Electron Transport Coefficients in a Helium Xenon Mixture

ISSN 1068-3356, Bulletin of the Lebedev Physics Institute, 2014, Vol. 41, No. 10, pp. 285 291. c Allerton Press, Inc., 2014. Original Russian Text c S.A. Mayorov, 2014, published in Kratkie Soobshcheniya po Fizike, 2014, Vol. 41, No. 10, pp. 20 30. Electron Transport Coefficients in a Helium Xenon Mixture S. A. Mayorov Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia; e-mail: mayorov sa@mail.ru; may4536@yandex.ru Received June 5, 2014 Abstract Electron drift characteristics in a helium xenon mixture under an electric field strength E/N =1 100 Td are calculated and analyzed taking into account inelastic collisions. It was shown that a minor xenon additive to helium, beginning with percent fractions, has a significant effect on the discharge, in particular, on the characteristics of inelastic processes. The effect of the percentage of helium and xenon on electron drift, i.e., the diffusion and mobility coefficients, ionization frequency, etc. is studied. DOI: 10.3103/S1068335614100030 Keywords: electron, drift, electric field, inelastic collisions, elastic collisions, inert gas mixture, energy balance. Introduction. The electron drift in gas mixtures has essential features which can be used for various purposes. For example, a small additive of a heavy, easily ionized gas to an inert gas with a high ionization potential can radically change ion flux parameters [1, 2]. After all, the ion composition in this case will be controlled by the easily ionized additive, and the heavy ion motion in an extrinsic gas leads to the formation of a supersonic ion flux [2]. Therefore, the consideration of the electron drift kinetics in a mixture of helium with xenon is of particular interest. The use of the gas-mixture discharge in studying dusty plasma shows that the dust component has a number of interesting features. A small argon [3, 4], krypton [5], or xenon [6] additive to helium resulted in a significant change in dust component properties. The major factors affecting changes in the dust component characteristics when adding a heavier gas to helium are: (i) a decrease in the electron temperature and a decrease in the dust particle charge, (ii) the formation of a highly anisotropic distribution of ions and a significant increase in the ion drag force, (iii) an increase in the ion flux on walls and a decrease in the surface potential, (iv) a change in the plasma parameters of the gas discharge, i.e., the density, temperature, ionization frequency, diffusion length, and others. We note that a decrease in the atomic temperature in cryogenic discharges also leads to a change in the ioncompositiondueto heliumionconversion into molecular ions. Therefore, experiments with dusty plasma at cryogenic temperatures also show a significant change in dust component characteristics [7]. In this study, we consider electron drift in a helium xenon mixture with the aim of studying the effect of the xenon concentration (fraction) on the electron transport coefficients. The numerical experiment is based on the consideration of an ensemble of noninteracting electrons whose motion is controlled by given fields and instantaneous collisions with atoms. The collision model is based on random number generation, i.e., the Monte Carlo-type method. Implementation of electron atom collisions by the Monte Carlo method allows the consideration of the electron energy balance based on elementary events, including those during inelastic collisions. We note that the solution to the Boltzmann equation, rather than that to the two term approximation, is in fact found in such a numerical experiment. Furthermore, such problem statement makes it possible to determine other electron drift characteristics in a given field. The drift velocity, average electron energy, characteristic Townsend energy, relation between energy losses in elastic and inelastic collisions, and Townsend ionization coefficient were calculated. Furthermore, electron diffusion coefficients along and across the electric field were calculated, and regions of the strong manifestation of the anisotropic diffusion effect were determined. 285

