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Electron mobilities in binary rare gas mixtures Leung, Ki Y. 1990

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E L E C T R O N MOBILITIES IN B I N A R Y R A R E GAS M I X T U R E S by KI Y . L E U N G B.Sc.(Hon.), Queen's University, 1987 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Chemistry We accept this thesis as confirming to the requirement standard The University of British Columbia September 1990 © Ki Y. Leung, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract This thesis presents a detailed study of the composition dependence of the thermal and transient mobility of electrons in binary rare gas mixtures. The time independent electron real mobility in binary inert gas mixtures is calculated versus mole fraction for different elec-tric field strengths. The deviations from the linear variation of the reciprocal of the mobility of the mixture with mole fraction, that is from Blanc's law, is determined and explained in detail. Very large deviations from the linear behavior were calculated for several binary mixtures at specific electric strengths, in particular for He-Xe mixtures. An interesting effect was observed whereby the electron mobility in He-Xe mixtures, for particular compositions and electron field strength could be greater than in pure He or less than in pure Xe. The time dependent electron real mobility and the corresponding relaxation time, in particular for He-Ar and He-Ne mixtures are reported for a wide range of concentrations, field strengths (d.c. electric field), and frequencies (microwave electric field). For a He-Ar mixture, the time dependent electron mobility is strongly influenced by the Ramsauer-Townsend minimum and leads to the occurrence of an overshoot and a negative mobility in the transient mobility. For He-Ne, a mixture without the Ramsauer-Townsend minimum, the transient mobility increases monotonically towards the thermal value. The energy thermal relaxation times 1 /PT for He-Ne, and Ne-Xe mixtures are calculated so as to find out the validity of the linear relationship between the 1/Pr of the mixture and mole fraction. ii A Quadrature Discretization Method of solution of the time dependent Boltzmann-Fokker-Planck equation for electrons in binary inert gas mixture is employed in the study of the time dependent electron real mobility. The solution of the Fokker-Planck equation is based on the expansion of the solution in the eigenfunctions of the Fokker-Planck operator. iii Table of Contents Page Abstract ii Table of Contents iv List of Tables v List of Figures vi Acknowledgements x Chapter 1 Introduction 1 Chapter 2 Theory 11 A. Introduction 11 B. Fokker-Planck Equation For a Binary Mixture 14 C. Quadrature Discretization Method 25 D. Transient Mobility Coefficients 30 Chapter 3 Thermal Mobilities of Electrons in Binary Inert Gas Mixtures 33 A. Introduction 33 B. Calculations and Results 37 C. Results and Discusion 39 Chapter 4 Transient Mobilities of Electrons in Binary Inert Gas Mixtures 68 A. Introduction 68 B. Calculations and Results 70 C. Interpretation of Results 84 Chapter 5 Summary 89 Reference 92 iv List of Tables: Table Description Page 1 Xenon and Krypton Cross Sections 41 2 Relaxation times versus helium mole fraction, Xjie for a He-Ne mixture; 81 the initial delta function is at uo=3.0 and T B=300.0K. T(0.9) is the time in units of 10 1 1 sec cm~3 required for the real mobility to decay to within 0.9 of the stationary value. 3 Relaxation times versus Xj{e for a He-Ar mixture,the initial delta 82 function is at u0=3.0 and TB=300.0K. T=(0.9) is the time in units of 10 1 1 sec cm~3 required for the real mobility to decay to within 0.9 of the stationary value. Asterisks denote values of r for relaxation to 1.1 of the stationary value. v List of Figures Figure Description Page 1 Momentum transfer cross sections for electron-rare gas collisions. 40 2 Average electron energy in the pure rare gases versus density reduced 42 electric field strength. 3 Average electron mobility in the pure rare gases versus density reduced 43 electric field strength. 4 Mole fraction dependence of the percent deviation from Blanc's law for 47 electrons at zero field. X\ is the mole fraction of the first cited gas of the mixture. (A) (a)Kr-Xe, (b)Ar-Xe, (c)Ar-Kr. (B) (a)He-Ne, (b)He-Xe, (c)He-Kr, (d)He-Ar. (C) (a)Ne-Xe, (b)Ne-Kr, (c)Ne-Ar. 5 (A)Variation of with mole fraction for Kr-Xe. E/N in Td is equal 52 to (a)0.05, (b)0.1, (c)0.2, and (d)0.4. The other graphs are for the variation of the mole fraction dependence of the percent deviations from Blanc's law with E/N in Td equal to (B) (a)0.011, (b)0.013, (c)0.015, and (d)0.016; (C)(a)0.018, (b)0.05, (c)0.06, and (d)0.08; (D) (a)0.07, (b)0.08, (c)0.1, and (d)0.40. 6 Mole fraction dependence of the percent deviation from Blanc's law for 53 A r - K r mixtures: (A)E/N in Td equal to (a)0.004, (b)0.006, (c)0.008, (d)0.01, (e)0.02, and (f)0.03. {B)E/N in Td equal to (a)0.04, (b)0.1, (c)0.5, (d)l , and (e)1.2. vi Figure Description Page 7 Mole fraction dependence of the percent deviation from Blanc's law for 54 Ne-Ar mixtures: {A)E/N in Td equal to (a)0.005, (b)0.007 (c)0.009, (d)0.02, and (e)0.05. (A) E/N in T d equal to (a)0.1, (b)0.25, (c)0.4, (d)0.8, and (e)1.4. 8 (A)Mole fraction dependence of the percent deviation from Blanc's law for 55 He-Xe mixtures; (A)E/N in Td equal to (a)0.01, (b)0.02, (c) 0.04, (d)0.06, (e)0.08, (f)0.10, and (g)0.2. Mole fraction dependence of l//x with E/N in Td equal to (B)0.04, (C)0.1 and (D)0.2 9 (A)Mole fraction dependence of the percent deviation from Blanc's law 56 for Ne-Xe mixtures; E/N in Td equal to (a)0.001, (b)0.002, (c)0.005, (d)0.008, and (e)0.012. (B) Mole fraction dependence of l//x with E/N in Td equal to (a)0.04, (b)0.1, (c)0.2, and (d)0.3. 10 (A)Variation of 1/cr with reduced speed for He-Xe with Xfje 63 equal to (a)0, (b)0.2, (c)0.4, (d)0.6, (e)0.8, and (f)l . (B)Variation of 1/cr with reduced speed for Ne-Xe with X^e equal to (a)0, (b)0.2, (c)0.4, (d)0.8, (e)0.95, (f)0.98, and (g)l. 11 Variation of the mobility versus field strength: 66 (A) He-Xe with XHe equal to (a)0.04, (b)0.2, and (c)0.5. (B) He-Ar with XHe equal to (a)0.02, (b)0.06, (c)0.1, (d)0.2, and (e)0.5. 12 Time variation of the real mobility with E/N=l x 10~5 Td and 72 W/N=l x 1 0 - 5 Hz. Initial delta function distribution at UQ equal to (a)2.0, (b)3.0, and (c)4.0, and T f c=300K; (A)Pure helium, (B)Pure argon. vii Figure Description Page 13 (A)Time variation of the real mobility vs E/N for pure argon. Initial delta 73 function distribution at u 0=4.0, W/N=l x 10" 5 Hz, and T t =300K, E/N in T d equal to (a)O.OOOOl, (b)0.001, (c)0.005, and (d)0.01. (B)Time variation of the real mobility vs W/N for pure argon. Initial delta function distribution at u o=3.0, E/N = 0.01 Td , and r t =300K, W/N in Hz equal to (a)0.00001, (b)2.5, (c)10.0, and (d)25.0 14 Time variation of the real mobility for a He-Ar mixture with initial delta 75 function distribution at u0=4.0 and E/N=l x 10" 5 Td , W/N=l x 10" 5 Hz r 6 =300K. (A) Mole fraction of helium equal to (a)0.0, (b)0.05, (c)0.1, and (d)0.2. (B) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l . 15 Time variation of the real mobility for a Kr-Xe mixture with initial delta 76 function distribution at u 0=3.9, TB=300K, W/N=l X 1 0 - 5 Hz and E/N in Td equal to (A)E/N=1 x l O " 5 , (B)E/N = 0.01. Mole fraction of krypton equal to (a)0.0, (b)0.25, (c)0.5, (d)0.75, and (e)1.0. 16 Time variation of the real mobility for a He-Ne mixture with initial delta 78 function distribution at u0 = 3.0 , W/N=l x 10~5 Hz, and E/N in Td equal to (A)E/N=1 x 10- 5 , (B)E/N = 0.01. Mole fraction of helium equal to (a)0, (b)0.25, (c)0.5, (d)0.75, and (e)1.0 at T 6=300K 17 Time variation of the real mobility for a He-Ar mixture with initial delta 79 function distribution at u 0 = 3.0, T B =300K, W/N=l x 1 0 - 5 Hz, and (A) (B)£/AT=1 x 10" s Td , (C)(D)£/ iV=0.01 Td : (A) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.1, and (d)0.2. (B) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l . (C) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.1, and (d)0.2. (D) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l . vi i i Figure Description Page 18 Time variation of the real mobility for a He-Ne mixture with initial delta 80 function distribution at u 0 =3.0, T 6=300K, E/N=0.01 Td , and WfN = 10.0 Hz: Mole fraction of helium equal to (a)0, (b)0.25, (c)0.5, (d)0.75, (e)1.0. 19 Time variation of the real mobility for a He-Ar mixture with initial delta 80 function distribution at u 0=3.0, T 6=300K, E/N = 0.01 Td , and W/N =10.0 Hz: ( A ) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.10, (d)0.20. (B ) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, (d)1.0. 20 The composition dependence of the energy relaxation times, 1/Pr for 83 mixtures, with initial delta function distribution at u 0=4.0, T),=300K, E/N=l x l O " 5 Td, and W/N=l x 10" 5 Hz (A ) 1/PT in He-Ne mixture vs helium mole fraction, (B ) 1/PT in Ne-Xe mixture vs neon mole fraction. Acknowledgements I wish to express my sincere thanks to my research superviser, Dr. B. Shizgal. It has been a pleasure to have worked with him and his support, direction and assistance will always be greatly appreciated. Thanks are also due to the various members of Dr. Shizgal research group for contributing to an enjoyable working environment. Special thanks go to Dr. J. Barrett for the assistance of solving the thermal mobility problem, and to G. Statter for some assistance with computing. Dr. C. S. Kim, Dr. K. Kowari, and Dr. L. Demeio are also thanked for helpful discussions. Finally, I wish to express my great appreciation to my parents, Kim Leung, and W. F. Chan for all their patience and encouragement. This thesis is dedicated to them. x \ C H A P T E R 1 INTRODUCTION In the last 20 years, the growing interest in a large number of fields including radiation physics and chemistry [1], radiobiology [2], laser systems [3], discharge devices [4], and at-mospheric applications requires an understanding of the motion and the thermalization of electrons through gases. A number of investigations of the thermalization process of low energy electrons has been made both theoretically and experimentally. The theory has been considered by several people and developed via different methods. These include (i) the displaced pseudo Maxwellian approximation introduced by Mozumder [5]; (ii) the Monte Carlo simulations used by Koura [6]; (iii) the approach based on the Fokker-Planck equation developed by Shizgal and coworkers [7,8,9]. On the other hand, the experimental method involves the irradiation of a gas sample in a cavity with nanosecond pulsed X-rays. Electrons produced through ionization of the gas in the cavity have high kinetic energies and lose their energies through collisions with the gas atoms or molecules. If a weak uniform electric field is applied throughout the gas, a steady flow of the induced electrons along the field lines will develop. The velocity of the center of mass of the electron cloud or equivalently the average velocity of electrons, is called the drift velocity w, and this 1 velocity is directly proportional to the electric intensity E, provided that the field is kept weak. Thus w=fiE, where the constant of proportionality /i is called the mobility of electrons. With the definition of the electron mobility, one can define the electron conductivity as We/z, where N is the total gas density and e is the charge of a electron. A number of experimental techniques, involving the measurement of microwave electrical conductivities [10], [11] and electron transient mobilities [12] has been observed. Both theoretical and experimental results provide for a useful comparison. Suzuki and Hantano reported pulsed microwave thermalization times for a helium-krypton mixture [11]. With the assumption that the reciprocal of the thermal relaxation times, 1/Pr for the mixture varies linearly with the mole fraction of helium gas, Xue, the relaxation time for pure helium was obtained from a linear fit of measurements of 1/Pr for the mixture at small X / / e and extrapolation to Xtf e =l- The theoretical I/PT value is determined from the Pr(l . l ) and PT(1.01) for electrons to decay within 10% and 1% of the thermal value, where a pure exponential decay is assumed. These were taken from the work by Shizgal and McMahon. The linear relationship can also be very useful for predicting the relaxation time for the mixture by simply knowing the relaxation time for each component. However, recent calculations by Shizgal and Hatano [13] have shown that the linearity in density dependence of the thermalization times in helium-krypton mixtures is not correct. My interest is to study the composition dependence of the thermal and transient electron 2 mobilities in binary rare gas mixtures. With the assumption that electrons are dispersed dilutely in a large excess of a mixture of rare gas moderators at equilibrium, only the col-lisions between electrons and moderator species are included. Furthermore, only elastic collisions are considered. The average electron mobility and the velocity distribution func-tion are determined from the approach based on the Fokker-Planck theory and the numerical