Trak can handle multiple charged-particle species in a single run. The program can even determine self-consistent space-charge-limited emission for different particle types from multiple emission surfaces. When responding to a user question about run setups, I realized that there was no example to illustrate the capability in the Trak library. This example describes the example BIPOLAR that I added to the code distribution.
To demonstrate the validity and accuracy of a program, the best example is one with a known answer. In this case, I modeled bipolar flow, counter-streaming electrons and protons in a planar gap. Section 6.4 of my book Charged Particle Beams reviews the theory. It states that 1) the ratio of current densities is jp/je = √(me/mp) and 2) the current density of each species is 1.86 times the standard Child-law prediction because of mutual space-charge cancellation.
I set up a planar solution (variations in x-y, infinite length in z) with a gap d = 1.0 cm from x = 0 cm to x = 1.0 cm. The applied voltage was V0 = 10.0 kV. The solution volume extended from y = 0.0 cm to y = 1.0 cm with Neumann boundary conditions at the top and bottom. Regions 2 and 3 represented the physical cathode and anode and Regions 4 and 5 were emission line regions on their surfaces. There was one tricky point in creating a 1D solution — Trak does not accept emission points on the solution-volume boundaries. As a workaround, I added thin elements on the top and bottom and defined the emission surfaces so they extended only to the node just inside the boundary.
The other issue was to use the long form of the Dt command for efficient tracking of particles with a large difference in mass:
Dt DtRef Mass
The quantity DtRef is the time step for a particle with mass 1 AMU (the proton). The velocity of a proton at 10 keV is 1.38E6 m/s. The time to cross an element of width 0.0001 m was 7.24E-11 s. I used a value DtRef = 5.0E-11 s..
The Trak calculation gave a total current (per meter in z) of 433.2 A. Using the formulas in Charged Particle Beams, the Child limit for bare electron and proton beams is je = 2.33 A/cm2 and jp = 0.0544 A/cm2. The expected total current per meter is therefore
1.86*(2.33+0.0544)*100.0 = 443.5 A/cm2,
within 2.2% of the code result. Using filtering by x position in GenDist to separate electrons and protons, I found the following values for the proton and electron current per meter: Ip = 10.119 A and Ie = 430.34 A. The ratio of currents was Ip/Ie = 0.0235, close to the theoretical value of 0.0233.
This zip file contains the input files for this example: bipolar.zip.
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