286 MAYOROV Problem statement for Monte Carlo simulation of electron drift in gas. Let us consider the electron drift in a steady-state spatially uniform electric field. In the case typical of the gas discharge, the average electron energy significantly exceeds the atomic energy. Then the energy gained by the electron from the electric field is lost in elastic collisions with colder atoms and is expended to atomic level excitation and ionization. Moreover, electrons lose or gain energy during collisions with excited atoms, recombination, etc. (see, e.g., books and reviews [8 12]). Due to Joule heating during drift in a dc and uniform electric field, the electron gains on average the energy per unit time Q EW = eew, (1) where e is the electron charge, E is the electric field strength, and W is the drift velocity. The electron energy balance can be written as Q EW = Q ea + Q exitation + Q ionization + Q recombination, (2) where the right-hand side presents the corresponding average electron energy losses per unit time for elastic collisions, energy expenditures for excitation, ionization, and recombination. The electron kinetics can be strongly complicated by such effects as step ionization, the presence of metastable atoms, resonant radiation transport, superelastic collisions, and others. For example, the electron can gain energy during recombination; recombination heating plays a decisive role in the energy balance of supercooled plasma [13]. When simulating electron atom collisions, we make the following assumptions. (i) Gas atoms have the Maxwellian velocity distribution and do not change their temperature in collisions with electrons. (ii) Elastic electron atom collisions occur as hard sphere ones, i.e., the isotropic scattering in the center-of-mass system takes place at collisions but the collision cross section is assumed to depend on the energy of their relative motion. (iii) Electron losses for excitation of atomic levels are irreplaceable, i.e., it is assumed that excited atoms lose the excitation energy in the mode of spatial de-excitation, metastable atoms rapidly diffuse over boundaries of the volume under consideration and do not affect the electron energy distribution. (iv) During electron-impact ionization, the electron incident on the atom loses the energy equal to the sum of the ionization energy and the kinetic energy of the knocked out electron. (v) The processes of electron and atom recombination, excited level quenching, and resonant radiation transport do not change the electron energy. (vi) The ionization and excitation probabilities are defined by the reaction cross sections for which linear approximation is used, beginning with the reaction threshold [9 12]. The electron energy distribution function defines many properties of electron drift and is the most important characteristic of the gas discharge. For its determination, various theoretical models were developed; however, the two term approximation (TTA) of the Boltzmann equation is very often used for its numerical solution. In the case where the electron drift in a dc and uniform field is controlled only by elastic collisions with atoms, and the field strength is such that the average electron kinetic energy greatly exceeds the energy (temperature) of atoms, the solution to the two term approximation of the Boltzmann equation, i.e., the electron velocity magnitude distribution function, is written as f 0 (v) =A exp 3m ( ) mn 2 v c 2 σel 2 M ee (c)dc, (3) 0 where m and M are the electron and atom masses, σ el is the elastic collision cross section, and A is a constant determined from the normalization condition 1=4π c 2 f(c)dc. In the case of the powerlaw velocity dependence of the cross section σ el (c) =σ 0 (c/c 0 ) r, the integral in Eq. (3) is calculated in 0

ELECTRON TRANSPORT COEFFICIENTS 287 explicit form [10, 11]. At a constant collision frequency, the velocity dependence of the cross section has the form σ el (c) =σ 0 (c/c 0 ) 1/2 ; then Eq. (3) transforms to the Maxwellian distribution f Maxwell (ε) ε 1/2 exp( ε/t ). (4) At the constant cross section σ el (c) =σ 0, the mean free path is independent of velocity, and distribution (3) transforms to the Druyvesteyn distribution f Druyvesteyn (ε) ε 1/2 exp[ (ε/ε D ) 2 ]. (5) The velocity dependence of the collision cross section has a complex shape, and the approximation of the constant free path length is most appropriate for it in the energy range 0 <ε E 1,I. Therefore, the experimental electron energy distributions in the gas discharge are usually better described by the Druyvesteyn distribution, rather than the Maxwellian distribution. The Druyvesteyn and Maxwellian distributions are often used when considering various problems of gas discharge physics. However, all these models disregard electron creation and annihilation; therefore, they are in principle inapplicable to, e.g., the case of the stratified discharge in a tube at a reduced pressure. On the contrary, the pipeline model considers the zero-energy electron creation, drift in the kinetic energy with a constant diffusioncoefficient, and instantaneous annihilationupon reaching theionization or excitation energy [9, 12], f pipe line (ε) =1.5(1 ε/i) 1/2 I. (6) As a result of the use of these approximations for distribution functions, only the drift velocity is often determined; other coefficients important for simulating the gas discharge kinetics, such as the diffusion coefficient along and across the field, the energy and first Townsend ionization coefficients are not determined due to the problem complexity. Therefore, the numerical experiment is almost a single reliable tool for studying electron drift characteristics, in particular, for gas-mixture discharges, where small additives can have a significant effect on the discharge. Calculated results and discussion. Collisions were simulated using a Monte-Carlo-based algorithm developed for simulating ion and electron drift in gas [14, 15]. Collisions were simulated taking into account the known energy dependences of collision cross sections [10 12, 16]. The detailed calculation results for electron drift in all inert gases are given in [17]. Figure 1 shows the dependences of the electron drift characteristics on the reduced electric field strength E/N: (a) the solid curve showing the electron drift velocity corresponds to drift in pure helium; the solid curve with closed circles is drift in pure xenon; the dashed curve with open circles, dash-dotted and dashed curves correspond to the drift in helium with 5%, 10%, and 50% of xenon, respectively; (b) the reduced Townsend ionization coefficient determined by the number of pairs produced per 1 cm and related to the numerical density of atoms: solid curves correspond to drift in pure helium and xenon, the dashed curve with circles, dash-dotted and dashed curves corresponds to calculations for0.1%, 1%, and 2% of xenon in helium, respectively. The dependences of the drift velocity and ionization frequency show that helium dilution with xenon to 10% does not lead to a noticeable change in the drift velocity in the region E/N > 10 Td in which significant ionization necessary for maintaining the Discharge occurs. However, even 0.1% of xenon at 10 Td < E/N < 20 Tdresults in an approximately hundredfold increase in the ionization frequency! Figure 2 shows the energy characteristics of the electron drift as a function of the reduced electric field strength E/N: (a) the solid curve showing the characteristic Townsend energy defined by the ratio of the transverse diffusion coefficient and mobility ed /μ corresponds to the drift in pure helium; the solid curve with closed circles corresponds to the drift in pure xenon; the dashed curve with open circles, dash-dotted and dashed curves correspond to 5%, 10%, and 50% of xenon; (b) the average electron energy; the notations are identical to those of Fig. 2(a).

