Setting up initial conditions for thermionic cathodes

When a thermionic cathode is called for the program creates a finely divided 'cathode space-charge region' in front of the cathode. The appropriate space-charges are deposited in this region, and are limited, if necessary, according to Child’s Law (for zero kT) or Languirs relationships (for non_zero kT) and these are automatically modified in an empirical (but accurate and effective) way to take account of any curvature that the cathode has. (For more detailed information see publication number 60 on the simulation of thermionic cathodes.)

This note deals with the following topics on setting up a thermionic cathode (look for red headings):

1. Rays from the cathode.

2. Initial temperature or current per unit area.

3. Side of the cathode electrode from which the electrons are emitted.

4. Angle and energy distributions.

5. Cathode depth.

6. Extra detailed information on convex cathodes (inner and outer regions).

7. Other information.

For information on setting up a cathode iteration see notes on cathode iterations.

For information on user-defined thermionic cathodes see note on user-defined cathodes.

For technical information see publication number 60 on the simulation of thermionic cathodes.

For emitting particles that are not electrons see the 'constant mass' option.

For information on setting up a space-charge see notes on cathode iterations.

The 3D thermionic cathode also includes a fixed energy option (in databuilder) with a lambertian distribution of directions).

1. Rays from the cathode.

A single ray will start from the centre of each of the N segments. The current from a cathode segment is proportional to the area of the segment and also depends of course on the field at the segment.

More precisely, for a thermionic cathode the rays start from the outer edge of the cathode region of depth ‘d’, see below.

Note that the current will be shared between the rays, each of which represents a model particle that moves in the total electric field as if it were a single electron or ion but carries the charge and current of many adjacent electrons or ions (this model particle has sometimes been labelled a super-electron or ion), since the ray current is usually greater than that of an individual electron.

Some of the rays might carry currents that are very much smaller than the average current per ray. The distribution of rays in the screen plots is therefore not representative of the current distribution.

2. Initial temperature or current per unit area.

The user has the choice of specifying either

(1) the temperature, the work function (in eV) and the initial Richardson’s constant (in A/sq cm), or

(2) the initial current per unit area (in mA/sq mm), from any part of the cathode surface.

(The program uses the number in the ‘temperature/current density’ box to recognise which choice is required -greater than 200 gives the temperature choice.)

If the temperature and other constants are entered then the program calculates the current density using the Richardson-Dushman formula, which in the units used by CPO3DS is

j = 10*R*T^2*exp(-11604W/T)

where R is Richardson’s constant for the material of the cathode, W is the work function and the factor 10 converts the units to those required by the program.

Note that the ‘initial’ values of R, W and T used here are not usually the physical values for the cathode, but are used only to define the initial j, in obtain the fastest possible convergence of the iterations -they usually do not affect the final current. If the value is too large then too much space-charge will be deposited in the first iteration and the second iteration will then fail because the potential near the cathode will be too repulsive. On the other hand if the value too small then more iterations will be needed before convergence is reached.

The user is also asked to specify either the final Richardson’s constant or the final current per unit area -see setting limits to final parameters. The final values have two uses. The first is to obtain the fastest possible convergence of the iterations, without affecting the final current -these final values will prevent the currents becoming too large during the iterations. The second is to provide, if necessary, a physical limit to the current, caused for example by a temperature-limited cathode.

In general the initial Richardson’s constant should be chosen to be significantly lower than the final value, to give a gentle start to the cathode iterations. If using current densities, the initial value should be significantly lower than the final value, for the same reason.

The user has to establish the best values of the temperatures or current densities empirically (see experimenting with parameters).

It is usually advisable to disable the 'zero total charge' option when the potentials near the cathode are critical (which they usually are) and the anode voltage is very large compared with the potential differences near the cathode.

Start by doing a quick approximate simulation to establish the approximate asymptotic value of the cathode current density (mA per square millimetre) and hence the final temperature. Then use this density or temperature as the initial value. When kT is non-zero try to set the maximum current emitted for the cathode surface at approximately 1.5 times the value of the space-charge limited current -see the footnotes of the example files xmpl3d11 and xmpl3d21. A series of runs may be needed to establish the most appropriate values to use.

