Journal of Physics D: Applied Physics You may also like Calculation of ion mobilities by means of the twotemperature displaced-distribution theory To cite this article: P G C Almeida et al 2002 J. Phys. D: Appl. Phys. 35 1577 View the article online for updates and enhancements. - Ion mobility spectrometry coupled with multi-capillary columns for metabolic profiling of human breath Jörg Ingo Baumbach - Theoretical investigation of the decay of an SF6 gas-blast arc using a twotemperature hydrodynamic model WeiZong Wang, Joseph D Yan, MingZhe Rong et al. - Thermodynamic properties and transport coefficients of a two-temperature polytetrafluoroethylene vapor plasma for ablation-controlled discharge applications Haiyan Wang, Weizong Wang, Joseph D Yan et al. This content was downloaded from IP address 149.162.213.221 on 19/09/2022 at 15:19 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 35 (2002) 1577–1584 PII: S0022-3727(02)35564-5 Calculation of ion mobilities by means of the two-temperature displaced-distribution theory P G C Almeida1 , M S Benilov1 and G V Naidis2 1 Departamento de Fı́sica, Universidade da Madeira, Largo do Municı́pio, 9000 Funchal, Portugal 2 Institute for High Temperatures of the Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412, Russia Received 5 April 2002 Published 18 June 2002 Online at stacks.iop.org/JPhysD/35/1577 Abstract Electric-field dependent ion mobility is calculated in the framework of the two-temperature displaced-distribution theory. The considered approach provides a better accuracy than the conventional approach based on (the first approximation in) the two-temperature theory and is better justified for the case of heavy ions, while being not much more complicated than the conventional approach. Calculated ion mobilities are in good agreement with experimental data. 1. Introduction Motion of ions in an applied electric field in a uniform weakly ionized gas is characterized by the ion mobility, which, in a general case, depends on the electric field strength. If the gas in question is hot, in most cases there is no experimental information in the literature and one must resort to the kinetic theory in order to evaluate the mobility in terms of a potential of the ion–neutral particle interaction (which in many situations is known or may be estimated relatively easily) or in terms of experimental data available for the same gas under different conditions (measurements of mobilities at the room temperature are available for many gases). The simplest tool of the kinetic theory available in the literature for a rapid evaluation of the mobility of ions with account of a dependence on the local electric field is the first approximation in the framework of the so-called twotemperature theory [1]. Such an approach is based on the assumption that the ion distribution function is close to the Maxwellian one, with a temperature that may be different from the temperature of neutral particles (atoms). This assumption implies, in particular, that the drift velocity of the ions is much smaller than their thermal velocity. The latter assumption holds when the mass of ions, mi , is much smaller than the mass of atoms, ma , and is violated for mi /ma O(1). In order to overcome this difficulty, one can assume that the ion distribution function, being close to the Maxwellian function with a temperature that may be different from the 0022-3727/02/131577+08$30.00 © 2002 IOP Publishing Ltd atomic temperature, is displaced relative to the atoms, i.e. characterized by a non-negligible mean velocity of the ions. Such an approach may be termed the two-temperature displaced-distribution theory. The next step is to introduce in the (displaced) ion distribution function two different temperatures, one in the direction along the electric field and another in the transversal directions. The latter approach is usually termed the three-temperature theory, and it is to this approach that most of the attention has been paid in the literature [1]. Obviously, the three-temperature theory is more complex than the two-temperature displaced-distribution theory. On the other hand, it is not quite clear whether the introduction of different temperatures in the