methodology developed by Shizgal and coworkers [7,8,9]. The velocity distribution function /(v ,t) is given by a linear Fokker-Planck equation derived from the space independent Boltzmann equation [3,14]. The orientation of the velocity vector v relative to the direction of the external field E, is given by an angle 0. The electron velocity distribution /(v,t) is expanded in the Legendre polynomials, P/(cos#). With the substitution of this expansion into the Fokker-Planck equation, an infinite set of coupled differential equations is obtained for the expansion coefficients fi(v,t) (see Chapter 2, Eq. (5)). In order to solve this infinite set of equations, the expansion needs to be truncated at some finite number of terms. Owing to the small electron rare gas moderator mass ratio m e /m , (where rae is the mass of electron and m, is the mass of ith moderator), the randomization of the electron velocity will occur faster than the rate of energy exchange. The electron velocity distribution function will therefore become isotropic in a very short time scale compared with the energy exchange. Consequently, only the two terms with /=0 and 1 are retained, and the 3 set of coupled equations reduces to the two equations for / 0 and f\. This procedure is refered to as the two-term approximation. If inelastic collisions are important, then the anisotropy of the electron velocity distribution function is larger and more terms in the expansion of the distribution are needed. Furthermore, for a steady field as discussed in detail in Chapter 2(A), the relaxation of fi to its steady value occurs on a time scale of the order of m e /m , shorter than the time scale for fo to approach equilibrium. Consequently, I set dfi/dt=0 and express f\ in terms of fo. With the replacement of / i for / 0 , a single partial differential equation for the isotropic component of the velocity distribution function, fo{v,t) is obtained. On the other hand, for a microwave electric field, the procedure to reduce the coupled set of equations to a partial differential equation for fo is somewhat more involved and is discussed in Chapter 2(B). An important assumption in this case is to average over one cycle of the microwave field such that the resulting differential operator is time independent. This approximation should be very good at sufficiently high microwave frequencies for which the distribution fo does not vary appreciably over one cycle of the microwave field. The steady solutions at infinite time fo(v, oo) can be calculated from this Fokker-Planck equation by setting dfo/dt=0 and referred to as the thermal electron velocity distribution M(v). For a d.c. electric field, it is the Davydov distribution, whereas for a (time averaged) microwave field, it is known as the Margenau distribution [8,9]. These distributions can 4 be easily calculated and involve simple integrals over the electron-moderator collision cross-sections. One of my studies is to investigate the application of Blanc's law [15] to electron thermal mobility in mixtures. Blanc's law was introduced long ago in connection with the deter-mination of the mobility of ions in gaseous mixtures. It states that the reciprocal of the ion mobility varies linearly with the gas composition. A great deal of work has been done [16]-[20] to study the range of validity and the deviations from Blanc's law for ion mobility in mixtures. Blanc's law has been shown to be accurate [20] when the mean ion energy remains the same in the gas mixture as in the single gas and also the ion velocity distribution is not much disturbed by changing the gas composition. For this to be so the energy which is acquired from the electric field must be negligible compared to the thermal energy. When the mean ion energy is no longer equal to the thermal energy and is balanced between the energy acquired from the electric field and energy lost by collisions, Blanc's law break down. In this thesis, a comprehensive study of the composition dependence of the electron mobility in mixtures is carried out that complements the previous work on ion mobilities. The study of deviations from Blanc's law for ions requires a calculation of the ion velocity distribution function from the Boltzmann equation. Since the ion-rare gas mass ratio is close to unity, the differential operator approximation of the integral Boltzmann collision operator is not applicable, and the two-term approximation is inaccurate. [21]. Consequently, 5 the ion velocity distribution function cannot be expressed in a simple, explicit form as can be done for the electron velocity distribution function. The electron thermal velocity distribution function can be written in terms of simple integrals so that the thermal mobility can be determined directly with a numerical integration performed with Simpson's rule. The electron thermal mobilities are calculated for a wide variety of electric field strengths and concentrations, and deviations from Blanc's law is studied in detail. The linear mixture rule can be very useful for the electron transport properties in gases for which the properties of the mixture are related to the properties of the individual components of the mixture, if one knows the details and the range of the validity of such linear relationships. An important factor that determines the magnitude of the departures from Blanc's law, which depends strongly on the applied field strength and the concentration of the mixture, is the nature of the energy dependence of the cross-section ratio C i / o ^ , where G\ and cr2 are the electron-moderator collision cross-sections for components 1 and 2, respectively. Varying either the electric field or the composition of the mixture changes the energy region over which the collision cross-sections are sampled, and hence affect the deviations from Blanc's law. The cross-sections employed are those reported by Nesbet [22] for helium, by O'Malley and Crompton [23] for neon, by Mozumder [5,24] for argon, krypton and xenon. Both helium and neon have cross-sections that are approximately independent of the electron energy, while argon, xenon, krypton gas have cross-sections that vary rapidly with electron energy 6 and have a distinctive minimum at low energies. The minimum in the collision cross-section for the heavier gases(Ar,Kr,Xe) is known as the Ramsauer-Townsend minimum. The effect of the Ramsauer-Townsend minimum on deviations from Blanc's law can be understood by studying different pairs of gases such as (i)He-Ne, a mixture for which both gases have no Ramsauer-Townsend minimum, (ii)Kr-Xe, a mixture for which both gases have Ramsauer-Townsend minima, (iii)He-Ar, a mixture for which one gas has a Ramsauer-Townsend minimum, the other gas does not. For a nearly zero field strength case (E/N~Q), the magnitude of the deviations from Blanc's law is due to the departure of the cross-section ratio from the mobility ratio u\j u2 (where \i\ and \i2 are the thermal mobility for components 1 and 2, respectively) (see Chapter 3 (B)). If the cross-sections are constant, the deviations from Blanc's law are equal to zero which agree with the results reported earlier by Petrovic [25]. If the cross-sections are nearly equal and have a similar energy dependence, then the deviations from Blanc's law are small. On the other hand, for a higher field strength, the magnitude of the deviations from Blanc's law is due to the departure of the cross-section ratio from the appropriate mass fraction ratios 2M\ and 2M2 (where Mi = m\l(m\ -f m2) and M2 = m2/(m,i + m 2) are the mass ratio of components 1 and 2, respectively) (see Chapter 3 (B)). If the cross-section ratio is closer to the mass ratio, the deviations from Blanc's law are small. In order to calculate the transient mobility, the velocity distribution function is written as 7 a general Fokker-Planck equation of the form —Lg=dg/dt, where L is the Fokker-Planck op-erator and g(v, t) is defined as fo(v, t) = M(v)g(v, t) [8,9]. The solution of the Fokker-Planck equation is determined from the expansion of the electron velocity distribution function in the eigenfunctions of the Fokker-Planck operator. The average electron mobility is expressed as a sum of exponential terms with each term characterized by a different eigenvalue of the Fokker-Planck operator. Since the reciprocal of the eigenvalues are characteristic relaxation times, for sufficient long times the decay of the electron mobility will be a pure exponential with the reciprocal of the smallest eigenvalue, 1/Ai equal to the relaxation time when the velocity distribution is close to the thermal distribution. For a gas mixture, the relaxation time is governed by the eigenvalues of the Fokker-Planck operator which is the sum of the corresponding Fokker-Planck operators for each gas weighted with the appropriate mass and density factors. However, for a mixture, the eigenvalues of the Fokker-Planck operator representing the mixture will not in general be given by the sum of the corresponding eigenvalues representing each gas seperately. It is clear that the extrapolation of the results to small helium densities and the linear dependence of the 1 / P r on mole fraction for helium-krypton suggested by Suzuki and Hantano can be wrong. A numerical method known as the QDM (Quadrature Discretization Method) [7,8,9] which is based on the DO (Discrete Ordinate) representation, the representation of a func-8 tion by its values at the set of quadrature points, is employed for solving the Fokker-Planck eigenvalue problem. The eigenfunctions are determined at a discrete set of points which co-incide with the points of a quadrature procedure based on speed polynomials B„(x) (where x = v/v0 with v0 = (2fcT/m e)2, is the dimensionless speed) orthogonal with respect to the weight function, x2e~*2 (see Chapter 2(B)). This representation of the eigenfunctions is equivalent to the representation in terms of the coefficients in the polynomial expansion, as there exists a1 unitary transformation between the two representations. However, the operator L is self-adjoint with respect to the steady solution x2M(x) and not with respect to the weight function. As discussed by Blackmore and Shizgal [8], the symmetric repre-sentative of L can be constructed by evaluating the matrix representative in the basis set of functions orthonormal with respect to the steady solution, and transforming to the DO representation with the appropriate unitary transformation(see Chapter 2(C)). The result is that the eigenvalues and corresponding eigenfunctions are determined from diagonalizing the DO representation of the Fokker-Planck operator. An important advantage of the QDM is to calculate the matrix representation of the Fokker-Planck operator without requiring the explicit integral evaluation of the matrix elements. Also, the eigenfunctions are determined at the set of quadrature points appropriate for the evaluation of integrals for the calculation of the electron mobility. As with Blanc's law, a study of the validity of such linear rules for relaxation times 9 is important. Some of my work has been done on testing the validity of the linear rela-tionship between the 1/PT value and mole fraction for other mixtures. The linear rule is valid for mixtures with energy independent cross-sections (i.e.without Ramsauer-Townsend minimum), whereas for mixtures with strongly energy dependent cross-sections (i.e.with Ramsauer-Townsend minimum ), the linearity is lost. To extend the study of the effect of Ramsauer-Townsend minimum on the transient mobilities and their corresponding relax-ation times, a wide variety of field strengths, concentrations, and oscillatory frequencies for mixtures with Ramsauer-Townsend minimum and mixtures without Ramsauer-Townsend minimum are considered. It is important to mention that the results are very sensitive to the region of the electron-moderator collision cross-section that is sampled and whether a Ramsauer-Townsend minimum occurs in this energy region. Chapter 2 outlines the theoretical approach for the calculation of the thermal and tran-sient mobilities which includes the Fokker-Planck theory, the Q D M numerical method, and the explicit formulae for the mobility. A detailed discussion of the thermal mobilities and their deviations from Blanc's law is presented in Chapter 3, whereas Chapter 4, provides information for the composition, frequency and field strength dependence of the transient mobilities and the effect of the Ramsauer-Townsend minimum. 