In an array of N point charges, the space charge density at position
r is
However, Equation 6-3 is inconvenient to use for the following reasons:
For example, instead of allocating degrees of freedom for 1012 electrons, it will often suffice to model 10
4 particles, each of which has a
Charge multiplication factor of 10
8, meaning that it represents 10
8 electrons.
The treatment of particle trajectories as paths in a constant-current beam is determined by the Particle release specification setting in the settings window for the Charged Particle Tracing physics interface. If
Specify release times 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
Specify current 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 Specify current and
Specify release times particle release specification is thus analogous to the difference between integration over
Elements and time and integration over
Elements 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 Specify release times is selected from the
Particle release specification list, this does not require special consideration because the trajectories and fields are computed simultaneously. If
Specify current 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:
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
Particle release specification set to
Specify release times, 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
If instead the Particle release specification is
Specify current, 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