direction along the field and in the transversal directions results in a substantial enhancement of accuracy of the theory. It should be emphasized in this connection that there are indications [1,2] that an account of the velocity displacement plays a more important role than the temperature anisotropy. In [3], the two-temperature displaced-distribution description has appeared in a natural way as a part of the fluid description of ion motion with account of variable ion temperature under conditions of non-uniform plasma, where gradients are strong and the distribution function of the ions is far from equilibrium with a local electric field. It should be emphasized that rates of exchange of momentum and energy between the ion and atom fluids, calculated within this two-temperature displaced-distribution description, take exact values in the case when the mean ion velocity is much higher than their thermal Printed in the UK 1577 P G C Almeida et al velocity (i.e. the ions constitute a beam in the neutral gas). This is essential for a correct description of ion motion both in uniform plasmas in the case mi ma and in non-uniform plasmas, where the ions may be accelerated up to velocities much higher than their thermal speed. It is of interest to apply this two-temperature displaceddistribution description to the calculation of ion mobilities in spatially uniform plasma and to compare high-field mobilities of ions calculated in such a way with other theoretical results and with experimental data. It will give an idea of accuracy of this approach as is used in the fluid description of ion motion and may even result in a method of rapid evaluation of mobility which, while being not much more complicated than the first approximation in the two-temperature theory, would give more accurate mobility values and would be better justified for the case mi /ma O(1). 2. Theory 2.1. The model Consider steady-state motion of ions in a uniform neutral gas in a uniform electric field E. Equations of conservation of momentum and energy of the ion fluid may be written in the form eni v i · E = ria(e) , eni E = r(m) ia , (1) where ni and vi are the number density and the mean (drift) (e) velocity of ions and r(m) ia , ria are the rates of loss per unit volume of, respectively, momentum and energy of the ions due to (elastic) collisions with neutral particles. (e) The rates r(m) ia and ria are expressed in terms of distribution functions of neutral particles and ions and of the crosssection for momentum transfer between the ions and the neutral particles. If the ionization degree is low enough, the distribution of neutral particles is not disturbed by the presence of ions and is described by the Maxwellian function. The distribution of ions will be approximated by a shifted Maxwellian function 3/2 mi (ci − vi )2 mi fi(0) = ni , (2) exp − 2kTi 2πkTi where Ti is the temperature of the ion species. Then, one can use results [3] and arrive at the following expressions: r(m) ia = ria(e) eni vi , µi (3) eni [3χk(Ti − T ) + mi vi2 ], = µi (mi + ma ) (4) m i T + m a Ti , mi + m a (5) where mia = m i ma , mi + m a µi = η= 1578 Tia Ti e 3π kTi 16 1/2 Tia = Q(m) ia 2kTi πmia , (1,1) Q̄ia 1/2 1 nQ̄(1,1) ia η χ= Q(e) ia Q(m) ia , 2 − M 2, 3 (6) (7) Q̄(1,1) ia Q(m) ia Q(e) ia × M= =e = −M 2 ∞ 5 −x x e 0 2 Q(1) ia ∞ 5 −x 2 x e 0 Q(1) ia 2kTi x dx, mia 2kTia x mia F (m)(2Mx) dx, F (e)(2Mx) mia vi , 2kTia (9) F (m) (y) = 3 F (e) (8) y cosh y − sinh y , y3 sinh y . (y) = y (10) Here, T and n are the temperature and number density of (1) neutral particles and Q(1) ia = Qia (g) is the velocity-dependent cross-section for momentum transfer in collisions of ions with neutral particles. Note that mia is the reduced mass, is quantity Tia may be called the reduced temperature, Q̄(1,1) ia the conventional average cross section for momentum transfer evaluated at the temperature Ti , µi is the mobility of the ions in the gas of neutral particles, η and χ are dimensionless coefficients expressed in terms of the weighted cross-sections (e) Q(m) ia and Qia . M may be termed the Mach number of the ion motion; note that it can be written also in the form M= vi , (11) u2a + u2i √ √ where ui = 2kTi /mi and ua = 2kT /ma