Multi-term Solution Research Articles

Periodic structures in the Franck-Hertz experiment with neon: Boltzmann equation and Monte-Carlo analysis

The Franck-Hertz experiment with neon gas is modelled as an idealised steady-state Townsend experiment and analysed theoretically using (a) multi-term solution of Boltzmann equation and (b) Monte-Carlo simulation. Theoretical electron periodic electron structures, together with the ‘window’ of reduced fields in which they occur, are compared with experiment, and it is explained why it is necessary to account for all competing scattering processes in order to explain the observed experimental ‘wavelength’. The study highlights the fundamental flaws in trying to explain the observations in terms of a single, assumed dominant electronic excitation process, as is the case in text books and the myriad of misleading web sites.

Electric and Spectroscopic Analysis of Surface Corona Discharges in Ambient Air and Comparison With Volume Corona Discharges

This paper is firstly devoted to experimental electrical analysis of a dc surface corona discharge in ambient air at atmospheric pressure and room temperature. The surface corona discharge is established between a positive pin and a grounded plate electrode, set on the same side of an insulating plate made either in polymethyl methacrylate (PMMA) or in polytetrafluoroethylen (PTFE). The interelectrode distance, the tip radius of the pin, the applied voltage to the pin, and the nature of the dielectric have been parameterized, and discharge currents are analyzed. It is shown for instance that the surface corona discharge can be ignited over a PMMA dielectric for an applied voltage 20% lower than the PTFE case. This is coherent with calculations of electric fields which are, in the pin region, 25% higher over PMMA than over PTFE. Electrical characteristics of dc surface corona discharge are then compared with those obtained in the case of volume corona discharges generated in the same ambient atmospheric air by using the same positive pin in front of a copper cathode plane. It is observed for the same conditions (applied voltage, gap distance, and tip radius) that the surface corona discharge develops with a higher discharge current even if the UV-visible spectra of both surface and corona discharges remain quasi-similar. This better electric efficiency of surface corona discharge in comparison to volume corona is confirmed from calculations of electric fields and corresponding production rates of certain active species near the pin using multiterm Boltzmann equation solution and literature electron-molecule collision cross sections for electrons impacts with ambient air.

On the approximation of transport properties in structured materials using momentum-transfer theory

In this paper, we present a fluid model for electrons and positrons in structured and soft-condensed matter utilizing dilute gas phase cross-sections together with a structure factor for the medium. Generalizations of the Wannier energy and Einstein (Nernst–Townsend) relations to account for coherent scattering effects present in soft-condensed matter are presented along with new expressions directly relating transport properties in the dilute gas and the structured matter phases. The theory is applied to electrons in a benchmark Percus–Yevick model and positrons in liquid argon, and the accuracy is tested against a multi-term solution of Boltzmann's equation (White and Robson 2011 Phys. Rev. E 84 031125).

Spatially resolved transport data for electrons in gases: Definition, interpretation and calculation

The spatiotemporal evolution of electron swarms in the presence of electric and magnetic fields is investigated to facilitate understanding temporal and spatial non-locality in low-temperature plasmas. Using two independent techniques, a multi-term solution of Boltzmann’s equation and a Monte Carlo simulation technique, the synergism of an applied magnetic field and non-conservative collisions (ionization and/or electron attachment) is demonstrated as a means to control the non-locality of relaxation processes. In particular, oscillatory features in the spatial and temporal profiles are demonstrated, and shown to be enhanced or suppressed through the magnetic field strength, the angle between the electric and magnetic fields, and the degree of ionization. Finally we discuss the impact of field configurations and strengths on the transport properties, highlighting the distinctions in the measured transport properties between various experimental configurations when non-conservative processes are present.

Multiterm solution of a generalized Boltzmann kinetic equation for electron and positron transport in structured and soft condensed matter

In this paper, we generalize the semiclassical Boltzmann kinetic equation for dilute gases to consider highly nonequilibrium electrons and positrons in soft condensed matter, accounting rigorously for all types of interactions, including positronium formation, and allowing for both coherent and incoherent scattering processes. The limitations inherent in the seminal paper of Cohen and Lekner [M. H. Cohen and J. Lekner, Phys. Rev. 158, 305 (1967); Y. Sakai, J. Phys. D 40, R441 (2007)] are avoided by solving the kinetic equation using a "multiterm" spherical harmonic representation of the velocity distribution function, as well as formulating a necessarily nonperturbative treatment of nonconservative collisional processes such as positronium formation. Numerical calculations of transport properties are carried out for a Percus-Yevick model of a hard-sphere system, and for positrons in liquid argon. New phenomena are predicted, including structure-induced negative conductivity and anisotropic diffusion.