10 C H A P T E R 2 THEORY (A) Introduction In this chapter, the theoretical approach for the determination of the composition depen-dence of the thermal and time-dependent electron transport properties in binary rare gas mixtures under the influence of steady (d.c) or oscillatory (a.c) electric fields is presented. The approach is analogous to the one employed by Viehland et al. for a single component gas system [26]. Here the extension to a gaseous mixture is made, and the composition depen-dence of the electron velocity distribution function and of the electron transport properties is studied. The electron velocity distribution function and the transient transport properties are determined by solving the Fokker-Planck equation which is derived from the Boltzmann kinetic equation. If the orientation of the velocity vector v of an electron relative to the electric field is given by angle 0, then the velocity distribution function can be expanded in a set of Legendre polynomials. where P/(cos#) are the Legendre polynomials. With the substitution of this expansion in oo (1) 1=0 11 the Boltzmann equation, a set of coupled differential equations for the quantities fi(v,t) is obtained as discussed Chapter 1. The relaxation of the electron velocity distribution function can be divided into two time domains, termed time domain (i) and (ii). The time scale ratio of domain (i) and (ii) is equal to m e/m,-, where m,- is the mass of iih moderator and so domain (i) is much smaller than domain (ii). For small electron-rare gas mass ratios, the anisotropic components of the velocity distribution function, i.e., fi(v,t),l > 0, decay quickly to zero in time domain (i), while the isotropic component, fa(v,t), which involves energy exchange, approaches the thermal distribution much more slowly in time domain (ii). If only the elastic collisions between electrons and rare gas moderators are considered, owing to the small electron-rare gas mass ratio, electrons will leave a collision with almost the same speed that it enters and the velocity of the neutral atom will not be changed. Electrons could be expected to suffer large directional changes of velocity. Thus the distribution of electron velocities ought to be nearly isotropic in velocity space, and therefore only the two terms fo and / i are retained. With this two term approximation, the Fokker-Planck equation reduces to two equations for / 0 and f\. Since I am concerned with the transient behaviour in time domain(z), dfi/dt =0, and the coupled equations for fo and f\ reduce to a single differential equation for fo. However, when inelastic collisions are important, the anisotropy of the electron velocity distribution function is larger and more terms in the expansion of 12 the distribution are needed. Consequently, the isotropic properties of the electron velocity distribution will be destroyed. The use of the two-term approximation has been shown by Lin et al. [21] to be an excellent approximation for elastic electron-moderator collisions. In Section 2, the details of the Fokker-Planck equation are presented and I show that, (2) where L is the Fokker-Planck operator. The solution of Eq. (2) can be written in terms of the eigenvalues and eigenfunctions of the operator L, as is also discussed in Section 2. In Section 3, I described the quadrature discretization procedure (QDM) [7,8,9,27] employed to determine the eigenfunctions and eigenvalues. The QDM yields the value of the eigenfunctions at a set of quadrature points, allowing the calculation of integrals over the distribution as quadrature sums. The expressions for the real and imaginary components of the transient mobilities in terms of these quadrature sums are presented in Section 4. 13 (B) Fokker-Planck Equation For a Binary Mixture The methodology for calculating the transient transport properties follows closely the formalism in the earlier papers by Shizgal and McMahon [7,8,9], and the general theory of electron motion in gases has been presented by Kumar et al [14]. The major assumption of the work is that the anisotropic, spatially homogeneous, electron velocity distribution function / (v, t) in the rare gas moderators is governed by the spatially independent Boltzmann kinetic equation, f + f . V v / = E * [ / ] , (3) where e and m are the electron charge and mass, respectively, and E is the external d.c. or a.c. electric field. «7j is the collision operator which takes into account the effect on the distribution function of electron collisions with the ith moderator [14]. The neutral species are assumed to be present in large excess and characterized by Maxwellian velocity distributions. We express v in spherical coordinates [3] to obtain, d£ + eE dt m 'idf + (i-e)d/ = J2Mfl (4) dv v d£ where v is the electron speed and £ — cos 6 is the cosine of the angle between the velocity vector v and the direction of the electric field. The electron distribution function in Eq. (3) is expanded in Legendre polynomials, Pi{(), /(v,o = £;/,(«, wo- (5) /=0 14 I substitute Eq. (5) into Eq. (4) to obtain, i at m E * § * + i p < i - ? > f ; and then I multiply Eq. (6) by Pi>(£) and integrate over £. Using the orthogonality and recurrence relations of the Legendre polynomials, a set of coupled differential equations for the quantities / / is obtained [14], dfl + eE dt m ( ' ) d ) f +(l + 2 + <l + 1 ) d \ f Representation of the distribution function as a series of differential equations is useful only if a small number of terms can be retained and the rest can be rejected. Owing to the small electron-moderator mass ratio, the randomization of the electron velocities will occur at a faster rate than the rate of exchange of energy. It is useful to define two time domains [7,8]. Time domain (?) is the time scale for the decay of the anisotropic components of the distribution function fij > 0. Time domain (ii) is the time scale for energy relaxation to occur and is characterized by a time T 2 equal to the reciprocal of the electron moderator collision frequency. The time scale characterizing time domain (i), T\, is (rne/rrii)T2 (where m, is the mass of ith moderator) and so is much shorter than r 2 . As the initial anisotropic distribution decays very quickly in time domain (i) before appreciable energy relaxation, the electron velocity distribution will become isotropic on a time scale very short in comparison with the time for energy exchange [28]. Consequently, only the first two terms with / = 0 15 and / = 1 in Eq. (7) need to be retained. The two lowest terms of the collision operator are [14], J? = m rriiV2 dv v\(v)(l + ¥±£-) mv dv fo, J} = -ni?) fl-it) (9) In Eq. (8), Tj, is the temperature of the moderator, k is the Boltzmann constant. The colli-sion frequency for an electron colliding with component i is Ui(v)=Ni<Ti(v)v where <r, is the momentum transfer collision cross-section, and iV, is the number density of the component i. For a gas mixture, the effect of electron collisions with each gas i in the mixture on the distribution function is additive and the set of coupled equations, Eq. (7), with the two-term approximation reduces to two equations for fo and f\ which are given by [14], dfp | eE ( ^ 2 / i } _ 1 d dt 3mv2 dv v2 dv t '3 ( E - ^ ) ) ( i + — | o m, mv dv fo, (10) dfx eE'dfo ^ . . (11) For a d.c. field, E is the steady electric field and for an a.c. field, E —Epexp(iut), where Ep is the amplitude, is the frequency of the external field, The mobility is defined by /z(r) = w(t)/E, where w(t) is the net drift velocity, defined as [8], w (0 = f~f{v,t)v2dv • (12) 16 If I substitute Eq. (5) into Eq. (12), I have, Multiplying both denominator and numerator by P 0 , and integrating over £, I obtain, where the orthogonality of the Legendre polynomials has been used. For the a.c. case, since the electric field varies as exp(iu>t), we assume that f\ also has a time variation of the form exp(iut) [6,29]. If we let vc = YLi vii Eq. (11) can be rewritten as, iuMv,t) + + uMv,t) = 0, (15) from which, m(vc + IUJ) Inserting Eq.(16) into Eq.(14) and multiplying both denominator and numerator by (vc — iu), I find that the mobility of a mixture can be written as two parts, a(t) = aR(t) + im(t), (17) the real component, 3m Jo vt + ov Jo 17 and the imaginary mobility /// given by, Nit)=~r 2(v* affiwrf0v*dv. (19) 3m Jo vl(v) + ui2 ov Jo With an integration by parts in Eq. (18) and (19), it can be shown that, M t ) = f - r / o# [ ! c ( y,w r f°v2dv> (2°) 3m Jo ai> v£(v) + u2 Jo Hii*) = ^ fojrl 2 ( \ , t]dvf / / o ^ . (21) 3m Jo f^(u) + Jo In some instances Eq. (20) and (21) give more accurate numerical results, since the derivative of uv3/(v2 + LL)2) can be more accurately calculated than dfo/dv. In the work by Viehland et al. [26], it was found that better results could be obtained if the calculation of dfo/dv can be avoided. With the form of E = Ep sinu;£, as done previously [3,26,30], the general solution of Eq. (11) is, / ! ( « , *) = exp(-M)/ i iv , 0) - ( ^ ) exp(-i/ e<) / ' exp(i/ei') sin (ut')[dfo{^ * V ' t (22) m Jo which can be substituted in Eq. (10) to obtain a single integro-differential equation for fo. In order to reduce the equation for fo to a linear equation with a time independent linear operator [3,26], three approximations are made. The first approximation is to assume that the initial value is arbitrary and could be set to zero, i.e., /j(i;,0)=0. This is because of the lack of information and experimental results concerning the initial state for short times in 18 domain (i) that might be sensitive to this initial condition. The second approximation is to assume that the largest contribution to the t' time integral in Eq. (22) is for t' values equal to r, so that the velocity derivative of / 0 can be taken to be constant with t'. Therefore, I am led to approximate Eq. (22) by, ffat) = _ £ ^ d / o ( M ) ] e x p ( _ M ) / • t e x p ( l / c < ' ) s i n ( a ; < ' )^ . (23) m ov Jo The indefinite integral in Eq. (23) can be evaluated and I have, / i (».0 = T~T~t~2\ksinut - u c o s ^ P ^ r ^ - (24) m(i / c Hw 2 ) ov . I now substitute Eq. (24) for / i into Eq. (10) and make the third approximation which is that /o does not vary appreciably over one cycle of the microwave field [3,26,30]. This approximation is reasonable when the oscillations of the electric field are much faster than the time scale for energy relaxation. The differential equation for the spherical component of the distribution function is then found by averaging the result from the substitution of Eq. (24) into Eq. (10) over one cycle of the microwave field. I obtain, df0 e2E2 d 2 vc .a/0, 1 d iv (..7 , ..J-arl = 3^: « 3 ( E - ^ ( « ) ) a + — h • m, mv ov fo, (25) dt 3m2v2 dv i/2 + u2 dv v2 dv where E2=\E2p is the mean square field strength. For a binary mixture, I introduce <7avg, the average cross section and creff, the effective cross section, which are defined as, °~avg = Xl&l + X2O-2, (26) 19 = { ^ — ) X 1 a 1 + ( ^ - ) X 2 c r 2 , (27) m 1 - r - m 2 / * * ' 'mi + m 2 ' where X\ and X2 = 1 — X\ are the mole fractions of gas 1 and gas 2, respectively. Since vc = YJ,-1/,, I also have that uc(v) = (iViaj + N2<r2)v. (28) I now introduce the reduced mass, M = , and the dimensionless speed, x = u/t>0 with vo = \J2kT/m. The total number density of gas is N=N\ + N2 and the temperature T differs from the bath temperature, TJ,. The quantity s2 = T/Tf, is a parameter used to scale the points in the discrete ordinate method, as discussed elsewhere [8,9]. I also introduce the dimensionless time, t' = t/to with, (29) Nm<rn /2kTh ' 2 M V rn and the dimensionless cross sections £ e / / = C e / z / ^ o and aavg = (Tavg/ao , where o~o is some convenient hard sphere (constant) cross section. With these definitions, Eq. (25) can be rewritten in the form of, dfo _ s_d_ dt' x2 dx (a/sy - ( <s2 xaavg[1 + u}2/(Nvocr0xaavg)2'_ 2 ) i0~x~2dx 2&effX4 + 4 x 3 q e / / 5 S2 dx /o. (30) After some algebra, I get that, (31) 20 where B(x) is given by, B(x) = xae}} + . W*? — j - , (32) x<Tavg[l + u>z/(Nv0o-0xcravg) j and depends on the ratio of the frequency of the microwave field to the moderator density. For a steady external electric field, u;=0, Eq. (32) reduces to, B(x) = xaeff + t ^ J l (33) The quantity a is a parameter that is a measure of the electric field strength and is given by, From Eq. (31), the steady solution attained at t' = oo is easily shown to be given by, /o(x,oo) = M(x) = C exp 2 5 2 r ^ i i d x ' Jo B(x') (35) 1(x') where C is a normalization constant. The function M(x) is the steady, thermal electron distribution at equilibrium. In the absence of an electric field, B(x) is equal to xfrefj and M(x) is a Maxwellian distribution. For a steady d.c. electric field, M(x) is the Davydov distribution[9], whereas for a microwave field averaged over a cycle of the oscillatory field, M(x) is known as the Margenau distribution [26]. The real and imaginary components of the electron thermal mobility, / i ^ and fi[h, are defined as in Eqs. (20) and (21) with / 0 = / 0(x,oo) given by Eq. (35). With the ther-mal electron velocity distribution, I can calculate the electron average thermal mobility by 21 numerical integration. The composition dependence of the electron thermal mobility will be discussed fully in the Chapter 3. Here, a general development of the calculation of the time-dependent electron mobility is presented. Numerical results for the transient mobility will be discussed in Chapter 4. In order to calculate the transient mobility, Eq. (31) is written in the form of a general Fokker-Planck equation. I write, f0(x,t') = M{x)g0(x,t'), (36) and substitute Eq. (36) in Eq. (31), and use Eq. (35) to obtain, dg0 _ 1 dt' ~ s ( o c 2 , . 2 A i 2B(X) , dB(x) dgQ d2g0 (37) Eq.(37) can be written in the form of a Fokker-Planck equation, where, | ? = - i » c (38) I - | ( - < + J K . ) | ) , (39) with a, x n 2 2- 2B(x) dB(x) A(x) = 2s2x2aeff ^ - i - i . (40) X ax Eq. (39) defines the Fokker-Planck operator L, a differential operator in the speed variable which is parameterized by the electron-moderator momentum transfer cross sections, the 22 composition of the gas mixture, and the external electric d.c. or (time averaged) a.c. field. The method of solution of Eq.