are characteristic thermal velocities of the ions and the atoms. Using equations (3) and (4), one can rewrite equation (1) as vi = µi E, 3 1 ma vi2 3 . kTi = kT + χ 2 2 2 (12) The first equation in equation (12) is the conventional expression for the drift velocity. The second equation relates the ion temperature to the temperature of neutral particles and the ion drift velocity and represents an analog for the considered model of the well-known Wannier equation; note that the classical Wannier equation can be obtained from the second equation in equation (12) by setting χ = 1. It may be shown also that the ratio of the ion temperature to the reduced temperature is given by the equation 2M 2 Ti . =1+ Tia 3χ (13) Now the model is complete. In order to determine the ion drift velocity and temperature as functions of E/n and T , one needs to solve equations in equation (12) for vi and Ti . It should be emphasized that the form of the expressions for the ion mobility and temperature, equation (6) and the second equation in equation (12), has been chosen in such a way that the structure of these equations is identical, to the accuracy of coefficients η and χ, to that of the respective expressions in the first approximation in the twotemperature theory (e.g. [1, equations (6-2-12) and (6-2-13)]). (Note that what appears in place of Ti in the formulae of Calculation of ion mobilities 2.2. Properties of coefficients η and χ 3 8 (1) 3πχ Qia (vi ) + ···, 2 Q̄(1,1) ia χ= γ1 + 4 + ···, 3 (14) where γ1 is a coefficient given by equation (39) of appendix. For the particular case of a power-like dependence of the cross-section for momentum transfer on the relative velocity, one finds 6+n C 2kTi n/2 , γ1 = n. (15) = Q̄(1,1) ia 2 2 mia Thus, equation (14) in this case assumes the form η= 3 4 π(n + 4) 2 n+4 2 n/2 χ 1.3 1.2 1.1 1 η 0.9 Coefficients η and χ, as all other quantities characterizing ion transport, are governed by E/n and T . From the point of view of calculations, however, it is possible and convenient to consider formally these coefficients as functions of M and may be Tia . (Note that Ti in the definitions of η and of Q̄(1,1) ia expressed in terms of Tia by means of equation (13).) In the particular case of a power-like dependence of the cross-section for momentum transfer on the relative velocity, Q(1) ia (g) = Cg n , where C and n are constant quantities, these coefficients are independent of Tia and depend only on M. In the limiting case M → 0, one finds η → 1, χ → 1, which follows from the fact that F (m) (0) = F (e) (0) = 1. A second approximation asymptotic behaviour of the cross (e) sections Q(m) ia and Qia in the opposite limiting case M → ∞ is found in appendix. (It should be emphasized that the and Q(e) asymptotic behaviour of Q(m) ia ia must be found to a second approximation in order to determine a first approximation asymptotic behaviour of the coefficient χ.) Using equation (40) of appendix, one finds the following asymptotic behaviour in the limit M → ∞: η= 1.4 χ, η the first approximation in the two-temperature theory is the temperature Teff defined by an equation formally similar to the above definition of Tia . However, if one removes T from the definition of Teff by means of the (classical) Wannier equation and takes into account that the temperature Tb [1], being related to the total mean ion energy, is different from the ion temperature Ti of the present work, then Teff will become equal to Ti ). In other words, if coefficients η and χ are set equal to unity, then the formulae of the present model will coincide with the respective formulae of the first approximation in the two-temperature theory. 1 + ···, ((6 + n)/2) (16) n+4 χ= + ···. 