Institute of Physics Publishing Plasma Sources Science and Technology Plasma Sources Sci. Technol. 14 (2005) 751-756 doi:10.1088/0963-0252/14/4/014 Theoretical study of pulsed microwave discharge in nitrogen Z Bonaventura, D Trunec, M Meško, P Vašina and V Kudrle Department of Physical Electronics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic E-mail: zbona@physics.muni.cz Received 7 March 2005 Published 18 October 2005 Online at stacks.iop.org/PSST/14/751 Abstract A pulsed microwave discharge burning in pure nitrogen was studied theoretically. The time-dependent Boltzmann equation for electrons was solved numerically in multi-term approximation. It was assumed that the discharge was ignited by a lOOkW microwave (/ = 9.4 GHz) pulses with 2.5 /xs duration; the repetition frequency was 400 Hz. It was shown that the electron distribution function approaches very quickly the steady state distribution function after a change of the amplitude of electric field intensity. The steady state time averaged values of electron mean energy, diffusion and rate coefficients and drift velocity were calculated for different values of electric field intensity. With these values the actual values of electric field intensity from a previous experiment were determined from the measured time dependence of electron concentration. The calculated values were compared with previous experimental results. 1. Introduction Microwave excited discharges are often used because of the lack of electrodes and the comparatively high density of charge carriers and radicals. Pulsed discharges have additional parameters, i.e. pulse duration and repetition frequency, which enable us to control the parameters of produced plasma. Recently the pulsed microwave discharges in nitrogen were studied experimentally by Baeva et al [1] and Mesko et al [2]. Baeva et al [3] also presented a theoretical study concerning their experimental results. They solved the Boltzmann equation for electrons in the effective field approximation (EFA) coupled with the master equations for vibrational levels of nitrogen. Furthermore, they used a two-term approximation in the expansion of the distribution function in Legendre polynomials. Other references on modelling of discharges in nitrogen can be found in [3]. The main aim of this paper is to complete and interpret the experimental data in our previous work concerning pulsed microwave discharges in nitrogen [2]. The plasma in this experiment was produced in a spherical glass vessel with the inner diameter, R, of 50 cm. From one side the vessel was irradiated by a horn antenna, and microwaves were focused by the dielectric lens in the centre of the vessel. An aluminium foil was placed on the opposite side wall in order to reflect and focus the microwaves. Microwaves (x-band) were generated by pulse magnetron (peak power reaches lOOkW). The pulses had a 2.5 /xs duration and a repetition frequency 400 Hz. The pulses had a rectangular shape with the rising and falling time of the edges of 10 ns. The time dependence of electron concentration was measured by a microwave method, which was already successfully used for electron concentration measurement in afterglow plasmas [4]. The time resolved optical emission spectroscopy and transmitted power measurement were also performed. The electron kinetics in the discharge was described by the Boltzmann equation for spatially homogeneous plasma. Because the microwave power was supplied to the discharge in short pulses, we considered first the fully time-dependent Boltzmann equation. A new simple numerical technique to solve the time-dependent Boltzmann equation was used. However, it was found that the distribution function relaxes very quickly to the steady state distribution function after a change and that the changes of macroscopic parameters during one period of the microwave power are small under our conditions. This allowed us to use time averaged values of collision frequencies, mean energy, etc for further calculations of electric field intensity. The results of the calculation were compared with the experimental data. 0963-0252/05/040751+06$30.00 © 2005 IOP Publishing Ltd Printed in the UK 751 Z Bonaventura et al 2. Boltzmann equation The time-dependent distribution function of electrons in the discharge was calculated using the spatially homogenous Boltzmann equation: eo dt E dv (1) where F(v,t) is the distribution function of electrons and me and —eo are the mass and the charge of electron. Cel and C™ are elastic and inelastic collision integrals for the collisions of electrons with neutral gas particles (mass M and density A'). The gas particles are assumed to be at rest. For simplicity the elastic and all inelastic collisions were assumed to be isotropic. The electric field E was chosen to be parallel to the z-axis, i.e. E = E(jSmcotez. In this case the distribution function has reduced velocity dependence F(v, t) = F(v, vz/v, t), and it can be expanded in Legendre polynomials Pn(vz/v) and approximated by an arbitrary number I of terms: F(V, Vz/V, t) = J2 Fn(V> t)Pn(vz/v). (2) Substituting this expansion to the Boltzmann equation we obtain the hierarchy of partial differential equations: 1/2 a — fo ~ — dt 3 \meU ou U~ll2W (f/3/2^£vd(f/)/o) + I>^)/o E u 1/2 v™{U + U™)MU + U™)=0, (3) dtfn n ( 2 V?2 --- — e0E(t) In — 1 \ me I Ull2 — fn-i dU 1 1 2n + 3 1/2 2 e0E(t) 2 , n 9 n + U1/2-fn+i +- dU 2 u-l'2fn+l (v*l(U) + YJ<(U)Sjfn=0, 1 < n < I - 2 (4) -f -- dtJl 1 21 1 / 2 \1/2 — — e0E(t) -3 Vffio / U ' -fl-2 dU I -u-^2fl for the expansion coefficients fn{U, t)=2it 3/2 F„(v(U),t). (5) (6) Here U = mev2/2 is the electron energy and U™ is the energy loss in the with inelastic process. In deriving this hierarchy a further expansion of each collision integral with respect to the ratio me/M of the electron-to-gas particle masses has been performed; only a leading term has been taken into account. The quantities, / 2 \ 1/2 vel(U) ={ — ) U1/2NQm(U), \meJ / 2 \1/2 (7) (8) are the collision frequencies for elastic and inelastic collisions, where Qm and Q™ denote the cross sections for momentum transfer and with inelastic collision. The cross section for electron-nitrogen collision were taken from [5]. Usually only the first two terms are taken into account in expansion (2). This approach to nitrogen can lead to inaccurate results because there are large inelastic cross sections at low electron energies [6]. Also the time-dependent problems need to consider more terms in order to obtain a correct solution [7]. In our calculation 150 terms in expansion (2) were taken. Review papers on theoretical studies of electron behaviour in different electric and magnetic fields can be found in [8-10]. 3. Numerical procedure The system of equations (3)-(5) given above was solved as an initial-boundary value problem. The boundary conditions are given by relations: fn(U = 0,t) = 0 n^l, (9) f„(U ^Uoo,t)=0 n^l, where condition is sufficiently large energy, together with the dU :0. A detailed derivation of these boundary conditions can be found in [11]. A new numerical method for the solution of time-dependent Boltzmann equation was used as follows. The system of equations (3)-(5) was discretized in energy space on a uniform mesh £/,-: Ui = (i-l)AU i = l,...,k AU = — (10) k with the density of ten points per one electronvolt. The equations were solved for the values of Eq from 0.25 kV cm-1 to 2.0kVcm_1; Uoa was then set from 25 to 300 eV, and the number of mesh point was in the range 250-3000. A second order central finite difference formula was employed for the discretization approximating dfn(Ui)/dU by (/„,,+! — /n,,-_i)/(2AlO, where fnJ = /„(!/;). Thus the system of I partial differential equations was transformed into a system of kl ordinary differential equations. This system of ordinary differential equations for the values of /„ in the mesh points was solved by an explicit Runge-Kutta method of order 5(4) (DOPRI5 procedure [12]). This method for solution of time-dependent parabolic partial differential equations is a standard 752 Pulsed microwave discharge in nitrogen method known as the method of lines (MOL). For a detailed introduction to the MOL see, e.g. [13]. The MOL is easy to implement in comparison with the method from [11] (for an overview of different implementations of MOL see [14]). The used MOL also does not exhibit any artificial oscillations at the same testing examples as the method from [11], which additionally uses an iterative treatment. The estimated error of the solution by our method was less than 0.1%. The described method was already successfully used for the study of electron relaxation in afterglow plasma [15]. However, the calculation of the full time-dependent solution of the equations given above for our conditions is not possible even with current computers. Assuming ionization to be a conservative process, fortunately the electron distribution function approaches the steady state distribution function very quickly after a change of iio, as is shown below. So for a given iio the calculation was performed only until the steady state was reached. Because the changes in the rate coefficients during one period of the microwave power were less than 10%, for further calculation the rate coefficients were time averaged over one period. The values of time averaged rate coefficients as functions of iio were the results of these calculations. These rate coefficients were then used for the calculation of electron and excited nitrogen concentrations from the rate equations. A similar approach was used by Baeva et al [3]. The electron concentration can then be calculated from the equation: dne(t) dt : ne(N(ki) ~ = (D)/A2, (D) is the time averaged diffusion coefficient and A = R/n is the diffusion length. The losses of the electrons due to recombination can be neglected for the short times after the beginning of the microwave pulse. The electric field Z?b corresponding to the breakdown can be obtained from the relation N(ki)(Eh) = (vd)(Eh). (12) In the case of a pulsed discharge with a small repetition frequency (or a single pulse), breakdown occurs when the pulse duration tp is sufficiently long so that the electron density equals the cut-off density before the end of the pulse [16]. The breakdown electric field is then obtained by solving the following equation: N{ki)(Eb) - Figure 1. The time dependence of the mean electron energy and the ionization rate coefficient after the beginning of the microwave pulse. The nitrogen pressure was 100 Pa and the amplitude of electric field intensity Eq was 1 kV cm-1. > 1.2 0.8 0.6 0.4 0.2 \ 1 ! 