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xmpl3d11, 11th 'example' data file for CPO3D

Space-charge limited planar diode, with thermal energies.

The set-up is as described in file xmpl3d10.dat, except that the thermal energy is non-zero. The effect on the directions of the rays is apparent. The potential minimum in front of the cathode can also be seen in the contour plots. For example when kT = 0.2eV and the emitted current density is put at 100 mA/mm^2 (a factor of 3600 higher than the saturated current) the potential of the virtual cathode is 1.64 volts lower than that of the real cathode and its position is 0.18mm in front of the real cathode.

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

(Useable only with the space-charge version, CPO3DS).

(This is a 2-dimensional simulation and so can be solved with higher accuracy by CPO2D).

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

The saturated current density (j) depends on the thermal energy (kT) and the thermionic current density (j0) emitted from the cathode surface. The current (i) given by the program is j*A/8, where A is the area of the cathode (25*pi) and the factor 8 is present because only an octant of the circle is used. The following table shows values of j (and hence i) calculated from the eta-xi data of Langmuir and Adams (Phys. Rev. 21, 419, 1923), and compares with the values of i computed with CPO3D (and taking the average of the innermost 4 rays and putting the voltages of the bounding cylinders at the correct calculated values):

j0 is in (mA/mm**2), and the current in the penultimate column is the computed current. The results in the last row have bee obtained using 'direct' ray tracing, with ray inaccuracy of 0.001, 5 interpolation points in the cathode region, 10 iterations and a damping factor of 0.7 -and see below for comments on the potential minimum in front of the cathode.

The large errors that occur when j0 is much larger than j are due partly to the limited accuracy used in this example, but are caused mostly by fringe field effects (which allow the current density to be too large near the boundary, thus building up space charge and suppressing the current density in the centre of the cathode, and requiring more iterations for convergence).

The potential minimum in front of the cathode:

Also, when j0 is much larger than j the virtual cathode (that is, the effective origin of the electrons that form the final beam) has a potential much lower than that of the real cathode itself, so causing most of the emitted electrons to return to the cathode surface. For example, when kT = 0.2eV and j0 = 100 mA/mm**2 (a factor of 3600 higher than the saturated current) the potential of the virtual cathode is 1.64 volts lower than that of the real cathode (and its position is 0.18mm in front of the real cathode). This minimum exists in the computations, but to see it in the contour plots at the interactive stage the user would have to

(a) use the 'direct' method of ray tracing (or else the contour plots would be derived from the potentials at mesh points that have the spacing originally specified and would therefore be unlikely to be closely enough spaced),

(b) choose a small ray inaccuracy (0.001 was used in the example marked by a * above),

(c) use 5 or more interpolation points in the cathode region (10 is the maximum number), and

(d) use a large number of closely spaced grid points when the contour option is called.

But the CPO3D program is not intended for accurate modelling of this region. Also, no attempt has been made in this example file to reproduce this type of potential minimum in the voltages applied to the boundary cylinders.

Without these precautions, instabilities and errors tend to occur when j0 is large. Such instabilities have also been noticed for spherical cathodes when j0 is much larger than j.

IT IS STRONGLY ADVISED therefore to use a j0 that is only slightly larger (that is, 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.

Transition to temperature limited current:

Using parameters that are typical for tungsten, kT = 2.3 eV and maximum temperature limited current = 16 ma/mm^2, we obtain the following dependence of the current on the anode voltage:

The current starts by being approximately proportional to V^1.5, and then saturates at the temperature-limited value (in fact the asymptotic value of the current is less than expected because some of the outer rays exit from the sides of the diode, and do not then contribute). The transition from the space-charge limited to the temperature limited regimes is smooth, but the program does not claim to give an accurate current in the intermediate region.