A multi-term solution of the nonconservative Boltzmann equation for the analysis of temporal and spatial non-local effects in charged-particle swarms in electric and magnetic fields

A multi-term solution of the Boltzmann equation has been developed and used to investigate the temporal and spatial relaxation of charged-particle swarms and associated phenomena induced by non-local effects under the influence of electric and magnetic fields crossed at arbitrary angles when nonconservative collisions are operative. The hierarchy resulting from a spherical harmonic decomposition of the Boltzmann equation in both the hydrodynamic and non-hydrodynamic regimes is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a variety of Maxwellian based weighting functions. Temporal and spatial relaxation profiles of various charged-particle swarm transport properties are presented for certain model and real gases over a range of field strengths and angles between the fields. It was found that the magnetic field strength and angle between the fields have an ability to control the relaxation process: in general, these parameters can be used to enhance or suppress the oscillatory features in the relaxation profiles of various transport properties. The explicit and implicit effects of nonconservative collisions on the drift and diffusion elements in varying configurations of radio-frequency electric and magnetic fields are considered using physical arguments.

Basic data for atmospheric pressure non-thermal plasma investigations in environmental and biomedical applications

The aim of this paper is to discuss some aspects of the optimization of the active species generated by corona or DBD discharges at atmospheric pressure which are very useful in the field of plasma environmental and biomedical applications. For such an optimization, this paper targets, in particular, the use of discharge modeling tools and the problem of accuracy of the required basic data. First of all, an overview on the different experimental diagnostics used for the characterization of these non-thermal plasmas is given followed by a short description of the different models (streamer dynamics, gas dynamics and chemical kinetics coupled with models of basic data calculation) required for complementing such experimental investigations. Then, emphasis is placed on the basic data of charged particles (electrons and ions) needed for streamer dynamics modeling and particularly on the necessity to use accurate and validated basic data in order to have a quantitative (not only qualitative) description of the phenomena and processes occurring in such discharges. An overview is given on the calculations and the fitting methods of collision cross sections and swarm coefficients of the data of charged particles and their validation using, in particular, pulsed Townsend measurements for experimental comparisons. Swarm coefficients are calculated from a multi-term solution of the Boltzmann equation or from Monte Carlo simulation. Some illustrative results are given in the case of the simulations of a dc positive point-to-plane corona discharge in air at atmospheric pressure. The effect of consideration of some basic data, particularly those of polyatomic ions, is shown on the discharge development and the radical production.

Non-equilibrium transport of positron and electron swarms in gases and liquids

The transport properties of positron and electron swarms in gases and liquids find application in many and varied fields. In this paper we present a time-dependent multi-term solution of Boltzmann's equation valid for electrons and positrons, and benchmark it against an independent Monte-Carlo simulation where possible. The transport properties of positrons in dilute gaseous and liquid argon are considered and compared. The sensitivity of the macroscopic transport properties to the anisotropic nature of elastic scattering is highlighted. The temporal non-locality of diffusion for electron swarms in gases under the influence of time-dependent electric and magnetic fields is addressed through the consideration of their associated relaxation profiles.

Benchmark calculations of nonconservative charged-particle swarms in dc electric and magnetic fields crossed at arbitrary angles

A multiterm solution of the Boltzmann equation has been developed and used to calculate transport coefficients of charged-particle swarms in gases under the influence of electric and magnetic fields crossed at arbitrary angles when nonconservative collisions are present. The hierarchy resulting from a spherical-harmonic decomposition of the Boltzmann equation in the hydrodynamic regime is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a Maxwellian velocity distribution at an internally determined temperature. Results are given for electron swarms in certain collisional models for ionization and attachment over a range of angles between the fields and field strengths. The implicit and explicit effects of ionization and attachment on the electron-transport coefficients are considered using physical arguments. It is found that the difference between the two sets of transport coefficients, bulk and flux, resulting from the explicit effects of nonconservative collisions, can be controlled either by the variation in the magnetic field strengths or by the angles between the fields. In addition, it is shown that the phenomena of ionization cooling and/or attachment cooling/heating previously reported for dc electric fields carry over directly to the crossed electric and magnetic fields. The results of the Boltzmann equation analysis are compared with those obtained by a Monte Carlo simulation technique. The comparison confirms the theoretical basis and numerical integrity of the moment method for solving the Boltzmann equation and gives a set of well-established data that can be used to test future codes and plasma models.

