1 MAGNETIZED DIRECT CURRENT MICRODISCHARGE, I: EFFECT OF THE GAS PRESSURE Dmitry Levko and Laxminarayan L. Raja Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas 78712, USA Following the Paschen s law, electrical breakdown of gaps with small pd, where p is the gas pressure and d is the interelectrode gap, requires extremely high voltages. This means that the breakdown voltage for low-pressure microdischarges is of the order of a few kilovolts. This makes impractical the generation of low-pressure dc microdischarges. The application of dc magnetic field confines electrons in the cathode-anode gap. This leads to the significant decrease of the breakdown voltage because each electron experiences many collisions during its diffusion toward the anode. However, as was obtained experimentally, magnetized low-pressure microdischarges experience numerous instabilities whose nature is still not completely understood. In the present paper, we study the influence of magnetic field on the low-pressure microdischarges. We use self-consistent one-dimensional Particle-in-Cell Monte Carlo collisions model which takes into account the electron magnetization while ions remain unmagnetized. We obtain striations in the discharge. We show that these striations appear in both homogeneous and non-homogeneous magnetic field. We find simple expression for the instability growth rate which shows that the instability results from ionization processes.
2 I. Introduction Microdischarges are discharges that the leverage pd of conventional glow discharges to operate on a continuous basis with dimensions of tens to hundreds of micrometres. 1,2 These discharges have applications in electric propulsion, plasma medicine, reconfigurable photonic crystals, plasma etching and film deposition (see Refs. 1-6 and references therein). Atmosphericpressure microdischarges attract attention due to the possibility to generate rather dense (~10 12-10 14 cm -3 ) stable non-equilibrium plasma in the large volume. For instance, such dense plasmas can serve as reconfigurable structural elements for the control of electromagnetic wave propagation. 5,6 Also, these plasmas are chemically reactive and can be used, for instance, for synthesis of nanoparticles 7 and other useful chemicals. 4 In atmospheric-pressure microdischarges both electrons and ions are dominated by collisions. Indeed, the ion mean free path in such microplasmas is ~1 μm which is smaller than the typical sheath thickness. 8,9 This means that ions lose the energy during their propagation toward the electrodes. This complicates the control of ion energy distribution function which is required, for instance, in plasma etching and deposition. 10 In order to make ions collisionless the background gas pressure can be decreased. However, as it follows from the Paschen's law 10 the decrease in gas pressure results in the significant increase of the breakdown voltage. High operating voltages can accelerate ions to energies >1 kev which is undesirable in many applications. For example such energetic ions can heat and even damage the substrate. Therefore, other methods must be developed in order to use the microplasmas for better control of ion energy distribution function. One possible method of microplasma generation at low gas pressures (0.1-10 Torr) was studied in Refs. 11-15. In this method, a dc magnetic field was applied in parallel to the electrodes (gap 1-2 mm). The amplitude of dc magnetic field was 1 T which allows confining
3 the electrons while ions remained unmagnetized. Indeed, the Larmor radius of electrons in such magnetic field is estimated as ~10 μm while the ion Larmor radius is >1 mm. The use of magnetic field allows for high plasma density (~10 13 cm -3 ) at relatively low gas pressure (~0.1 Torr). Such high densities are more typical of atmospheric-pressure microplasmas generated using microwaves or microhollow cathodes. 1,2 The authors 12,14 have also shown that the application of magnetic field significantly decreases the breakdown voltage which means that microdischarges can operate at relatively low powers. These experiments 11-15 have shown the excitation of various instabilities during the operation of low-pressure magnetized microdischarges. In spite of a wide interest in these instabilities, their nature is still not understood well. These instabilities influence significantly the electron transport through the magnetic field. Also, they influence the ion heating in the plasma. In this paper, we study using one-dimensional Particle-in-Cell Monte Carlo Collisions model (1D3V PIC/MCC) the influence of dc magnetic field on the breakdown voltage of microgap. Also, we study the influence of magnetic field on the microdischarge dynamics and parameters. We obtain the excitation of instabilities moving from the anode to the cathode. We analyze the nature of these instabilities. II. Numerical model Here, we use the PIC model detailed in our previous papers. 8,9 This model resolves one spatial coordinate (along electric field) and three components of velocity. Below, we present the key features of this model: 1. Particle weighting on the numerical grid in order to define electron and ion densities. The cathode-anode gap is divided into 1000 equal intervals which allows us to resolve the cathode sheath and waves obtained in our simulations. 2. Solution of the Poisson's equation for given electron and ion densities and potential
