Electron random walk and collisional crossover in a

PHYSICS OF PLASMAS 20, 042118 (2013)

Electron random walk and collisional crossover in a gas in presence of electromagnetic waves and magnetostatic fields Sudeep Bhattacharjee,1 Indranuj Dey,2 and Samit Paul1 1 2

Department of Physics, Indian Institute of Technology, Kanpur 208016, Uttar Pradesh, India Kyushu University, Kasuga Kouen 6-1, Kasuga City, 816-8580, Japan

(Received 12 January 2013; accepted 4 April 2013; published online 19 April 2013) This paper deals with random walk of electrons and collisional crossover in a gas evolving toward a plasma, in presence of electromagnetic (EM) waves and magnetostatic (B) fields, a fundamental subject of importance in areas requiring generation and confinement of wave assisted plasmas. In presence of EM waves and B fields, the number of collisions N suffered by an electron with neutral gas atoms while diffusing out of the volume during the walk is significantly modified when compared to the conventional field free square law diffusion; N ¼ 1:5ðK=kÞ2 ; where K is the characteristic diffusion length and k is the mean free path. There is a distinct crossover and a time scale associated with the transition from the elastic to inelastic collisions dominated regime, which can accurately predict the breakdown time (sc) and the threshold electric field (EBD) for plasma initiation. The essential features of cyclotron resonance manifested as a sharp drop in sc, lowering of EBD and enhanced electron energy gain is well reproduced in the constrained random walk. C 2013 AIP Publishing LLC [http://dx.doi.org/10.1063/1.4802191] V

I. INTRODUCTION

An understanding of the electron random walk and the ensuing dynamics of electrons in a gas in presence of electromagnetic (EM) waves and magnetostatic (B) fields, leading to initiation of a discharge, is crucial in many areas of fundamental and applied plasma physics research employing high frequency electromagnetic waves1–4 and intense short pulsed lasers.5–7 This is essentially a problem where the classical field free random walk needs to be evaluated in presence of constraints imposed to the electron motion by the EM waves and B fields. We show that there is an intricate connection of the physics of this problem to the problem of finding out the plasma initiation time (breakdown time) and critical threshold electric fields. A review of the literature shows that, although the breakdown phenomenon has received numerous interpretations both in theory and experiments,8–18 still it is one of the least understood subjects8,9 owing to multiplicity of determining methods and their limitations, as elaborated in Ref. 8 and reproduced below for completeness. The Boltzmann equation was earlier expanded employing spherical functions in the mean free path and moderate amplitude limits, this however provides a regime of breakdown for fields of weak to moderate amplitudes.12 Analytical criterion dealing with particle balance equations is more related to discharge maintenance, because it assumes that breakdown and steady state occur almost at the same time.14 In pulsed discharges, the use of averaged fields has some limits of validity because the statistical process is smeared out; moreover, breakdown actually occurs much earlier than the time required to attain the critical (cutoff) density (x ¼ xp), that is conventionally employed to describe breakdown in pulsed discharges.15–18 In dc discharges, many authors have employed the exponential growth criterion like the “Townsend criterion” for 1070-664X/2013/20(4)/042118/6/$30.00

obtaining breakdown time where the electron density (n) is taken to increase by an arbitrary factor say, 108 from its initial value, i.e., when n ¼ no ehi it , becomes 108  no, where no is the initial density, h i i is the average ionization rate.17 In experiments, often output of an optical signal (flashover) is employed to identify the breakdown time.18 However, in reality, a sufficient number density of electrons is required to produce a detectable level of optical output, implying that breakdown has already occurred much earlier in time. Therefore, despite several works on the subject, the interpretation of the threshold breakdown field and breakdown time has remained debatable. It must be mentioned that the underlying physics of the subject (at least in wave induced discharges) was pioneered by Allis,10 Brown,11 and McDonald,12 who developed an extensive understanding of the mechanism of gaseous breakdown in presence of microwaves, both by experiments and mean field theory.10–13 For explaining the optimum breakdown in high frequency gas discharges, McDonald12,13 employed a statistical relation N / (K/k)2, originally obtained by Kennard,19 on electron random walk, which relates the average number of collisions (N) that an electron encounters in escaping from a spherical volume of radius K ¼ 1.5 cm (evaluated for an experimental plasma chamber 30 cm long and 7.2 cm in diameter8,9) to the collision parameter (K/k). However, the simple (field free) Kennard’s relation for N was originally deduced in the absence of EM and (or) B fields and therefore leads to (i) inaccurate prediction of threshold fields when evaluated using the breakdown criterion from the ran ¼ Ui ; where U is the average dom walk; defined as UN energy gain by an electron during the flight between two successive collisions and Ui is the ionization energy. (ii) The field free relation for N could only successfully explain experimental breakdown curves over a limited region (low pressure side) of the Paschen-like breakdown curve and breakdown in