288 MAYOROV Fig. 1. (a) Dependences of the electron drift velocity on E/N and (b) dependences of the reduced Townsend ionization coefficient in He/Xe mixtures. Fig. 2. Dependences of the electron energy characteristics on E/N in He/Ne mixtures: (a) the characteristic Townsend energy ed /μ and (b) the average electron energy.

ELECTRON TRANSPORT COEFFICIENTS 289 Fig. 3. Electron energy distribution function for E/N =15Tdat various xenon fractions in the helium xenon mixture in the logarithmic and linear scale. Figure 3 shows the electron energy distribution functions at various percentages of xenon atoms in the helium xenon mixture: the solid curve and the solid curve with closed circles correspond to drift in pure helium and pure xenon, respectively; the dashed curve with open circles correspond to the drift in helium with 0.1% of xenon; dash-dotted curves correspond to 1%, 2%, 5%, 10%, and 50% of xenon. For all calculations, E/N =15Td. In the left figure, the distribution function is given on a log scale to show distribution function tails; the right figure shows it on the linear scale to show the effect of the xenon fraction on the distribution function body. The insets shows the average electron kinetic energy K = ε. The calculation results give a rather complete picture of the mechanism of the effect of small xenon additives on gas discharge characteristics. The most interesting and practically important fact is a significant increase in the ionization frequency at a minor (of the order of percent fraction) xenon additive. Furthermore, it should be noted that xenon atoms will be mostly ionized in this case; hence, xenon ions will be mostly presented in the discharge. Figure 4 shows the electron energy distribution functions at 1% of xenon in the helium xenon mixture at E/N =15Td; for comparison, the Maxwellian (4) and Druyvesteyn (5) distributions with an average energy calculated by the Monte Carlo method and the pipeline distribution (6) (infinite sink model) are also shown. This figure clearly demonstrates the significant difference of the actual electron energy distribution from the commonly used Maxwellian and Druyvesteyn distributions. A detailed analysis of the distribution functions shows that they cannot be described at all by any one-parameter function with an effective temperature defined by the relation K = ε =1.5T eff.inthe actual distribution function, we can distinguish several characteristic energy ranges, the distribution in which is controlled by the dominance or competition of various processes. (i) The region of subthermal energies ε<t eff ; the distribution in this region is controlled in many respects by excitation and ionization events after which electrons appear in the low-energy region. (ii) The thermal energy region ε<e 1,I; the distribution in this region is controlled by the drift in the energy space with a diffusion coefficient defined by the elastic collision cross section. (iii) The energy region E 1 <ε<i; the distribution in this region is controlled by the drift in the energy space and the line slope in the linear approximation of the excitation cross section. (iv) The energy range I<ε<I+3T eff ; the distribution in this range is controlled by the velocity drift in the energy space and the line slope in the linear approximation of the ionization cross section.

290 MAYOROV Fig. 4. Electron energy distribution function (EEDF) in a mixture with 1% xenon content at E/N =15Td;for comparison, the Maxwellian, Druyvesteyn, and infinite-sink-approximation EEDFs are shown. (v) The energy region ε>>i+3t eff ; the distribution in this region is controlled by the runaway electron effect. The presented division of characteristic energy ranges is quite arbitrary, its main objective is to indicate exactly the multifactoriality in the electron energy distribution formation in various energy spectrum ranges, which does not allow us to apply the temperature notion to the electron component of the gas discharge. The main objective of this study is to present the new calculated data on electron drift characteristics in the helium xenon mixture, which can be useful when designing experiments with dusty plasma. Theresultsofcalculationsallowtracingtheeffect of the percentage in the helium xenon mixture on electron drift in a dc uniform electric field with strengths in the range from 1 to 100 Tdcharacteristic of dc discharges at a reduced pressure. Of interest is the problem of the maximum energy efficiency of the discharge maintenance. As calculations show,at E/N =10Td, the largest energy fraction is expended by the electron for ionization at 1% of xenon; at E/N =20Td, the maximum fraction of energy expenditures for ionization is achieved at 2% of xenon. Conclusions. As noted previously in [1 5, 14], the discharge in the helium xenon mixture has many features which can be useful in the search for new methods for controlling the gas discharge. For example, in the present time experiments with dusty plasma in dc discharges are performed in pure gases with current and pressure as controlled discharge parameters [18]. In inductive and HF discharges, the situation is similar: the pressure and deposited power are varied. Therefore, the possibility of varying significantly the discharge parameters by choosing the mixture composition and percentage seems very interesting. The significant change in discharge characteristics by small easily ionized impurity additives can be used in the search for new active media for various plasma technologies in microelectronics, material processing, thin film deposition, light source development, plasma panels, and medicine [8, 19]. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research, project no. 14-02-0512-a.

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