Three-temperature Theory Research Articles

Mobility of carbon ions and C+(4P) in helium computed from quantum-mechanical transport cross sections

On the light of a previous work and its recommendations by Matoba et al. [J. Phys. B 41, 145205 (2008)], this paper proposes to look at the mobility of ground and metastable-excited states of C+ ions moving into a helium buffer gas. The calculations, based on the three-temperature theory for solving the Boltzmann kinetic equation, are accomplished with quantum-mechanical transport cross sections at the low temperatures 4.3 and 77 K. The obtained mobility results are contrasted with the available theoretical data and experimental measurements, which show acceptable agreements.

Monte Carlo simulation studies on the validity of the Gram-Charlier calculations of velocity distributions of Na+ swarm in neon gas

Abstract The Gram-Charlier analytical/iterative approach tailor-designed to account for the skewness, kurtosis and correlation of mutually orthogonal velocity components of ion swarm velocity distributions is tested by intensive Monte Carlo simulation of the motion of Na + ion swarm drifting in Ne gas. Comparisons are made of the longitudinal and transverse ion temperatures and the shape parameters of the velocity distribution functions obtained by the two approaches. More detailed comparisons of the velocity distribution functions show that the analytical/iterative method significantly underestimates the high-velocity tail of the velocity distribution function. While the Gram-Charlier method is an improvement over the standard three-temperature theory, it still suffers from the difficulty arising from the truncation of velocity expansion terms at high velocities.

Transport properties of Rb + ions in Ne and Ar gases

The ratio of the transverse diffusion coefficient to mobility, D T/ K, has been experimentally measured for Rb + ions drifting in Ne and Ar gases in the presence of a uniform electric field. Results at 303 K are reported at electric field to neutral gas density ratio, E/ N, values ranging from 5 to 500 Td for both Rb +Ne and Rb +Ar systems. The reduced mobility K O and the ratios of the longitudinal and transverse diffusion coefficients to mobility, D L/ K and D T/ K respectively, were calculated for both Rb +Ne and Rb +Ar systems with a Monte Carlo simulation (MCS) technique which used the interaction potential of Koutselos, Mason and Viehland (KMV) (J. Chem. Phys., 93 (1990) 7125) as input. The accuracy of this potential has been evaluated by comparing the D T/ K results to the experimentally determined values. The KMV potential appears to be a good representation of the actual potential in the whole E/ N range (5–500 Td) for Rb +Ne but only in the 5–350 Td range for Rb +Ar. Values of D L/ K and D T/ K were also derived using K O obtained from the present MCS calculations and from experimental data available in the literature, employing the generalised Einstein relations (GER) based on the three-temperature theory. In the E/ N range studied for both Rb +Ne and Rb +Ar systems, the accuracy of the experimental measurements of D T/ K is estimated to be ±3% and that of the MCS calculations to be ±2.5%.

Measurements of DT/K of Rb+ ions in nitrogen and carbon dioxide

Values of the ratio DT/K, the transverse diffusion coefficient to mobility, for Rb+ ions drifting in nitrogen and carbon dioxide gases are experimentally measured for the first time at E/N ranging from 5 to 600 Td and from 5 to 800 Td respectively. Results were obtained with estimated accuracies better than 3% at about 303 K. The drift tube system which is capable of mass identification of the ions and of variation of drift distance enhances the reliability of the experimental results. In addition, calculations of DT/K are made using the generalized Einstein relation technique based on the three-temperature theory.

Longitudinal and transverse diffusion coefficients for Li+ ion swarms in Kr gas.