3 One can see that in the particular case of a power-like dependence of the cross-section for momentum transfer on the relative velocity coefficients η and χ tend in the limit M → ∞ to finite values. For an arbitrary cross-section for momentum transfer, coefficients η and χ must be calculated by means of numerical integration. For some model cross-sections, they may also be found analytically [3]. For completeness, we present here expressions for the cases when ions interact with neutral 0.8 0 2 4 6 8 10 M Figure 1. Coefficients χ and η for the model of rigid spheres. - - - -: limit values at large M. particles as Maxwell molecules or rigid spheres. In accord to the above, the coefficients η and χ are independent of Tia in these cases and may depend only on M. For Maxwell molecules (Q(1) ia (g) = C/g, the model of constant collision frequency), one gets η = 1, χ = 1. (17) As it should have been expected, this equation conforms to the limit values given by equation (16) for n = −1. For the model of rigid spheres (Q(1) ia (g) = C, the model of constant mean free path), one gets 1/2 √ 1 π 3 3χ 2 erf M 1+M − η= 4M 2 M 8 3χ + 2M 2 1 2 + 1+ e−M , (18) 2M 2 √ 2 8M 2 [ π (1 + 2M 2 )erf M + 2Me−M ] χ= √ . 3[ π (4M 4 + 4M 2 − 1)erf M + 2M(2M 2 + 1)e−M 2 ] (19) Graphs of these functions are shown in figure 1. Also shown are limit √ values given by equation (16) for n = 0, which are η = 9π/32 ≈ 0.94 and χ = 4/3 (as it should have been expected, these values coincide with those given by equations (16) and (19) in the limiting case M → ∞). One can see that the coefficient η is slightly non-monotonous. While this coefficient is (for the model of rigid spheres) not very different from unity, coefficient χ may exceed unity by more than 30%. 2.3. Limiting cases of low and high electric fields If the electric field is low enough, the ion temperature is close to the atomic temperature, Ti − T T . Using the second equation in equation (12), one can see that the last inequality is equivalent to vi ua , meaning that the lowfield limit corresponds to the ion drift velocity being much smaller than thermal velocities of neutral particles, in accord to the criterion of the low-field behaviour [1, equation (5-2-28)]. 1579 P G C Almeida et al It follows from equation (11) that M 1 in the limiting case considered. According to the above, η → 1, χ → 1 in this limiting case and the value of the ion mobility given by equation (6) coincides with the one evaluated by means of the first approximation in expansion in the Sonine polynomials in the method of Chapman–Enskog at the gas temperature. In the opposite limiting case of high electric field, Ti T , or, which is equivalent, vi ua , it follows from the second equation in equation (12) and from equation (11) that M2 = 3χ mi . 2 ma (20) (Note that in the case of heavy ions, mi ma , this equation applies in the range Ti /T mi /ma .) One can conclude that the Mach√number in the limiting case of high field is of the order of mi /ma , i.e. M 1 if mi ma , M = O(1) if mi /ma = O(1), and M 1 if mi ma . If χ is independent of Tia and may depend only on M, which is the case for a power-like dependence of the crosssection for momentum transfer on the relative velocity, equation (20) may be considered as a closed equation for M. It follows that the Mach number which can be attained by ions in such a case is given by the root of equation (20). In the case of intermediate electric fields, Ti − T = O(T ), or, which is equivalent, vi = O(ua ), one finds from equation (11) that M 1 if mi ma and M = O(1) if mi /ma O(1). 2.4. Approximations of the model An approximate character of the considered model stems from the fact that the ion distribution function is replaced by the shifted Maxwellian function (2) while calculating the rates of momentum and energy exchange between ions and neutral particles. The model should provide exact results in particular cases when a difference between the exact distribution and the shifted Maxwellian function does not affect the rates. Two such cases can be indicated. First, this is a case of heavy ions, mi ma , in high electric fields. In such a case vi ui ; the ion swarm is close to a monoenergetic particle beam and all collisions occur at approximately the same relative velocity vi . The difference between the exact distribution and the shifted Maxwellian function is insignificant (both distributions are close to the delta-function), hence the model should give exact values for the rates of momentum and energy exchange. Indeed, employing the first equation in equation (14) (note that M 1 in the case considered), one arrives at the exact value of the ion mobility for the case considered, µi = e/mia nvi Q(1) ia (vi ). Note that the value of