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Figure from:Modelling of a dipolar microwave plasma sustained by electron cyclotron resonance

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Figure 1. Model geometry of a dipolar plasma source antenna. Metal parts are in white. The left boundary (x = 0) represents the metal chamber wall, the other boundaries are open. The actual simulation domain continues beyond the figure until x = 0.24 m and r = 0.12m. The permanent magnetic field lines are shown by thin lines, the ECR surface by a thick line and the magnetic field strength in grey scale with logarithmic intervals.

Figure 1 Model geometry of a dipolar plasma source antenna. Metal parts are in white. The left boundary (x = 0) represents the metal chamber wall, the other boundaries are open. The actual simulation domain continues beyond the figure until x = 0.24 m and r = 0.12m. The permanent magnetic field lines are shown by thin lines, the ECR surface by a thick line and the magnetic field strength in grey scale with logarithmic intervals.

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Table 1. Reactions taken into account in the model. the net power density lost by the electrons in collisions. In these equations it is assumed that the electron pressure is isotropic and that the electron energy distribution is Maxwellian. We have used same equations previously to model magnetized electrons in Hall-effect thrusters [31] under conditions similar to the present ECR-plasma conditions, for which they give approximate but very reasonable results. an alternative approach to include non-local effects in a fluid model by an effective plasma permittivity and conductivity deduced from the dispersion relation for right-hand polarized waves.
Table 1. Reactions taken into account in the model. the net power density lost by the electrons in collisions. In these equations it is assumed that the electron pressure is isotropic and that the electron energy distribution is Maxwellian. We have used same equations previously to model magnetized electrons in Hall-effect thrusters [31] under conditions similar to the present ECR-plasma conditions, for which they give approximate but very reasonable results. an alternative approach to include non-local effects in a fluid model by an effective plasma permittivity and conductivity deduced from the dispersion relation for right-hand polarized waves.
For electrons we combine the continuity equation (7) with the following momentum equation, energy equation and closure relation for the heat flux: where 7, is the electron temperature in eV, Q,. is the electron heat flux vector, ve is the momentum transfer frequency for collisions with the background gas, P is the absorbed microwave power per electron given by equation (5), a is a normalization factor to fix the total absorbed power and I] is
For electrons we combine the continuity equation (7) with the following momentum equation, energy equation and closure relation for the heat flux: where 7, is the electron temperature in eV, Q,. is the electron heat flux vector, ve is the momentum transfer frequency for collisions with the background gas, P is the absorbed microwave power per electron given by equation (5), a is a normalization factor to fix the total absorbed power and I] is
Figure 2. Two-dimensional (axial—radial) grid used in the simulations, with the arrangement of the different electromagnetic field components for the FDTD method.
Figure 2. Two-dimensional (axial—radial) grid used in the simulations, with the arrangement of the different electromagnetic field components for the FDTD method.
and similar for the other components. The electron mean velocity is advanced in time from the central difference discretization of the momentum equation (4)
and similar for the other components. The electron mean velocity is advanced in time from the central difference discretization of the momentum equation (4)
For electrons we solve the momentum equation (11) in drift— diffusion form. Due to the tensor nature of the magnetized mobility, the axial and radial components of the electron flux contain cross terms involving gradients in the transverse direction. Using standard finite difference discretization, we define the axial gradients (and the axial vector components) at points (i + 1/2, 7) and radial gradients at different points (i, 7 + 1/2); hence the cross terms require averaging of 4 transverse gradient points, e.g. for the axial electron flux: where 2 = eB/me.v.. Substitution of the discretized electron flux in the continuity equation (discretized by the control volume method) then yields a system of linear nine- point equations for the electron density, which we solve by the MSI procedure. Although this discretization scheme can produce large numerical errors in the electron transport across the magnetic field, it turns out to work rather well for the ambipolar plasma diffusion in the present model. The electrons diffuse predominantly along the magnetic field lines because these are connected with the conducting chamber walls (the current being closed in the walls), so that errors in the perpendicular transport are of minor importance as long as they stay negligible with respect to the parallel transport. This is essentially different from cases where an electric field is applied across the magnetic field (magnetrons, Hall effect thrusters, etc) and perpendicular electron transport controls the discharge operation; the above scheme is then not appropriate [20].
For electrons we solve the momentum equation (11) in drift— diffusion form. Due to the tensor nature of the magnetized mobility, the axial and radial components of the electron flux contain cross terms involving gradients in the transverse direction. Using standard finite difference discretization, we define the axial gradients (and the axial vector components) at points (i + 1/2, 7) and radial gradients at different points (i, 7 + 1/2); hence the cross terms require averaging of 4 transverse gradient points, e.g. for the axial electron flux: where 2 = eB/me.v.. Substitution of the discretized electron flux in the continuity equation (discretized by the control volume method) then yields a system of linear nine- point equations for the electron density, which we solve by the MSI procedure. Although this discretization scheme can produce large numerical errors in the electron transport across the magnetic field, it turns out to work rather well for the ambipolar plasma diffusion in the present model. The electrons diffuse predominantly along the magnetic field lines because these are connected with the conducting chamber walls (the current being closed in the walls), so that errors in the perpendicular transport are of minor importance as long as they stay negligible with respect to the parallel transport. This is essentially different from cases where an electric field is applied across the magnetic field (magnetrons, Hall effect thrusters, etc) and perpendicular electron transport controls the discharge operation; the above scheme is then not appropriate [20].
Figure 3. Steady-state self-consistent model solution for 10 W total absorbed power and | Pa argon pressure. The maximum values are indicated on the graphs. The greyscale in graphs (a)—(d) is in log scale covering two decades (log2d) or four decades (log4d); the other graphs are in linear scale starting from zero.
Figure 3. Steady-state self-consistent model solution for 10 W total absorbed power and | Pa argon pressure. The maximum values are indicated on the graphs. The greyscale in graphs (a)—(d) is in log scale covering two decades (log2d) or four decades (log4d); the other graphs are in linear scale starting from zero.
Figure 4. Spatial profiles of the absorbed power density and the plasma density in different conditions. The greyscale is logarithmic in all eraphs.
Figure 4. Spatial profiles of the absorbed power density and the plasma density in different conditions. The greyscale is logarithmic in all eraphs.