Charged particle trajectories in QuickField

Version 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:

Calculated electrostatic field,
Particle attributes: charge, mass, initial velocity or energy; Initial velocity might point outside of the calculation plane,
Emitter attributes: coordinates (starting point of all beam trajectories), limits for the angle between initial velocity and horizontal axes, and the total number of trajectories in the beam.

Viewing calculation results, you see:

Projections of beam trajectories on the calculation plane,
Kinematic parameters in every trajectory point,
velocity,
acceleration,
the path length and the time spent on the way to any trajectory point.

Calculating trajectories QuickField uses following assumptions:

There are no relativistic effects,
Electrostatic field inside any finite element is linear relative to coordinates,
The beam space charge field can be ignored in the equations of motion ("infinitely small current" approximation),
Distinctive emitter physical features can be ignored, so that all beam particles have the same starting point and kinetic energy.

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:
Charged particle trajectories equation

We reorganize this system of three second degree equations into six first-degree equations and append the following additional equation:
length of the trajectory covered by the particle

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.