the ion temperature given by the model in this particular case need not be exact: on the one hand, it does not affect the ion mobility; on the other hand, it is governed by the first term in square brackets on the right-hand side of equation (4), which is negligible relative to the second term and is not calculated quite correctly. Another case when the difference between the exact distribution and the shifted Maxwellian function does not affect the rates of momentum and energy exchange between ions and neutral particles is the case of Maxwell molecules. A form of the distribution function produces no effect whatsoever on the 1580 rates in this case. Indeed, substituting η = 1 into equation (6), one arrives at the exact expression for the ion mobility for the model of Maxwell molecules, µi = e/mia ngQ(1) ia (g); the classical Wannier equation, to which the second equation in equation (12) is reduced with χ = 1, is exact for the model of Maxwell molecules regardless of the distribution function provided that 23 kTi is understood as the mean kinetic energy of the ion chaotic motion [1, p 153–4]. Apart from the case of heavy ions in high electric fields, another particular case exists in which the difference between the exact distribution and the shifted Maxwellian function (2) is small. This is the case of low electric field, in which both the exact distribution and the shifted Maxwellian function are close to the non-shifted Maxwellian function with the atomic temperature. (Strictly speaking, this does not apply to heavy ions since the inequality vi ua does not guarantee validity of the inequality vi ui in this situation, however, both the exact distribution and the shifted Maxwellian function are close to a non-shifted delta-function in such a situation.) However, the rate of momentum exchange in this case is governed just by an anisotropic part of the distribution. Hence, the difference between the exact distribution and the shifted Maxwellian function, in spite of being infinitesimal in this particular case, can produce a finite effect on the ion mobility. This is why the model gives in this case the above-mentioned (approximate) value coinciding with the first approximation in the method of Chapman–Enskog rather than an exact value. However, the difference between the two values is very small in most cases, so the accuracy of the model in the limiting case of low field is quite high. 3. Comparison with other theoretical results Consider first the case of light ions, when M 1 regardless of the electric field. The shift in the ion distribution function (2) is inessential and the model considered should be equivalent to the first approximation in the two-temperature theory. Since η = χ = 1 at M → 0, this is indeed the case. Note that µi in this case coincides with the value of ion mobility evaluated by means of the first approximation in expansion in the Sonine polynomials in the method of Chapman–Enskog at the temperature Ti . In [4], the high-field mobility was calculated by a moment method for the model of rigid spheres. In accord to the above, coefficient χ is independent of Tia for this model and is a function of M only, hence equation (20) can be numerically solved for M at different values of mi /ma . The solution is illustrated by figure 2, in which the (normalized) root of this equation is shown. The dashed lines represent the one-term asymptotic behaviour at small and large mi /ma , which reads mi 3 mi M= , M= 2 , (21) ma 2 ma respectively. The reduced high-field ion drift velocity for the model of rigid spheres is given in the framework of the approach considered by the expression vi f x 1/4 . = 1/2 (1 − x)1/2 (aλ) (22) Calculation of ion mobilities 1/nQ(1) ia Here a = eE/mi is the ion acceleration, λ = is the mean free path between collisions of an ion with neutrals, x = mi /(mi + ma ), and f is a function of x defined by the equation 27πχ 1/4 f = , (23) 128η2 where coefficients η and χ are evaluated in terms of the highfield Mach number which is governed by equation (20) and shown in figure 2. In figure 3, the reduced high-field velocity calculated by means of equation (22) is compared with accurate results from [4] for various values of mi /ma . Also