Computer Simulation of Technological Electron – Optical Systems
1. Trajectory analysis of the beam formation in electron guns
The progress in electron beam technologies requires further improvements to the design as well as optimization of the electron guns, producing intense beams. In this respect the computer simulation of formation of the beams is a powerful means to analyze and optimize electron-optical systems of the technology electron guns.
In most of computer programs a general algorithm is used (see Figure 57) enabling the potential field, electron trajectories as well as the space charge distribution to be self-consistently obtained. Its basis steps are: (i) Dividing of discrete parts of the appropriate boundary conditions and the space of gun for calculation of electrostatic potential distribution by means of suitable mesh system; (ii)solution of Laplace‘s equation; (iii)calculation of the emission current density applying the law of Child-Langmuir to the virtual elementary diodes in the vicinity of the cathode emitting surface;(iv) calculation of a finite number of electron trajectories through the obtained electric field; (v)allocation of the space charge carried by separated trajectories to the grid nodes; (vi)solution of Poisson‘s equation for the newly determined space charge distribution; (vii)reiteration of above procedure from step (iii) to
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step (vi) until a self consistent solution(stable values of potentials, electrical field and current and position of separated trajectories ) is obtained.
Recently several authors proposed a number of improvements concerning these basic steps. Kasper [26,27]developed a space charge allocation method based on analytic formula for the space charge density and local divergence or convergence of the beam. Kumar and Kasper [28] proposed incorporation of new version of finite-difference method and interpolation procedure for calculation of the electric field in space charge limited electron beam. The thermal velocities of electrons and possible distinct appearance of the potential minimum in front of the cathode (virtual cathode) are also included in their theory. Weber [29], Ninomiya [30] and Monro [31] improved taking into account the thermal velocity effects on beam formation. Van den Broek [32] developed a method in which the cathode current is evaluated using Langmuir‘s law instead of Childs law and the space-charge density is calculated with a fitting technique. All these improvements substantially increase the accuracy and adequacy of the simulations.
Figure 57. Flow chart of beam trajectory and current computer simulation
Information of such numerical experiments and interpretation of the data is performed mainly by analyzing the trajectory (ray)tracing. The adequacy obtained results are determined considerably by choice of region for calculation of the potential distribution and boundary conditions, division of the region of calculations on sub-regions and accepted net steps values.
Mathematically, the trajectory analysis models can be described by the following basic equations:
Poisson's equation governing the electrostatic potential U relatively to the space charge density
in axially-symmetrical beam is given by:
2U
0
, (112)where 2is the Laplace operator in cylindrical coordinate system,
0is the vacuum dielectric permittivity. Due to cylindrical symmetry, the potential need only to be determined in a half plane of the gun electrode configuration (from r=0 to r=rmax).Two types of boundary conditions in addition to (112) render the problem well posed:
Neumann boundary condition along the axis of the region considered (i.e. the radial component of the potential distribution
0
r
U
) and Dirichlet boundary conditionsalong the rest of the boundary (potential in the end points of the mesh is potential of electrodes Uj ; in the gaps between electrodes the potential is assumed to be distributed logarithmically in radial direction and linearly if boundary is chosen parallel to the axis z.
The motion of the electrons in these conditions is given by the Newton equations:
i
directions (i=1,2,3). In (113) are assumed (i) that the beam is non-relativistic (m=Const) and (ii)self-magnetic field of the beam is negligible.
The solution of differential equation (21) after exclusion of t to obtain trajectories of particle motion can be find using standard Runge-Kutta methods. In same cases for the increasing the accuracy of calculations near the cathode and/or electrodes a net with more fine pitch is required.
Instead many thousands electron trajectories (due to the limited computer resources) the calculated tracks are usually restricted to some tens. For that are used virtual big charged
―particles‖ containing the charge of emitted by a cathode segment current (methods of cell or current tube method [26,27] . The current, obeying Child-Langmuir‘s equation is determined
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for every near to flat plane diode in vicinity of a chosen in that way cathode segment. After that is carried out allocation of the space charge transferred by the calculated trajectories to the net nodes. The next step is solution of the Poisson‘s equation for the newly determined space charge distribution. At repeatedly reiteration of above procedure are obtained beam simulation results, describing complex electron gun characteristics, utilizing as an base at experimental improvement of its design.
As one example let we shows two EBW guns (electrostatic parts) with indirect healing L a B6 cathodes and very similar electrode geometry –see Figure 58.
a)
b)
Figure 58. Geometry parameters of two electrostatic partsof EBW guns, The difference is only cylindrical or conical inner wall of the control electrode
The emitter is from La6B tablet. All dimensions of electrodes(emitter, control electrode and anode are the same. The difference is only in shape of control electrode shape-in variant a) this wall is a cylindrical one, as well as in the case b) there are a conical shape.
Results of trajectory analysis of generated beam at various voltages on control electrode M are shown on Figure 59 and Figure 60. The accelerating voltage K-A is 30 keV in all cases.
At comparison of beams shown on Figure 59 and Figure 60a one can understand qualitative character of the trajectory analysis. Every trajectory presented carry different electron current and exact comparison of beams after mixing the trajectories originating from central emitter area and from emitter periphery is impossible. In ref. [33] are calculated statistical values of emittance and brightness at distances z equal to 3,4 and 5 cm from the emitter surface ( for three control voltages : 0,-500 and -1500 V) and definitively the second case of gun (caseFigure 58b) with conical inner wall of control electrode was chosen due to lower emittance values and bigger brightness.