11 1 i » 1 W ■ 1 fill! W 1 A ■' ill/ \aJ '■>•*■' ' " ■Mi 1 J \ I Jf^ri v.' v W w 2t ---------- j W 4t .................... . 20t - * MC 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time (ns) 0.8 Figure 2. The time dependence of the mean electron energy after the beginning of the microwave pulse for different numbers of terms / in the distribution function expansion. The nitrogen pressure was 100 Pa and the amplitude of electric field intensity Eq was lkVcm-1. Dot-dashed line: / = 1, dotted line: / = 3, full line: / = 19. Full circles represent the results of Monte Carlo simulation. The time course of the electric field intensity is shown in the upper part of this figure. 4. Results and discussion The time dependence of the mean electron energy and the ionization rate coefficient is shown in figure 1. The electric field E = Eq sin cotez was switched on at t = 0, and the initial distribution function was chosen to be isotropic Maxwellian with a mean electron temperature of 300 K. As can be seen from figure 1 the mean electron energy reaches the steady state after ~20 ns. The time evolution of the electron distribution function together with a solution of electron particle balance was studied by Winkler and Wuttke in [ 17] for pulsed discharge in a mixture of Ne/Xe/HCl. They also came to the conclusion that the electron distribution function follows the changes of electric field intensity within tens of nanoseconds. The influence of the number of terms I in the expansion (2) on the accuracy of the calculated electron mean energy is demonstrated in figure 2. For a comparison, the values of the electron mean energy were obtained from a Monte Carlo simulation with 20 million electrons, and these values are 753 Z Bonaventura et al (a) 10 8 6 (eV) 4 2 0 0.3 0.25 0.2 0.15 0.1 > 0.05 3- 0 0J 1 0 1 1 1 vz/v ^ 1 1 1 1 1 Figure 3. The electron distribution function in steady state for different electric field phase: full line: / = 7 and dashed line: 1 = 1. The nitrogen pressure was 100 Pa and the amplitude of electric field intensity Eq was 1 kV cm-1. (a) Maximum of electric field and (b) zero electric field. represented by full circles in figure 2. It can be seen from this figure that the use of the usual two-term approximation leads to large errors in the calculated electron mean energy, and these errors increase with time. However, the values of the electron mean energy obtained from the solution of Boltzmann equation with I = 19 agree very well with the Monte Carlo values. Also in [18] it was found that the multi-term approach is necessary for the solution of the time-dependent electron Boltzmann equation in rf fields. The higher Eq is, the higher the I to be chosen. Therefore the large value I = 150 was chosen for all the calculations. The electron distribution function in steady state for different electric field phase is shown in figure 3. In this figure the function Ul^2F(v(JJ), vz/v, t) is plotted because it has a meaning of distribution function for U and vz/v. An eight-term approximation is sufficient for the steady state; however, a two-term approximation is not sufficient and it exhibits artificial oscillations, see figure 3. For the steady states at different values of Eq the macroscopic parameters (mean energy, rate coefficients, etc) were calculated, and their values were averaged during one period of microwave power. The averaged values of ionization rate coefficient together with the averaged values of mean energy are plotted in figure 4; there are also the results obtained by the conventional EFA [19]. It can be seen that this approximation exhibits large discrepancies (30% for ionization rate coefficient) primarily for higher values of Eq in comparison with the 'accurate' multi-term approximation. The averaged values of collision frequency and diffusion 1 1.5 E0 (kV/cm) Figure 4. The dependence of the steady state time averaged mean electron energy and ionization rate coefficient on the amplitude of electric field intensity in microwave discharge at nitrogen pressure 100 Pa. Mean electron energy: circles; ionization rate coefficient: triangles. Crosses and stars represent the results obtained by (two-term) EFA. E o O E0 (kV/cm) Figure 5. The dependence of the steady state time averaged collision frequency and diffusion coefficient on the amplitude of electric field intensity in microwave discharge at nitrogen pressure 100 Pa. Collision frequency: circles; diffusion coefficient: triangles. 