Ekaterina M. Vinogradova

The electrostatic quadrupole lens mathematical modeling

In this paper a mathematical model of the electrostatic quadrupole lens is presented. The quadrupole lens consist of four equiform electrodes of an infinite length and each of these electrodes is a part of the circular cylinder. The variable separation method is used to solve the boundary-value problem for the Laplace equation in plane polar coordinates. The solution of initial value-boundary problem is reduced to the solution of linear algebraic equations system. The electrostatic potential distribution is calculated for the whole region of the system.

Calculating a multipole cylindrical electrostatic system

The results of simulations of a multipole electron-optical system has been considered. The electrodes of the system consist of an even number of identical parts of one circular cylinder that is cut parallel to the generatrix with infinite length. To determine the potential distribution, the Laplace equation has been solved by the method of variable separation in the polar coordinates. All of the geometrical dimensions of the system and the number of electrodes are parameters of the problem.

The Infinitely Thin Field Emitter Mathematical Modeling

In this work an axisymmetric diode electron-optical system based on a field emitter is simulated. The field emitter in the form of a thin filament of finite length is located on the flat substrate with the dielectric layer. The anode is a plane. The electrostatic potential distribution was found in an analytical form — in the form of Fourier-Bessel series in the whole area of the system under investigation. The coefficients of Fourier-Bessel series are the solution of the system of linear equations with constant coefficients.

The Multipole Lens Mathematical Modeling

In the present work the mathematical model of the multipole system is presented. The multipole system is composed of arbitrary even number of the uniform electrodes. Each of the electrodes is a part of the plane. The potentials of the electrodes are the same modulus and opposite sign for neighboring electrodes. The variable separation method is used to solve the electrostatic problem. The potential distribution is represented as the eigen functions expansions. The boundary conditions and the normal derivative continuity conditions lead to the linear algebraic equations system relative to the series coefficients. INTRODUCTION Electrostatic multipole systems are widely used in the accelerator technology for the charged particle beams transport [1]– [3]. In this paper the mathematical modeling of the electrostatic multipole system is presented. The multipole system consist of the even number uniform plate electrodes of the same shape and size. Fig. 1 shows a schematic representation of the multipole system. The similar system was investigated in [4]. A quadrature expression was obtained for the field potential and the constraints imposed on the electrode potentials, under which such a solution is possible, were determined. In our work a system with an arbitrary even number 2N of electrodes is modeled. The variable separation method [5]– [7] is used in plane polar coordinates (r, α) to solve the boundary-value problem for the Laplace equation [8]. The multipole potential distribution has the planes of symmetry α = (πk)/N and planes of antisymmetry α = π/(2N) + (πk)/N , k = 0, N − 1. An additional plane r = R2 can be introduce to limit the area of the problem under consideration without loss of generality. Thus it suffices to consider sector 0 ≤ α ≤ π/2N , 0 ≤ r ≤ R2 to find the electrostatic field. Schematic diagram of the multipole system sector is presented on Fig. 2 (α1 = π/2N ). The problem parameters are: (R1, 0) — the coordinate position of the multipole electrode’s edge, R2 — the radius of the area, α1 = π/2N — the boundary of the area (the plane of antisymmetry), ∗ e.m.vinogradova@spbu.ru Figure 1: Schematic representation of the multipole system. Figure 2: Schematic representation of the multipole system sector. U0 — the multipole electrode potential (α = 0, R1 ≤ r ≤ R2). MATHEMATICAL MODEL The electrostatic potential distribution U(r, α) in the area (0 ≤ r ≤ R2, 0 ≤ α ≤ α1) satisfies the Laplace equation and the boundary conditions Proceedings of RuPAC2016, St. Petersburg, Russia THPSC001 Magnetic and vacuum systems, power supplies ISBN 978-3-95450-181-6 535 C op yr ig ht

The field emission system with the thin emitter mathematical modeling

This paper is devoted to the mathematical modeling of the field emission diode system. The diode system is consists of the infinitely thin field emitter as a cathode and a plane anode. The emitter substrate is a plane. The interior of system domain is filled with two different dielectrics. The electrostatic potential distribution is found by the method of separation of variables for the Laplace’s equation in cylindrical coordinates as the Fourier- Bessel series.

The arbitrarily shaped field emitter mathematical modeling

This article is devoted to the field emitter of the arbitrary shape mathematical modeling. The axially symmetrical diode system with a field cathode on a plane substrate and a plane anode is under investigation. The effect of the field emitter is simulated using the charged circular lines to solve the boundaryvalue problem so that the zero electrostatic equipotential coincided with the cathode surface. The potentialal distribution as the Fourier-Bessel series is found an analytical form in the cylindrical coordinates.

Mathematical modeling of a diode system with a spherical field cathode

The rotationally symmetric diode system with a field cathode is under investigation. The shape of the field cathode is sphere-on-cone. The anode is a part of a sphere. In this paper two cases are considered, when the interior of system’s domain is filled with two and three different dielectrics. To calculate the electrostatic potential distribution the method of separation of variables is used. The solution of the boundary-value problem for the Laplace’s equation in the spherical coordinates is found an analytical form as the Legendre functions series.

The electrostatic octupole lens mathematical modeling

In this paper the mathematical modeling of the electrostatic octupole lens is presented. The octupole lens consist of eight electrodes of the same shape. Each of these electrodes is a part of the circular cylinder of an infinite length. To solve the boundary-value problem for the Laplace equation the variable separation method is used in plane polar coordinates.

Diode system on the basis of field emitter with semi-ellipsoid shape mathematical modeling

In this paper the mathematical modeling of the diode field emission system on the basis of field emitter as a semi-ellipsoid on a flat-surface substrate is considered. The anode is a semi-ellipsoidal surface. The internal area of the system is contained with two layers of different dielectrics. The boundaryvalue problem for the Laplace’s equation in the prolate spheroidal coordinates is solved to found electrostatic field. The electrostatic potential distribution in analytical form is calculated with the method of separation of variables as the Legendre polynomials expansion.

The sharp-edged field cathode mathematical modeling

In this work the axially symmetrical diode system based on a sharp-edged field cathode on a plane substrate is under investigation. Anode is a plane. All interior area of the system is filled with two dielectrics. To solve the boundary-value problem the variable separation method is used and an effect of the field cathode is simulated using the charged circular lines. The electrostatic potential distribution is represented as Fourier-Bessel expansion. The unknown coefficients in the expansion are a solution of the linear algebraic equations with constant coefficients.