3. Side of the cathode electrode from which the electrons are emitted.

The ‘Reference point’ defines which side of the electrode the electrons are emitted from. The coordinates of a suitable point P that lies either 'inside' or 'outside' the cathode are required. The program then knows which side is the emitting side. For convex or concave cathodes use 'inside' or 'outside' respectively.

For example, if the cathode is convex, so that electrons are emitted from its outer surface, then the letter 'i' must be used and an 'inside' point must be specified somewhere inside that surface. It is important to choose a sensible reference point -eg the centre of curvature. For a planar cathode the point could be at an arbitrarily large distance from the surface. The reference point must be consistently either always inside or always outside the cathode.

4. Angle and energy distributions.

In CPO2DS the particles all start from the cathode with zero energy, but in CPO3DS the thermal energy kT is required. (Non-zero thermal energies in CPO2DS require the user-supplied cathode version.) The thermal energy is given in units of electron volts. kT = 1eV when T = 11604K. kT can be zero. If kT is positive the rays will be given Maxwellian randomised thermal energy components in the directions perpendicular and parallel to the surface of the cathode. The distribution of angles therefore follows the Lambert cosine law. Another choice also exists: kT can be entered as a negative number, in which case its absolute value is used as a fixed energy for the particles and their directions are distributed according to the Lambert cosine law (but for 2D systems see the note on 2D Lambert distribution).

The randomisation can be the same each time (that is, a repeatable randomisation, always with the same sequence, see note on randomisation), or can be a different, uncontrolled randomisation each time the program is run. Yet another option is non-random, with the rays starting normal to the surface of the cathode with an energy kT (but note that this is non-physical)

When kT is non-zero (in CPO3DS) a potential minimum can appear in front of the cathode, causing the electrons of lower energy to be reflected back to the cathode, which at the same time of course builds up the space-charge in this region. The space-charge cloud at this minimum then constitutes the 'virtual cathode', which is the effective origin of the electrons that form the final beam. All this is automatically and accurately dealt with in the program by using the methods described by I. Langmuir and E. Q. Adams (Phys. Rev., vol 21, pp 419-435, 1923) for the planar diode, with appropriate empirical adaptations for non-planar cathodes. The randomised thermal momentum components are added to the particle at the beginning of the ray tracing, at the outer edge of the specified cathode region.

When kT is entered as negative, to trigger the option of a fixed energy with a lambertion distribution of directions, the method of Langmuir and Adams is still used, to give an approximation for the effects of the space-charge cloud. In principle the method should be modified for this case, but there do not appear to be any papers on this in the literature (and we have not yet attempted to do it ourselves).

When kT is entered as negative and the ‘random’ option is selected as non-random (see above), then the rays start normal to the surface of the cathode with a constant energy abs(kT) (but note that this is non-physical).

The time taken to traverse the cathode region is adjusted to allow for the thermal energy, but no correction is made for any curvature that the cathode might have, nor for any potential minimum that might exist in front of the cathode.

For important advice and further information please see notes on current densities and starting iterations.

5. Cathode depth (or distance).

The user is asked to specify a distance 'd' over which Childs Law or Langmuir's relationship will be used, and an integer 'n' that gives the number of interpolation points inside the distance 'd' (see publication number 60 on the simulation of thermionic cathodes.).

There is a cloud of electrons in front of a thermionic cathode. This cloud creates a negative potential that reflects most of the emitted electrons back into the cathode surface. The ray tracing therefore cannot start from the cathode surface itself because most of the ray tracing would be wasted. Therefore ray tracing is started at the distance 'd' and either Child's Law (for kT = 0) or Langmuir's relationships (for non-zero kT) is used to determine the space charge density in the inner region.

The distance 'd' is taken in the direction normal to the plane of the segment. The charge density in this region and the cathode current will depend on the values of the potential and field that are found to exist at d. Outside the cathode region the charge density will be handled as described in note on ray space charges.

A suitable value for 'd' is in the region of the average width of the cathode segments. The program does not allow 'd' to be less than 0.2*s, where s is the length of the segment in CPO2D or the maximum distance between the middle of the segment and a corner in CPO3D. A convenient way to find a suitable small value of ‘d’ is to set it to zero: the program will then tell you the minimum allowed value.

Warning: the program might allow the distance 'd' to be so small that the rays start inside the space-charge cloud in front of a thermionic cathode, but this must be avoided.