4 boundary conditions for the left and right electrodes. 3. Propagation of the electrons and ions into new positions using the Boris solver 16 and check whether the particle leaves the interelectrode gap. If the particle remains in the gap, check the possibility of collision of this particle with other particles using the Monte Carlo algorithm. In the present model, we take into account electron-neutral (elastic, excitation of electronic states and ionization), electron-ion Coulomb and ion-neutral collisions. 4. If ion leaves the cathode-anode gap, check the possibility of the secondary electron emission (SEE) with the coefficient of =0.02. 5. Return to step 1 for the next time step. The time step is defined by the Courant condition Δ Δ /2, where Δ is the space step (1 μm in our simulations), and is the largest electron velocity obtained between electrodes. We assume that 2 /, where is the cathode potential, is the elementary charge, and is the electron mass. The cathode-anode gap is = 1 mm, the working gas is argon, gas pressure varies in the range 1-100 Torr and gas temperature is 300 K. The 1D simulation domain is shown in Fig. 1. dc magnetic field is parallel to the electrodes and decays as the linear function from the left to the right, i.e. 1 / with = 1 T (red line in Fig. 1). These parameters correspond to the parameters of experiments described in Refs. 11-15. FIG. 1. Sketch of the one-dimensional simulation domain used in our studies.
5 III. Results and discussion A. Influence of magnetic field and gas pressure on the breakdown voltage Initially, we seed the interelectrode gap with the quasineutral plasma having the density of 10 7 cm -3. In this paper, we define the breakdown voltage as the voltage for which plasma density increases ~3 orders of magnitude on the reasonable time scale (~100 ns). Such choice is dictated by the Child-Langmuir law 10 which gives that the applied voltage ~500 V is screened by the plasma having the density 10 10 cm -3 on the length scale of 1 mm (interelectrode gap in our studies). FIG. 2. Breakdown voltage as the function of the background gas pressure; =0.02 First, we studied the influence of the magnetic field on the breakdown voltage. We applied the negative dc voltage to the left electrode (cathode). The right electrode (anode) is grounded. Figure 2 shows the comparison between the breakdown (Paschen) curves obtained with and without magnetic field. We see that the magnetic field significantly reduces. For instance, for the gas pressure of 0.7 Torr the application of magnetic field allows obtaining breakdown at ~ 700 V while without magnetic field one has 20 kv. The results shown in Fig. 2 are explained by the influence of magnetic field on the electron energy relaxation length. For ionizing collisions, it is defined as Λ /, where
6 is the electron drift velocity in the direction of electric field, and is the ionization frequency. In zero magnetic field Λ, while in high magnetic field such that the Hall parameter h 1 Λ,. Here is the electron mobility in zero magnetic field and is the electron-neutral momentum transfer collision frequency. The comparison between Λ, and Λ, shows that in zero magnetic field Λ, while in high magnetic field Λ, and does not depend on the gas density. For instance, we get for 0 T Λ, 1.4 cm and for 1 T Λ, 1 μm. FIG. 3. Breakdown voltage as the function of background gas pressure for three values of the secondary electron emission coefficient. Here, it is important to note that through this paper we use the assumption of classical diffusion in magnetic field. This means that the electron diffusion coefficient in magnetic field scales as rather than which is true for anomalous transport.10 The critical magnetic field separating classical and anomalous transport can be estimated if we equalize classical and anomalous diffusion coefficients in the direction perpendicular to the magnetic field:, and,, respectively.10 For instance, for 1 ev electrons moving through the argon gas at 5 Torr we estimate 110 m 2 V -1 s -1. Then, the critical magnetic field is estimated as ~0.1 T. We see that the magnetic field considered in our studies is far above this value.