(37) follows the work in the previous papers by Shizgal and coworkers [7,8,9,27]. The solution can be written, gQ(x, 0 = exp(-Lt)go(x, 0), (41) where go(x,0) is the initial value. If fa and A* are the eigenfunctions and associated eigen-values of the Fokker-Planck operator L, that is, Lfa = Xkfa, (42) then I can expand the initial value go(x,0) in the eigenfunctions fa and rewrite go(x,t') as, oo g0(x, t') = a>kfaexp(-\kt'), (43) k=o where the ak quantities are the expansion coefficients of the initial distribution in the eigen-functions (f>k which form a complete set, that is, roo a k = x2M(x)fa(x)go(x,0)dx. (44) Jo Using Eq. (36) for / 0 , with g0 given by Eq. (43), the average transient mobilities, defined by Eq. (20) and (21) can be written in the form, tiR(t') = f2tfexp(-\kt'), (45) 23 and, /*/(*') = Erfesj>(-A f c t ') , (46) k=o where the coefficients /z* and /zj[ are expressed in terms of the coefficients a* and integrals with the eigenfunctions <f>k [7,8,9] by, - ? r x « ) w . ) ^ - i i " ( v v + V (47) o Jo 3mu0 xa-avgF ^ = j l M ( x ) ^ ( x ) 3 ^ ( i V ) [ x^T - 1 ^ - (48) 24 (C) Quadrature Discretization Method My objective is to determine the real and imaginary components of the electron mobil-ity given by Eq. (45) and Eq. (46), which requires the evaluation of the eigenvalues and eigenfunctions of the Fokker-Planck operator, L (Eq. (39)). Hence, the theoretical problem is reduced to the determination of the eigenvalues and eigenfunctions of L. A particularly useful numerical method for the solution of the Fokker-Planck eigenvalue problem has been developed by Blackmore and Shizgal [27]. The method is the Quadrature Discretization Method (QDM) [8,9,27] and is based on the well-known quadrature method for approxi-mating the integral of some function f(x) over the interval [0, oo] with the weight function where the iw, and Xi are the weights and the points of the quadrature procedure. Here I choose B„(x). These were introduced to study electron transport properties [7,27] because a better convergence of the eigenvalues is obtained than with "classical" polynomials (such as the Laguerre polynomials). The discrete points are the roots of the polynomial of order N, i.e. B^i^i) = 0. The polynomials are orthogonal with respect to the weight function, that is, w(x) (49) w(x) = x2 exp(—x2) so that the weights and points are based on a set of "speed" polynomials (50) 25 The QDM is based on the representation of a function by its values at the set of quadra-ture points Xi, known as the D0(Discrete Ordinate) representation. This DO representation is entirely equivalent ,to the PB (Polynomial Basis) representation, which is given in terms of the coefficients an in the polynomial expansion, Because of the orthogonality relation, Eq. (50), the expansion coefficients in Eq. (51) are given by, and Eq. (52) can be evaluated numerically with the quadrature rule, Eq. (49), that is, N an = Y^wiBn{xi)f(xi). (53) i=i This is a matrix equation that transforms the DO representation to the PB representation. However, it does not appear that the transformations are symmetric owing to the presence of Wi in Eq. (53) and not in Eq. (51). In order to symmetrize the two transformation, Eq. (53) and (51) can be rewritten as N f{x) = £ anBn(x). (51) n = l (52) N (54) »=i and N (55) n = l 26 respectively. These two equations give a unitary transformation between the DO and PB representations. The two transformation matrices are T and respectively, with elements, Tin = y ^ B „ ( x , ) , (56) T$ = y/wlBn{Xi). (57) The values of the eigenfunctions are determined by diagonalizing the Discrete Ordi-nate representation of the Fokker-Planck operator as described by Shizgal and McMahon [8,9] However, the Fokker-Planck operator is self-adjoint with respect to the equilibrium distribution x2M(x) and not with the weight function w(x). A symmetric representa-tion of L = [w(x)x2M(x)}~*L[w(x)x2M(x)]* can be constructed by evaluating the matrix representation in the basis set of functions orthonormal with respect to w(x), and trans-forming to the DO representation with the appropriate unitary transformation where L = [x2M(x)]-1L[x2M(x)]. The DO matrix representation of the Fokker-Planck operator can be evaluated and is given by [9,27], 1 N La = - £ B(xk)[Dik + h(xk)Sik][Dkj + h(xk)6jk], (58) s k=i where Dik is the Discrete Ordinate (DO) matrix representation of the derivative operator d/dx, and the function h(x) is, h ( x ) - J ^ l _ WM(x)]>  [ X ) ~ 2w(x) 2x2M(x) ' ( 5 y ) 27 which arises owing to the choice of a weight function w(x) differing from the steady ther-mal electron distribution x2M(x). Since Lij given by Eq. (58) is clearly symmetric, the eigenvalues A* are all real. The corresponding eigenfunctions are, fa = x2M(x) . . . ^•M*)- (60) ^ w(x) Dik is easily found by using the unitary matrix T to transform the derivative operator from the Polynomial Basis (PB) representation to the DO representation. Thus, A ? ° = E ^ f l ( n i , (6i) where T is the unitary transformation matrix defined by Eq. (55), T* is the transpose of T, and DPB is the Polynomial Basis representative derivative operator [28], where, (T% = Tjh (62) TOO Dki = / w(x)Bk(x)B[(x)dx. (63) Jo Using the quadrature rule, Eq. (49), I can write Eq. (63) in the form, DpklB ^J2^Bk(xn)Bl(xn). (64) n Using Eq. (64) in Eq. (61), and the definition Eq. (56) and (57), I find that, DS° ~ E E E M W ^ W B I W ^ , ^ ) . (65) ( 28 Due to the orthonormality of the polynomial set, I have that wnY,Bk{xi)Bk(xn) = 6in. (66) k and Eq. (65) can be reduced to, Dg° ~ y/w&j'EBl{xi)Bl{zi). (67) It is important to note that the eigenfunctions determined with the diagonalization of Eq. (58) are proportional to the zeroth eigenfunction, V > o ( x ) = *>M(x) m ^ w(x) ' with eigenvalue Ao=0. The steady solution is thus determined without explicit evaluation of the integral in Eq. (35). Another important advantage of the QDM, apparent from Eq. (58), is that the evaluation of the matrix represention of the Fokker-Planck operator does not require the explicit integral evaluation of matrix elements and the eigenfunctions are determined at the set of quadrature points appropriate for the evaluation of the integrals in Eqs. (47) and (48). 29 (D) Transient Mobility Coefficients For a binary mixture, the collision frequency in Eqs. (18) and (19) is given by vc = Naavgv which depends on the total density of the gas mixture, N, and the dimensionless average collision cross-section aavg. By introducing the dimensionless speed x and the frequency strength parameter F into Eqs. (18) and (19), given by* xaavgNv0 I can rewrite the real and imaginary components of the transient mobilities for a binary mixture, and after some algebra I get that, • 2 r ° ° * 2 df0 (70) ^ A ' 3mN2 Jo &*vgF dx 1 [ 0 Jo J° (71) If I let, S = / x2f0(x,t)dx, (72) Jo the real mobility fiR and imaginary mobility \i\ can be written, 3mv0S Jo cravgF ox N ^ = ^ r t - r % d x - ( 7 4 ) 30 Integrating Eqs. (73) and (74) by parts, I find that, W) = » / o ( » , 0 [ 2 " a ( V ^ + g)]&, (75) " ^ - Z ^ s W J o / o ( *'* ) [ ~*lg~F ]rfx' ( ? 6 ) where F' and a1 are the first derivatives of F and aavg with respect to x. From Eqs. (75) and (76), I see that the calculation of the time dependence of the mobility coefficients involves integrals of the form [8], -1 t°° P { t ' ) = SJo fo{x^)p(x)x2dx, (77) where for the real component of the mobility, p ( x ) = e 2 /a + ^ Smvo xaavgr and for the imaginary component of mobility, " W ( J V " ^ i j '• ( 7 9 ) v _ e ti; l - a ( 2 ^ „ f l / g a „ g + ^ ) 1 ' / O 1 V A r / I „ 2 i.9 E l J Using Eq. (36), I have that 1 Z - 0 0 P(t')=sJ0 M(x)g0{x,t')p(x)x2dx. (80) With the expansion of go(x, t') given by Eq. (43), I find that the coefficients u£ in Eq. (45) and pl in Eq. (46) are defined by Eq. (47) and (48) in the form of ak f°° pk = — I M(x)4>k(x)p(x)x2dx. (81) O JO 31 In terms of the eigenfunctions ipk(x) given by Eq. (68), this integral can be written in a form appropriate for the quadrature rule of Eq. (49), °fc f°° / \ x2M(x) Pk = -=• / w(x). T - r - V f c ( « ) p ( « ) d o Jo \ w(x) N x2M(xi) ll>k(Xi)p(Xi). (82) S fe N W(Xi) For a pure gas, the real and imaginary components of the transient mobilities have the same form as for the binary system but with a single collision cross section, a, instead of the average collision cross section, aavg. 32 C H A P T E R 3 T H E R M A L MOBILITIES OF ELECTRONS  IN BINARY INERT GAS MIXTURES (A) Introduction Linear mixture rules for which a property of a binary mixture depends linearly on the mole fractions of the constituents and the values of the property for the pure components can be very useful. However, such a linear dependence on composition is often only approximate and a study of the range of validity of such linear laws is important. There are linear mixture rules for both thermal and nonthermal quantities. For example, Boyd [31] discussed the applicability of a linear mixture rule with regard to rate coefficients in reactive systems, and Pritchard et al [32] considered rotational and vibrational relaxation times for mixtures. Shizgal and Hatano [13] have demonstrated the failure of such a linear mixture rule as it applies to electron thermalization times in He-Kr mixtures. Recently, Hwang and Su [33] applied a linear mole fraction relation for the diffusion coefficient of a gaseous mixture. There has been considerable work on the linear mixture rule of the composition dependence of the mobility of ions in a mixture of neutral gases. In 1908, Blanc [15] derived a phenomenological relation between the mobility a of ions in a mixture containing mole fraction Xi of component 33 i and the mobilities /z; of the ions in pure gas component i , 1 = E ~ - (83) Equation (83) is known as Blanc's law and is accurate provided the ion-neutral collision frequency is independent of ion energy. This is the case for ions in atomic systems, where the ion-neutral interaction is approximately an induced dipole interaction. There has been considerable work on the deviations from Blanc's law [16] — [20], and its modification to give useful mixture rules. Much of the effort in this regard is based on the momentum transfer theory [34]-[37] which assumes that the ion-neutral collision frequency varies very slowly with energy. One of the main assumptions of this theory is that averages involving products of the collision frequency and some property of the ion energy or mo-mentum can be replaced by products of the collision frequency evaluated at the average ion energy and the ion property at the average energy or momentum [38]. Momentum transfer theory provides a simple way of obtaining ion transport properties and its predictions for ion drift velocities can often be in excellent agreement with values from solutions of the Boltzmann equation [20]. The purpose of the work of this chapter is to present a detailed study of the composition dependence of the mobility of electrons in rare gas mixtures as a function of electric field strength. The electron transport problem could be considered as a special case of the ion problem in the limit of very small charged carrier to neutral species mass ratio. The trans-34 port properties of electrons in gases are important in a wide variety of practical applications including laser systems, radiation detectors, and discharge devices for which the properties of the mixture in relation to the properties of the individual components is of considerable interest. In the present work, we are particularly interested in the range of validity of Blanc's law in comparison to previous work on ion mobilities. The advantage provided by the elec-tron problem is that the distribution function, with the two-term approximation, can be written down in terms of simple quadratures, and provides a good approximation if only elastic collisions are important. The study of deviations from Blanc's law for ions is more difficult since the ion distribution function must be calculated from the Boltzmann equation and cannot be expressed in as simple a fashion as the electron distribution function. Since the ion to neutral mass ratio is of the order of unity, the differential operator approximation of the integral Boltzmann equation is not applicable. It is important to mention that there are many physical situations for which the electron transport properties in gas mixtures [39]-[42] depend on inelastic collision processes as well as ionisation or attachment. For such situations, the two term approximation is known to be inaccurate. Although the electron distribution function for elastic collisions in the two-term approx-imation is well-known [43], the concentration dependence of the electron mobility in binary mixtures has not been explored fully previously and presents some interesting features which we present and explain. Ogawa et al. [41] have presented some theoretical and experimental 35 results for electron transport properties in rare gas mixtures, assuming only elastic scatter-ing. However they did not fully examine the dependence on mole fraction of gases in the mixture, nor did they provide any explanation for their results. Much of the work on devia-tions from Blanc's law has been for ion mobilities for which the distribution function cannot be written down explicitly. For this case, the distribution function is given by the solution of the Boltzmann equation [42] and the departures from linearity cannot be analyzed as can be done for electron mobilities. In this chapter, electron mobilities in rare gas mixtures are calculated for a wide variety of electric field strengths and concentrations and deviations from Blanc's law, Eq. (83), are shown and explained. The theoretical expression for the electron mobility in mixtures is derived in Section 2, based on the well-known two-term approximation for the electron distribution function. In Section 3, results for 1/// and the percent deviations from Blanc's law as functions of mole fraction for a variety of binary rare gas mixtures for different electric field strengths are shown and explained. 