Charged Particle Tracing
The Charged Particle Tracing interface () is found under the AC/DC branch () in the Model Wizard. It is primarily used to trace ions and electrons under the influence of electric and magnetic forces. These electric and magnetic forces can either be entered explicitly as user-defined expressions or computed using another physics interface. The magnetic force can also be taken from built-in data for the Earth’s magnetic field.
Motion of trapped protons in the Earth’s magnetic field. The protons follow helical paths that oscillate between the northern and southern hemispheres.
Included in the Electric Force and Magnetic Force features are built-in options to modulate AC fields. In this way, it is easy to include combinations of stationary and time-harmonic forces in a single model.
Quadrupole mass filter: Argon ions propagate through a quadrupole arrangement of electrodes (left). The stability of the design is determined by the magnitude of the DC and AC potentials used to guide the ions through the filter and reported as a function of the dimensionless Mathieu coefficient (right).
Monte Carlo Collision Modeling
In addition to electric and magnetic forces, energetic ions and electrons are often subjected to collisions with the molecules in a rarefied background gas. At sufficiently low pressure, these collisions can be modeled stochastically using a Monte Carlo approach. In each time step taken by a time-dependent solver, each model particle has a chance to collide with a background molecule and be scattered in a random direction.
The collision probability and the probability distribution function of scattered particle velocities are based on the particle kinetic energy, type of reaction, background gas density, and collision cross section data. Because the collision cross section is usually a nontrivial function of energy, it is common to load cross section data from a separate file and use it to define an interpolation function.
Some of the built-in collision types can release secondary particles of the same or a different species. For example, ionization reactions can yield secondary electrons, and charge exchange reactions can yield new combinations of ionized and neutralized species.
Because of its Monte Carlo collision modeling capability, the Charged Particle Tracing interface is sometimes extremely useful in modeling the motion of neutralized species, not just ions or electrons, in spite of the interface’s name.
Charge exchange cell: A fast neutral beam is produced by accelerating protons (light gray) and passing them through a vacuum chamber containing a rarefied argon gas. Collisions with the background gas produce energetic neutral particles (dark gray). The remaining protons are deflected by an electrostatic field between two oppositely charged plates (red and blue).
Multiphysics Interfaces for Particle Field Interaction
If the number density of charged particles is sufficiently low, the Coulomb interaction between ions or electrons can sometimes be neglected, since it is likely to be dwarfed in magnitude by the external electric and magnetic fields. Such models use a unidirectional coupling from the electric and magnetic fields to the particle trajectories. In other words, first the fields are computed, and then these fields are used to exert forces on the charged particles.
Einzel lens: The electrostatic force from the electrodes, illustrated by isosurfaces of electric potential, is much greater than the Coulomb interaction between the electrons, making this a unidirectionally coupled model.
Coulomb Interaction
If the density of charged particles is suitably high, then it may be necessary to include the Coulomb force that acts between the particles. The built-in Coulomb force model for the Particle-Particle Interaction node computes the instantaneous force on each charged particle based on the position and charge number of all other particles in the model.
When particle-particle interactions are included in a model, the computational requirements to evaluate the force increase quadratically with the total number of particles because each particle’s motion is affected by the position of every other particle in the model. When including the Coulomb force, it is often best to start with a small number of particles, solve the model, and then assess whether or not the effect is important.
The Coulomb force is well-suited to models in which the interaction between particles may be significant but the total population of particles in the model remains constant over time. For beam simulations with space charge effects, however, ions or electrons are not just affected by particles that are released simultaneously, but also particles that are released earlier or later in time. This can lead to a very large number of model particles being required to accurately capture the Coulomb interaction. A two-way coupled particle-field interaction, described in the next section, might be a better choice.
Bidirectional Particle-Field Interaction
An alternative to directly applying the Coulomb force is to allow particles to contribute to the space charge density in the domain for the purpose of computing the electric potential, then to use this modified electric potential to compute a more accurate electric force on the particles. This is called a bidirectional coupling between the particles and fields. The bidirectional coupling is well-suited to simulations in which ions or electrons are released continuously over time such that the electric potential at any point in space remains approximately constant. The dedicated Particle Field Interaction, Nonrelativistic interface () combines the Charged Particle Tracing () interface with the Electrostatics () interface to quickly set up the two-way interaction in which the particles perturb the electric potential in their surroundings.
A beam of relativistic ions or electrons can generate sufficient current to produce a significant magnetic force on the beam. This self-induced magnetic force can be implemented using the Particle Field Interaction, Relativistic interface (), which automatically combines the Charged Particle Tracing (), Electrostatics (), and Magnetic Fields () physics interfaces. This multiphysics coupling requires both the AC/DC Module and the Particle Tracing Module.
Relativistic electron beam: despite being released at the beam waist, the beam electrons (gray) diverge because the outward force exerted by the electric field (red) is greater in magnitude than the inward force exerted by the magnetic field (blue).
Releasing Nonlaminar Ion and Electron Beams
The Charged Particle Tracing interface includes a dedicated feature for releasing ion and electron beams. You can specify quantities such as the beam emittance and Twiss parameters to characterize the particle position and velocity distributions.
When one or more charged particle beams are released in a model, quantities such as beam emittance are automatically computed along the nominal beam trajectory and are available for postprocessing.
A beam of electrons is focused by a magnetic lens (left). The beam emittance can then be plotted as the thickness and color along the nominal beam path (right).