7 Therefore, more accurate analysis requires anomalous theory. Also, we carried out the simulations for different values of SEE coefficient. Figure 3 shows the breakdown curves obtained for =0.02, 0.06 and 0.1. We see that the increase in results in the decrease of the breakdown voltage. Our simulation results have shown (see below) that the secondary electron emission is crucial for this discharge. Secondary electrons accelerated in the cathode sheath are the main source of plasma. Therefore, the increase in results in the injection of more electrons in the gap which increases the rate of plasma generation. The comparison between the results of our studies shown in Figs. 2 and 3 with the results of experimental studies 14 for 1 mm argon microplasma shows qualitative agreement for the right branch of the Paschen s curve and disagreement for the left branch. Namely, experimental results 14 show weak dependence of on the gas pressure while our results show increasing for decreasing gas pressure. This difference can be explained by the two-dimensional effects such as E B drift or radial dependence of magnetic field. This difference cannot be explained by the influence of magnetic field on because the increase in B results in the decrease of. 17 Thus, increasing B requires higher breakdown voltage than that predicted by the model with fixed. Two-dimensional effects will be studied elsewhere. B. Plasma parameters at B 0 = 1 T and pressure 10 Torr Next, we studied the plasma parameters for various parts of the breakdown curve (Fig. 2). We started from 10 Torr for which we obtained the minimum of (Fig. 2). Figures 4 and 5 show the results of simulations obtained in the steady-state for the gas pressure of 10 Torr ( - 290 V). We see the formation of the non-homogeneous plasma. This plasma is supported by the secondary electrons emitted from the cathode due to ion impact. In our model, electrons are emitted with the velocity ~10 6 m/s. The Larmor radius of these electrons in the vicinity of
8 the cathode is ~ 10 µm and Hall parameter is ~34. Thus, 1, i.e. the plasma electrons are magnetized. The Hall parameter of ions is much smaller than unity, i.e. ions remain unmagnetized at the considered conditions. Moreover, the ion mean free path is comparable with the cathode-anode gap. Therefore, in further analysis we assume that ions are collisionless. FIG. 4. One-dimensional profiles of (a) electron density, and (b) potential and electric field, insert shows zoomed in potential and electric field; B 0 = 1 T, gas pressure is 10 Torr and γ = 0.02. When secondary electrons enter the cathode-anode gap, they experience three drifts in the plasma sheath. These are the drift in field of plasma sheath which is perpendicular to the cathode, E B-drift which results in the velocity 10 1, (1) and drift due to the gradient of magnetic field which results in the velocity 10. (2) Both and are perpendicular to the simulation domain. For the results shown in Fig. 4 we get ~3 10 6 m/s and ~10 3 m/s, i.e. and the latter can be neglected in our analysis. We conclude that electrons in the cathode sheath gain an additional energy due to the E B motion which significantly exceeds the ionization threshold of argon ~ 15.8 ev [see
9 Fig. 7(b)]. Here we recall again that in large magnetic field electron energy relaxation length is defined as Λ,. For instance, using electric field in the vicinity of the cathode [Fig. 4(b)] we estimate for 100 ev electrons Λ, ~30 μm. Electrons move in the decaying electric field. Hence, Λ, decreases when electrons approach the edge of the sheath and when these electrons enter the plasma bulk they dissipate almost all their energy on the short length. This explains peak of the electron density obtained at the edge of the cathode sheath [Fig. 4(a)] and most efficient plasma generation there [Fig. 5(b)]. FIG. 5. Phase diagram of (a) electron heating and (b) number of ionizations per unit of time obtained for B 0 = 1 T, gas pressure of 10 Torr and γ = 0.02. In the plasma bulk (x > l sh ), electric field is small. Therefore, E B motion becomes inefficient and electrons diffuse toward the anode with the velocity in the ambipolar electric field. Electrons cannot gain enough energy in this electric field to ionize the gas and plasma is not generated by electrons created in the plasma bulk. Figure 5(b) shows only a few ionizations per unit of time in the plasma bulk. These ionizations are by the energetic electrons being