A suitable value for the interpolation number 'n' is usually 2. The results are usually not sensitive to this number, which controls the degree of detail that the program uses to determine the space charge density in the inner region.

Warning.

The cathode region must not overlap with any other electrodes, or else Child's Law and Langmuir's relationships are invalidated. This is not checked by the program.

Extra detailed information:

The local curvature of the cathode surface is automatically taken into account by the program (for references, see the notes at the ends of the files test3d10.dat and test3d11.dat or see publication number 60 on the simulation of thermionic cathodes). The correction for the curvature (spherical or cylindrical) is essentially exact, except when

(1) the surface is conical (the program then takes it to be cylindrical at the local point for the purposes of this correction),

(2) the cathode is concave and the ratio of the distance d to the radius of curvature R is greater than 0.9,

(3) the cathode is convex and d/R > 10, or

(4) the temperature is non-zero (since the curvature corrections for Langmuir's relationships do not seem to have been studied yet).

If any of these circumstances arise and an accurate answer is required then a small value of d would have to be used.

For a concave cathode the distance d should normally be greater than the spacing of the outer mesh points, but should also normally be smaller than the radius of curvature of the cathode surface.

It is also desirable in principle for 'd' to be significantly greater than the cell spacing 's' if the cell method of assigning space-charge is being used, although in practice this has not always been found to be strictly necessary (see for example files test2d11.dat, xmpl2d09.dat and test3d10.dat).. These conditions are sometimes difficult to satisfy simultaneously, and the user might then have to compromise on the inaccuracy of the total cathode current.

End of extra detailed information:

6. Extra detailed information on convex cathodes (inner and outer regions):

For convex cathode surfaces the above condition (at the end of detailed information) is usually too difficult to satisfy, and so the program adopts a different method of solution, by dividing the region of thickness d into two distinct regions (see publication number 60 on the simulation of thermionic cathodes). The first of these has the cathode surface as a boundary, and is called the 'inner cathode' region. Here Childs Law is used (or Langmuir's relationship), appropriately corrected for the curvature of the cathode. The second region is called the 'outer cathode' region, and is created only for convex cathodes. Outside the 'outer cathode' region is what can be called the 'ray' region, in which the space charges of the rays are assigned to mesh cells or tubes, as described in the note on ray space charges.

The purpose of the 'outer cathode' region is to bridge the gap between the 'inner cathode' and 'ray' regions. This region is necessary when the spacing of the mesh cells is comparable to, or larger than, the dimensions of the cathode, as will frequently happen for convex cathodes. If the 'outer cathode' region did not exist it would usually be necessary to use a much smaller mesh spacing, with a consequent large increase in computing time. The creation of the outer region allows the study of cathodes that are much smaller than the dimensions of the rest of the system. An example is given in file test3d11.dat.

The distance d specified by the user is the overall depth of the cathode region (that is, the 'inner' plus 'outer' cathode regions). The depth of the 'inner' region becomes d/8. To define the 'outer' region the program finds the 'radius' R of the cathode, using the 'inside' point that the user has defined -and so this should be carefully defined- and then places the outer region between the radii R+d/8 and R+d. For a non-spherical cathode R is the minimum distance from the 'inside' point to the mid-point of any of the cathode segments.

7. Other information:

The interpolation number 'n' controls the number of interpolation points (see publication number 60 on the simulation of thermionic cathodes) used in the cathode region.

Increasing the interpolation number gives higher accuracy (but increases the computing time, and the required memory space). A small value, in the range 2 to 4, is usually adequate for concave cathodes, unless the potential minimum referred to above is deep. Similarly, the values 3 or 6 are usually adequate for convex cathodes. The maximum value is 20.

The calculated ray information is automatically saved at the end of the binary part of the processed dat file, file, and is available for re-use, provided that the present input data file is identical to the one used previously and the processed data file is also the one generated previously. Only the results of the last iteration are saved, and the contouring option is not available.

Some of the rays might carry currents that are very smaller than the average current per ray (although some of these might be suppressed). The distribution of rays in the screen plots is therefore not representative of the current distribution.

Extra detailed information:

When the cathode region is divided into two for convex cathodes, the 'inner' region is given n/3 interpolation points and the 'outer' region 2*n/3.