Figure 59. Trajectory analysis of electrostatic part of EBW gun shown on Figure 58 a) at control electrode(M) potential -1,5 kV to the emitter electrode (K)
Figure 60a. Trajectory analysis of electrostatic part of EBW gun shown on Figure 58 a) at same conditions as Figure 59.
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Figure 60b,c. Trajectory analysis of the generated beam in electrostatic part of EBW gun shown on Figure 58 b) at control electrode voltage -0,5 kV and 0 kV
The welding system used at Leybold-AG, Hanau, (now PTR GmbH, Dörnigheim) is a triode electron gun with two focusing coils and a deflection system [34]. The gun itself has a small square directly heated cathode, situated in the circular aperture of a Wehnelt electrode, which is biased negatively with respect to the cathode with voltages of –300 to –3000V. At – 300V a maximum of current is drawn, while at –3000V the electron emission is suppressed completely.
The basic approximation is the assumption that a round cathode in the simulations will give results, which agree well with experimental results obtained with a square shaped cathode. While this has turned out to be true, the explanation may be seen in the 3D interpenetration of electrical fields, which is stronger at the edges of the square cathode, hence reduces there the emission. By this effect the cathode will be ―effectively round‖, which then becomes obvious by the shape of the beam spot on the work piece.
The second problem is that due to the limited mesh resolution the cathode becomes invisible and the results are questionable. The simulation of equipotential lines shown on Figure 61 is used to calculate a field-line (black and dashed) between Wehnelt and Anode in order to cut out the cathode part, using this field line as a slanted and curved Neumann boundary for the simulation in Figure 62.
Figure 61. Calculation of electric field distributions in the famous Steigerwald electron gun, used in former times as EBW gun [34]
Figure 62. Calculation of electron emission in the gun part of Figure 61 with 10 times smaller mesh size, using the curved Neumann boundary, shown in Figure 61 to close the boundary
A more detail explanation of the problem follows. For computer simulations with a finite difference method (FDM) Poisson solver such a gun presents a substantial difficulty, because a cathode of typically 1 mm radius is situated in a anode housing with a radius of 80-100 mm.
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A good simulation of the electron beam, however requires that the mesh size is much less than the cross-over radius of the beam, which is in the order of 1/10 mm. In response to this, about 10 000 meshes will be needed in radial direction. This is impossible, even for to days fastest PCs and only attainable on super computers, not everywhere available. The problem can be solved principally by a non-uniform mesh, best introduced by a logarithmic transformation of the radial coordinate [35]. This procedure, however, needs too many program modifications for well established programs, while developing a new program, which incorporates such a transformation, will require too much development to include all required features of well established programs. For the existing programs of the EGUN family it has been simplest to subdivide the problem by the calculation of a field line in an appropriate position – see Figure 61 – and to use this field line as a slanted and curved Neumann boundary for the calculation of the cathode part of the gun. The field line is written on a file with proper syntax for the direct inclusion into a input file. For the inner part of the whole gun the position and kind of curvature of this Neumann boundary represents all electrostatic influences from the much larger outer part.
From Figure 61 to Figure 62 the mesh size has been reduced by a factor of 10. Only by this the close vicinity of the strip cathode inside the bore of the Wehnelt electrode becomes visible. The trajectory end data from this calculation then can be used to calculate the beam through the lens and deflection system, which will not be performed here. Another point is more important for the optimization of such a gun. This is the reduction of surface fields on those surfaces where electrons could start and be accelerated to full power. A program provides a special tool for this, consisting of a plot of the geometry in connection with a plot of potentials and surface fields shown in Figure 63.
Figure 63. Electric field (full) and potential (dashed) along the boundary with numbers indicating maxima of surface field, synchronized with Figure 61, showing their locations. Dangerous for sparking is maximum No 2, because electrons from there will be accelerated to the anode
0 40 80 120 160 200 240 280 320 360 400 440 480
From an inspection of Figure 61 and Figure 63 it becomes clear at once, that the field maximum No. 2 on top of the Wehnelt electrode of about 160 kV/cm should be reduced by increasing the radius of the electrode curvature there. The field maxima No. 5 and 6 are located at the anode and do not need cure, because no electrons can be accelerated from there.
By removing the anode disk and increasing the radius of curvature at the Wehnelt tip the surface field strength could be reduced to about 60 kV/cm. This improvement has been essential for the continuous welding of aluminum parts over more than 100 hours.
The problems of computer simulation of electron guns with point (hairpin) emitters are typical for analyzing electron beams but also in low power EBW and EB machining guns such emitters are utilized. As example space charge limited emission from the emitting tip of a shaped as ―V‖ tungsten direct heated wire was simulated in [41]. There, after a suitable choice of the suitable calculation mesh the current density emitted from such thermionic emitter surface is iteratively established from the potential distribution near this surface.
Instead conclusion it can be seen that an inherent drawback of the trajectory analysis is its qualitative character. From the representation of the beam as a set of trajectories not a single quantitative characteristic of the beam structure which is of paramount importance in technological applications can be found. As individual trajectories carry different space charge it is difficult to evaluate their contribution to the beam formation as well as to study how the structure of the beam as a whole evolves along its axis.