10 accelerated in the plasma sheath which do not dissipate all their energy at the cathode sheath edge. We conclude from Fig. 4(b) (see insert) that the electric field in the plasma bulk is always directed toward the cathode, i.e. it always accelerates electrons. This means that the plasma has negative charge with the respect to the anode. Only in the vicinity of the anode (x > 0.8 mm) we obtain positively charged plasma (compare with the results shown below in Fig. 13 for the homogeneous magnetic field). Here, it is important to note that we obtained the discharge stratification for larger applied voltages (see discussion below). However, the amplitude of these striations was much smaller than that obtained at lower gas pressure (see Sect. III.C). C. Plasma parameters at B 0 = 1 T and pressure 5 Torr Our simulation results have shown that the decrease in the gas pressure leads to the excitation of instability even for the breakdown voltage. Figures 6-9 show the results of simulations obtained for the gas pressure of 5 Torr and cathode voltage -350 V. We see striations in the plasma column. Each of these striations is the double layer in which the potential drop ~10 V is obtained. These striations move from the anode to the cathode with the velocity ~4 10 3 m/s. Similar behavior was obtained for smaller gas pressure (not shown here) but with larger amplitude of each striation. The frequency of oscillations having the largest amplitude is ~10 MHz [see Fig. 9(c)] which is much smaller than the electron cyclotron frequency ( ~2 10 11 Hz). Thus, obtained oscillations are not the Bernstein modes reported, for instance, in Refs. 18,19. The time of flight of argon ions from the anode to the cathode sheath is ~ 10-6 s, i.e., which means that the obtained instability is not associated explicitly with the ion motion.
11 FIG. 6. Phase diagrams of (a) electron density, (b) ion density, (c) electric field, and (d) ion flux velocity obtained for B 0 = 1 T, gas pressure of 5 Torr and γ = 0.02. In Appendix A we have derived the following expression for the instability growth rate:. (3) This equation shows that the instability is caused by the gradient of the ionization frequency. But it is stabilized by the electron diffusion. We see from Eq. (3) that the larger magnetic field the smaller the second term and, as a consequence, the larger the instability growth rate. The latter means that in non-homogeneous magnetic field increases in the direction of magnetic field gradient. Using Eq. (A19), we find the critical magnetic field necessary for the excitation of ionization instability. At 5 Torr, it is estimated as 0.25 T. In non-homogeneous magnetic field, it is reached only at the distance of 0.25 mm from the anode. Here, it is important to note that Fig. 9(c) shows the excitation of another instability having frequency ~ 15 MHz. This instability can be explained by the excitation of two-ion stream instability. 20 Indeed, striations accelerate not only electrons but also ions. This leads to the formation of ion beam which propagates through the plasma and excites oscillations. 20 The
12 increment of this instability is estimated as 21 /, (4) where is the ion plasma frequency and is the ratio between ion beam and plasma density. For the plasma with ~10 12 cm -3 [anode region in Figs. 8(a,b)] and ~10-3 obtained in our simulations we get ~15 MHz, i.e. indeed ~. FIG. 7. Phase diagrams of (a) electron heating profile, (b) average electron energy, and (c) number of ionizations per unit of time obtained for B 0 = 1 T, gas pressure of 5 Torr and γ = 0.02. The motion of striation can be explained as follows. Each of the double layers shown in Fig. 8 consists of the neighbor regions of excess of negative and positive charge. Electrons from the negative charge region move toward the positive charge region neutralizing it and leaving
13 behind uncompensated positive charge. Also, the electrons been accelerated in the double layer generate quasineutral plasma in it. Thus, the region of uncompensated positive charge shifts towards the cathode. We see from Figs. 6(a) and (b) the periodic spikes of the plasma species densities at the cathode sheath edge. These spikes appear when the striation moving from the plasma bulk reaches the cathode sheath edge. Indeed, each of these striations brings high positive space charge at the cathode sheath edge. This increases the ion current to the cathode which increases the current of secondary electron emission. The latter increases the number of energetic electrons arriving at the cathode sheath edge from the cathode. This increases the rate of plasma generation there. The spike of plasma density at the edge of the cathode sheath disappears when the ions leave this region through the sheath. FIG. 8. One-dimensional profiles of (a,b) electron density and (c,d) electric field obtained at three different times; B 0 = 1 T, gas pressure of 5 Torr and γ = 0.02. Figure 7(a) shows that the most efficient electron heating is obtained in the proximity of the cathode where the electron drift velocity in the E B-direction has the largest value. Also, we see from Fig. 7(a) efficient electron heating at the edge of the cathode sheath. Figure 7(b) shows
