Space charge kinetic treatment in Langmuir probes with cylindrical geometry

In this paper, an analysis of the space charge build up in the interelectrodic region of a velocity analyzer with cylindrical symmetry is performed using kinetic theory. Thus the present treatment includes temperature effects. Azimuth symmetry is also assumed. A detailed and comparative analysis, between planar and cylindrical electrodes, is carried out, showing the advantages of each kind of symmetries.


INTRODUCTION
Space charge formation is one of the main factors limiting the current obtained in the collector grid of a velocity analyzer, thermoionic diodes and other engineering devices (Langmuir, 1913(Langmuir, , 1923;;Langmuir and Blodgett, 1923;Page and Adams Jr., 1958;Braun et al., 1973;Martin and Donoso, 1989;Varney, 1982;Wheeler, 1980).This problem was first treated for plane electrodes (two grids) using fluid equations by Langmuir andChild (Langmuir, 1913, 1923;Langmuir and Blodgett, 1923), obtaining the so called Langmuir-Child current (Page and Adams Jr., 1958).Treatments using kinetic theory were developed later for planar electrodes (Braun et al., 1973;Martin and Donoso, 1989;Varney, 1982;Wheeler, 1980) and in this way the effect of temperature were also studied.A modified Langmuir-Child equation, including both temperature and relativistic effects, was also derived and studied using kinetic theory (Qian et al., 1994).The cylindrical and spherical probes have also been treated amply, beginning with the pioneer work of Bohm and Massey (1949) but there is no need to go through a long list of publications, because there are very good reviews in this theme, thus referring only to some of them (Parrot et al., 1982;Estes and Martín, 2000).Previous authors followed essentially two lines, one based in orbital theory and the other one by considering two regions in the probe plasma denoted by sheath and presheath.The important parameters in that analysis were the size of the orbits and the limits of each region.A weak point of most of these treatments is the precise limits of the sheath and presheath, which are not well defined.Another point is that they also usually assume that the amount of particles with "trapped orbits" is zero, which is not clearly justified for cylindrical probes.The treatment we are now presenting is very ample, and although kinetic theory is used, our results are rather simple and easy to calculate.The procedure here followed is an extension of the technique used by Martin and Donoso (1989) for plane electrodes.
The case of cylindrical and spherical electrodes has been also considered in recent works (Bohm and Massey, 1949;Parrot et al., 1982;Estes and Martín, 2000;Estes and Martín, 2000).Treatments have been also developed for cylindrical and spherical probes and other collectors in collision less plasmas, in the limit where the ratio of Debye length to probe radius vanishes (Estes and Martín, 2000).Here we will analyze in detail the space-charge formation in velocity analyzers with cylindrical grids.Numerical integration of the corresponding differential equation will be performed.
Here the accuracy of the approximations we have found is not so satisfactory as in the case of plane electrodes (Martin and Donoso, 1989;Estes and Martín, 2000).Thus in order to find reliable results, it is better to use direct computer calculations of the second order nonlinear differential equations coming from the Poisson equation, once the right distribution functions have been introduced.In this way, our analysis includes temperature effects.The build-up of interelectrodic space charge is discussed for velocity analyzers with plane and cylindrical symmetries.The results of each geometry will be compared.

THEORETICAL ANALYSYS
The velocity analyzers with cylindrical electrodes here considered are shown in Figure 1.The discriminating grid G2 is a hollow tube of radius .The entrance grid G1 (radius ) is biased to a negative potential or allowed to float with zero current.In this way most of the electrons are repelled by this grid.Therefore in the region, between the entrance (G1) and discriminating (G2) grids, there are not electrons, but only ions.This is the region of interest in this paper.We assume that all the ions that go through the discriminating grid are collected by the central collector, which is set up to a negative potential.The grid G2, of radius c, is biased to a positive potential in order to repel some of the ions, and this is called discriminating grid.Here , ( ) and , ( ) are the radius of the entrance and discriminating grids.The potentials and of and are given and we look for the interelectrodic potential between both grids as it is shown in the lower part of the figure .Here, c is choses as the unit length, and this will also be the normalization unit for the draws in velocity analyzer with cylindrical electrodes.
The theoretical analysis is in somewhat similar to the case of plane electrodes, (Martin and Donoso, 1989).

