I. Description of the model
The presented project is directed to solving the problem of the dynamics of plasma boundary in a planar diode with a plasma cathode. The planar diode with the plasma cathode can be presented by simple scheme:
Without plasma the planar diode is merely two parallel metal plates with square S and gap d between the plates, which are placed inside the vacuum chamber. Let us assume that at the moment t=0 we can anyhow produce a plasma at the surface of the cathode plate. For example, we can use a metal grid instead of solid cathode plate and some external plasma source, from which we direct the plasma into the gap through the cathode grid. In such a case the moment t=0 is correspondent to arrival of plasma boundary to the cathode grid.
From this moment plasma begins to expand toward the anode with ion thermal velocity. When a positive voltage pulse is applied to the anode, an electron current appears in the diode because of electron emission from the plasma, which has the cathode potential when its density is large enough. In experiment we can measure all of the parameters of applied voltage pulse U(t), such us the amplitude of the voltage pulse, Ua, the duration of the pulse, tu, the rise/fall time, tr, and the time delay of the pulse application with respect to t=0, t .
The current density in the diode is limited, from one side, by the plasma electron saturation current density, jpl, and from another side, by the space charge limit current density, jcl. The equilibrium of these two currents determinates the motion of plasma boundary. When jpl > jcl, the plasma continues to expand toward the anode; when jpl < jcl, the plasma erosion happens and plasma boundary moves back toward the cathode; the equilibrium between jpl and jcl corresponds to stationary plasma boundary.
Child-Langmuir law determines the limit of electron current density in a vacuum diode as:
Here x(t) is the position of plasma boundary at time t, x(t=0) = d. Bohm law for maximal electron current density that can be extracted from a plasma with known parameters (ne and Te):
Here VTe is the average electron thermal velocity,
In order to simplify the calculation one can assume thermal and electrical equilibrium of the plasma as well as spatial uniformity of the plasma density. From these two equations it is easy to obtain the differential equation for x(t):
Initial conditions are: x(t=0) = d, dx(t)/dt|(t=0) = -VTe.
Electron plasma density, ne, electron plasma temperature, Te, the anode-cathode gap, d, the square of the electrodes, S, the amplitude of the voltage pulse, Ua, the duration of the pulse, tu, the rise/fall time, tr, and the time delay of the pulse application with respect to t=0, t , are the parameters of calculation.
II. Numerical methods
The fifth order Runge-Kutta method, invented by Fehlberg, with adaptive step size, was selected for solving the equation (See Ref. 1). The reasons of the selection were excellent accuracy and good stability under solving a nonlinear equations (see Ref. 2). This method allows to check the accuracy on each step and to change the step size in order to achieve the desired accuracy. Nevertheless, it was needed to change the suggested in Ref. 1 method of step size estimation by simple division/multiplica- tion method because of appearance of instability in step size under some conditions.
III. Program realization
The program was written down on C language under UNIX-V language compilance. The Runge-Kutta method was realized and tested for solution of sets of equations, that allows to add new components (such us circuit equation) into the model, or use the subroutines for solving another problems.
Debugging and testing of the program were done on PC placed in Plasma Lab. The Runge-Kutta method was tested by solving of known sets of equations. For example, the results of solving the set
are presented in Fig 2:
After finishing of project debug the program was replaced to the TX and prepared for demonstration.
PGPLOT graphics calling from C program was used for realizing of user interface and for presentation of obtained results. It needs to be note that PGPLOT is originally FORTRAN graphics library and therefore can be linked with C program only after compilation. It can be done by the use of the script PGPLOTCCL, that provides compilation of the C program by usual C compiler and link with PGPLOT and X11 libraries by f77 compiler.
Another note concerning the syntax of calling of PGPLOT subroutines from C code. It can be done either by adding the underscore ‘_’ character after name of subroutine (like pgbeg_() for calling PGBEG() subroutine) or by adding ‘c’ character before name (like cpgbeg()). At the last case the header file “cpgplot.h” with function prototypes is needed. Both of approaches are supported by TX software.
IV. Obtained results
Temporal behavior of the current in the diode, of the position of the plasma boundary, and of the diode impedance is calculated. Obtained results are presented on screen immediately after ending of calculations. The results are shown in form of standard waveforms, that allows easy comparison with experimental data.
The comparison of the calculated current (yellow line) with current measured in the experiment (red line) is presented in Fig. 3. Experimental data were obtained under experiments with ferroelectric plasma cathodes, which are under investigations now in the Plasma Lab. The waveforms of current (Ia, red line) and voltage (Ua, blue line) were registered by digitizing oscilloscope Tektronix 640 and saved on floppy disk. Calculated current was saved in output file instead of output graph on screen. All data were united in Microsoft Excel. One can see good agreement of calculations with the experiment.