1.4 r 1.2 ■ ' —> 1 ■ CT> 'o 0.8 ■ 0.6 ■ LU > 0.4 ■ 0.2 ■ 0 ■ 1 1.5 E0 (kV/cm) Figure 6. The dependence of the steady state time averaged rate coefficient for the excitation to C state and (ev,iE) on the amplitude of electric field intensity in microwave discharge at nitrogen pressure 100Pa. Rate coefficient: triangles; (ev,iE): circles. coefficient are shown in figure 5 and the averaged values of excitation rate coefficient for C state and ev&E are shown in figure 6; all these quantities are shown as the functions of the amplitude of electric field intensity Eq. For the calculation of 754 Pulsed microwave discharge in nitrogen 0.4 0.6 0.8 time (lis) Figure 7. The comparison of measured and calculated electric field intensity. «e is the measured electron concentration from [2]. macroscopic quantities (mean energy, absorbed power, etc) at the real conditions of the experiment, it is necessary to know the corresponding value of Eq. Unfortunately, this value is not usually known. In our previous experiment [2] the time dependence of electron concentration was measured. So we decided to calculate the amplitude of electric field intensity just from the time dependence of electron concentration using equation (11). The measured time dependence of the electron concentration was differentiated, and the electric field intensity was then determined from equation (11) using calculated values of (k\) and (v^). The results are shown in figure 7. The breakdown field calculated for breakdown time t\> = 0.2 /xs from (13) is 1.67 kV cm-1. The time dependence of the calculated electric field from figure 7 has the maximum value of 1.25 kV cm-1, and it is also shifted to longer times in comparison with measured electric field. In the experiment the discharge was ignited in the centre of the vessel, where the highest electric field intensity was, and then the plasma expanded. Finally the plasma ball was created. The microwave power was absorbed in the plasma, alternatively it could be reflected. The electric field was measured by an antenna behind the plasma ball, where the microwaves were dumped by the plasma. Also the measured electron concentration represents the average value of electron concentration across the plasma layer between the measuring antennae. Therefore the decrease of calculated electric field has to be shifted to longer time in comparison with the time development of the measured electric field. The plasma in the experiment had a radius of approximately 7.5 cm and the average electron concentration during the pulse was 1012 cm-3. Then the power absorbed in the plasma at the electric field intensity 0.8 kV cm-1 calculated using the values from figure 6 is about lOOkW. The intensity of SPS of nitrogen was also calculated from the electric field intensity and electron concentration data in figure 7. However, in this case the peak (the so-called overshoot [20]) in the intensity of SPS was not observed. So we tried to calculate the intensity of SPS for an artificially chosen time dependence of electric field intensity. The initial value of Eq of this time dependence was roughly chosen from the experiment, and the decrease of the electric field was shifted to longer times so that the calculated values approximately correspond to the measured data. However, we did not try to fit them completely. The results of this calculation are shown /Y\nc I A ne J ^ \-~~ 1.5 E > 0.5 0.2 0.4 0.6 time (|xs) 0.8 Figure 8. The time dependence of the intensity of SPS of nitrogen, (it is directly proportional to ric) and electron concentration. The time dependence of E0 was chosen as initial data for the calculation. The time dependence of intensity of nitrogen SPS measured in experiment [2] is plotted by crosses. The maximum value of measured intensity was set to the maximum value of Hern figure 8. The C state and the electron concentrations were calculated using equations (11) and (14); the initial electron concentration was chosen as 109cm-3. The maximum of the calculated intensity of SPS is shifted by 25 ns to longer times in comparison with the maximum of measured intensity. This fact and the more gradual increase of measured intensity indicate that the initial electric field intensity was higher than was chosen. 5. Conclusion The pulsed microwave discharge burning in nitrogen under the conditions described in [2] was studied theoretically. The electron distribution function was obtained by the numerical solution of time-dependent Boltzmann equation in multi-term approximation. It was shown that the macroscopic quantities (mean energy, collision frequencies, etc) do not change significantly during one period of microwave power, and therefore their time averaged values can be used for further calculation. It was also shown that the electron distribution function relaxes very quickly to its steady state, at which the electron distribution function does not depend on the previous history and depends only on the electric field amplitude Eo for a given nitrogen pressure. So the time averaged rate coefficients were calculated as the functions of Eo only. Then the continuity equation for electrons could be solved, and for the electron concentration obtained from the experiment the corresponding electric field amplitude was determined. The breakdown electric field amplitude was also calculated. The obtained electric field amplitudes are in qualitative agreement with experimental values. Finally, the calculated intensity of nitrogen SPS shows an analogous time course to the measured intensity. Acknowledgment We are grateful to the Czech Science Foundation for financial support under contracts No 202/03/0827 and 202/03/H162. 755 Z Bonaventura et al References [1] Baeva M, Luo X, Pfelzer B, Schäfer J H, Uhlenbusch J and Zhang Z 1999 Plasma Sources Sei. Technol. 8 142 [2] Meško M, Bonaventura Z, Vašina P, Kudrle V, Tálský A, Trunec D, Frgala Z and Janča J Plasma Sources Sei. Technol. in press [3] Baeva M, Luo X, Pfelzer B and Uhlenbusch J 1999 Plasma Sources Sei. 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