other applications. A more general approach is the kinetic Vlasov equation with respect to the distribution function [7], supplemented by a system of Maxwell’s equations for electromagnetic fields. This model leads to a partial differential equation of the first order; it is a mathematical formulation of the well-known Liouville theorem on the conservation of phase volume [8]. The properties of the medium, such as particle concentration, charge density, average velocity, etc. are moments of the distribution function. For the numerical solution of the kinetic equation, methods such as Particlein-Cell (PiC) and Cloud-in-Cell (CiC) [9] are used. In these methods, the medium is replaced by a set of a finite number of particles possessing macrocharge that interact with each other. Each particle is attributed to the characteristics of the medium: charge, mass, momentum, energy, etc. The average values of these quantities are calculated as the sum of all model particles located in the considered region. Macroparticles have a finite size, within which the spatial distribution of charge, mass, etc. is set. Most often, this distribution is chosen piecewise constant. In this case, the geometric dimensions of all particles are considered the same. In some works, more complex form-factors of the particles are considered. The motion of macroparticles obeys the equations of Newtonian mechanics (or relativistic Lorentz equations). This leads to a system of ordinary differential equations (ODEs) for the coordinates and velocities of particles and a system of the Maxwell’s equations for electromagnetic fields. For this system, the «leap-frog» scheme is traditionally used. First, electromagnetic fields are set and the change in the coordinates and velocities of the macroparticles is calculated in one time step. Then, according to the changed coordinates and velocities, the electromagnetic fields are refined. After that, coordinates and velocities are calculated at the next time step, etc. Based on this approach, Tarakanov developed the KARAT [10] code, which was widely applied to solving various problems of plasma physics. Among them are formation of a virtual cathode, formation of an electron beam in a coaxial diode with magnetic isolation, dynamics of a laser target and the initiation of deuterium-deuterium reactions, focusing of an electron beam and the development of hose instability, anisotropic Waibel instability and many others. We also note the works of Borodachev (see [11] and other works of this author). He proposed several improvements to this approach and performed calculations of a large number of tasks. The main difficulty of the particle (cloud) method in a cell is the need to introduce space-time discretization separately for particles and separately for electromagnetic fields. This leads to a number of numerical artifacts. Among them are the stroboscopic effect (the onslaught of the phase of the electromagnetic field when its frequency does not match with the sampling frequency in time), non-conservativeness (either the momentum conservation law or the energy conservation law is fulfilled, but not simultaneously), the grid dispersion of the medium, the parasitic increase of shot and grid noise, and some others. This limits the accuracy of this method. In the present work, a new method for calculating the emission problem of a coaxial diode with magnetic isolation is proposed. Instead of particles of finite size, it uses point macroparticles. Their motion is described by relativistic Lorentz equations. The electromagnetic field of the beam is calculated in a static approximation based on the instantaneous position of the particles: the electrostatic field is taken according to the Coulomb law and the magnetic field is according to the Biot-Savart-Laplace law. Edge effects at the cathode boundary are considered insignificant. Test emission calculations are performed and the beam velocity and current are compared with the well-known Fedosov’s law. This comparison shows excellent accuracy of the proposed method: the discrepancy between the calculation and the specified theoretical law is no more than 1%. Such accuracy is obviously sufficient for applied calculations. 2. Problem statement Consider the problem of infinite electron emission in the model of a coaxial diode with magnetic isolation (CDMI) in a strong magnetic field [12]. A solid cylindrical conductive cathode with a radius of[M. V. Kuzelev et al., “Plasma relativistic microwave electronics,” Plasma Physics Reports, vol. 27, no. 8, pp. 669-691, 2001. DOI: 10.1134/1.1390539.][S. P. Bugaev, E. A. Litvinov, G. A. Mesyats, and D. I. Proskurovskii, “Explosive emission of electrons,” Physics Uspekhi, vol. 18, no. 1, pp. 51-61, 1975. DOI: 10.3367/UFNr.0115.197501d.0101.][O. T. Loza and I. 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