charged particle trajectories, calculation, trajectory, electrostatic
QuickField 5.0 introduces the possibility to study trajectories of charged particles movement in the plane-parallel and axisymmetric electrostatic fields. It utilizes our original approaches based on our implementation of Finite-Element technology¹ and modern computational algorithms².
Trajectory calculation uses the following data:
Viewing calculation results, you see:
Calculating trajectories uickField uses following assumptions:
According to these assumptions, we can describe the trajectory x(t), y(t), z(t) of a charged particle in two-dimensional electrostatic field E(x,y) with Newton's system of differential equations:
d²x/dt² = (q / m) · Ex(x,y)
d²y/dt² = (q / m) · Ey(x,y)
d²z/dt² = 0
We reorganize this system of three second degree equations into six first-degree equations and append the following additional equation:
dl/dt = √(dx/dt)² + (dy/dt)² + (dz/dt)²
Defining the length l(t) of the trajectory covered by the particle in time t. We integrate the resulting system using the Runge-Kutta-Merson method with automatically defined integration step. Numerical integration stops immediately before the finite element's boundary, the step leading outside of the element being excluded. At the last point in the element, we extrapolate the trajectory with cubical segment of its Taylor series relative to time and solve the resulting equation using Tartaglia-Cardano formula and taking into account possible decrease of the equation's degree in homogeneous or zero fields.
¹ D.V.Grigorjev, V. Podnos, "Finite-element software application for electron optics" Report at The 5th All-Russian Seminar PROBLEMS OF THEORETICAL AND APPLIED CHARGED PARTICLE OPTICS, 14 - 15 November 2001, Moscow, Russia
² Synthesis of electrode configurations applied to electron spectrometers. Yu.K. Golikov, D.V. Grigorjev, N.K. Krasnova, K.V. Solovjev, A.D. Lubchich report at The 5th All-Russian Seminar PROBLEMS OF THEORETICAL AND APPLIED CHARGED PARTICLE OPTICS, 14 - 15 November 2001, Moscow, Russia