Analytical description of the potential of electrostatic multipole systems based on a conducting circular cylinder

Abstract

An urgent task of corpuscular optics and scientific instrumentation is the creation of new methods for calculating the physical and instrumental parameters of mass spectrometers. Leveraging the increased capabilities of computational technology, these methods provide a solid basis for the design and calculation of instruments with improved analytical capabilities. In this work, a method was developed for calculating the electrostatic field of multipole systems based on a conducting circular cylinder. This method uses the broad analytical capabilities of the theory of functions of a complex variable. Analytical expressions are found for the electrostatic field potential of a quadrupole system for the case of infinitely narrow gaps between the electrodes. Analytical expressions for the derivatives of the potential are also obtained. A study was carried out of the influence of the finite size of the gaps between the electrodes on the field configuration. For this purpose, numerical simulations of planar electric fields satisfying the Laplace equation were carried out. The calculation of the electrostatic field potential was carried out using the boundary element method in two stages. First, the distribution of electric charge at the boundary was calculated according to the known boundary potential distribution, that is, the “inverse” problem was solved. Then, using the found values of the charge distribution and the found potential values, the “direct” problem was solved. To solve this problem, a method was developed for solving integral equations with singular and quasi-singular kernels, which provides high accuracy in field calculations for electronoptical systems with rectilinear boundaries. The inaccuracies in the calculations resulted solely from rounding mistakes. In the case of an Equation of State electron-optical systems characterized by curved boundaries, the precision is dependent exclusively on how accurately the boundaries are represented using linear segments. For the purpose of segmenting the curved borders of the second-order electron-optical systems, these boundaries were broken down into arcs with an angular measurement not exceeding one degree.