Electron Swarm Coefficients in $\hbox{CO}_{2}$–$\hbox{N}_{2}$ and $\hbox{CO}_{2}$–$\hbox{O}_{2}$ Mixtures

The electron swarm coefficients (i.e., drift velocity, longitudinal diffusion coefficient, and effective ionization coefficient) have been measured using a time-resolved pulsed Townsend technique in O2, N2, and CO2 and their mixtures CO2-N2 and CO2-O2, with CO2 shares of 20%, 50%, and 80%. Simultaneously, these swarm data have been calculated from an adequate multiterm solution of the Boltzmann equation, using previously validated sets of electron-molecule collision cross sections. We have found good agreement between measurements and calculations, both for the pure gases and mixtures, particularly for the case of the calculated mean electron drift velocity Wm, which takes into account the effects of longitudinal diffusion and effective ionization. An effort has been placed in covering as high an E/N range as possible (0.01 Td up to 1000 Td) on specific electron swarm data such as transverse and longitudinal characteristic energies, which are also needed for a better understanding of the electrohydrodynamic and kinetic behavior of nonthermal atmospheric pressure electrical discharges.

Relaxation processes of electrons and positrons in gases in electric and magnetic fields crossed at arbitrary angles

A multi-term solution of the Boltzmann equation has been developed and used to investigate the temporal relaxation of charged-particle swarms under the influence of electric and magnetic fields crossed at arbitrary angles. A Monte Carlo simulation technique has been used to verify the Boltzmann equation analysis under hydrodynamic conditions and for an independent study of spatial relaxation of the electrons under non-hydrodynamic conditions. We present results for model and real gases highlighting the explicit influence of the magnetic field and non-conservative collisions on temporal and spatial relaxation characteristics, including the existence of transiently negative electron diffusivity. We present results for thermalization of the positrons in nitrogen highlighting the applicability of our theory for contemporary kinetic studies in the domain of positron physics.

On the existence of transiently negative diffusion coefficients for electrons in gases in E × B fields

A recent study (Raspopović Z et al 2000 J. Phys. D: Appl. Phys. 33 1298) reported the existence of negative diagonal diffusion tensor elements for electrons in a neutral gas under the influence of crossed radiofrequency electric and magnetic fields. In this paper we demonstrate, using a time-dependent multi-term solution of the Boltzmann equation and time-resolved Monte Carlo simulation, that this phenomenon is a transient relaxation effect obtainable under dc crossed field conditions.

Time dependent multi-term solution of Boltzmann's equation for magnetised low-temperature plasmas

We present details of the first time dependent multi-term solution of Boltzmann's equation for charged particles in gases under the influence of electric and magnetic fields. A Sonine polynomial representation of the velocity distribution function requires a time dependent weight function to accommodate arbitrary initial conditions and/or time dependent fields. The theory and code are benchmarked against an independent time resolved Monte Carlo simulation for both DC and RF orthogonal electric and magnetic fields. The results support the numerical integrity of the technique.

Radio-frequency electronegative gas discharge behaviour in a parallel-plate reactor for material processing

In this paper, a second-order continuum model for radio-frequency glow discharge plasma between parallel-plate electrodes is presented. Simulations have been performed for discharges in argon and O2. The basic data needed for different collision processes are explicitly given with a high degree of accuracy. These data are rigorously calculated by using a multiterm Boltzmann equation solution for electrons and a Monte Carlo simulation technique for ions. The specificity of the present work is the use of a second-order fluid model solved by using a powerful finite-element scheme associated with basic data rigorously calculated. Detailed results for two types of discharge are presented and contrasted. The phenomena associated with the transition of the discharge into different operating regimes at relatively low and high pressures are demonstrated and discussed.

