# Moderately converging ion and electron flows in two-dimensional diodes

## Abstract

Flow of particles in diodes is solved selfconsistently assuming an approximated system of flow lines, that can be easily represented by an analytic transformation in a complex plane, with assumed uniformity in the third spatial direction. Beam current compression is tunable by an angle parameter {alpha}{sub 0}; transformed coordinate lines are circular arcs, exactly matching to the curved cathode usually considered by rectilinear converging flows. The curvature of flow lines allows to partly balance the transverse effect of space charge. A self-contained discussion of the whole theory is reported, ranging from analytical solution for selfconsistent potential to electrode drawing to precise numerical simulation, which serves as a verification and as an illustration of typical electrode shapes. Motion and Poisson equation are written in a curved flow line system and their approximate consistency is shown to imply an ordinary differential equation for the beam edge potential. Transformations of this equation and their series solutions are given and discussed, showing that beam edge potential has a maximum, so supporting both diode (with {alpha}{sub 0}{approx_equal}{pi}/3) and triode design. Numerical simulations confirm the consistency of these solution. Geometrical details of diode design are discussed: the condition of a zero divergence beam, with the necessarymore »

- Authors:

- INFN-LNL, viale dell'Universita n.2, 35020 Legnaro (Italy)

- Publication Date:

- OSTI Identifier:
- 22093974

- Resource Type:
- Journal Article

- Journal Name:
- Review of Scientific Instruments

- Additional Journal Information:
- Journal Volume: 83; Journal Issue: 11; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0034-6748

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; 46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; ANALYTICAL SOLUTION; APPROXIMATIONS; BEAM CURRENTS; COMPUTERIZED SIMULATION; EXTRACTION; IONS; MAGNETIC FIELDS; NUMERICAL ANALYSIS; PARTICLE BEAMS; POISSON EQUATION; RELATIVISTIC RANGE; TRANSFORMATIONS; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Cavenago, M.
```*Moderately converging ion and electron flows in two-dimensional diodes*. United States: N. p., 2012.
Web. doi:10.1063/1.4766461.

```
Cavenago, M.
```*Moderately converging ion and electron flows in two-dimensional diodes*. United States. doi:10.1063/1.4766461.

```
Cavenago, M. Thu .
"Moderately converging ion and electron flows in two-dimensional diodes". United States. doi:10.1063/1.4766461.
```

```
@article{osti_22093974,
```

title = {Moderately converging ion and electron flows in two-dimensional diodes},

author = {Cavenago, M.},

abstractNote = {Flow of particles in diodes is solved selfconsistently assuming an approximated system of flow lines, that can be easily represented by an analytic transformation in a complex plane, with assumed uniformity in the third spatial direction. Beam current compression is tunable by an angle parameter {alpha}{sub 0}; transformed coordinate lines are circular arcs, exactly matching to the curved cathode usually considered by rectilinear converging flows. The curvature of flow lines allows to partly balance the transverse effect of space charge. A self-contained discussion of the whole theory is reported, ranging from analytical solution for selfconsistent potential to electrode drawing to precise numerical simulation, which serves as a verification and as an illustration of typical electrode shapes. Motion and Poisson equation are written in a curved flow line system and their approximate consistency is shown to imply an ordinary differential equation for the beam edge potential. Transformations of this equation and their series solutions are given and discussed, showing that beam edge potential has a maximum, so supporting both diode (with {alpha}{sub 0}{approx_equal}{pi}/3) and triode design. Numerical simulations confirm the consistency of these solution. Geometrical details of diode design are discussed: the condition of a zero divergence beam, with the necessary anode lens effect included, is written and solved, as a function of beam compression; accurate relations for diode parameters and perveance are given. Weakly relativistic effects including self-magnetic field are finally discussed as a refinement.},

doi = {10.1063/1.4766461},

journal = {Review of Scientific Instruments},

issn = {0034-6748},

number = 11,

volume = 83,

place = {United States},

year = {2012},

month = {11}

}