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the high pressure regime remained unexplained by the aforementioned definition. Moreover, there was no known method for the prediction of breakdown times during that time. This leads to an interesting conjecture; how would N be modified in presence of EM waves and/or in presence of magnetic fields. If this can be done then it would be possible to overcome the limitations of the earlier model. In this investigation we determine the nature of electron random walk in presence of EM and B fields and develop a consistent breakdown criterion. We show that the criterion naturally emerges during the random walk of the electrons as a consequence of the crossover from elastic to inelastic collisions dominated regime in the presence of EM waves and B fields. The research can, therefore, address the commonly encountered situations for plasma initiation in wave generated magnetically assisted discharges in a unified way. The article is organized as follows. A model for the Monte Carlo simulations is presented in Sec. II. Results are presented in Sec. III. And finally a summary and conclusions is presented in Sec. IV. II. MODEL

In the present study, we address the situation where the electromagnetic waves in this case microwaves are launched perpendicularly to the magnetic field in the (k?B) mode, where k is the wave vector. The dynamics of a classical electron (point charge) in space can be described by the equation of motion given by me

r ~ d2~ ¼ F tot ; 2 dt

(1)

where me is the electron mass, ~ r ¼ xi^ þ yj^ þ zk^ is the electron position vector at any instant of time t defined with ~tot is the respect to a right handed coordinate system, and F total force acting on the electron, which may include electrostatic, magnetostatic, electromagnetic, or pressure gradient forces. For an ensemble of free electrons moving in the background of neutral atoms, collisional interaction comes into play. The dynamics of such an ensemble of electrons is conventionally addressed in the kinetic theory with the Boltzmann transport equation which incorporates the collision term implicitly.20 Particle number and momentum averaging of the Boltzmann equation leads to the fluid equation where the collisions occur as the Langevin drag force.10,21 In view of the statistical phenomena such as microwave breakdown of gas, in contrast to the fluid-average method, a discrete approach is more appropriate where the motion of each electron in the ensemble is followed by solving Eq. (1) numerically using a Runge-Kutta 4th order (RK4) algorithm, and the phenomenon of the random collisions is incorporated explicitly by considering actual random collisions of the electrons with neutrals by employing a Monte-Carlo based code. All the electron collisions, excitation and ionization cross sections are taken into account.21,22 The angle dependent scattering cross section was investigated from the available models,23 and found that this can sometimes lead to overestimate differential scattering cross section with

