Working on electrostatic problems you can calculate and view trajectories of charged particles in electric field. To do it, choose Particle Trajectory from View menu. 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:
| 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)2defining 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.
Related Topics
Particle Properties
Emitter Properties
Particle Kinematic Parameters