Poisson Solver in SIMION¶

Overview¶

In SIMION 8.1.0, the electromagnetic field solver (Refine function) has been extended to support solving the Poisson Equation. As before, the Refine function can calculate potentials inside a volume defined by electrodes with potentials, but the new Poisson solving capability allows it to also handle the case when a known, arbitrary distribution of space-charge density fills that volume (such as due to high current charged particle beams or charged insulators). Dielectrics are also supported.

Applications¶

Taking into account space-charge density can be important for certain problems. Space-charge effects tends to be most severe for a slowly moving, high current beam confined in a small volume, for this causes a large space-charge density and large periods of time over which the fields from that space-charge density can have an effect on charged particle motion. A typical example is the beam near the surface of a thermionic cathode operated at maximum (space-charge-limited) current, for here the beam is not just at high current but also at low energy (thermal energies as not yet having been accelerated by the anode).

The Poisson solver can also be useful for other situations in which a charge density distribution is known a priori, such as in the case of a charged insulator material with known charge distribution. This is simpler to handle because it does not require particle flying to do the Refine.

Other applications of the Poisson solver include

Dielectric materials, with known dielectric constants. See Dielectrics.
Solving magnetic vector potential (and hence magnetic fields) from arbitrary distributions of current density in space. See Magnetic Potential.
Solving current density from arbitrary distributions of material conductivity in space and boundary conditions on electric potential. See Current Density.
Floating conductors. See Floating Conductor

Version Requirements¶

Compatibility Warning: The Poisson solving capability is available in SIMION 8.1.0. It is not in previous release versions. The Poisson solving capability is one of the major new features in 8.1.0.

Previous versions of SIMION (8.0 and below) assumed that space-charge density was negligible. That is, it solved only the Laplace equation. However, 7.0 and 8.0 do have some rough “charge repulsion” methods that are useful in approximating space-charge effects under certain conditions (see the SIMION Example: repulsion in SIMION 8.0.4 or above for full details on this). Unlike the SIMION 7.0 charge repulsion methods, the Poisson solver takes into account the effect of space-charge on the distribution on surface charges on electrodes (important when high space-charge density is near electrode surfaces).

Math¶

The Poisson Equation is formally

$\nabla^2 \Phi = -\rho / \epsilon_0$

or in the presence of dielectrics,

$\nabla \cdot \left( \epsilon_r \nabla \Phi \right) = -\rho / \epsilon_0$

where $\phi(x,y,z)$ is electrostatic potential as a function of position $(x,y,z)$ , $\rho(x,y,z)$ is space-charge density as a function of position, $\epsilon_r(x,y,z)$ is the relative dielectric constant as a function of position, and $\epsilon_0$ is the permittivity of free space constant. To solve for the potentials ( $\phi$ ) inside some volume, which is done by the Refine function, SIMION must in general know not only $\phi$ on the boundary of the volume but also $\rho$ and $\epsilon_r$ throughout the inside of that volume. In the case of no space charge ( $\rho = 0$ ) and dielectric constant of 1 outside electrodes ( $\epsilon_r = 1$ as is the case of vacuum and approximately in air), the equation reduces to the Laplace Equation, which earlier versions of SIMION solved.

Note

See these additional sections in this topic in the offline SIMION 8.1.0 help file:

SIMION Array Modeling
Effect of Scaling After Refining
Calculation Method
Usage
Comments on Fast Adjust (.PA#) Files
Comments on magnetic arrays (experimental)
Computing charge density from trajectories
Charge density smoothing

Performance¶

The performance of the Poisson solver will naturally likely be slower than the regular Laplace solver, though not substantially so. The charge density array also uses addition memory. The Pierce gun examples with a non-continuous beam have run times on the order of a few minutes. Using a lower resolution charge density array might speed things up, or at least use less memory.

Examples¶

See the SIMION Example: poisson for examples on using the Poisson solver. Note that the Poisson solver is an advanced feature. It currently requires Lua user programming to effectively make use of it, but it can be very powerful and flexible. It is recommended that you study the source code.

Also, the Gauss’s Law example may be useful when experimenting with the Poisson solver.

Current Limitations / Caveats¶

The Poisson solving currently works with basic arrays (.PA files) not fast adjust arrays (.PA# files), but the latter shall be supported in the future, as discussed above.

The Poisson solving currently is only intended for electrostatic arrays, not magnetic arrays, as discussed above.

The Poisson solver is currently not accessible via the GUI but rather via the Lua programming API.

The Poisson solver is fully compatible with Anisotropically Scaled Grid Cells in SIMION (2D planar, 2D cylindrical, and 3D planar).