scattering angle. For example, for low energy electrons of 5 eV impinging on Argon atoms where we expect an isotropic scattering, while the differential elastic scattering cross section of the above reference, suggests a six fold higher cross section at 0 compared to 180 . In this study, we have therefore employed a simpler model by considering appropriate experimental scattering cross section with isotropic scattering. The details of the simulation procedure have been outlined in earlier works,8,9 the important points are reproduced here for completeness. A number of seed electrons and ions n are taken to be present initially in a spherical volume of radius K of the test gas (Argon). These electrons (ions) may arise from cosmic rays or other stray electron and radiation sources and usually lie in the range 103 to 104 per cm3.24 Typically, for a spherical volume defined by a radius K ¼ 1.5 cm, n is obtained to be 104. The electrons, neutrals, and ions follow the assumptions of kinetic theory of gases, and obey Maxwell’s velocity and energy distribution laws. The ratio of electron mass (me) to neutral (Ar) mass (Mn) is me/Mn  104, therefore the energy loss of electrons in the elastic collision process is 102 times smaller compared to the thermal energy (0.03 eV). At the same energy, the velocity ratio of an electron (ve) to a neutral (vn) (or an electron to an ion) is ve/vn  103 hence in the timescale of the simulation, which is of the order of a few nanoseconds, the neutrals and the ions do not move appreciably from their positions as compared to the electrons and may be considered as a static background. An estimate of the internal electric field due to the 104 randomly distributed electrons and ions is calculated at 104 random electric field points in the volume of interest and the most probable field (Eint) is determined to be 4.7  105 V/cm. The magnitude of this field being much smaller than typical applied electric field 103 V/cm in experiments, the space charge effects may be neglected. When ionization occurs, the number of electrons increase and so does the ions which is expected to contribute to the space charge. The ions are bulky and do not respond to the high frequency wave field. Since the modeling is limited to the predischarge regime where the number density (no  107–108 cm3) of the charged particles is much smaller than the cutoff density no  Nc (the cut-off density corresponding to the 2.45 GHz EM wave frequency is 7.44  1010 cm3), the effect of the space charge may be considered as negligible, and the electrons can be treated as point particles which follow the assumptions of the kinetic theory for low density gases. The process of collision between the electrons and neutrals is implemented by considering their collision crosssections as a function of electron energy.8,9,21,22 The mean free path (k), which an average electron traverses between two successive collisions, is given by the relation k ¼ 1=ng r, where ng is the neutral gas density given by the relation p ¼ ngkBT, p being the neutral gas pressure, kB is the Boltzmann’s constant, T  298 K (room temperature); r is the total electron-neutral collision cross-section summed over all the cross-sections of possible collisions (relastic þ rinelastic) at a given energy U.9,21,22 The electron-electron collisions are neglected due to their lower collision cross-section compared to the electron-neutral collisions. Also, since the density of

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electrons and ions is quite small compared to neutral atoms, the chances of electron–electron and electron–ion collisions are negligible compared to electron–neutral collisions. The calculation of the mean free path k has been described in detail in Ref. 8. In the simulation, the implementation of k is achieved by considering a randomized Gaussian spread with a width w  1% in the value of k calculated at a particular neutral density and electron energy, to incorporate the fact that the electron may travel any length of free path from 0 to 1, which when averaged over large number of collisions will give the mean free path k at that energy and density. A collision is counted if the distance travelled by the electron exceeds the free path determined from the randomized Gaussian distribution. The outline of the algorithm of the program for studying the electron random walk is as follows. The program starts at some particular pressure with one of the electron out of the 104 seed electrons available in the spherical test volume of radius K. The initial position (xo, yo, zo) of the electron is chosen randomly in the volume such that x2o þ y2o þ z2o  K2 . An initial Maxwell–Boltzmann velocity distribution of electrons is taken in accordance with Ref. 8. Moreover, the randomization of the direction of motion of the electron after each collision ensures that the velocity components along the x, y, and z directions are uniformly distributed in velocity space, in the absence of an external field: Vx ¼ V(1  r12)1/2 cos /, Vy ¼ V(1  r12)1/2sin /, and Vz ¼ Vr1, where r1 is a random number (1  r1  1) and / is the azimuthal angle. The final velocity distribution of the temporally evolving electrons is naturally chosen by the electrons themselves as they evolve with an initial Maxwell–Boltzmann distribution. The electron is then allowed to evolve with time step dt ¼ 103 nanoseconds subject to the total external force of ~tot (Eq. (1)). Depending upon the nature of the force field F (in this work EM and B), the electron evolves in space and may collide elastically or inelastically. The change in velocity and energy after each time step dt and the free path is evaluated and the collision criterion is checked. The collisions help to randomize the phase of the electrons and they gain energy. The collisions remain elastic up to 11.55 eV which is a metastable (4s) state of Argon9,21 and the ionization process may start at the first ionization threshold 15.76 eV.9,21 On each collision, the nature of the collision (elastic/excitation/ionization) is decided by the respective collision cross-section ratio (relstic:rexcitation:rionization). After an inelastic collision, the electron is tagged as an inelastic electron otherwise the electron is tagged as an elastic electron. On the other hand, the new born secondary electron due to ionization process is labeled as an elastic electron until it collides inelastically. In the event of an inelastic collision, the inelastic threshold energy is subtracted from the energy of the electron, and it is allowed to evolve in time and gain energy again. In case of ionization, a new elastic electron is introduced and the remaining energy is divided equally between the parent inelastic electron and the new born elastic electron. When an inelastic electron undergoes an inelastic collision again, its label does not change, hence the total number of inelastic electron remains unchanged. So, the electron motion is followed until the electron escapes from

