28th example file for CPO3DS.

The system approximates to a Pierce gun. The cathode emits potassium 39 ions at a temperature of 1500C.

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 anode is at 100V and is 8mm from the cathode, which has a radius of 2mm. The cathode is surrounded by a short cone at the Pierce angle (67 degrees).  The outer cylinder has a radius of 2.5mm and is divided into sections that are given a potential proportional to z**(4/3). The system therefore represents a rough approximation to an axiallysymmetric Pierce gun.

Zero temperature:

Child's Law for electrons is w*V**1.5/l**2, where w = 0.002334mA/mm**2, so with V = 100, l = 8mm, the current density becomes .03647mA/mm**2, and the total current before symmetry reflections should be 0.05728mA and the average cathode brightness should be 0.001161A/(ster*cm**2).

For Potassium 39 singly charged ions w has to be divided by sqrt(39*1836). With V = 100, l = 8mm, the current density becomes .0001363 mA/mm**2, the total current before symmetry reflections should be 0.0002141mA, and the average cathode brightness should be 4.338microA/(ster.cm**2).

In fact CPO3DS reproduces these results to within a few percent.

Non-zero temperature:

It can be seen from the table in the notes to xmpl3d11 that for  j0 = 0.1mA/mm**2, the dependence of the values of the Langmuir current density

j on the temperature of the cathode is (for electrons):

kT (eV)       0.0001    0.1          0.2         0.1528

j (Langmuir)  0.02334  0.02567  0.02676  0.02640

where the last row has been obtained by interpolation (and kT = 0.1528eV corresponds to T = 1500C). The numbers quoted above for zero temperature should therefore be multiplied (in the case of electrons) by .02640/.02334 when T is 1500C and j0 is 0.1mA/mm**2 (an arbitrary current density).  For electrons the current density should therefore be 0.04125mA/mm**2, and the total current before symmetry reflections should be 0.06479mA.  For potassium 39 singly charged ions, taking j0 to be 3.74E-4mA/mm**2 (obtained by scaling the value 0.1mA/mm**2 for electrons), the space charge limited current density j should therefore perhaps be 1.542E-4mA/mm**2, and the total current before symmetry reflections should perhaps be 2.422E-4mA. The value of j depends on the value of j0, which in turn depends on the work function of the surface.

The results obtained for the total current in the present iterations are:

2.016E-04 3.908E-04 4.221E-04 3.734E-04 3.791E-04 3.591E-04 mA

Average cathode brightness =  7.371E-06 A/ster.cm**2

Beam brightness =  4.824E-05 A/V.ster.cm**2

Using average of inner 4 rays: total current = 3.054E-4 (should be 2.42E-4?).  (The inner rays are used here because they are least affected by the inexactboundary conditions -see the comments in xmpl3d11.dat.)

Higher accuracy:

To obtain higher accuracy, most of the critical parameters have been increased or decreased by a factor of 2 or 3 (eg number of cathode segments doubled, maximum step length halved, ray tracing inaccuracy decreased from .003 to .0001, etc) and the 'direct method of ray tracing has been used. Then the  results obtained for the total current in the iterations are:

2.028E-04 3.931E-04 4.206E-04 3.586E-04 3.631E-04 mA

Average cathode brightness  =  7.408E-06 A/ster.cm**2

Beam brightness  =  4.848E-05 A/V.ster.cm**2

Using average of inner 8 rays: total current = 3.099E-4 (should be 2.42E-4?). The results are therefore essentially the same as with the lower accuracy.

The relevant space-charge limited current density is therefore 1.97E-4mA/mm**2 = 1.97E-5A/cm**2, which can be compared with the value 1.54E-5 obtained by scaling the value 4.13E-3 expected for electrons.

Conclusions:

When the maximum emissivity j0 of the cathode is chosen (arbitrarily) to be 3.74E-5A/cm**2 (which corresponds to 1.0E-2A/cm**2 for electrons), the space-charge limited current density j for potassium 39 ions emitted at a temperature of 1500C is 1.97E-5A/cm**2. This value of j can be compared with the value 1.54E-5A/cm**2 obtained by scaling the space charge limited current density 4.13E-3A/cm**2 expected (from the Langmuir relationships) for electrons -which seems to show that simple scaling does not apply.  The value of j depends on the value of j0, which in turn depends on the work function of the surface, but other values of j0 have not been used in the present study.