14 the average energy of electrons in the sheath ~250 ev. The ionization cross section for this group of electrons is the decreasing function of energy. The energy relaxation length decreases when electrons approach the edge of the cathode sheath (see discussion above). As a consequence, the electron energy decreases and ionization cross section attains its peak value when these electrons enter the plasma bulk. This explains the largest plasma production at the cathode sheath edge [see Fig. 7(c)]. We also conclude from Fig. 7 electron heating and plasma generation at the striations. FIG. 9. Time evolution of (a) electron density, (b) electric field and (c) Fourier spectrum of this electric field obtained in three different locations of the cathode-anode gap; B 0 = 1 T, gas pressure of 5 Torr and γ = 0.02. Our simulation results showed that the potential drop in each double layer is ~10 V which is smaller than the ionization threshold of argon ( ). The length of each striation is ~20
15 μm (Fig. 8) which is smaller than the electron Larmor radius in the local magnetic field. The fact that can be explained by the additional gain of electron energy due to the E B motion. Indeed, we conclude from Fig. 8 that the electric field in striations is ~2-4 10 5 V/m. Magnetic field in the center of the cathode-anode gap is 0.5 T. Thus, E B motion, as it follows from Eq. (1), provides additional ~2 ev to the electrons. These two motions (in crossed electric and magnetic field and in electrostatic field of the double layer) accelerate thermal electrons diffusing from the edge of the cathode sheath to the anode up to the ionization energy of argon. Figure 9 shows the time dependence of the electron density and electric field, and Fourier spectrum of this electric field in three different locations of the cathode-anode gap. We see that the amplitude of oscillations increases toward the cathode. We conclude from Fig. 9(c) the excitation of two harmonics with comparable amplitudes and frequencies ~10 MHz and ~15 MHz (their nature is discussed above). When these two waves move toward the cathode, the frequencies of these harmonics slightly increase. This can be explained by the spatial dependence of the instability growth rate (3). Indeed, the second term of Eq. (3) decreases for increasing magnetic field. Therefore, the stabilization term in Eq. (3) decreases toward the cathode which leads to the increase in. Also, we see the generation of higher harmonics at the distance of 0.25 mm from the cathode. The generation of higher harmonics near the cathode can be also explained by the wave propagation through the plasma with varying density. In order to prove that the discharge stratification is caused by high magnetic field (see discussion above and Appendix A), we carried out the simulations for zero magnetic field and homogeneous magnetic field of 1 T. In both cases cathode voltage was -350 V and gas pressure was 5 Torr. Figure 10 shows the steady-state profiles for zero magnetic field. We see that the plasma is stable. Also, the sheath thickness is larger in comparison with the case of nonhomogeneous electric field (compare with Fig. 8). The latter is explained by much smaller
16 plasma density for zero magnetic field. FIG. 10. Steady-state profiles of (a) plasma species densities, and (b) potential obtained for B 0 = 0 T, gas pressure of 5 Torr and γ = 0.02. FIG. 11. Phase diagrams of (a) electron density, and (b) electric field obtained for homogeneous magnetic field of 1 T, gas pressure of 5 Torr and γ = 0.02. Figures 11-13 show the results of simulations obtained for the homogeneous magnetic field of 1 T. We see that plasma dynamics differs from that obtained for non-homogeneous