Phys. Plasmas 20, 042118 (2013)

the volume or participates in an inelastic event. During this evolution, a situation arises when the percentage of inelastic electrons exceeds the percentage of elastic electrons, the cross over point indicates the breakdown time and breakdown is initiated.8 The instant, when the crossover occurs, is therefore defined as breakdown time (sc). The above process is repeated for all the 104 electrons at various pressures. Since elastic collision cross-section is much higher than inelastic below 100 eV electron energy, the elastic collision mean free path is much smaller than the inelastic. Parameters like the average number of collisions for the 104 electrons (N), mean free path travelled (k), energy gained per unit time (DU), average energy (hUi) are recorded. In the earlier works,8,9 MC simulations of the random ~tot ¼ 0 walk of the electrons for the field free case, where F was carried out and compared with the law deduced by Kennard.19 A good agreement was obtained. The work was further extended in presence of high frequency EM fields.8 The focus of the present article is to investigate the electron random walk when a magnetostatic field B is also present along with the EM wave, which has not been reported earlier. As case studies, we consider several values of the B field (i) supra-ECR (Electron Cyclotron Resonance at 2.45 GHz wave frequency) (1000 G), (ii) at ECR (875 G), and (iii) subECR (500 G). Additionally, for the perpendicular launch (k?B), two cases of electric field polarization: (a) parallel (E||B) and (b) perpendicular (E?B) are considered. When a B field is imposed upon the seed electrons, perpendicular to the wave vector of the high frequency EM field, their dynamics becomes all the more interesting due to the cyclotron motion of the electrons, which increases the probability of collisions. The external force term now becomes the Lorentz ~ þ~ ~ ~ and two principal situations may ~tot ¼ eðE v  BÞ, force F ~ field lines with arise depending upon the orientation of the B ~ ~ polarizations, namely respect to the wave electric field (E) ~B ~ B ~ ~ and E? ~ ~ cases as mentioned before. For the parallel E|| ~ and B ~ ~ along i^ direction, the equations of case, i.e., both E motion (Eq. (1)) becomes d2~ r ^ ¼ Co cosðko z  xtÞi^  xce vz j^ þ xce vy k; dt2

(2)

where xce ¼ eB=me is the electron cyclotron frequency, Co ¼ eEo =me is the maximum acceleration and the electronic charge is given by e. The magnetic field affects the motion along the j^ and k^ directions further destroying the random nature of the phenomenon; however, the dynamics is dominated by the electric field along i^ direction, and the variations of sc, DU are similar to the ones obtained for the only EM wave case. For an individual electron, since the magnetic field is parallel to the electric field, there will be a helical motion of the electron and an oscillatory motion because of the time varying electric field vector (i^ direction). There will be no effect even if the EM wave frequency becomes equal to the cyclotron frequency corresponding to the magnetic field. ~ along j^ direction and For the perpendicular case, with B ~ ~ ^ E along i direction, the equation of motion (Eq. (1)) becomes

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d 2~ r ^ ¼ ½Co cosðko z  xtÞ þ xce vz i^  xce vx k: 2 dt

Phys. Plasmas 20, 042118 (2013)

(3)

The electron motion in this case is strongly affected by the B field and when the electron cyclotron frequency equals the EM wave frequency, resonant transfer of energy occurs from the EM wave to the electrons, accelerating the electrons to high velocities, and hence bringing about the onset of discharge much earlier in time. We next describe the two limiting cases. A. Limiting case 1: Absence of EM wave and magnetic field

In the absence of any EM wave and magnetic field, the electrons do not gain any energy and all the collisions are elastic. It is found that the simulation results of the average number of collisions suffered by an electron in escaping from a given spherical volume can be fitted by8 N ¼ aðK=kÞ2 :

(4)

The value of the coefficient a arises as a natural consequence of the electron motion (particularly their origin). It is found that, if all electrons originate at the center, a ¼ 1.5, and if their appearance is chosen to be random (which is more realistic), i.e., at any arbitrary point inside the volume, a ¼ 1.25. The reproduction of the conventional square law originally predicted by Kennard19 verifies the assumptions made in the study and the Monte-Carlo recipes based on the random numbers used in the simulation code. B. Limiting case 2: Presence of an EM wave alone

