Info: Poisson Equation

From SIMION

SIMION 8.1 will support solving the Poisson equation, in which zero space-charge density distribution (rho) is permitted when the potentials and field is solved (refined):

 Del^2 V = -rho/epsilon_0

This capability has already been implemented (for 2D arrays), and 8.0 users may preview it. For further details on the new Poisson capabilities, see Issue I482 (http://www.simion.com/issue/482). A draft [description of the solver (http://www.simion.com/docs/poisson.html)] is online. Some screenshots of examples are shown below.

Examples

Computation of field with known space-charge density distribution

The following shows a conductive rectangular tube (2D planar) with a known (sinusoidal) space-charge density inside. This tube is modeled with a potential array, as usual, and the charge density is modeled with a charge density array that defines charge density (C gu^-2 mm^-1) in each grid cell. A user program created these arrays and refined the potential array while passing the charge density array as an option. As shown, the space-charge density causes local minima and maxima in the potential plot where there is high negative and positive charge density respectively.

Computation of field and trajectories together

The following is a simulation of a Pierce gun (2D planar or cylindrical). A beam of electrons travels from the cathode to anode. In the absence of space-charge (low current beam) the potential contour lines are curved inward, and the beam diameter narrows. In the presence of space charge (high current beam), the space-charge of the particles trajectories alter the potential contours so that they become straight and the beam travels horizontally. In the below simulation, whenever the beam advances approximately 1 grid unit, a workbench user program updates the charge density array, rerefines the potential array given the charge density array, and also updates the display of the contour lines. The simulation with space-charge density disabled (in which the beam narrows) is also shown for comparison.

See also:

• Laplace Equation