shown are the data corresponding to the first approximation in the twotemperature theory, which are given by equation (22) with χ = η = 1. One can see that in the case of heavy ions, the accuracy of the present approach is better than the accuracy of 1.44 results obtained in the framework of the first approximation in the two-temperature theory. Note that function f = f (x) may be evaluated, to the accuracy within 0.7%, by means of the explicit formula f = A= 27π , 128 M/(mi/ma)1/2 1.28 1.24 1.2 0.1 1 mi/ma 10 Figure 2. High-field value of√the Mach number for the model of rigid spheres normalized by mi /ma . - - - -: asymptotic behaviour for light and heavy ions. , (24) (2A − 1.7)A , 1−A 2.3A − 2 D= . 1−A B= C = A − 1, (25) This formula has been constructed in such a way as to give correct values of the function f 2 and of its derivatives in the particular cases of infinitely light ions, x = 0, and of infinitely heavy ions, x = 1: 27π 1/2 f 2 (0) = (26) , f 2 (1) = 1, 128 d(f 2 ) 3 (0) = dx 10 1.32 1/2 where 1.4 1.36 A + Bx + Cx 2 1 + Dx 27π 128 1/2 , d(f 2 ) (1) = 0. dx (27) Note that the values of the derivatives are obtained by expanding equations (18) and (19) at M → 0 and M → ∞ and making use of equation (21). A model of ion–atom interaction that involves inelastic collision process is considered in [1, p 360], the total momentum transfer cross-section being the sum of elastic and inelastic collision cross-sections. The cross-section of elastic process is assumed to be constant (independent on the ion energy), while the cross-section of inelastic process is taken equal to the same constant for relative velocities higher than some threshold value vth and equal to zero for velocities lower than vth . Data on the reduced high-field ion drift velocity calculated by means of various methods are given for different values of the reduced field strength 2 aλ/vth at mi = ma . In figure 4, the data from [1, p 360] are compared with results given by the present approach. One can see that the present results are close to those given by 4 deviation (%) 0 –4 –8 –12 0 0.2 0.4 0.6 x = mi/(mi+ma) 0.8 1 Figure 3. Deviation of the high-field ion mobility for the model of rigid spheres from results [4]. : first approximation in the two-temperature theory. +: present work. Figure 4. Reduced ion drift velocity vs reduced field strength. ——: first and fifth approximation in the two-temperature theory. - - - -: present work. Points: Monte Carlo simulation. 1581 P G C Almeida et al the Monte Carlo simulations and by the fifth approximation in the two-temperature theory, and are more accurate than the ones given by the first approximation in the two-temperature theory. Note that these calculations have been performed for mi = ma ; it should be emphasized that with an increase of the ratio mi /ma the present two-temperature displaceddistribution approach will become still more accurate, while accuracy of the first approximation in the two-temperature theory will deteriorate. It is also of interest to compare the ion energy in high electric fields calculated by means of the present method with results [4] obtained by means of a moment method on the basis of the three-temperature theory, and with those provided by the first approximation in the two-temperature theory. The average chaotic ion energy is given in the framework of the three-temperature theory by the formula [5] kT (28) + kT⊥ , 2 where T and T⊥ are the ion temperatures in the direction along the electric field and in a transversal direction, respectively. In the present model, ε = 23 kTi and one finds in the high-field limit ε 1 = . (29) 2χ ma vi2 In the first approximation in the two-temperature theory, one finds in the high-field limit ε = 1 ε = . 2 2 ma v i (30) A comparison of results of the present approach with results [4] and with those given by the first approximation in the two-temperature theory is shown in figure 5 for the model of rigid spheres for different ion–atom mass ratios. One can see that both the present approach and the first approximation in the two-temperature theory provide a relatively good accuracy for light ions. For heavy ions, the accuracy of the first approximation in the two-temperature theory is about 50%, while the accuracy of the present approach is better than 15%. 