Once the electromagnetic field is computed, particle trajectories can be evaluated by solving the equations of motion:
 | (5–178) |
where:
| m = mass of particle |
| q = charge of particle |
| {E} = electric field vector |
| {B} = magnetic field vector |
| {F} = Lorentz force vector |
| {a} = acceleration vector |
| {v} = velocity vector |
The tracing follows from element to element: the exit point of an old element becomes the entry point of a new element. Given the entry location and velocity for an element, the exit location and velocity can be obtained by integrating the equations of motion.
The particle tracing algorithm is based on Gyimesi et al.([230]) with these assumptions:
No relativistic effects (the velocity is much smaller than speed of light).
Pure electric tracing ({B} = {0}), pure magnetic tracing ({E} = {0}), or combined {E-B} tracing.
Electrostatic and/or magnetostatic analysis
Constant {E} and/or {B} within an element.
Quadrangle, triangle, hexahedron, tetrahedron, wedge or pyramid element shapes bounded by planar surfaces.
These simplifications significantly reduce the computation time of the tracing algorithm because the trajectory can be given in an analytic form:
parabola in the case of electric tracing
helix in the case of magnetic tracing.
generalized helix in the case of coupled E-B tracing.
The exit point from an element is the point where the particle trajectory meets the plane of bounding surface of the element. It can be easily computed when the trajectory is a parabola. However, to compute the exit point when the trajectory is a helix, a transcendental equation must be solved. A Newton-Raphson algorithm is used to obtain the solution. The starting point is carefully selected to ensure convergence to the correct solution. This can be quite involved, as about 70 sub-cases are considered by the algorithm. This tool allows particle tracing within an element with accuracy limited only by machine precision. This does not mean that the tracing is exact, since the element field solution may be inexact. However, with mesh refinement, this error can be controlled.
Once a trajectory is computed, any available physical items can be printed or plotted along the path (see PLTRAC, PATH, PAPUT). For example, elapsed time, traveled distance, particle velocity components, temperature, field components, potential values, acoustic pressure, mechanical strain, etc. Animation is also available.
The plotted particle traces consist of two branches: the first is a trajectory for a given starting point at a given velocity (forward ballistic); the second is a trajectory for a particle to hit a given target location at a given velocity (backward ballistics).