The ratio of the transverse diffusion coefficient to mobility, ${\mathit{D}}_{\mathit{T}}$/K at 309 K for ${\mathrm{Li}}^{+}$ ion swarms drifting in Kr gas in the E/N (electric field to neutral gas number density ratio) range of 5 to 170 Td, was experimentally determined with an overall accuracy of \ifmmode\pm\else\textpm\fi{}4%. The ${\mathit{D}}_{\mathit{T}}$/K results were effectively corrected for longitudinal end effects present appreciably in the drift tube by an analysis which requires the measurement of variance 〈${\mathit{x}}^{2}$〉 of the transverse ion-current density profile at different drift lengths z and the derivation of the magnitude ${\mathit{a}}_{2}$ of the end effects. Good agreement of the results with those calculated by Monte Carlo simulations (MCS) using an established interaction potential demonstrates the accuracy and reliability of the present ${\mathit{D}}_{\mathit{T}}$/K results. In addition, elaborate calculations of the reduced mobility ${\mathit{K}}_{0}$ and ratio ${\mathit{D}}_{\mathit{L}}$/K of the longitudinal diffusion coefficient to mobility of the ${\mathrm{Li}}^{+}$-Kr system are calculated with the MCS method. The accuracy of the MCS calculations is estimated to be \ifmmode\pm\else\textpm\fi{}2.5%. The calculated ${\mathit{D}}_{\mathit{L}}$/K values are compared with the experimental data available in the literature. Both ${\mathit{D}}_{\mathit{T}}$/K and ${\mathit{D}}_{\mathit{L}}$/K values for ${\mathrm{Li}}^{+}$ in Kr were also derived using reduced mobility ${\mathit{K}}_{0}$ data obtained from the MCS calculations and from experimental data, employing the generalized Einstein relations based on the three-temperature theory.

Transverse diffusion measurements and Monte Carlo simulation studies of Rb+ ions in Kr and Xe

The ratio of the transverse diffusion coefficient to the mobility (DT/K) for Rb+ ions drifting in Kr and Xe gases at about 303 K has been measured at electric field to neutral gas density ratio (E/N) values ranging from 5 to 500 Td. Calculations of the reduced mobility (K0) and the ratios of the longitudinal and transverse diffusion coefficients to mobility (DL/K and DT/K, respectively) for both Rb+–Kr and Rb+–Xe systems were made using a Monte Carlo simulation (MCS) technique and the interaction potential of Koutselos, Mason, and Viehland [J. Chem. Phys. 93, 7125 (1990)]. Furthermore, DL/K and DT/K values were derived from K0 values obtained from the present MCS calculations and from experimental results reported in the literature, using the generalized Einstein relations which are based on the three-temperature theory. For the Rb+–Kr system, the experimental values of DT/K were found to be 2%–4% higher than those from MCS calculations for E/N values above 120 Td. For the Rb+–Xe system, the experimental DT/K values were similarly higher than the MCS values in the range 90–220 Td. Given that the standard error of the experimental measurements of DT/K is estimated to be better than 3% and that of the MCS calculations is better than 2.5%, the agreement between the present experimental DT/K data and those from MCS calculations can be considered to be fairly good. The interaction potential of Koutselos, Mason, and Viehland therefore appears to represent well the actual potential for both Rb+–Kr and Rb+–Xe systems.

Transverse diffusion of Li+ ion swarms in Xe gas

In our transverse diffusion measurements of Li+ ions in Xe gas, appreciable longitudinal end effects were found throughout the whole range of E/N (electric field to neutral gas number density ratio) studied, i.e. 5-150 Td. By applying a known linear relationship of the measured variance (x2) of the transverse ion current density profile with drift length z, the magnitude a2 of the end effects can be numerically derived. The ratio of the transverse diffusion coefficient to mobility, DT/K, at 309 K for the Li+-Xe system were eventually corrected for end effects by an analysis which requires measured (x2) values and a2. The value of a2 is assumed to vary linearly with E/N. The corrected DT/K values are determined with an overall accuracy of +or-4%. The self-consistency and the good agreement of the results with those calculated by Monte Carlo simulations (MCS) bears evidence of the accuracy and reliability of the DT/K results. The accuracy of the MCS calculations is estimated to be +or-2.5%. The DT/K values for Li+ in Xe were also derived using reduced mobility K0 data obtained from the MCS calculations and from experimental data available in the literature, employing the generalized Einstein relations (GER) based on the three-temperature theory.

Ab initio calculation of gas-phase ion mobilities of third period atomic ions in helium

The interaction potentials for helium with the ground-state atomic ions of the third row of the periodic table, Na+ to Cl+, and the Si+(4P) excited state have been calculated at the MP4SDTQ/6-311 + G(3df,3pd) level of theory, with and without a counterpoise correction for basis-set superposition. The gas-phase ion mobilities of the atomic ions in helium have been calculated using three-temperature theory from these calculated interaction potentials. Where the atomic ion is a state, various ways of averaging the contribution from the Σ and π molecular states were investigated. In the few cases where experimental measurements are available, there is good agreement between theory and experiment.