17 magnetic field. However, we still obtain the discharge stratification although the amplitude of these striations in much smaller than in non-homogenous magnetic field [see electric field profile in Fig. 13(b)]. The frequency of ionization instability is ~10 MHz. Also, it is important to note different electric field profile. Namely, in homogenous magnetic field Fig. 13(b) shows the negative electric field in the plasma in the whole gap. FIG. 12. Phase diagrams of (a) electron heating profile, and (b) number of ionizations per unit of time obtained for the homogeneous magnetic field of 1 T, gas pressure of 5 Torr and γ = 0.02. Figure 12(a) shows that the most efficient electron heating is obtained again in the cathode sheath where the acceleration of secondary emitted electrons occurs. We conclude from Fig. 12(a) that the main mechanism of electron heating in the sheath is the heating of secondary emitted electrons due to E B motion in the vicinity of the cathode. As a consequence, we obtain the efficient plasma generation [Fig. 12(b)] and non-monotonic plasma density [Fig. 13(a)] at the edge of cathode sheath. Also, Fig. 13(b) shows the anode fall where the plasma electrons are accelerated up to 20 ev. The latter results in the second peak of plasma source term in the
18 vicinity of the anode [Fig. 12(b)]. Such behavior of electric field is attributed to the restraining effect of the magnetic field on the transport of electrons to the anode. Ions, which remain unaffected by the magnetic field, drift away from the anode much faster than electrons drifted towards it [see Fig. 13(c)], leaving a region of negative space charge adjacent to the anode. As a consequence, the electric field is directed away from the anode in order to maintain the electron current to the anode and provide the current continuity. FIG. 13. One-dimensional profiles of (a) electron density, (b) potential and electric field, and (c) average energy of electrons and ions obtained for the homogeneous magnetic field of 1 T, gas pressure of 5 Torr and γ = 0.02. The comparison between Figs. 8 and 13 shows that the peak of plasma species densities which is obtained at the cathode sheath edge are comparable. Also, on average the plasma
19 species densities look similar. However, we conclude that in the homogeneous magnetic field the potential drop ~ 80 V is realized on the length ~0.9 mm while in the non-homogeneous magnetic field ~ 40 V is realized only in 3-4 striations on the length scale ~60-80 μm. This can be explained by the more efficient heating of electrons in the sheath for the homogeneous magnetic field [compare Figs. 7(a) and 12(a)]. Indeed, we obtained larger electric field in the vicinity of the cathode for the homogeneous magnetic field. This means that E B-drift velocity of secondary emitted electrons is higher. Therefore, in order to support discharge at the given voltage there is no need in the large-amplitude striations in the plasma column in the homogeneous magnetic field. IV. Conclusions The influence of the gas pressure on the direct current microdischarge with magnetized electrons (magnetic field 1 T) and unmagnetized ions was studied by the one-dimensional Monte Carlo Collisions model. We obtained that the application of dc magnetic field significantly reduces the breakdown electric field which is explained by the increase of the effective gas pressure and decrease of the electron energy relaxation length. Also, the increase in the secondary electron emission coefficient due to ion impact led to the decrease of the breakdown voltage. Our simulation results showed that on the left branch of the Paschen s curve plasma column is stratified for all applied voltages. On the right branch, the stratification was obtained only for voltages significantly exceeding the breakdown voltage. The quantitative analysis of the instability growth rate showed that this instability is of ionization nature. Also, we found that the instability can be stabilized by the increase of the plasma density or decrease of the magnetic field. We also carried out the simulations for the homogeneous magnetic field and obtained small-amplitude striations. These striations did not play any noticeable role on the discharge
20 dynamics. In addition, the excitation of second instability was obtained. Its frequency exceeded the frequency of ionization instability. We associated this instability with the excitation of two-ion stream instability in the striations electric field. Acknowledgements This work was supported by the Air Force Office of Scientific Research (AFOSR) through a Multi-University Research Initiative (MURI) grant titled Plasma-Based Reconfigurable Photonic Crystals and Metamaterials with Dr. Mitat Birkan as the program manager.