4. Comparison with experimental data In the case of ions moving in parent gases, ion mobility and diffusion are controlled by charge exchange. The total crosssection of charge exchange may be accurately represented, for a wide range of energies, by an expression of the form [1] Qex = (a − b ln )2 (31) where a and b are constants and is the relative energy of colliding particles. The energy-dependent momentum transfer cross-section Q(1) ia is related to Qex as [1] Q(1) ia 2Qex . (32) Using these formulae, drift velocities have been calculated for He+ in He, for Ne+ in Ne, for Ar+ in Ar, for Kr + in Kr, and for Xe+ in Xe. The set of constants (a, b) in equation (31) used in the calculations was (4.88, 0.299) for helium, (5.7, 0.37) for neon, (7.0, 0.6) for argon, (8.2, 0.57) for krypton, and (9.2, 0.6) for xenon, where a and b are in 10−10 m and is in eV. Note that the data for He have been taken from [1], for Ar from [6] and for Ne, Kr, and Xe from [7]. Results of calculations are compared with experimental data in figures 6 and 7. Agreement between the theory and the experiment is good. Also shown in figure 6 are results for drift velocities in argon obtained with the use of momentum transfer crosssections given in [9]. One can see that the effect of the variation of the cross-section is not considerable. 5. Concluding remarks There are different ways of application of the results obtained. The most direct way consists in numerically solving equation (12) for vi and Ti for given T and E/n. This can be performed by iterations. An initial approximation may be obtained with the use of the low-field value of the mobility (which is given by equation (6) with η = 1 and Ti = T ). 100 H e +-He v i (1 0 4 c m /s ) Deviation (%) 40 20 0 A r + -A r 77 K 1 0 0.2 0.4 0.6 x = mi/(mi+ma) 0.8 1 Figure 5. Deviation of the normalized high-field chaotic ion energy for the model of rigid spheres from results [4]. ——: results [4]. : first approximation in the two-temperature theory. +: present work. 1582 10 N e +-Ne 10 100 E/n (Td) 1000 Figure 6. Drift velocity of He+ in He (77 K and 300 K), Ne+ in Ne, and Ar+ in Ar (both 300 K). ——: theoretical. Points: experiment [5, 8]. - - - -: cross section from [9]. Calculation of ion mobilities in the square brackets on the right-hand side of equation (34) and η = 1 in the other terms, the equation obtained, Kr+– Kr 10 1 = vi2 vi(104 cm/s) Xe+–Xe 1 100 1000 E/n (Td) 10000 Figure 7. Drift velocity of Kr+ in Kr and Xe + in Xe at 300 K. ——: theoretical. Points: experiment [5, 8]. The iterations converge rapidly in all the cases considered in this paper, including at high fields (about 12 iterations are required to achieve accuracy within 10−6 for E/n = 3 × 104 Td). A program that performs these calculations for the above-mentioned ion–atom systems is available in [10]. In the framework of the first approximation in the twotemperature theory, an analytical formula for the ion drift velocity for arbitrary electric fields may be obtained in the model of rigid spheres. It is interesting to note that a similar formula may be derived also in the framework of the approach considered. The procedure is as follows. Writing equation (6) for the model of rigid spheres, expressing from this equation Ti in terms of ηvi /E, substituting the result in the second equation in equation (12) and formally solving the obtained equation for vi , one arrives at 1/2 aλ 2 3χkT 3χkT 2 27πχx 2 + , (33) − vi = 2ma 128η2 1 − x 2ma Equation (33) may be transformed to 1 = vi2 η4 + 4 4vLF η ηHF 2 χHF 1 4 χ vHF 1/2 + η2 , 2 2vLF (34) where vLF is the drift velocity evaluated in the low-field approximation and given by the equation vLF = 3π mi 16 kT 2kT πmia 1/2 aλ, 1 1 + 4 4 4vLF vHF 1/2 + 1 , 2 2vLF (36) will provide correct values in the liming cases of low and high fields and may be considered as an interpolation formula for the intermediate case. It should be emphasized that equation (36), supplemented with equations (22), (24), and (35), represents a closed explicit expression for the drift velocity, in contrast to, e.g. equation (33), in which the right-hand side depends on vi and Ti through