Validity of the effective temperature scale for a dilute Li+ swarm in a Xe gas.

The effective-temperature concept arising from the two-temperature (2T) kinetic theory for the nonreactive migration of a dilute ion swarm in a neutral gas under the action of a uniform electrostatic field has been tested over a wide temperature range by Monte Carlo simulation (MCS). Such a test over the prescribed temperature span has not been made experimentally because of the inherent difficulty in designing an apparatus which permits the temperature of the ion drift chamber to be varied so widely. The MCS test is here applied to ${\mathrm{Li}}^{+}$ in Xe, which offers the most favorable conditions for the 2T theory to be a valid approximation of the more sophisticated three-temperature theory. Results established that for such cases it is possible to express the mobility ${\mathit{K}}_{0}$ and transverse diffusion ${\mathit{D}}_{\mathit{T}}$, but not the longitudinal diffusion ${\mathit{D}}_{\mathit{L}}$, in terms of a univariate effective-temperature scale instead of the separate parameters of gas temperature and the ratio of electric field to gas concentration. More remarkably, the ratio ${\mathit{D}}_{\mathit{T}}$/K was found to bear a simple linear relation with ${\mathit{T}}_{\mathit{e}\mathit{f}\mathit{f}}$ over a very wide range.

An appraisal of the three-temperature theory and the interaction potentials of K+-He and K+-Kr by Monte Carlo Simulation

Monte Carlo simulation calculations on a supercomputer have been made of the mobility of K+ ions drifting in pure He and Kr gases respectively under the action of a uniform electric field. Typical errors of 1% or less are achieved. By making direct comparisons with other theoretical and experimental data, these calculations have helped to verify both the three-temperature theory used to derive the theoretical mobility as well as the validity of various interaction potentials proposed. Results of these comparisons together with an earlier one for the K+-Ar system provide evidence that the three-temperature theory is generally applicable for the three different ion-neutral mass ratios studied, but yields a slight overestimation of the mobility around the peak mobility region.

Experimental measurements and theoretical calculations of the mobility of the N+ ion in helium

Abstract The mobility of the 3 P ground state of N + in helium has been calculated using the three-temperature theory of ion transport. The relevant adiabatic interaction potentials of the HeN + species have been calculated at the MP4SDQ/6-311G + (3df, 3pd) level of theory. Two limiting methods for calculating the mobility of an atomic ion in an atomic gas with orbital angular momentum greater than zero are developed and discussed.

Generalized Einstein relations for ions in molecular gases

Generalized Einstein relations (GER) connecting ion diffusion coefficients with the mobility are extended to include internal degrees of freedom and inelastic collisions. The Boltzmann equation is solved by means of a four-temperature theory that includes as parameters the gas temperature, two translational ion temperatures, and one internal ion temperature. The results reduce in the limit of elastic collisions to those obtained by Waldman and Mason from a three-temperature theory. In addition, an analytic procedure is developed for the calculation of the unobservable ion temperatures and correction terms that appear in the GER. The analytic procedure is checked by comparison with numerical results and experimental data for the elastic-collision limit. The general results are tested numerically with an inelastic two-state model and with a semiempirical analysis of experimental results on K + in H 2 and N 2. More details on ion-molecule inelastic collisions are needed for further progress.

Laser-induced fluorescence measurements of drift-velocity distributions for Ba+ in Ar: Moment analysis and a direct measure of skewness