36 (B) Calculation and Results The physical situation considered here is that of electrons of mass m and charge —e drifting through a binary mixture of inert gases at temperature T under the influence of a uniform electric field strength E. The procedure to reduce the coupled set of equations to a partial differential equation for fo which is solved subject to the boundary condition that f(v, 6) vanishes for v —• oo has been discussed in Chapter 2(A). The result is conveniently written in terms of the dimensionless variable x = v/vo, where = 2kT/m, as, /o(x) = exp(-d(x)), (84) where d(x)= fD{x')dx', (85) Jo and 2x D { X ) = l + ay[x>o-eff(x)aavg(x)Y ( 8 6 ) In Eq. (86), the cross-sections, o~avg and <re//, have been defined in Eqs.(26) and (27), and where M = m i m 2 / ( m i + 7712) is the reduced mass,and N is the total number density of gas atoms. 37 The electron drift velocity in the direction of the electric field is given in terms of the distribution function by Eq.(14), where the Legendre polynomial expansion of f(v,0) has been employed and the angular integrations performed. With the replacement of dfo/dv for fx in Eq. (14) and changing variables from v to x, we obtain the following equation for the mobility fi — w/E, oCmvoJo o-avg(x) ox where C is a normalization constant given by, C= exp[-d(x)]x2dx. (89) Jo In general fo and Nfi are found by numerical integration of Eqs. (85) and (88), respectively. The results, Eqs.(84)-(86) and (88), are analogous to those reported previously [44] — [46] and form the basis of the study reported in this chapter. 38 (C) Results and Discussion A detailed study of the thermal electron mobilities in binary rare gas mixtures was carried out. The mobility in the mixtures is given by Eqs. (85) and (88). The momentum transfer cross section for helium calculated by Nesbet [22] and the cross sections for xenon and krypton reported by Mozumder [24] and the xenon cross section provided by Koizumi et al [47] were employed in the calculations. The energy dependence of these cross sections is shown in Fig. 1. A comparison of the sets of xenon and krypton cross sections is shown in Table 1. The integrations in Eqs.(85) and (88) were performed with Simpson's rule. The mobility versus mole fraction and electric field strength calculated in this way are shown in the figures that follow. The basic objective is to study the deviations from the simple linear mixture rule as given by Blanc's law, Eq.(83), and to provide a detailed interpretation of the results obtained. The number of parameters in this study, which includes the pair of gases, the concentration and the electric field strength is moderately large, so that a complete reporting of all the calculations that have been carried out is not practical. I report in this thesis results which illustrate the different behaviour that does occur and which provide for a useful interpretation of the effects. The average electron energy and electron mobility for the pure gases are shown versus density reduced electric field strength in Figs. 2 and 3, respectively. The extent of the heating of electrons serves as a useful way of interpreting the variation of the electron distribution 39 204— 1 L 1 1 E(eV) Figure 1: Momentum Transfer Cross Sections for electron-rare gas collisions. 40 0.300 3.633 5.310 1.648 1.385 0.350 2.234 3.520 1.073 0.899 0.400 1.593 2.299 0.871 0.530 0.450 1.258 1.514 0.640 0.266 0.500 1.053 0.936 0.568 0.151 0.550 0.920 0.579 0.549 0.104 0.600 0.827 0.451 0.562 0.111 0.650 0.814 0.407 0.585 0.141 0.700 0.866 0.383 0.596 0.187 0.750 0.937 0.389 0.612 0.247 0.800 1.037 0.438 0.651 0.320 0.850 1.211 0.535 0.711 0.407 0.900 1.433 0.717 0.781 0.517 0.950 1.668 0.963 0.847 0.647 1.000 1.902 1.222 0.900 0.785 Cross section in A1 from Mozumder, ref 26. (fc) Cross section from Koizumi et al, ref 27. Table 1: Xenon and Krypton Cross Sections 41 2.0 - T 1 1 " 1 " r 0.0 0.1 0.2 0.3 0.4 E/N (Td) Figure 2: Average electron energy in the pure rare gases versus density reduced electric field strength. 42 E/N (Td) Figure 3: Average electron mobility in the pure rare gases versus density reduced electric field strength. 43 function and the mobility in mixtures versus E/N. In the study of the mobility of ions in mixtures [20,42], the mean relative energy of the ion with respect to each component in the mixture in comparison with the mean relative energy in the pure component is an important consideration. This provides a qualitative description for the change of the ion distribution function with changes in composition. In the present case, the heating of the electrons in the different pure gases can be understood by considering a hard sphere cross section and the high field limit (large a) for which the distribution function is the Druveystyn form [43]. For large c t , the term in the denominator in Eq. (86) dominates and with the definition of a in Eq. (87), it can be shown that the average energy, Eavg = (kT/C) /0°° exp(—d(x))x4dx reduces to, where T=300K, Eavg,cr,mi and E/N, are in units of eV., A2, amu and Td , respectively. The result for the mobility from Eq. (88) is, where Nfii is in units of V _ 1 c m _ 1 s _ 1 , and T=300K. The mass mi refers to the mass of the pure component under consideration. The E/N dependence given in Eqs.(90) and (91) is qualitatively correct for helium and neon with cross sections that are nearly independent of energy. The heating in neon is much greater than in helium owing to the larger mass and 1.82 E (90) (91) 44 smaller cross section. The results in Fig. 2 for these gases can be fitted to straight lines with cross sections equal to 1.9A2 and 6.SA2 for electrons in Ne and He, respectively. The mobility in neon is also greater than in helium by a factor involving the ratio of the cross sections and the square root of the mass ratio which is approximately 1.5. The nonlinear behaviour for the average energy and the maxima for the mobilities for electrons in the heavier gases is a manifestation of the Ramsauer-Townsend minima that occur in the cross sections for these gases. For sufficiently high fields, the average energy and the mobility tend asymptotically to the dependence given by Eqs.(90) and (91). In this limit, the average energy is nonthermal and arises from the transfer of energy to the electrons from the field. With the elimination of EjN in Eqs.(90) and (91), we find that \mw2/'Eavg = 0.945 at T=300K. If the thermal energy (ffcT) is added to both Eqs.(90) and (91), the resulting ratio varies from unity at E/N=Q to 0.945 at high EjN. This result as well as Eqs.(90) and (91) apply reasonably well to electrons in He and Ne but not in the other gases owing to the rapid variation of the momentum transfer cross sections of the heavier gases with energy. It is useful to mention that this result appears to be consistent with momentum transfer theory [34] — [38]. Momentum transfer theory assumes that the rates of collisional transfer processes that involve averages over the distribution function and the collision frequency can be expressed as the product of the collision frequency and some other physical quantity of interest evaluated at the average energy. For collision frequencies that vary rapidly with 45 energy, the momentum transfer theory is not expected to yield useful results [38]. As mentioned previously, departures from Blanc's law are anticipated even at zero electric field strength because the electron-atom collision frequencies are energy dependent. It is generally believed that Blanc's law implies that collisions in a mixture are to be counted as simple addition of cross sections [20]. However, as can be seen from Eqs.(26) and (88), since the cross sections are in general energy dependent and appear in an integral, the electron mobility in the mixture does not result from the simple addition of cross sections. In Fig. 4, the percent deviations from Blanc's law, which are negative are shown for several binary gas mixtures for E/N=0. As can be seen, there is a large variation in the percent deviations for different gas mixtures ranging from less than 1% for Kr-Xe in Fig. 4A, curve a, to almost 25% for Ne-Ar in Fig. 4C, curve c. The results in Fig 4. can be understood by expressing the mobility for the mixture in the following way, N a = e — r - f^dx (92) * 3mv0CJo [X,+ X2{f + F(x))]a1 dx ' ^ e f°° x2 df0 where F(.) = 2 - £1. (93) With Eq. (93) in Eq.(92), I recover Eq.(88). If the factor X1+X2n1/u2 is factored out of the denominator, Eq. (92) can be rewritten so that Blanc's law is obtained explicitly together 46 Figure 4: Mole fraction dependence of the percent deviation from Blanc's law for electrons at zero field. X\ is the mole fraction of the first cited gas of the mixture. (A) (a)Kr-Xe, (b)Ar-Xe, (c)Ar-Kr. (B) (a)He-Ne, (b)He-Xe, (c)He-Kr, (d)He-Ar. (C) (a)Ne-Xe, (b)Ne-Kr, (c)Ne-Ar. 47 with a correction factor P , that is, P = P\LBL, (94) where P = — / T 1—<fx, 95 and a similar expression with subscripts 1 and 2 reversed and P(x) replaced with G(x) given by, G(x) . 2L - ( 9 6 ) a2 /xa The Maxwellian distribution obtained from Eqs.(84)-(86) with a = 0 has been used in Eq.(95). The percent deviation is denoted by %Dev = 100(P - 1). For small X 2 , %Dev is given approximately by, lim %Dev « -100 * L (97) x2^o Cp whereas for small Xi we have that Urn %Dev « -100-^- f°° x3e~x'^-dx. (98) * i-0 Cp2 Jo o-2 The percent deviations from Blanc's law for several different gas mixtures shown in Fig. 4 can be interpreted with Eqs.(97) and (98). The magnitude of the deviations reflect the extent of the energy dependence of the cross section ratios in G(x) and F(x) and the departure from the mobility ratio. If the cross section ratio is constant, then %Dev=0. If the cross 48 sections are nearly equal and have a similar energy dependence, than the deviation from Blanc's law is small. This is clearly shown in Fig. 4. For example, the cross sections for Kr and Xe vary in a similar fashion with energy at low energies and the %Dev shown in Fig. 4A, curve a, is very small, less than 5%. Similarly, the He and Ne cross sections are nearly independent of energy and the %Dev shown for this mixture in Fig. 4B, curve a, is also relatively small. At the other extreme are the results in Fig. 4C for Ne and the heavier gases showing large deviations from Blanc's law owing to the large ratios for the Ne cross section to the cross sections of these other gases in the lower energy region sampled at E/N = 0. In Fig. 4C, the initial slopes of the curves for XN C — * 1 are large owing to the large ratio of the cross sections of these gases to the Ne cross section and the large contribution to the integral from F(x) in Eq.(97). The size of the slopes for the different curves, i.e. c<b<a are in the same order as the cross section ratios c^/cfte- The initial slopes on the left for the curves for XN C — • 0 are smaller owing to the small ratio of the Ne cross section to the cross sections of the other gases. The magnitude of the integral in Eq.(98) depends on G(x) and the initial slopes are in the reverse order, i.e. a<b<c. This ordering of the initial slopes on the right and left of Fig. 4C arises from the large values of F(x) and small values of G(x). This interpretation of the behaviour in Fig. 4 is qualitative since the magnitude of the effects depend on integrals over F(x) or G(x), and these functions have positive and negative portions. For example, the initial slopes for Fig. 4A for X i —• 0 are in the order b < a < c 49 whereas in the X\ —• 1 limit these initial slopes are in the order a < c < b, which is not exactly in the opposite order. In Fig. 4B, the %Dev for He-Ne (curve a) is very small for this pair with nearly constant cross sections. The ratio CT1/02 is about 3.5 so that the initial slope on the right is much greater than the initial slope on the left. The results for He-Xe (curve b) and He-Kr (curve c) are similar since the Xe and Kr cross sections are similar. The He cross section is smaller than the cross sections for the other two gases in the low energy region sampled when E/N = 0. By contrast, the ratio of the He cross section to the Ar cross section is greater than 1, and the initial slope on the left for this case (curve d) is much greater than the others, and also smaller on the right hand side of the graph, except for neon. In Fig. 4C, since the neon cross section is smaller than the He cross section, the variation near X\ —> 1 is much more rapid than in Fig. 4B. This general behaviour was verified with calculations, which we do not report here, based on model power law cross sections. Of particular interest, is that in the study of deviations from Blanc's law for the ion mobility in mixtures it is generally accepted that there are no deviations at zero field [20,42]. Presumably there are deviations but they are very small owing to the near independence of the collision frequencies with energy for the ion-molecule systems. The situation for non-zero electric field is more complicated since the concentration de-pendence enters through the distribution function in terms of the mole fraction weighted cross sections, Eqs.