Here, the linearly polarized plane EM wave with the ~ ¼ E cosðk z  xtÞi^ is ~ electric field vector of the form E o o imposed on the seed electrons in the same test volume. The frequency of the EM waves is 2.45 GHz, which provides x ¼ 1.54  1010 rad/s and ko ¼ 0.081 cm1. Eo is the amplitude of the EM wave, and is varied from 5 kV/m to 100 kV/m, approximately corresponding to a power range of 1 kW to 200 kW superimposed on a circular cross-sectional area of radius 36 mm. Usually this is the value of the wave frequency and power range used in the experiments, e.g., see Refs. 25 and 26. Equation (1) can then be written as d 2~ r ^ ¼ Co cosðko z  xtÞi: 2 dt

(5)

The electrons are allowed to evolve in time similar to the field free case, but with a constraint of motion along the axis of electric polarization (i^ direction), and are random along the other two directions j^ and k^ directions. Therefore, the motion may be referred to as a constrained random walk.8,9 III. RESULTS

The variation of N with K/k at E ¼ 10 kV/m for five different magnetic fields including B ¼ 0 G and one case at E ¼ 30 kV/m for B ¼ 875.5 G (ECR at 2.45 GHz frequency) ~ B ~ ~ polarization are shown in Fig. 1, where it is and for E?

FIG. 1. N vs (K/k) plot in the absence and presence of magnetostatic (B) field.

observed that the variation is no more quadratic, but has a strong linear component at larger values of K/k. The variation may be fitted with an equation of the form,9 N ¼ bðK=kÞ þ cðK=kÞ2 ;

(6)

where b and c are coefficients that are functions of the applied electric field. The magnetic field imposes one more constraint over the random nature on the electron motion. Hence, the value of N decreases due to the application of magnetic field as shown in Fig. 1. The magnetic field has ~ and resembles the very little effect when it is parallel to E, case of EM wave induced dynamics in absence of a magnetic field. Unless otherwise mentioned, all the cases, from henceforth, have perpendicular polarization. In case of perpendicular polarization, the electron dynamics is highly affected by the magnetic field and as resonance is approached the value of N decreases, since the electrons are highly accelerated and may leave the volume without enough collisions. Figure 2(a) shows the percentage of the number of elastic electrons and the inelastic ones. It is observed that the percentage of the number of elastic electrons decreases and that of inelastic ones increases with time as expected. The two curves meet at a crossover point sc which provides a new definition of breakdown time. Figure 2(a) shows the cross over for the four magnetic field conditions for E ¼ 50 kV/m, at 1 Torr. The cross over time (sc) for the parallel case is about an order higher 26 ns similar to the magnetic field free case indicating that the parallel magnetic field does not affect the breakdown much. Another interesting observation is the occurrence of the cross over for E ¼ 10 kV/m at resonance (57 ns), which was not obtained in the absence of a magnetic field. This indicates that at resonance the threshold electric field required for plasma initiation is lowered down quite a bit. The cross-over for the 500 G, 875.5 G (ECR), and 1000 G for E ¼ 50 kV/m are quite close (3 ns) at 1 Torr pressure. Figure 2(b) shows the expanded view of the cross over for these cases, where it is observed that the cross over time (sc) decreases from 500 G (3.56 ns) to 1000 G (3.28 ns).

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FIG. 3. (a) Variation of cross over time (sc) with gas pressure (p). (b) Average energy gain per electron per nanosecond (hDUi) versus p in presence of magnetostatic (B) field. Symbols: Sub ECR-500 G (black square), ECR-875.5 G (red circle), supra ECR-1000 G (magenta inverted triangle), at 50 kV/m and 875.5 G (blue up triangle) at 10 kV/m.

FIG. 2. (a) Plot of the percentage of elastic (solid lines) and inelastic (dashed lines) electrons with time at 1 Torr. (b) Expanded view of (a) in the time range 3 to 3.8 ns for 500, 875.5 (ECR), and 1000 G.