Velocity distributions for Ba+ ions drifting in argon under the influence of an electric field are measured using single-frequency, laser-induced fluorescence techniques. A moment analysis of these data yields values for the ion mobilities, temperatures, and, for the first time, skewness parameters of the ion velocity distributions. Doppler profiles of the drifting ions are obtained with the laser beam propagating both parallel and perpendicular to the electric field direction as a function of E/N up to 148 Td. The velocity-component distributions perpendicular to the electric field direction are nearly Maxwellian at all E/N values studied, but the parallel distributions become noticeably skewed toward high velocities at the largest electric field strengths. The reduced mobilities for Ba+ in Ar, obtained from the first moments of the parallel velocity-component distributions, increase from 1.80±0.16 cm2 V−1 s−1 at zero field to 2.38±0.11 cm2 V−1 s−1 at 148 Td. The second central moments, which describe the widths of the distributions, increase monotonically with increasing E/N values for both the parallel and perpendicular distributions. Temperatures derived from the second central moments agree well with those calculated from a parametrized version of the three-temperature theory [M. Waldman and E. A. Mason, Chem. Phys. 58, 121 (1981)]. The third central moments of the parallel velocity-component distributions characterize the asymmetry of the distributions. Skewness parameters derived from the third central moments increase with electric field strength over the range of E/N studied.

Laser probing of ion velocity distributions in drift fields: Parallel and perpendicular temperatures and mobility for Ba+ in He

Measurements of ion velocity distributions are presented for Ba+ drifted in helium under well characterized conditions using single-frequency laser-induced fluorescence probing. We present the reduced mobilities and the Doppler profiles parallel and perpendicular to the electric field vector as a function of the ratio of the field strength (E) to the buffer gas density (N) up to 33.5 Td. The reduced mobility decreases monotonically with increasing E/N from the zero-field value of 16.7±0.4 cm2 V−1 s−1 at 313 K. The parallel and perpendicular ion temperatures are in very good agreement with both a repulsive Maxwell model and a parametrized version of the three-temperature theory of Lin et al. The parallel temperature is always higher than the perpendicular one. Effects of optical pumping on the Doppler profiles are also presented.

Influence of resonant charge transfer on ion diffusion and generalized Einstein relations.

A three-temperature theory of ion diffusion, and related generalized Einstein relations (GER) connecting diffusion and mobility, are shown to remain valid even in the presence of resonant charge transfer (RCT) which strongly distorts the ion velocity distribution function electric fields. Extreme models of RCT for which exact solutions are known, are used for testing. A previous semi-empirical parametrization scheme for the GER is refined to include the case of RCT and compared with the calculations of Sinha, Lin, and Bardsley for the system He + in He. Good results are obtained for both the diffusion coefficients and the anisotropic ion temperatures as a function of electric field strength.

Generalized Einstein relations from a three-temperature theory of gaseous ion transport

Generalized Einstein relations between mobility and diffusion coefficients are derived within the framework of a recently developed three-temperature kinetic theory of gaseous ion transport. A previously assumed connection between diffusion and differential mobility is firmly established within low-order approximations of the kinetic theory. Equations are obtained for the approximate calculation of the unobservable parallel and perpendicular ion temperatures, and for two higher moments of the ion velocity distribution function that appear as correction terms in the generalized Einstein relations. The present theory is tested on inverse-power potentials and on several alkali ion-noble gas systems, and is compared to two previous semi-empirical generalized Einstein relations. Simple procedures are recommended for the estimation of the parameters and correction terms that occur, to permit approximate calculation of gaseous ion diffusion coefficients from mobility data without the need of a computer or specific knowledge of the ion-neutral interaction potential.

Application of the three-temperatue theory of gaseous ion transport

The three-temperature theory of gaseous ion transport is generalized to allow arbitrary choices for the parallel and perpendicular ion temperatures and the velocity displacement. A truncation-parametrization sheme is proposed which simplifies the numerical computation. Results obtained in this way compare favorably with exact and Monte Carlo results available for special cases. Finally, the theory is used to test a proposed interaction potential for Cs +-Ar by comparison of calculated and measured mobility and diffusion coefficients.

Three-temperature theory of gaseous ion transport

A three-temperature kinetic theory of gaseous ion transport through neutral atomic gases, valid for electric fields of arbitrary strength without restriction on the ion—atom mass ratio or interaction potential, is presented. The theory is based on the use of a set of basis functions in which the ions are allowed to have different temperatures parallel and perpendicular to the field, neither of which is necessarily equal to the gas temperature, and for which the ion velocity distribution function is displaced from the origin. Although the theory is more difficult to use than is the two-temperature theory of Viehland and Mason and of Burnett, it has the important advantage of yielding accurate results in low orders of approximation to quantities that are intrinsically anisotropic, notably ion diffusion coefficients.