(26) and (27). The effect of an increase of the electric field strength on 50 the mole fraction dependence of the deviation from Blanc's law is shown in Figs. 5-9. The variation of \/\i versus -Y]<r for Kr-Xe mixtures is shown in Fig. 5A for four E/N values. The dashed curve is the linear behaviour given by Blanc's law. It is clear that the departures from Blanc's law are not very large for this situation. The changes in the mole fraction de-pendence of the percent deviations with increasing field strength are shown in Figs. 5B-D. The interesting effect is that the %Dev (as a function of E/N) is initially negative, goes positive (Fig. 5B) goes through a maximum at about 0.06Td (Fig. 5C, curve c) and then becomes negative again for sufficiently large E/N (Fig. 5D). The variation of the percent deviation from Blanc's law is shown for Ne-Ar and Ar-Kr mixtures in Figs. 6 and 7, respectively. The effect of positive and negative departures is also seen for these systems but since the cross sections for these pairs of gases are very different, the %Dev can get very large, up to about 90% for Ar-Kr in Fig. 6 and 15% for Ne-Ar mixtures in Fig. 7. The largest effects are observed for mixtures with helium and xenon. In Fig. 8A, we show the mole fraction dependence of %Dev whereas we show the variation of the reciprocal of the mobility, 1/fz, versus X n e for He-Xe mixtures in Figs. 8B-D. Of particular interest is the vary large oscillatory behaviour of l//x versus X'He> and that the mobility in the mixture can be larger than the mobility in either pure gas, see the maximum and minimum in Fig. 51 Figure 5: (A)Variation of 1/u with mole fraction for Kr-Xe. E/N in Td is equal to (a)0.05, (b)0.1, (c)0.2, and (d)0.4. The other graphs are for the variation of the mole fraction dependence of the percent deviations from Blanc's law with E/N in Td equal to (B) (a)O.Oll, (b)0.013, (c)0.015, and (d)0.016; (C)(a)0.018, (b)0.05, (c)0.06, and (d)0.08; (D) (a)0.07, (b)0.08, (c)0.1, and (d)0.40. 52 o.te ' i.b Figure 6: Mole fraction dependence of the percent deviation from Blanc's law for Ar-Kr mixtures: (A) E/N in Td equal to (a)0.004, (b)0.006, (c)0.008, (d)0.01, (e)0.02, and (f)0.03. (B) E/N in Td equal to (a)0.04, (b)0.1, (c)0.5, (d)l, and (e)1.2. 53 Figure 7: Mole fraction dependence of the percent deviation from Blanc's law for Ne-Ar mixtures: {A)E/N in Td equal to (a)0.005, (b)0.007 (c)0.009, (d)0.02, and (e)0.05. (A)E/N in Td equal to (a)0.1, (b)0.25, (c)0.4, (d)0.8, and (e)1.4. 54 Figure 8:(A)Mole fraction dependence of the percent deviation from Blanc's law for He-Xe mixtures; (A)E/N in Td equal to (a)0.01, (b)0.02, (c) 0.04, (d)0.06, (e)0.08, (f)0.10, and (g)0.2. Mole fraction dependence of 1/fi with E/N in Td equal to (B)0.04, (C)0.1 and (D)0.2 55 0.4 0.0 i i i i i i i i i B b.o 0.2 0.4 o.'e O.'B ""VO Figure 9: (A)Mole fraction dependence of the percent deviation from Blanc's law for Ne-Xe mixtures; E/N in Td equal to (a)0.001, (b)0.002, (c)0.005, (d)0.008, and (e)0.012. (B)Mole fraction dependence of 1/u with E/N in Td equal to (a)0.04, (b)0.1, (c)0.2, and (d)0.3. 56 8C. This behavior should be compared with the smaller effects obtained for Ne-Xe mixtures in Fig. 9. An important factor that determines the magnitude of the departures from Blanc's law is the extent of the heating of the electrons in the different binary mixtures as summarized in Fig. 2. The heating is the least efficient for He for all E/N and large for Ar and Kr at low and moderate E/N. The heating in Ne and Xe is somewhat intermediate, at least for low E/N. The heating of electrons is greater in Kr than in Xe for the same E/N owing to the smaller Kr cross section and deeper Ramsauer-Townsend minimum for Kr. An important feature is that for the heavier gases (Ar, Kr and Xe) the electron distribution function (rr2/o) can be bimodal. The range of E/N for which a bimodal distribution function occurs is approximately (0.015 - 0.05)Td for Kr and (0.03 - 0.115)Td for Xe, respectively. With an increase in electric field strength the electrons are more rapidly heated in Kr than in Xe. This region of bimodality in the distribution function is where Eavg shown in Fig. 2 increases rapidly with increasing E/N. For Ar, the range of E/N for which a bimodal distribution occurs is approximately (0.005 - 0.02)Td, which is at lower E/N than Kr or Xe owing to the smaller cross section for Ar and a Ramsauer-Townsend minimum occuring at lower energies. The behaviour shown in Figs. 5-9 can be understood by considering an expansion of P valid for small Xi and small X2. Analogous to the procedure employed for the zero field 57 case (Eq.(95)), we can write, P = — T 1—dx 99 and a similar expression with the subscripts 1 and 2 interchanged and in place of F(x). With the expansions D(x) = D^ix) + X2D[1)(x) + (100) d(x) = "4 0 )(x) + ^ d P o o + (101) C = C f + X 2 Cj ( 1 ) + (102) in Eq.(99), we have that in the limit of small X2, Hm %Dev » " 1 0 0 ~ f - j f [^ (D _ ^(oyo + F ( x ) + c f ) / C f ) ] dx. (103) With expansions valid for small Xi analogous to Eqs.(100)-(102), we also have that, U m ^ o D e v w - l O O ^ - j T [D^ - Z t f V ? * + G(x) + C f V c f > ) ] dx. (104) In Eqs.(103) and (104), ( 1 ) 2 « V i y a / a 1 - 2Ma) ^ ( X ) " (a2 + x2M2o-2y ' ( 1 0 5 ) and 58 The important aspect is the dependence on the quantitites {o2/<J\ — 2M2) and (<Xi/cr2 — 2M\). If these factors vanish, then all quantities in square brackets in Eqs.(103) and (104) are zero and since the effects vanish to first order, they can be expected to be small. This was verified in several calculations for a fictitious model system with <7Kr = 0.78oxe and o~\T = 0.64&7j<r where the numerical factors arise from the appropriate mass fractions. The % Dev for these model systems, which are not reported, were of the order of 1%, considerably smaller than the deviations shown in Figs. 6 and 7. Hence, quite generally, the size of the deviations from Blanc's law is given qualitatively by the departure of the cross section ratios from the appropriate mass fractions, 2Mi or 2 M 2 . The variation with E/N in Eqs.(103) and (104) enters through the distribution functions of the pure components (x2 exp(—d20 )^ or x2 exp(—d^)) and the quantities in square brackets. With increasing electric field strength, the distributions peak at higher energies and can be bimodal as discussed previously. The quantity in square brackets, which can be positive and negative, is sampled at different energies with increasing field strength. The change in the initial slopes versus E/N near mole fraction equal to 0 and 1 is determined by the change in the distribution function of the pure component with field which can be qualitatively described in terms of the extent of the heating and the range of bimodality. It is clear that the dependence on mole fraction and field strength is not simple, because the quantities d^ and d^ are integrals (see Eq.(85)), the normalization of the distribution functions changes 59 with field strength and the % Dev is an average over energy. This average involves positive and negative contributions which can cancel. We here present a qualitative explanation for the results obtained that provides a useful physical interpretation of the behaviour shown and which can be used to predict the results for other situations. In Fig. 5B, the initial slope is large for X]<r —• 1 and varies rapidly with E/N owing to the smaller Kr cross section relative to the Xe cross section and the more rapid displacement of the distribution function to higher energies for Kr, i.e. the electrons are hotter in Kr than in Xe for the same E/N. By contrast, the initial slopes for XK F —* 0 are smaller and do not appear to vary as rapidly with E/N. For the higher E/N values shown in Fig. 5C, the curves are more nearly symmetrical about X]<r = 0.5 reflecting the fact that at higher energies the two cross sections are more nearly equal, although the ratio is not equal to the mass fraction, so that there is a sizeable deviation from Blanc's law. The % Dev decreases with increasing field strength and goes negative as shown in Fig. 5D. The smaller effects arise because the cross section ratios are approaching the mass fractions. The deviations are negative because the negative portion of the quantity in brackets in Eqs.(103) and (104) dominates the integrals at higher E/N. In Fig. 6A, the large initial slopes for XA F —• 1 arise owing to the larger cross section ratio, 0"Kr/o"Ar) and the greater heating of the electrons in Ar for this smaller range of E/N. The initial slopes near XAr = 0 are much smaller, owing to the small cross section ratio 60 °Ar/cKr a n u the change with E/N is not as rapid. For the higher E/N values shown in Figs. 6A and 6B the variation with XA r is more symmetrical about XA r=0.5 and the deviations are much less owing to the near equality of the cross sections at these higher energies. Note that the Ar and Kr cross sections are more nearly equal at higher energies and differ from the Xe cross section. The behaviour shown in Fig. 7A is consistent with this interpretation. The rapid variation of the initial slope for XN c —* 0 (in particular, curves a and b) and the large slopes arise from the rapid change in the electron distribution function in Ar and the significant difference of the cross section ratio cr^e/o'Ar from 2MAf for this energy range. The variation of the initial slopes versus E/N for XN C —^ 1 is not initially as rapid as it reflects the decreased heating of the electrons in Ne. For larger E/N, the higher energy portion of the cross sections is sampled where the cross section ratio is closer to 2MA f or 2MNC SO that the %Dev decreases. The very large deviations from Blanc's law for He-Xe mixtures are shown in Fig. 8. The results in Fig. 8A should be compared with the results for Ne-Xe mixtures shown in Fig. 9A which show smaller deviations and a different field dependence at the mole fraction limits, X i —* 0 and X i —» 1. An important aspect of the He-Xe system is the disparity in the heating of the electrons in He relative to the heating in Xe as shown in Fig. 2. By contrast, the heating in Ne is qualitatively of the same magnitude as in Xe although the detailed variation with E/N is different. The large mass disparity in the He-Xe system as compared 61 with the Ne-Xe system and the existence of bimodal distributions in pure Xe and hence in He-Xe mixtures at low He mole fraction play an important role. The variation of the % Dev shown in Fig. 8A is analogous in form to the Ne-Ar system shown in Fig. 7A except the effects are much larger in He-Xe and range from -60% for E/N=0.04 in Fig. 8B to 180% for E/N=0.2 in Fig. 8D. A useful way to interpret some of these results, that complements Eqs.(103) and (104), is to consider an integration by parts in Eq.(88) so that the mobility is given by, The quantity a is in units of a cross section which can be positive and negative depending is shown in Fig. 10A for He-Xe and Fig. 10B for Ne-Xe. If a > 0, there is clearly a negative contribution to Nu. The overall thermal mobility is positive although there are nonequilibrium situations for which the mobility can be negative [8,12]. For a hard sphere cross section, there are no negative contributions to Nfi. For pure He and Ne, there are small negative contributions. The rapid change in the %Dev with E/N in Fig. 8A and the decrease in n (increase (107) where (108) on the sign of a' . The variation oi 1/cr with reduced speed for several mole fractions 62 t ' W " 4 x 6 1 10 Figure 10: (A)Var ia t ion of 1/a w i th reduced speed for He -Xe wi th XHC equal to (a)0, (b)0.2, (c)0.4, (d)0.6, (e)0.8, and ( f ) l . (B)Var ia t ion of 1/a w i th reduced speed for Ne-Xe wi th Xse equal to (a)0, (b)0.2, (c)0.4, (d)0.8, (e)0.95, (f)0.98, and (g ) l . 63 in 1/u) in Fig. 8B for small XH c is a result of the large change of the electron distribution from the form in pure Xe on the addition of small amounts of He with little change in o in Eq.