Valdeblanquez and Martin 61
However now the radial distance  replaces the distance x, and the operator 2  has to be written in the corresponding cylindrical coordinates.Looking in detail the case of cylindrical symmetry, the distribution function for the coordinates  (radial),  (angular) and z (along the axis), will be Here the temperature is given in electron volts and the radial symmetry is considered, thus  and z does not appear in the distribution function.The end effects have been also neglected.In this analysis the radial velocity is the important one, and the integration in  and z can be carried out straightforward from - to +  giving: The ions with radial kinetic energy larger than qVp (Vp maximum potential at p    ) will go through the maximum potential reaching the discriminating grid, and then they will collected by the collector grid.The ions with radial kinetic energy lower than qVp will be reflected.In this work, the entrance velocity are considered positive, this allows a simple comparison with planar electrodes.Therefore the radial velocity will be considered positive when they go toward the axial of the cylinder and negative in the other way.Now as it was explained in Equation 2 of Martin and Donoso (1989); the distribution function in the interval (c, p), can be written as:

Velocity analyzer with cylindrical electrodes
In the interelectrodic region between the maximum potential VP and the discriminating grid at =c, there is no reflected particles, because we assume that all the ions arriving to the discriminating grid are collected by the collector.Therefore, the distribution function will be: For  = p, ( ) and both distribution functions are coincident.Thus the continuity of the current is assured.Using now

RESULTS
The integration of Equation 18 has been carried out using fourth order Runge-Kutta algorithm.The analysis is simplified, if we give the values of  p and p  , and the potentials V 0 and V R , corresponding to  0 and  R , are determined from  p and p  .
In Figures 2 and 3, the interelectrodic potential for cylindrical and plane velocity analyzers are shown.The procedure here found more convenient, is to give values to the maximum potential as well as the position and to determine all the other values from these quantities.In Figure 2, the same values than in Figure 3 are used for p  and the position of the maximum, but the analysis is performed for electrodes with cylindrical symmetry, instead of a plane one.The scale factor potential in Figure 2 is double than in Figure 3.For for cylindrical symmetries.One advantage of cylindrical electrodes is that the current can be increased easily using longer electrodes.In Figure 3, we show the case of planar electrodes for the corresponding values of  p = 0.75 and (0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4).Now the interelectrodic distance is measured from the discriminating electrode instead of the central axis.Thus this interelectrodic distance goes from zero to four, instead of one to five as in previous case, and the values are one unit smaller.The same phenomena, that happened when we move from cylindrical to planar electrodes, happens now, that is, for  R = 0.75, the absolute charge effect begins when a  = -1.42 for planar electrodes.This value is smaller than the absolute value of a  ( a  = -3.8) in the case of cylindrical electrodes.Therefore using planar electrodes there is a better repelling of electrons.The advantage of using cylindrical electrodes could be in the facility of increasing the collector current by increasing the length of the cylinder.Furthermore the plasma disturbing, because of introduction of a probe, could be less important in this case.We want to point out that the distances in Figures 2  and 3, cannot be compared well, because the unit in Figure 3, D  , instead of the radius c of the discriminating grid in Figure 2. When p  changes, there is also changes in the pattern of the characteristics curves.In Figure 4, we have

Conclusion
In this paper space charge effects in velocity analyzers for cylindrical geometries, using kinetic theory, and therefore including temperature effects, have been analyzed.Our analysis shows that the repelling of the electrons is more effective for planar electrodes than for cylindrical geometries.This kind of velocity analyzers seems appropriated to be installed in space ships to characterize the outside plasmas.The advantage of cylindrical grids with respect to planar ones could be in the facility to collect larger currents with a less plasma disturbance.The equations here presented include temperature effects, since kinetic theory is used to determine them.However, no simple equation has been  found for the electric current, for instance, similar to Equation 29 in Martin and Donoso (1989) for planar electrodes, generalizing the Langmuir Child current.Here the problem is more complicated, since no first integration of Poisson equation can be performed, as it was done in Martin and Donoso (1989).
is -3.8 volts (/c=5) in Figure2, compared with -1.4 volts in Figure3.In the case of planar electrodes, the repelling of electrons is more efficient than in the case of cylindrical electrodes.For p =0.75,

Figure 2 .
Figure 2. Interelectrodic potentials in a cylindrical velocity analyzer.Numerical calculations were performed keeping the same interlectrodic distance and changing the potentials in the electrodes.In the actual calculations the procedure was to give the value of the maximum 75 .0   p and taking five different values for the position maximum positions   c p / 1,

Figure 3 .Fig. 2 :Figure 4 .
Figure 3. Interelectrodic potentials in a plane velocity analyzer for a maximum potential 75 .0   p Space charge effect in circular cylindrical geometry