Breakdown electric field calculations of hot SF/sub 6/ for high voltage circuit breaker applications

The critical reduced electric field strengths of hot SF/sub 6/ corresponding to the dielectric recovery phase of a high voltage circuit breaker are calculated for a large temperature range (300-3000 K). Calculations are based on a multi-term Boltzmann equation solution using, in comparison to the literature works, improved cross section sets for the interactions of electrons with various SF/sub 6/ dissociated products. The obtained critical electric fields show a reasonable agreement with the available data. These results are then used in hydrodynamics simulations which correctly predicts the circuit breaker behaviors observed in the case of a successful breaking test as well as in a failed one.

Electric discharge fluid modelling with the contribution of convective and drift energy effects

A fluid model has been used in this work to analyze the electric and energetic behavior of a low-pressure DC glow discharge in Ar chosen as a gas test. The governing equations are the first three moments of the Boltzmann transport equations under their complete form without using the classical-drift-diffusion approximation for the momentum transfer equation while the energy conservation equation involves both thermal and drift energies. In the framework of the local energy approximation, the basic data needed more particularly in the collision source terms for both momentum transfer and energy equations are determined from a multi term solution of Boltzmann equation. Due to the strong coupling with electric field obtained from Poisson equation and the high sheath gradients, the transport equations are numerically solved using a powerful Galerkin finite elements method. This model, after a validation from comparison with literature results, is then used to analyze the convective and drift energy effects on the electric discharge characteristics. Present results show a large influence of the convective term in comparison to the drift-diffusion approximation, mainly on the electric field and charged density profiles due to the antagonist effect induced by this term on the electron and ion motion which reinforces the charge space. Present results show also the discharge characteristic changes mainly in the sheath due to the drift energy consideration.

Electron transport coefficients in O2 magnetron discharges

Electron transport coefficients required for the modelling of bulk electron transport in an O2 magnetron discharge are calculated from solution of the non-conservative Boltzmann equation. The influence on the transport coefficients (rate coefficients, drift velocity vector elements, diffusion tensor elements) of the electric and magnetic field strengths and their orientation with respect to each other have been systematically investigated over ranges consistent with practical operation. The results represent the first multi-term solution of the non-conservative Boltzmann's equation for electric and magnetic fields at arbitrary orientations.

Cross sections and transport coefficients for electrons in Zn vapour

In this paper we present details of the following: ab initio calculations of a set of electron impact cross sections for atomic Zn; andtransport coefficients and distribution functions for an electron swarm in Zn vapour, obtained from a multiterm solution of Boltzmann's equation using these cross sections, over a range of reduced fields, E/N, and gas temperatures of practical interest. Our work has been motivated, in part, by recent suggestions that zinc could be an attractive replacement for mercury in making high-pressure gas discharge lamps more environment-friendly (Born M 2001 J. Phys. D: Appl. Phys. 34 909; Born M 2002 Plasma Sources Sci. Technol. 11 A55). Current models of such lamps require a knowledge of the plasma electrical conductivity, which can be calculated from the (e, Zn) cross sections and mobility coefficients presented in this paper.

On the use of classical transport analysis todetermine cross-sections for low-energy e-H2 vibrationalexcitation

The long-standing discrepancy between the theoretically andexperimentally determined v = 0→1 vibrationalcross-section of hydrogen is addressed by analysing the transporttheory used to deconvolute electron swarm transport data. Theimplementation of the full energy and angular dependence ofquantum mechanically derived differential cross-sections in thesemiclassical transport theory (using both a multi-term Boltzmannequation solution and an independent Monte Carlo simulation) isshown to be unable to resolve the discrepancy. Assumptions andapproximations used in the original transport analyses arequantified and validated.

Development of swarm transport theory in radio-frequency electric and crossed electric and magnetic fields

The advancements associated with modern day technology demands incorporation of the best physical understanding and the most accurate modelling of charged particle motion in gases. In recent times there have been major advances in the fundamental swarm transport theory, and this is the subject of the present paper. We start from 1986, when “multi-term” solutions of Boltzmann’s equation for static electric fields had been developed to a sophisticated level, and proceed through to the present day, where the theory is motivated far more by application to industrial processes, which involve both electric and magnetic fields , either static or time varying. We present a unified time-dependent multi-term solution of Boltzmann’s equation, emphasising the common methods and techniques underlying the treatment of all these situations, whether they be for electron or ion swarms. New and significant numerical results are presented to highlight the rich and diverse range of phenomena which are observed.