Xmpl2d53, 53rd 'example' data file for CPO2D

2D cathode with thermal energies.

Space-charge limited beam of a planar diode, with thermal energies.

This cannot be a benchmark test because the beam is not infinite in the transverse direction.

See also xmpl3d11.dat.

The numbers of segments and rays are small, for a quick demonstration.

The set-up is as described in file xmpl3d11. The mean thermionic energy is 0.1eV and we have used 2 values of j0, the current density emitted by the cathode. The final mean current density j is limited by the program to 0.03 mA/mm**2, which is not much higher than the values of j given by the program (see below).

The voltages applied along the outer cylinder were determined iteratively in xmpl3d11. After a run the /contour/potential/grid option was used to find the potentials along the axis and then these potentials were applied to the outer cylinder for the next run. In this way the potential distribution in the z direction should approximate to that of a diode that is infinite in the transverse direction. These are the boundary voltages used:

j0 |
0.1 |
1.0 |

V at z=0.1 |
-0.13 |
-0.31 |

V at z=0.2 |
0.02 |
-0.18 |

V at z=0.3 |
0.29 |
0.06 |

V at z=0.5 |
1.04 |
0.75 |

V at z=0.7 |
1.95 |
1.59 |

V at z=1.0 |
3.50 |
3.02 |

V at z=1.5 |
6.54 |
5.85 |

V at z=2.0 |
10.06 |
9.15 |

V at z=4.0 |
27.58 |
26.01 |

V at z=6.0 |
48.97 |
47.39 |

V at z=8.0 |
73.27 |
72.31 |

The saturated current density (j) depends on the thermal energy (kT) and the thermionic current density (j0) emitted from the cathode surface (which is referred to in the program as the initial cathode current density).

The table below shows the results obtained, where I is the current after completion of the iterations (absolute values are given), j is the final average current density, Vmin is the measured depth of the potential in front of the cathode (caused by the space-charge cloud) and zmin is the position of that minimum. Analytical values calculated from the eta-xi data of Langmuir and Adams (Phys. Rev. 21, 419, 1923) are also shown.

cpo2ds langmuir

j0 |
I |
j |
Vmin |
zmin |
j |
Vmin |
zmin |

0.1 |
1.76 |
0.0224 |
-0.032 |
0.070 |
0.0257 |
-0.136 |
0.076 |

1.0 |
1.69 |
0.0215 |
-0.073 |
0.069 |
0.0259 |
-0.365 |
0.099 |

The values of j differ from the analytical values by approximately 15%. But it must be remembered that that the analytical results assume that the beam is infinite in width. Also the present simulation uses only a small number fo segments and iterations and also uses large step lengths, in order to give a reasonably quick calculation.

Some other details can be found in the footnotes to xmpl3d11.

When j0 = 1 is used it can be seen in running the simulation that there is a considerable fluctuation of the currents of the rays in the first few iterations. This type of instability is to be expected when j0 is more than about four times the final value of j.

The damping factor is of course vital in reducing the fluctuations. The limit imposed by the 'maximum final current density', which is set by the user, is also vital.

Here is the advice given in xmpl3d11:

WARNING************************

IT IS STRONGLY ADVISED therefore to use a j0 that is only slightly larger (ie by a factor of 4 or less) than the expected value of j. This roughly corresponds in practice to lowering the cathode temperature to nearly the point at which the current would be temperature limited. This will make most calculations more stable and accurate, without significantly affecting the final results.