Space Charge Density Calculation

In an array of N point charges, the space charge density at position r is

(6-3)

where

•

δ (SI unit: 1/m3) is the Dirac delta function,

•

qi (SI unit: m) is the position of the ith particle, and

•

qi (SI unit: C) is the charge of the ith particle.

However, Equation 6-3 is inconvenient to use for the following reasons:

•

The number of particles may be extraordinarily large, making it impractical to model all of them.

•

The charge density becomes infinite at the location of each idealized point source, making the electric potential calculation infeasible with most numerical methods.

In the following sections we discuss solutions to each of these problems.

Modeling a Representative Sample of Particles

Because the number of real particles, such as ions or electrons, may be too large for every particle to be modeled individually, a practical numerical approach is to release a representative sample of model particles, allowing each model particle to make the same contribution to the space charge density as an equivalent number of real particles.

For example, instead of allocating degrees of freedom for 1012 electrons, it will often suffice to model 104 particles, each of which has a of 108, meaning that it represents 108 electrons.

Simplification for Constant-Current Beams

If a beam of particles is released at constant current, then a full time-domain calculation of the coupled particle trajectories and electric fields may require particles to be released at a large number of time steps until a stationary solution for the electric potential is reached. This can be needlessly memory intensive and time consuming. An alternative approach is to release particles at time t = 0 and to allow each model particle to represent a continuous stream of real particles per unit time. The number of real particles per unit time represented by each model particle is denoted the effective frequency of release, frel.

The charged particles tend to contribute to a greater space charge contribution in regions in which they are moving slowly, causing particles that are released at successive times to be closer together. This behavior can be conveniently reproduced by defining an expression for the time derivative of the charge density, rather than the charge density itself:

(6-4)

The charge density can then be computed by integrating over time, as long as sufficient time is given so that the particle trajectories can be traced completely through the modeling domain.

The frequency of release can be computed using the current and number of model particles that are specified in release feature settings. For example, for an Inlet node with release current magnitude I (SI unit: A) and number of particles per release N (dimensionless), the effective frequency of release is

When particle beams are assumed to have constant current, then the space charge density at the last time step includes contributions from particles at every point along their trajectories in the modeling domain. Thus, it can be applied as the space charge density term when computing the electric potential.

The treatment of particle trajectories as paths in a constant-current beam is determined by the setting in the settings window for the Charged Particle Tracing physics interface. If is selected, the charge density is computed using Equation 6-3 and is determined by the instantaneous positions of all model particles. Thus, it is necessary to solve for the particle trajectories and electric potential in the time domain. If is selected, the charge density is computed using Equation 6-4 and is determined by the time history of the model particle positions.

The difference between the and particle release specification is thus analogous to the difference between integration over and integration over as described for the Accumulator (Domain) node.

At this point, the effect of a bidirectional coupling between the particle trajectories and fields has not been considered. If is selected from the list, this does not require special consideration because the trajectories and fields are computed simultaneously. If is selected, however, the trajectories and fields are computed using different study types, and an additional feedback mechanism is needed. The Bidirectionally Coupled Particle Tracing study step can be used to generate a solver sequence that does the following:

Set the space charge density contribution due to the particles to zero.

Compute the electric potential and other field variables using a Stationary solver, using the value of the space charge density computed in the previous step.

Compute the particle trajectories and the resulting space charge density in the time domain, using the field variables computed in the previous step.

Repeat steps 2 and 3 until a specified number of iterations has been reached, or until another user-specified convergence criterion has been satisfied.

If the number of iterations taken by the solver sequence is sufficiently large, the resulting solution will fully account for the bidirectional coupling between the particle trajectories and stationary fields.

Avoiding Infinitely Large Values of the Space Charge Density

The Electric Particle Field Interaction node defines a variable for the contribution to the space charge density by particles in each mesh element. This variable is discretized using constant shape functions that are, in general, discontinuous across boundaries between elements. For a mesh element j with volume Vj, and with the set to , the average space charge density ρj is

where ni is the charge multiplication factor of the ith model particle. The integral on the right-hand side is a volume integral over element j. The resulting charge density is the average charge density over the mesh element, which may be written more concisely as

where the sum is taken over all particles that are within mesh element j.

If instead the is , each model particle represents a number of particles per unit time, which follow along the same path, determined by the effective frequency of release frel. The space charge within the mesh element can then be expressed as the solution to the first-order equation