21 Appendix A. Derivation of the ionization instability growth rate Let us derive the instability growth rate following the method described in Ref. 22. At the conditions of our study inertia of plasma species can be neglected, i.e. their fluxed can be defined in the drift diffusion approximation: Γ. (A1) Here, is the species mobility, is the species densities, and is the species temperature. Also, 1 for positive ions, and 1 for electrons and negative ions. Let us neglect the current passing through the plasma due to secondary electron emission. Then, in the current-free plasma electron and ion flux balance and quasineutrality give the ambipolar electric field and flux, (A2) Γ, (A3) respectively. In these equations,, electron mobility is the mobility along applied electric field which is defined as, where is the mobility at zero magnetic field, is the elementary charge, is the electron mass, and is the electronneutral momentum transfer frequency. Then, in zero magnetic field we have, Γ. (A4) (A5) Substituting Eq. (A5) in the electron balance equation, (A6) we obtain
22. (A7) In Eq. (A6), is the ionization frequency. In Eq. (A7), we assumed that both electron and ion temperatures are constant. Linearization of Eq. (A7) (i.e. ) gives in the first order:. (A8) Now, let us present the perturbation in the form of plane wave, with frequency and wave vector. Then, we get the dispersion equation:. (A9) Frequency can be presented in the form. Then, the imaginary part is:. (A10) If it is negative, it is called the instability growth rate. Also, note that Eq. (A9) gives 0. Thus, in zero magnetic field the condition for instability is the following:. (A11) It follows from Eq. (A5) that is the coefficient of ambipolar diffusion of unmagnetized plasma. If take into account that in stable plasma in the steady-state the ratio between diffusion coefficient and ionization frequency is of the order of square diffusion length, we conclude that the instability develops only if its length exceeds diffusion length. In unmagnetized plasma, diffusion length is of the order of cathode-anode gap. Thus, we conclude that ionization instability does not develop in one-dimensional unmagnetized plasma. In high magnetic field, such that Eqs. (A2) and (A3) are simplified as:, Γ. (A12) (A13)
23 Then, the electron balance equation is. (A14) Here we again assumed constant electron temperature. Then, the local dispersion relation and instability growth rate, respectively, are., (A15) (A16) Also, we find the real part. (A17) Then, the wave phase velocity is. (A18) We can conclude from Eq. (A16) that in magnetized plasma like in unmagnetized one instability is developed if the wave length is of the order of the diffusion length. However, unlike in unmagnetized plasma the electron diffusion scale in magnetized plasma is of the order of the electron Larmor radius. The latter is much smaller in comparison with cathode-anode gap. Therefore, we indeed obtain the excitation of ionization instability in one-dimensional magnetized plasma (see discussion in the text). Also, it is important to note that this instability is excited in both homogeneous and non-homogeneous magnetic field. However, the field profile influences the phase velocity. Equation (A18) shows 0 in homogeneous field. Now, we can define the critical magnetic field necessary for the excitation of instability. Equations (A15)-(A18) are valid only when. Then, using the expression for electron mobility in high magnetic field, i.e., we find:
24. (A19)
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