-- Advanced electrostatics for space-charge, cathodes, and high accuracy

Cathode Emissions in CPO

Cathodes emissions are easy to specify in CPO-2DS and CPO-3DS because the Boundary Element Method is ideally suited to cathode and space-charge problems. In CPO, a cathode can be defined as the first N segments of an electrode and as such can have any shape or size. One ray of an appropriate current starts from the centre of each cathode segment. These rays interact by space-charge effects, which can limit the current emitted from the cathode.

CPO supports the following cathode types by version:

Note: Cathodes are only available in the space-charge versions of CPO (CPO-2DS and CPO-3DS).
CPO Version Cathode Types Supported
CPO-2DS thermionic (with zero emission energies) and cold field emission
CPO-3DS thermionic, field emission and extended Schottky emission, all including non-zero temperature.
CPO-3DS/Ca unconventional cathodes + cathodes in CPO-3DS

Figure 1: Spherical diode with very small inner thermionic cathode of radius 0.01 inside a spherical anode of radius 1. The program automatically creates an "inner" cathode region bounded by the cathode surface. Here Child's Law is used, appropriately corrected for the curvature of the cathode. Ray tracing starts at the outer edge of this region. A second 'outer' cathode region is created because the cathode is convex. In the present example the User has chosen the combined depth of the two cathode regions to be 0.05. After 4 iterations the current is reproduced to within 2.4% of the known analytical value.

Thermionic cathodes:

When a thermionic cathode is called for, a finely-divided 'cathode space-charge region' is created by the program in front of the cathode. The appropriate space-charges are deposited in this region, and are limited, if necessary, according to Childs Law (or Langmuir's relationship when kT is non-zero, which is available with the space-charge version CPO-3DS), and these are automatically modified to take account of any curvature that the cathode has.

The starting point of the rays is the outer edge of the cathode space-charge region.

For a convex cathode, such as one in which electrons are emitted outwards from a spherical surface, the program uses an additional refinement, to match the cathode to the space-charge cells or tubes that hold the ray space-charges. In this the cathode region of depth 'd' (which is specified by the User) is divided into two distinct regions. The first of these has the cathode surface as a boundary, and is called the 'inner cathode' region. Here Child's Law (or Langmuir's relationship for non-zero kT) is used, 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. The purpose of the 'outer cathode' region is to bridge the gap between the 'inner cathode' and 'ray' regions. The creation of the outer region allows the study of cathodes that are much smaller than the dimensions of the rest of the system.

Field Emission Cathodes

For field emission the Fowler-Nordheim equation is used calculate the current density as a function of the field at the surface of the cathode.

Schottky Emission Cathodes

For extended Schottky emission, a modification of the Richardson-Dushman equation is used.

Special User-defined Cathode Version:

A special 'User-defined cathode' version is also available. This version of the 2D and 3D space-charge programs allows the User to define the emission properties of new and unconventional cathodes. The User can for example define a thermionic temperature that depends on position, or new forms of cold-field or Schottky emitters, such as insulators.

As in the regular versions, the cathode is subdivided into segments and one ray starts from the centre of each segment. At the start of each ray the program sends information to the User-supplied routine on the positions of the cathode segments and on the strengths and directions of the electric fields at their surfaces. The routine then returns information on the emitted current densities. The rays are traced in the usual way, automatically taking care of any space-charge.

The User-defined cathodes fall into two types, depending on whether the space-charge in front of the cathode is significant or insignificant. In the first type (called type 1 in the databuilder) the User can for example define a thermionic temperature that depends on position. The program will then automatically take care of the space-charge. In the second type (type 2) the User can deal for example with new forms of cold-field or Schottky emitters, such as insulators. The choice of type is made in the databuilder.

The CPO package includes 2D and 3D examples (in C++) that can provide a convenient starting point for further editing by the User. In brief:

(1) For a type 1 cathode the example routine deals with a thermionic cathode which is flat and for which Child's Law is valid.

(2) For a type 2 cathode the example routine deals with conventional cold field emission.

A User-supplied data file could be used to define parameters such as the maximum current density of a thermionic cathode or the work function of a field-emission cathode, although this is not done in the present examples (where the values entered in the databuilder are used instead).

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