(108). The addition of He results in a rapid cooling of the electrons and for field strengths E/N less than approximately 0.05Td the mobility decreases, 1/u increases as in Fig. 8B. The decrease in u occurs because the cooling of the distribution displaces it to regions where l/o~ is small; see Fig. 10A. By contrast, the addition of Xe to He does not significantly alter the electron distribution function. There is a small heating of the electrons but the peak in the distribution function is at energies below the maximum in 1/<T. Hence, // decreases (1/u increases) owing to the decrease in 1/a at these energies. The %Dev is large at small He mole fractions and small at small Xe mole fractions owing to the large aue/oxe ratio and Small <7Xe/c"He-For E/N greater than about 0.05Td, the electron distribution function in pure Xe peaks at much higher energies beyond the maximum in 1/a in Fig. 10. Furthermore there is no negative contribution to u. With the addition of He there is a cooling of the electrons and a displacement of the distribution function from the RHS to the LHS of the peak in 1/a. Consequently, on addition of He, /x increases and 1/u decreases for this E/N as shown in Fig. 8C. With continued addition of He, u exhibits a maximum (1/u a minimum) and there is a decrease of u with increasing He. It is important to notice that the mobility at this maximum is greater than the mobility in pure He. The position of the minimum occurs at 64 higher X n e for higher E / N as shown in Fig. 8C. The increase in 1/fi beyond the minimum is a result of the decreasing 1/a as in Fig. 8B. The variation of the mobility versus E / N for different He mole fractions is shown in Fig. 11. The dashed lines show the variation for pure He and pure Xe, respectively. The mobility in He-Xe mixtures can be greater than in pure He and less than in pure Xe. A similar behaviour is shown for He-Ar in Fig. 11B for a lower E / N range. At still higher E / N the peak in the Xe distribution is displaced to higher energies. The addition of He causes a cooling of the electrons, displacement of the distribution to lower energies and there is a minimum in 1/u but displaced to higher X# e and broader. At higher E/N, the higher energy portions of the cross sections is sampled where the He and Xe cross sections are comparable and cross. The maximum in 1/u is barely discernible in Fig. 8D, whereas it dominates Figs. 8B and 8C. By contrast, the addition of Ne to Xe there is no large change in the electron distribution function since there is no great difference in the heating of the electrons in the two gases (see Fig. 2). The %Dev in the limit X ^ —> 0 is not large and does not vary rapidly with E / N owing to the small crjve/<7xe ratio. The large variation with E / N in the limit X ^ —* 1 arises from the large <7xe/c./Ve ratio. The graphs of 1/fi versus Ne mole fraction in Fig. 9B do not exihibit the minima shown in Fig. 8C and 8D. Consequently, graphs of fx for Ne-Xe mixtures versus E / N for different Ne mole fractions as in Fig. 11 are bounded by the mobilities of 65 I • • ' » 1 .0 0.1 0.2 0.3 E/N (Td) 1 6 0 1 i i 1 i ' i ' i 1 i 0°00 0.02 0.04 0.06 0.0B 0.10 E/N (Td) Figure 11: Variation of the mobility versus field strength: (A) He-Xe with X H T equal to (a)0.04, (b)0.2, and (c)0.5. (B) He-Ar with X H T equal to (a)0.02, (b)0.06, (c)0.1, (d)0.2, and (e)0.5. 66 the pure gases. C H A P T E R 4 TRANSIENT MOBILITIES OF ELECTRONS  IN BINARY INERT GAS MIXTURES (A) Introduction In recent years, there has been a growing interest in experimental and theoretical studies of the transient behavior and the thermalization of electrons in gaseous mixtures due to the importance to gaseous electronics [48], laser excitation [3] and discharge devices [4]. A number of experimental techniques involving the measurement of the microwave electrical conductivity (Warman and Sauer[10], Suzuki and Hantano[ll]) has been developed and provide a possible comparison with theoretical calculations. My main objective is to calculate the electron transient mobilities in binary mixtures and the corresponding thermalization times which is taken to be the time required for the decay to within 10 percent of the thermal value. The method that I employed is the Fokker-Planck theory derived from the Boltzmann Equation. The solution of the Fokker-Planck equation is obtained from an expansion of the electron distribution function in the eigenfunctions of the Fokker-Planck operator. The complete method has been fully discussed in Chapter 2. The representation of the transient electron mobility in terms of the eigenvalues of the 68 Fokker-Planck operator is a useful representation of the time dependence. In particular, it provides an estimate of the exponential approach to equilibrium and a possible comparsion with the Suzuki and Hantano experiments. The present studies are also directed towards a study of the effect of the presence of a Ramsauer-Townsend minimum on the time dependent mobilities and the electron relaxation times in inert gas mixtures. It is important to mention that the calculated transient mobilities are very sensitive to the Ramsauer-Townsend effect. Calculation of the transient electron mobilities and the corresponding relaxation times are presented in section 2, which is followed by a discussion in section 3. 69 (B) Calculations and Results The transient behavior of the real and imaginary components of the electron mobilities were written in the form MO = f>fcexp(-A f c<'), ( 1 0 9 ) fc=0 lu{t') = Y,u{exp{-\kt'). (110) The mobility coefficients uk for an initial delta function distribution were determined with Eqs.(73)-(82) in Chapter 2. The eigenvalues Xk and eigenfunctions (f>k were calculated by diagonalizing the D.O representation of the Fokker-Planck operator as discussed in Chapter 2. Specification of the cross-section defines the Fokker-Planck operator, and hence the eigen-values, that determine the time evolution as given in Eqs. (109) and (110). Fig. l shows the energy dependence of the electron-rare gas moderator momentum transfer cross-section. The cross-section for helium is from the calculation by Nesbet [22], and the neon cross-section is by O'Malley and Crompton [23]. The other cross-sections are taken from Mozumder [5], [24] Both helium and neon have cross-sections that are nearly independent of the electron energy, while argon, xenon, and krypton gas have cross-sections that vary rapidly with elec-tron energy. The minimum in the collision cross-section for argon, xenon, and krypton is known as the Ramsauer-Townsend minimum. For heavier gases, it is important to select the location of the initial energy of the delta function distribution in relation to the position of 70 the Ramsauer-Townsend mimimum in the cross-section. The variation of the transient mobilities for pure helium and argon with various initial energies and nearly zero field strength and frequency (E/N=l x 10~5, W/N=l x 10 - 5) is shown in Figs. 12A and 12B respectively. The time t' = t/t0 is in dimensionless units which are discussed in Chapter 2. Also the mobilities n(t') are shown relative to their thermal, infinite time values, (ith. For pure helium, which has a roughly constant cross-section, the u(t')/fith values increase monotonically to the thermal mobilities. For argon, however, a complicated dependence of the transient mobility versus initial energy is found. Particularly, in Fig. 12B(curve c) when the initial energy is above the Ramsauer-Townsend minimum, the mobility decreases below the initial value to a minimum, and then increases to a maximum which is followed by a slow approach to the thermal value. The duration of the negative mobility and the overshoot decreases in Fig. 13A with an increase in the d.c. electric field strength, whereas the extent of overshoot decreases in Fig. 13B with an increase in the oscillatory frequency. Both results shown in Figs. 12 and 13 are analogous to the results calculated by McMahon and Shizgal [8,9] . The average electron energy increases with field strength so that the peak of the electron velocity distribution occurs at a higher reduced speed. Consequently, the momentum transfer cross-section at the higher energy is sampled and the effect of Ramsauer-Townsend minimum 71 Figure 12: Time variation of the real mobility with E/N=l x 10~5 Td and W/N=l x 10~5 Hz. Initial delta function distribution at u 0 equal to (a)2.0, (b)3.0, and (c)4.0, and J6=300K; (A)Pure helium, (B)Pure argon. 72 5 4 -Figure 13:(A)Time variation of the real mobility vs E/N for pure argon. Initial delta function distribution at u0=4.0, W/N=l x 10 - 5 Hz, and rb=300K, E/N in Td equal to (a)O.OOOOl, (b)0.001, (c)0.005, and (d)0.01. (B)Time variation of the real mobility vs W/N for pure argon. Initial delta function distribution at uo=3.0, E/N = 0.01 Td, and rfc=300K, W/N in Hz equal to (a)0.00001, (b)2.5, (c)10.0, and (d)25.0 73 is reduced. The maximum of the peak occuring at higher energy decreases with an increase in the oscillatory frequency. Increasing the frequency reduces the extent of heating, so that a lower energy region of the cross-section is sampled. The electric field strength and oscillatory frequency dependence of the real transient mobilities for a gas system can be described from the combined effect between the average electron energy and which energy portion of the electron-moderator collision cross-section is sampled. For a helium-argon mxture, by adding a small amount of helium gas into the argon moderator, I obtain Fig. 14 which shows similar results to Fig. 13A. The addition of helium gas results in a rapid cooling of the electrons and a change in the average electron-moderator collision cross-section, and causes the disappearance of overshoot and the negative mobility. However, in Fig. 15, for a krypton-xenon mixture for which moderators have similar energy dependent electron-moderator collision cross-sections, the addition of krypton gas into xenon moderator causes no disappearance of overshoot. The occurence of overshoot and the negative mobility seems to be related to the presence of the Ramsauer-Townsend minimum in the gaseous mixture. The transient value of the real component of the average electron mobilities have been calculated for helium-neon(without Ramsauer-Townsend minimum) and helium-argon(with Ramsauer Townsend minimum) mixtures at 300K and initial energy Uo=3.0. The compo-sition and field strength dependence of the mobilities for helium-neon and helium-argon 74 5 Figure 14:Time variation of the real mobility for a He-Ar mixture with initial delta function distribution at uo=4.0 and E/N=l x 10"5 Td, W/N=l x 10"5 Hz Tfc=300K. (A) Mole fraction of helium equal to (a)0.0, (b)0.05, (c)0.1, and (d)0.2. (B) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l. 75 0.8 1.0 Figure 15: Time variation of the real mobility for a Kr-Xe mixture with initial delta function distribution at u0=3.9, 7B=300K, W/N=l X 1G~5 HZ and E/N in Td equal to {A)E/N=1 x lO" 5, {B)E/N = 0.01. Mole fraction of krypton equal to (a)0.0, (b)0.25, (c)0.5, (d)0.75, and (e)1.0. 76 mixtures are reported in Figs. 16 and 17 respectively, where the composition and frequency dependence of the mobilities for helium-neon and helium-argon mixtures are reported in Figs. 18 and 19 respectively. The most distinct feature in Figs. 17 and 19 is the appearance of the maximum at small helium density. On the other hand, in Figs. 16 and 18, the mobilities only increase mono-tonically to the thermal value. A summary of the electron mobility relaxation times and their variation with different gas composition and d.c. field strength for helium-neon and helium-argon mixtures is also presented in Tables 2 and 3 respectively. The relaxation time is defined as the time to relax to within 10 percent of the thermal value. Lastly, Fig. 20 shows the composition dependence of the thermal energy relaxation times 1 / P r for helium-neon, and neon-xenon mixtures at E/N=l x 10 - 5 . The 1/Pr value is determined from the Pr ( l . l ) and PT(1.01) for electrons to decay within 10% and 1% of the thermal value, where a pure exponential decay is assumed. 77 Figure 16:Time variation of the real mobility for a He-Ne mixture with initial delta function distribution at u 0 = 3.0 , W/N=l x 10 - 5 Hz, and E/N in Td equal to (A)E/N=1 x 10"s, (B)E/N = 0.01. Mole fraction of helium equal to (a)0, (b)0.25, (c)0.5, (d)0.75, and (e)1.0 at r6=300K 78 Figure 17:Time variation of the real mobility for a He-Ar mixture with initial delta function distribution at u 0 = 3.0, rB=300K, W/N=l x 10"5 Hz, and (A) (B)E/N=1 x 10"5 Td, (C)(D)£yiV=0.01 Td: (A) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.1, and (d)0.2. (B) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l. (C) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.1, and (d)0.2. (D) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, and (d)l. 79 t* Figure 18: Time variation of the real mobility for a He-Ne mixture with initial delta function distribution at u 0 =3.0, Tt=300K, E/N=0.01 Td, and W/N = 10.0 Hz: Mole fraction of helium equal to (a)0, (b)0,25, (c)0.5, (d)0.75, (e)1.0. Figure 19: Time variation of the real mobility for a He-Ar mixture with initial delta function distribution at u0=3.0, Tb=300K, E/N = 0.01 Td, and W/N =10.0 Hz: (A) Mole fraction of helium equal to (a)0, (b)0.05, (c)0.10, (d)0.20. (B) Mole fraction of helium equal to (a)0.25, (b)0.5, (c)0.75, (d)1.0. 80 E/N = 1 x l f r 5 Td E/N = 0.01 Td XHe T(0.9) ^ x 10 T(0.9) £ x 10 mobility mobility 0.00 1027 29.25 128.5 3.407 0.05 263.2 7.139 81.15 2.021 0.10 149.4 4.077 62.60 1.573 0.20 79.42 2.196 45.49 1.167 0.30 53.87 1.503 36.19 0.951 0.40 40.69 1.142 30.15 0.806 0.50 32.68 0.921 25.84 0.699 0.60 27.30 0.772 22.59 0.618 0.70 23.43 0.664 20.05 0.553 0.80 20.52 0.583 18.01 0.500 0.90 18.25 0.519 16.34 0.455 1.00 16.44 0.468 14.94 0.418 Table 2: Relaxation times versus helium mole fraction, A# e for a He-Ne mixture; the initial delta function is at uo=3.0 and 7B=300.0K. r(0.9) is the time in units of 10n sec c m - 3 required for the real mobility to decay to within 0.9 of the stationary value. 