Figure 3(a) shows the variation of sc with neutral pressure for the conditions shown in Fig. 2(a), except for the parallel case. It is observed that the minima of the curves lie between 0.1–1 Torr. The variation at resonance is quite interesting and different from the other variations. At lower pressures (105–103 Torr), where inelastic collisions are usually less due to unavailability of adequate number of neutrals, sc (resonance) is lower than sc (away from resonance). This is because at lower pressures, the electrons acquire a large energy (U > 100 eV) (cf. Fig. 3(b)) by resonant coupling which increases the probability of inelastic collisions and induces a faster cross over. As the neutral density increases, an optimal condition is realized 0.1 – 0.5 Torr, where there are just enough neutrals for efficient ionization leading to the steep minima. After the minima, sc (resonance) increases and converges with sc (no resonance) values since at higher neutral densities (higher pressures 1 Torr) [Fig. 3(a)]; the resonance mechanism is rendered ineffective due to increased elastic collisions and the electron energy gain also decreases. This is in conformity with Figs. 2(a) and 2(b), where the difference in the sc values at different magnetic fields is small at 1 Torr. Figure 3(b) shows the variation of the average energy gained by an electron per nanosecond (hDUi) versus pressure for the four conditions shown in Fig. 3(a). For the nonresonant cases, the maxima occur between 0.1 and 1 Torr, complimentary to the plot in Fig. 3(a) as expected. In the

resonant case, it is seen that hDUi is about two orders of magnitude higher in the low pressure range (105–102 Torr) confirming the fact that the electrons gain higher energies due to resonant coupling. At higher pressures, due to enhanced collisions, the variations converge with the non-resonant cases as expected. Figure 4 shows the variation of the breakdown electric field EBD with pressure in the presence of three different B fields at 500, 1000, and 875 G (ECR), for comparison a case without the magnetic field is also included. The threshold breakdown field (EBD) is defined as the minimum electric field below which crossover does not happen. To find out

FIG. 4. Variation of breakdown electric field EBD with pressure in the absence of a magnetic field and in the presence of 500, 875.5 (ECR), and 1000 G magnetic fields.

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whether crossover occurs at a given pressure, magnetic field, and EM wave electric field, the simulation is carried out for much longer time compared to the particle loss time scale (10 times). If crossover does not occur within this simulation time range, then cross over is not possible. At the low pressure regime, EBD decreases with pressure, then it reaches a minimum (30 kV/mm for B ¼ 0 G case) at 20 Torr and then increases again at the high pressure regime. The nature of the breakdown curve closely matches with the experimentally observed results.12,13,27 For example, the value of breakdown field varies in the range 10 kV/m to 300 kV/m for the value of K in the range 0.0275 to 0.0758 cm in case of hydrogen and the minima occurs in the pressure range 3 to 20 Torr. It may be noted that both sides of the Paschen-like minimum (as in dc discharges) are well reproduced in this new definition which incorporates the modified Kennard’s law for N in the presence of EM waves and B field. The values for N were directly used from the simulations. An interesting observation can be made that for off-resonance magnetic fields (500 G and 1000 G) the breakdown field increases on both sides of the minima. However, for the resonance magnetic field, EBD decreases monotonically with decrease in pressure. This is due to the fact that the electrons gain energy very quickly through collision less resonant coupling processes in the low pressure regime. As the pressure increases, the energy gain per unit time decreases due to collisions leading towards the increase of EBD. It can also be noted that for magnetic field assisted breakdown EBD is much smaller compared to the B ¼ 0 case. IV. SUMMARY AND CONCLUSION

The random walk of the initial seed electrons distributed randomly in space with background neutral gas atoms is studied, subject to externally applied forces of an EM wave and a B field, by a Monte Carlo based simulation algorithm. The conventional field-free dynamics is verified and the square law variation relating the number of collisions to the mean free path is confirmed. In presence of high frequency EM waves, the motion is found to be constrained and a new variation between N and K/k is obtained. A new discharge initiation criterion is defined from crossover of inelastic processes over elastic processes, and the cross over time is defined as the breakdown time. The threshold breakdown electric field is defined as the field below which there is no cross over. The procedure provides a unified way of addressing both the issues of breakdown time and threshold electric field. In presence of an EM wave and a B field, the cyclotron motion of the electrons is found to enhance the probability of collisions. At resonance, the discharge formation is found to

Phys. Plasmas 20, 042118 (2013)

occur at much lower pressures and also the threshold electric field is significantly lowered, confirming the observations of ECR plasma sources. ACKNOWLEDGMENTS

We gratefully acknowledge financial support from the Department of Science and Technology DST India under the Centre for Nano-Technology at Indian Institute of Technology IIT Kanpur. One of the authors S.P. would like to thank CSIR, India for providing Ph.D. scholarship. 1

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