General information on field emission and extended schottky emission cathodes

See also important information on setting up initial ray conditions for field emission.

For information on user-defined field-emission cathodes see note on User-defined cathodes

The field emission and schottky emission options are available in CPO2DS and CPO3DS.  See the note on the use of cathodes, and the files test2d13.dat and test3d12.dat for examples of setting up a field-emission cathode.  See xmpl2d61.dat and xmpl3d16.dat for extended Schottky emission, but note that we have not been able to provide benchmark tests for these cathodes.

For field emission the Fowler-Nordheim equation is used to calculate the current density as a function of the field at the surface of the cathode, as explained in the notes at the end of the benchmark test file test3d12.dat. For extended Schottky emission, a modification of the Richardson-Dushman equation is used, as given by P W Hawkes and E Kasper, Principles of Electron Optics (Academic Press, 1989), and as described in the notes at the end of the example file xmpl3d16.dat.

Warning: These equations have limited ranges of validity. If the program gives a message that the conditions are outside the allowed range this can mean that the field at the surface is too small or too large (for which might occur for example for rays that are near a sharp edge) or that the temperature is inappropriate. The number of the segment at which the conditions fail is given in the message.

The 'direct' method for tracing the rays must be used (because the 'mesh' method would not be accurate enough near the surface of the cathode) and so the program automatically ensures this.

The user starts by choosing whether to define the cathode by the first N electrodes or the first N segments (note that the program might change the number of segments in an electrode, see cathode segments for more details). The number N is then specified. If the choice is to define by the first N electrodes then the program will find the total number of segments in them. A single ray will start from the centre of each of the cathode segments. Nothing else is needed to define the shape of the cathode.

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.

Note that the total current is 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.

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.

The work function, in eV, of the material of the cathode is then required.

The user finishes by giving the thermal energy kT of the source electrons, in eV. It can be zero. In CPO3DS the rays will be given randomised thermal energy components in the transverse directions and randomised longitudinal components in the 'low temperature' limit, as given for field emission by R H Good and E W Muller, Handbuch der Physik, Vol 21, 176 (1956) and for both types of emission by P W Hawkes and E Kasper, Principles of Electron Optics (Academic Press, 1989).

The randomisation can be the same each time (that is, a repeatable randomisation, always with the same sequence), or can be a different, uncontrolled randomisation each time the program is run (see note on randomisation). 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)

For field emission the velocity components are non-zero even if kT is zero and are randomised . The energy spread is characterised by the acceleration energy 'd' gained over a distance lambda/(4*pi), where lambda is the de Broglie wavelength corresponding to the energy eW and W is the work function (see Hawkes and Kasper). In general d is an order of magnitude larger than kT at room temperature.

An 'artificial' option is also available (in CPO3DS) in which the electrons start with fixed energy (chosen by the user), normal to the surface.

If the effects of space charge in the beam are important at a focus of the beam then a space-charge iteration should be carried out. See the footnotes of file test2d13.

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.

Nano-cathodes (that is, very small sources): see note on setting up initial ray conditions for field emission.

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

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

For crashes and their solutions, see field emission crashes.

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 test2d13.dat, are:

field emission cathode

10 4.5 number of cathode segments, work function (eV)

i 0. 0. 'i' for 'inside' reference point, and (r,z) of point

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 test3d12.dat, are:

field emission cathode

22 4.5 number of segments, work function (eV)

i 0. 0. 0. 'i' for 'inside, and xyz of reference point

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

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 first line must start with 'fie', for 'field-emission cathode', or 'ext' for 'extended Schottky emission'.

The next line must contain 2 numbers, an integer giving the number N of segments that form the cathode and the work function, in eV, of the material of the cathode.

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.

The final line should contain the thermal energy kT of the source electrons, in eV.

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, see note on randomisation), 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. 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 ānā (but note that this is non-physical).

If the effects of space charge in the beam are important at a focus of the beam then the next line would start with 'ite' (for 'iteration', see below). Otherwise the next line should start with 'n', and the final line should be as defined elsewhere.