81 E/N = 1 x lO"5 Td E/N = 0.01 Td x H e T(0.9) i x 10 T(0.9) ± X 10 mobility mobility 0.00 1657* 20.88 9.142 0.05 180.1* 4.177 107.0* 3.974 0.10 85.47* 2.745 59.12* 2.323 0.20 10.62 1.684 . 11.24 1.380 0.30 18.20 1.227 16.89 1.018 0.40 22.78 0.968 20.19 0.816 0.50 22.61 0.800 19.97 0.684 0.60 21.37 0.682 19.03 0.590 0.70 19.97 0.595 17.92 0.520 0.80 18.66 0.527 16.83 0.465 0.90 17.48 0.473 15.84 0.420 1.00 16.44 0.430 14.94 0.384 Table 3: Relaxation times versus A# e for a He-Ar mixture,the initial delta function is at uo=3.0 and TB=300.0K. T=(0.9) is the time in units of 10n sec c m - 3 required for the real mobility to decay to within 0.9 of the stationary value. Asterisks denote values of T for relaxation to 1.1 of the stationary value. 82 Figure 20: The composition dependence of the energy relaxation times, 1 /Pr for mixtures, with initial delta function distribution at u0=4.0, 7t=300K, E/N=l x 10-5 Td, and W/N=l x 10"8 Hz (A) 1/PT in He-Ne mixture vs helium mole fraction, (B) l /Pr in Ne-Xe mixture vs meon mole fraction. 83 (C) Interpretation of Results The electron real transient mobilities (u(t')/uth) of a single gas and for a gas mixture depend strongly on the electron moderator collision frequency v = Navox, where for pure gas a is the momentum transfer collision cross section and for gas mixture cr is the average collision cross-section, defined a S 0~avg — X\0~\ -(- X.%0~2-Increasing either the electron speed or the collision cross-section increases the electron-moderator collision frequency, and therefore a decrease in the u(t')/uth value. All the features shown in section 2 can be described by considering a combination of these two effects. It is necessary to understand the behavior of transient mobilities in a pure gas before we consider the effects for the mixtures. In Fig. 12A, for pure helium, electrons lose energy through collision and thus attain a lower speed during the thermalization. Since the electron collision cross-section is nearly constant, a decrease in speed means a decrease in the collision frequency. This gives an increase in the mobilities with an increase in the time t'. In Fig. 12B, for pure argon, the appearance of the negative mobility and overshoot arise from the effect of the Ramsauer-Townsend minimum. The occurence of the negative mobility can be understood physically as follows: Electrons gain energy in the electric field direction, whereas lose energy in the direction opposite to the electric field. When the electron-moderator cross-section increases rapidly with energy, the forward moving electrons 84 have shorter free flight paths than the backward moving electrons. Hence, the average displacement is in favor of the backward motion and negative mobility is observed. On the other hand, the occurence of the overshoot appears to be associated with the higher peak of the electron distribution function which is close to the energy where the cross-section is at a minimum. Due to the small cross-section, electrons collide less frequently at time t' than they do at infinite time. So the ratio, n(t')/fith > 1, and an overshoot is found. The extent of the negative mobility and overshoot is less for a less pronounced Ramsauer-Townsend minimum. For a helium-heavier gas mixture, changing the helium composition of the mixture reduces the effect of Ramsauer-Townsend minimum owing to (i) the changes the average speed x and or (ii) the change in the average collision cross-section o~avg At low helium density, electrons move through the mixture with a small amount of lighter helium gas and a large amount of heavier argon gas. The helium gas is much more effective than the argon gas in causing electrons to lose energy on collision because electron energy loss is better for a smaller moderator mass. So for low helium density, a small increase in helium concentration will therefore cause a dramatic decrease in electron speed. Cooling the electrons reduces the effect of the Ramsauer-Townsend minimum as the lower energy regions of the momentum transfer cross section is sampled. 85 The addition of helium gas in argon moderator changes the strong energy dependence of the argon collision cross-section OAT into a less energy dependent average collision cross-section o~avg, and reduces the effect of Ramsauer-Townsend minimum. Combining the two effects, the composition dependent behavior of the real transient mobilities for a gas mixture at different fixed d.c. field strength(Figs. 16 and 17 ), and at low oscillatory frequency (Figs. 18 and 19 ) can be predicted. In Figs. 16 and 18, for helium-neon, a gas mixture without Ramsauer-Townsend minimum, the transient mobilities increase monotonically towards the thermal values and overshoot is not expected. On the other hand, for helium-argon, a gas mixture with Ramsauer-Townsend minimum, overshoot disappears with an increase in helium concentration which is shown in Fig. 17. In Figs. 17A and 17B (E/N=l x 10 - 5 Td), the addition of helium gas into argon moderator reduces the effect of Ramsauer-Townsend minimum by changing the electron average speed and collision cross-section. For a smaller Ramsauer- Townsend minimum, the extent of the overshoot is diminished and finally disappeared. With higher electric field strength (E/N=0.01 Td), Fig. 17C (curve b and c) shows the appearence of overshoot at small helium concentration owing to the counter effects between an increase in d.c. electric field strength and an addition of helium gas into argon moderator. For pure argon, increasing the d.c. electric field strength heats up electrons and reduces the effect of Ramsauer-Townsend minimun by sampling the higher energy regions of the collision cross-section; however such change can be offset by 86 adding helium gas into the argon moderator so that the lower energy regions of the cross-section is sampled. As a result, the effect of the Ramsauer-Townsend minimum is restored and overshoot is found, providing that the average collision cross-section is strongly energy dependent. Further increase in helium density causes the change in electron average speed and average collision cross-section, and hence the disappearance of the overshoot in Fig. 17D is shown. However, the details of the composition and frequency dependence of the transient mobility has not yet been understood. With increasing the helium concentration, the curve (b, c, d)in Figs. 19A and Fig.l9B vary in a manner which is similiar to the results for a d.c. field strength shown in Figs. 17C and 17D respectively. The addition of helium gas into argon moderators, which changes the electron average energy and collision cross-section, apparently minimizes the effect of the oscillatory frequency. At higher helium density, the frequency dependence is not as strong as for low helium density, and therefore only the field strength dependent mobility is found. For a helium-neon mixture, the addition of helium gas into neon moderator enhances the electron collision energy dissipation rate, and thus shortens the relaxation times. This is exactly the situation for the results to Table 2 , the relaxation times decrease with an increase in helium density. In Table 3, for helium-argon mixture, except where an overshoot occurs, the relaxation times attain a maximum and then decrease with further increase in helium concentration. Presumably, the appearance of the maximum in the relaxation time is 87 owing to the small cross-section near the Ramsauer-Townsend minimum, some electrons are temporally trapped in this energy range and the relaxation rate to the thermal distribution function is prolonged. In addition, the 1/Aj values are also reported in Table 2 and 3 as well. For sufficient long times, the decay of the electron mobility will be a pure exponential with 1/Ai equal to the thermal relaxation time. It is important to mention that the 1/Ai values are not very sensitive to the Ramsauer-Townsend minimum. The thermal relaxation generally decrease with increasing in helium density owing to the higher electron collision energy dissipation rate. The energy relaxation times 1/Pr of mixture are also calculated and plotted against the mole fraction of gas 1, X\. The purpose of this is to verify the linearity of the density dependence of the thermalization time in rare gas mixture suggested by Suzuki and Hantano. The linear rule is valid for mixture with energy independent cross-sections. This is shown in Fig. 20A, for a helium-neon mixture, the 1/Pr values vary linearly with the helium density which can not be seen in Fig. 20B, neon-xenon mixture, a mixture with a Ramsauer-Townsend minimum. Further investigation is in progress and a more detailed analysis of the concentration dependence of the transient mobilities(real and imaginary components) with larger field strengths and frequencies (d.c and microwave) will be carried out. 88 C H A P T E R 5  S U M M A R Y In this thesis, I have considered a kinetic theory calculation of the time dependent and time independent electron mobility in binary rare gas mixtures, based on solutions of the Fokker-Planck equation. With the two-term approximation, the steady electron velocity distribution can be written in terms of simple integrals and the time independent electron mobility can be determined directly with a numerical integration. The composition depen-dence of the thermal mobility in rare gas mixtures as a function of electric field strengths is calculated and studied in detail. The present results predict deviations from Blanc's law for electron thermal mobility in rare gas mixtures. The magnitude of deviations from Blanc's law depends strongly on the applied field strength, the concentration of the mixture, and the nature of the energy dependence of the collision cross-sections. Varying either the electric field strength or the composition of the mixture changes the energy region over which the cross-sections are sam-pled, and hence affects the deviations from Blanc's law. An interesting effect was observed that for certain composition and electric field, the electron mobility in He-Xe mixtures could be greater than pure helium and less than pure xenon. The results suggest that experimental measurement of the electron mobility in He-Xe mixtures would be very useful in comparsion 89 with the present theoretical results. In order to calculate the transient mobility, the velocity distribution function is written as the solution of a general Fokker-Planck equation. The solution of the Fokker-Planck equation is determined from the expansion of the electron velocity distribution function in the eigenfunctions of the Fokker-Planck operator. A numerical method known as the the Quadrature Discretization Method (QDM) is employed for solving the Fokker-Planck eigen-value problem. An important advantage of the Q D M is to evaluate the matrix representation of the Fokker-Planck operator without requiring the explicit integral evaluation of the ma-trix elements. The time dependent electron mobility and the corresponding relaxation time, in particular for He-Ar and He-Ne mixtures, have been calculated for a wide range of field strengths, concentrations, and frequencies. It is important to mention that the calculated results are very sensitive to the presence of the Ramsauer-Townsend minimum. For a He-Ar mixture, the transient mobility is strongly influenced by the Ramsauer-Townsend minimum, which leads to the occurence of an overshoot and a negative transient mobility. On the other hand, for a He-Ne mixture, the transient mobility increases monotonically towards the thermal value. Furthermore, the energy relaxation times for He-Ne and Ne-Xe mixtures are calculated so as to test the linear dependence on mole fraction suggested by Suzuki and Hantano. The linear rule is valid only for mixtures with collision cross-sections that are approximately en-90 ergy independent (eg. He-Ne mixture), whereas for mixtures with strongly energy dependent cross-sections (eg. Ne-Xe), the linearity is lost. The present work has been restricted to the calculation of the electron mobility at low field strengths and frequencies. For a complete investigation of the concentration dependence of the time independent and time dependent electron mobility in rare gas mixture, future work will be mainly concerned with: (i) a detailed analysis of the electron mobility and the corresponding relaxation times with larger field strengths and frequencies; (ii) a comparison of the theoretical results and experimental measurement of electron mobility in He-Ar, He-Ne, and He-Xe mixtures. Such a comparison could provide a sensitive test of electron momentum transfer cross-sections; (iii) a detailed study in the case of averaging over one cycle of the microwave field for the partial differential equation for / 0 such that the resulting differential operator is time independent. This study could give a useful information about the range of validity of such assumption. 91 References [1] M . C. Sauer Jr., Advances in Radiation Chemistry, Vol.5, p. 97. (Wiley-Interscience, New York, 1976). 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