coefficients χ and η. Thus, equation (36), being an exact result for the model of rigid spheres in the framework of the first approximation in the two-temperature theory, represents a natural interpolation formula in the framework of the present approach. Comparison of this formula (supplemented with equations (22), (24), and (35)) with the full (numerical) solution in the framework of the present approach shows that its accuracy is better than one per cent in the whole range of mass ratio and field strength values. The employed method, based on the two-temperature displaced-distribution theory, allows one to rapidly evaluate ion mobilities, including cases of a hot gas in a high electric field, without being much more complicated than the conventional approach based on (the first approximation in) the two-temperature theory. It is better justified for the case mi /ma O(1), in particular, it provides essentially better results for the ion energy. Ion mobilities calculated in such a way are in good agreement with experimental data and with results of more elaborate theoretical approaches. The fact that the application of the two-temperature displaced-distribution description to calculation of ion mobilities in spatially uniform plasmas has proved successful, is important for applications of this approach in the fluid description of ion motion under conditions of a non-uniform plasma, where gradients are strong and the distribution function of the ions is out of equilibrium with the local electric field. Acknowledgments The work was performed within activities of the project Theory and modelling of plasma-cathode interaction in high-pressure arc discharges of Fundação para a Ciência e a Tecnologia. (35) vHF is the drift velocity evaluated in the high-field approximation and given by equation (22), and ηHF and χHF are values of coefficients η and χ evaluated in terms of the high-field Mach number (i.e. the same values that are involved in equation (22)). The second term in the square brackets on the right-hand side of equation (34) is dominating in the limiting case of high fields, when coefficients η and χ tend to values ηHF and χHF , respectively. The other terms become dominating in the limiting case of low fields, when η and χ tend to unity. Therefore, if one sets η = ηHF and χ = χHF in the second term Appendix. Second approximation asymptotic behaviour of weighted cross-sections at large Mach numbers We rewrite equation (9) as ∞ Q(m) 2kTia −(x−M)2 (1) ia x e Qia = mia Q(e) 0 ia 3x 2 −4Mx −4Mx [2Mx(1 + e )−1+e ] 3 × 16M dx, 4 x (1 − e−4Mx ) 4M (37) 1583 P G C Almeida et al At large M, dominating contributions to the integral on the right-hand side is given by the vicinity of the point x = M. Introducing new integration variable z = x − M, expanding the integrand in 1/M and retaining three terms, one arrives at ∞ Q(m) 2 (1) ia e−z = Q (v ) i ia Q(e) −M ia z 3M + 3) 1 + (γ 1 M 8 1 1 2 2 2 + 3z γ1 + γ2 z + 3z − + · · · dz, × 2 M2 M3 z z2 1 + (γ1 + 4) + (4γ1 + γ2 + 6) 2 + · · · 4 M M (38) where d ln Q(1) ia (g) (vi ), d(ln g) (1) 1 g 2 d2 Qia (g) γ2 = 2 Q(1) dg 2 ia (g) γ1 = (39) . g=vi Replacing (to exponential accuracy) the lower integration limit in equation (38) by −∞ and evaluating the integral, 1584 one obtains 3M 3 3γ1 + γ2 + 2 + + · · · √ 8 16 M Q(1) = π 3 ia (vi ). + γ + 6 M 4γ 1 2 M + ··· + 8 4 (40) Note that the first term of the expansion on the right-hand side of this expression describes momentum and energy exchange between two monoenergetic particle beams and was derived in [3]. Q(m) ia Q(e) ia References [1] Mason E A and McDaniel E W 1988 Transport Properties of Ions in Gases (New York: Wiley) [2] Kumar K, Skullerud H R and Robson R E 1980 Aust. J. Phys. 33 343 [3] Benilov M S 1997 Phys. Plasmas 4 521 [4] Skullerud H R 1976 J. Phys. B: Atom. Molec. Phys. 9 535 [5] Viehland L A and Mason E A 1995 Atom. Data Nucl. Data Tables 60 37 [6] Benilov M S and Naidis G V 1998 Phys. Rev. E 57 2230 [7] Duman E L et al 1982 Institute for Atomic Energy, Moscow Preprint No 3532/12 (in Russian) [8] Ellis H W, Pai R Y, McDaniel E W, Mason E A and Viehland L A 1976 Atom. Data Nucl. Data Tables 17 177 [9] Phelps A V 1994 J. Appl. Phys. 76 747 [10] http://fisica.uma.pt/public/