Correction of the Mathematical Method for Studying a Cathode Lens with Two Planes of Symmetry

Abstract

The article is devoted to the theoretical problems of studying cathode lenses with two planes of symmetry. It is noted that the classical methods of studying conventional electron lenses are unsuitable for a detailed study of the focusing properties of cathode lenses because these methods do not take into account the specificity of the initial conditions of cathode lenses. For a cathode lens, it is inaccurate to assume that the trajectories of charged particles throughout have a small inclination to the main optical axis. In addition, in single and immersion lenses, the condition of the smallness of the spread of charged particle energies with respect to the values of the axial potential is satisfied throughout, while in the cathode lens, this condition is not satisfied in the cathode region. The method proposed in the article is based on the study of the parameters of the trajectory of an arbitrary particle in a beam of charged particles relative to the parameters of the trajectory of an axial particle with zero initial energy, the trajectory of which is chosen as a scalar motion. The results obtained in the article will expand the scope of the application of cathode lenses with two planes of symmetry.

Modeling and Improvement of Radio Frequency Mass Spectrometers for the Analysis of the Composition of Minerals and the Environment

“…The main disadvantages of a high-frequency mass spectrometer are associated with the presence of grid electrodes, which significantly impair its sensitivity and resolution. The theory of focusing charged particles in static electric and magnetic fields has been developed in sufficient detail (Ibrayev et al, 1981: 22-30;Ibrayev, 2015: 270-275;Ibrayev, 2023Ibrayev, : 1307Sikharulidze, 2004: 21-30;Szilagyi, 1990: 639;Yakushev, 2013: 147-247), and the problems of focusing in high-frequency fields have been considered quite a bit. Therefore, in order to solve theoretical problems of improving radio frequency mass spectrometers, this paper considers a method for modeling and analyzing the movements of charged particles in fields with a high-frequency component of the field in emission systems without grid electrodes.…”

Method for Constructing a Commutative Algebra of Hypercomplex Numbers

“…After successfully applying the algebra of complex numbers and functions of complex variables to solve a number of complex theoretical problems, many of the famous mathematicians in the world spent a lot of effort on solving the problem of constructing, first of all, an algebra of three-dimensional variables, in order to later construct algebras of multidimensional variables. Therefore, the quaternions proposed by W.R. Hamilton in 1843 gave rise to the rapid development of vector algebra and a number of other important sections of modern mathematics, which are an effective basis for building the fundamental foundations of theories and research methods in many areas of science and technology [1][2][3][4][5][6][7][8][9]. For example, vector algebra is the main mathematical tool for solving fundamental problems in the fields of mechanics [2,3,6,7], the theory of force fields [4,5], and the theoretical foundations for the formation of charged particle flows [8,9] in analytical instrumentation.…”

“…Therefore, the quaternions proposed by W.R. Hamilton in 1843 gave rise to the rapid development of vector algebra and a number of other important sections of modern mathematics, which are an effective basis for building the fundamental foundations of theories and research methods in many areas of science and technology [1][2][3][4][5][6][7][8][9]. For example, vector algebra is the main mathematical tool for solving fundamental problems in the fields of mechanics [2,3,6,7], the theory of force fields [4,5], and the theoretical foundations for the formation of charged particle flows [8,9] in analytical instrumentation. Algebras of quaternions, octonions, and other hypercomplex numbers enable effective mathematical modeling of various physical processes and distributions of different types of force fields in spaces with different types of symmetry.…”