It is usually (but not always) preferable for all the segments of a cathode to have approximately the same area. The reason for this is that the current that a ray carries is proportional to the area of the parent segment and it is also proportional to the current density (mA/mm**2) that is determined by the program (and that is often approximately constant over the active part of the surface of the cathode for each set of rays).

For example for a cathode on the axis of a 2D cylindrically symmetric system this is easily achieved by using the segment subdivision type '2' or '-2'.

Several of the types of 3D electrode available in this program are automatically divided into segments of equal, or approximately equal, area. For example, 'spherical' electrodes are divided equally if they have a closed end. Also 'evenly divided disc' electrodes are divided equally, and 'triangular', 'spherical triangle', 'end spherical triangle', 'end conical or cylindrical triangle', 'end disc triangle', 'rectangle', 'rectangle x,y,z', 'cylinder rectangle' and 'cylindrical' electrodes are divided up into segments of the same, or nearly the same, area, but the segments could sometimes be long and thin. 'Conical', 'conical triangle' and 'disc' electrodes are not divided into equal areas.

End of extra detailed information:

For users who are editing or constructing an 'input data file' without the use of the data-builder -that is, pre-processor:

But Manual editing is certainly not recommended -it is a relic from the time when the databuilder was not available.  All users are strongly encouraged to use the databuilder, which always gives the correct formats and which has many options for which the formats are not described or easily deduced.

CPO2D:

Typical data, taken from the 'benchmark test' file test2d10.dat, are:

thermionic cathode

10 0.032 number of segments, maximum current density,

o 0. 0. 'o' for 'outside', and (r,z) of reference point

0.10 2 distance for Childs Law, number of interpolation points

y n calculate space-charges?

0.1 mesh spacing for space-charges

iterate previous cathode rays

0.100 0.5 4 max current density, damping factor, no of iterations

To re-use ray data that was calculated previously the present file should be identical to the previous one except that the line

'y trace rays? (y/n)'

should be replaced by the line

'previously calculated rays'

(or any line that starts with 'p').

CPO3D:

Typical data, taken from the 'benchmark test' file test3d10.dat, are:

thermionic cathode

12 0.0001 number of segments, maximum current density

o 0. 0. 0. 'o' for 'outside, and xyz of reference point

0. kT, in eV (that is, thermal energy at cathode)

0.10 3 distance for Childs Law, number of interpolation points

y n calculate space-charges?

0.1 mesh spacing for space-charges

iterate previous cathode rays

To re-use ray data that was calculated previously the present file should be identical to the previous one except that the line

'start of ray information'

should be replaced by the line

'previously calculated rays'

CPO2D and CPO3D:

The next line must contain 2 numbers, an integer giving the number N of segments that form the cathode and the maximum current per unit area, in milli-Amp per square millimetre, from any part of the cathode surface (and there is no need to worry about the sign of this). Alternatively the initial temperature, work function and Richardson’s constant can be given after N.

The next line is used to define which side of the electrode the electrons are emitted from. The line should contain the letter 'i' (for 'inside') or 'o' (for 'outside') at the start of the line, followed by the coordinates of a suitable point that lies either 'inside' or 'outside' the cathode.

In CPO3DS the next line should contain either the temperature or the thermal energy kT, expressed in electron volts. Make kT negative if a fixed energy lambertian distribution is required. The randomisation is controlled by what is present in the 10th space on the line. If you want the same randomisation each time (that is, a repeatable randomisation, always with the same sequence), put the letter 's' in the 10th space. If you want to control and vary the sequence, put a 'c' in the 10th space and follow it with a controlling number (that is, a seed value), which can take any value. If this number is always the same, eg 1.1, then the same randomisation sequence will appear each time, but if the number is changed, eg to 1.2, then a different sequence will appear (see also note on randomisation). If you do not want to exercise any control, that is, if you want a different, uncontrolled randomisation each time the program is run, leave the 10th space blank, or put something there that is neither 's' nor 'c'. Yet another option is non-random, with the rays starting normal to the surface of the cathode with an energy kT, in which case use ‘n’ (but note that this is non-physical option).

The final line should contain the distance 'd' over which Childs Law or Langmuir's relationship will be used, and an integer 'n' that gives the number of interpolation points inside the distance 'd'.