The charge inside each cell/tube usually represents a large number of electrons. The force on any one electron is of course the sum of the coulomb forces due to the discrete charges of all the other electrons, but this is effectively the same as the force due to the continuous charges in the cells/tubes. The force causes the beam to spread, which is usually called 'global' or 'collective' space-charge spreading. The programs CPO2DS and CPO3DS deal with this type of spreading.

To put this in another way, each CPO ray is not a single particle but is usually a series of many particles. For example if there are 100 rays in a beam of 1 microAmp then each ray represents 10^-8 A, which is 6.10^12 particles per second. If the particles are electrons with an average energy of 100eV, velocity 6.10^6 m/s, flight path 0.1m, then the flight time is 1.6.10^-7 s and there are approximately 10^6 electrons along the length of the ray. The number of electrons inside each cell/tube is therefore significantly greater than 1 for each ray. We have to imagine that all 100 rays exist at the same time, each consisting of a train of electrons. So in this example the space-charge spreading is certainly 'global'.

Of course there can be a difference between the sum of the discrete forces and the continuous force. For example 2 electrons might come close to each other and scatter through a large angle. These individual electron-electron interactions occur randomly and are usually known as 'stochastic' interactions. These interactions can cause changes to the energies and directions of the individual electrons and so can be important in some situations, such as microscopy and lithography.

There is a special 'stochastic' version of CPO that deals accurately with stochastic interactions. We have published 2 papers that illustrate the accuracy of that version and also describe the operation of the program:

Monte-carlo calculation of Boersch energy spreading, by F. H. Read and N. J. Bowring, Nucl. Instrum. Meth. A519 (2004) 196-204.

The contributions of stochastic coulomb interactions and collective space-charge field aberrations to spatial spreading in charge particle projection systems, by F. H. Read and N. J. Bowring, Microelectronic Engineering 73-74 (2004) 97-105.

At the extreme limit of very small currents the number of particles per cell/tube will be much smaller that one and then the 'global' approach is not appropriate. But at this limit the global and stochastic spreadings are negligible.

There is an interesting intermediate regime but it seems that this has not yet been treated carefully in the literature. This regime is the subject of the example file xmpl3d74.dat, which is not yet finished.