Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron–neutral collisions and electron–electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.

https://www3.nd.edu/~sst/teaching/AME60637/reading/2005_PSST_Hagelaar_Pitchford_bolsig.pdf

Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models

Plasma–liquid interactions represent a growing interdisciplinary area of research involving plasma science, fluid dynamics, heat and mass transfer, photolysis, multiphase chemistry and aerosol science. This review provides an assessment of the state-of-the-art of this multidisciplinary area and identifies the key research challenges. The developments in diagnostics, modeling and further extensions of cross section and reaction rate databases that are necessary to address these challenges are discussed. The review focusses on non-equilibrium plasmas.

/pdf/plasma-liquid-interactions-a-review-and-roadmap-4kfoerpjnh.pdf

Plasma-liquid interactions: A review and roadmap

In this review we present the energies, configuration, and other properties of resonances (also called "compound states" and "temporary negative ions") in diatomic molecules. Much of the information is presented in the form of tables and energy level diagrams. Vibrational, rotational, and electronic excitation are discussed whenever these processes have given information on resonances; often these excitation processes proceed via resonances. The paper is divided according to molecular species (${\mathrm{H}}_{2}$, ${\mathrm{N}}_{2}$, CO, NO, ${\mathrm{O}}_{2}$), but the main conclusions are discussed by the nature of the processes involved.

Resonances in Electron Impact on Diatomic Molecules

We review the data and models describing the production of the electrons, termed secondary electrons, that initiate the secondary and subsequent feedback avalanches required for the growth of current during breakdown and for the maintenance of low-current, cold-cathode discharges in argon First we correlate measurements of the production of secondary electrons at metallic cathodes, ie the yields of electrons induced by Ar+ ions, fast Ar atoms, metastable atoms and vuv photons The yields of electrons per ion, fast atom and photon vary greatly with particle energy and surface condition Then models of electron, ion, fast atom, excited atom and photon transport and kinetics are fitted to electrical-breakdown and low-current, discharge-maintenance data to determine the contributions of various cathode-directed species to the secondary electron production Our model explains measured breakdown and low-current discharge voltages for Ar over a very wide range of electric field to gas density ratios E/n, ie 15 Td to 100 kTd We review corrections for nonequilibrium electron motion near the cathode that apply to our local-field model of these discharges Analytic expressions for the cross sections and reaction coefficients used by this and related models are summarized

https://iopscience.iop.org/article/10.1088/0963-0252/8/3/201/pdf

Cold-cathode discharges and breakdown in argon: surface and gas phase production of secondary electrons

A summary and critical review of available information on molecular quadrupole and higher moments is presented. A theorem is also proved which shows that only one independent scalar quantity is required to determine a molecular electric multipole tensor of rank p for molecules with an n-fold axis of symmetry where p < n.

Molecular multipole moments

Momentum-transfer cross sections for electrons in He, Ar, Kr, and Xe are obtained from a comparison of theoretical and experimental values of the drift velocities and of the ratio of the diffusion coefficient to the mobility coefficient for electrons in these gases. The theoretical transport coefficients are obtained by calculating accurate electron-energy distribution functions for energies below excitation using an assumed energy-dependent momentum-transfer cross section. The resulting theoretical values are compared with the available experimental data and adjustments made in the assumed cross sections until good agreement is obtained. The final momentum cross sections for helium is 5.0\ifmmode\pm\else\textpm\fi{}0.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for an electron energy of 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ eV and rises to 6.6\ifmmode\pm\else\textpm\fi{}0.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for energies near 1 eV. The cross sections obtained for Ar, Kr, and Xe decrease from 6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$, 2.6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$, and ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$, respectively, at 0.01 eV to minimum values of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.3 eV for Ar, 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.65 eV for Kr, and 1.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ at 0.6 eV for Xe. The agreement of the very-low-energy results with the effective-range theory of electron scattering is good.

Momentum transfer cross sections for slow electrons in he, ar, kr, and xe from transport coefficients,

Momentum transfer and inelastic collision cross sections for electrons in ${\mathrm{N}}_{2}$ have been obtained from electron transport coefficients for values of the electron energy between about 0.003 and 30 eV. The recently proposed polarization correction to the rotational excitation cross sections of Gerjuoy and Stein leads to less satisfactory agreement between theory and experiment than do the unmodified cross sections. The cross sections for vibrational excitation are consistent with those of Schulz provided the total cross section is normalized to 5.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ at 2.2 eV. Furthermore, a tail extending down to the threshold of 0.29 eV is postulated for the $v=1$ vibrational level. Electronic excitation is approximated by a set of six effective cross sections which, for the most part, are consistent with previous results. The ionization cross section of Tate and Smith was used. The mean energy of electrons in ${\mathrm{N}}_{2}$ subjected to high-frequency ac electric fields is found to be a single-valued function of the electric field $E$ divided by the ac radian frequency $\ensuremath{\omega}$, although there are regions of $\frac{E}{\ensuremath{\omega}}$ where the mean energy increases extremely rapidly with $\frac{E}{\ensuremath{\omega}}$.

Determination of momentum transfer and inelastic collision cross sections for electrons in nitrogen using transport coefficients.

Momentum-transfer and inelastic-collision cross sections for electrons in ${\mathrm{O}}_{2}$, CO, and C${\mathrm{O}}_{2}$ are calculated from measured values of the electron drift velocity, characteristic energy, attachment coefficient, and ionization coefficient. The experimental data for ${\mathrm{O}}_{2}$ are most consistent with vibrational excitation cross sections consisting of a series of resonances located at the vibrational energy levels of the negative ion and having values of cross section times energy half-width of the order of ${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ eV. The calculated effective dipole moment for rotational excitation of CO is in very good agreement with values obtained by other techniques. The vibrational excitation cross section for CO at electron energies below 1 eV is in agreement with theoretical predictions. The vibrational excitation cross sections required for consistency with the C${\mathrm{O}}_{2}$ data are very large [(2-5) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$] and include a peak very close to the vibrational threshold of 0.083 eV.

Momentum-Transfer and Inelastic-Collision Cross Sections for Electrons in O-2, CO, and C O-2

Rotational excitation and momentum transfer cross sections for low-energy electrons in hydrogen and nitrogen are obtained from a comparison of theoretical and experimental values for the mobility and the diffusion coefficient. The theoretical values of the transport coefficients were obtained by calculating accurate electron energy distribution functions using an assumed set of elastic and inelastic cross sections. The discrete nature of the energy loss occurring in a rotational or vibrational excitation collision was included in the theory, as were collisions of the second kind with thermally excited molecules. The resulting values of drift velocity and characteristic energy $\frac{D}{\ensuremath{\mu}}$ were compared with experimental data and adjustments made in the assumed cross sections until good agreement was obtained. The momentum transfer cross sections found in this manner agree well with several recent analyses valid in restricted energy ranges. The final values of the rotational excitation cross sections are about twice the values computed using the theory of Gerjuoy and Stein and the latest available value for the molecular electric quadrupole moments. In hydrogen, the analysis has been extended to energies for which vibrational excitation is important. A vibrational cross section with a maximum of roughly 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 3 eV is consistent with the measurements.

Rotational Excitation and Momentum Transfer Cross Sections for Electrons in H 2 and N 2 from Transport Coefficients

The lifetimes of the metastable states of the helium atom and molecule in pure helium have been determined by using a time-sampling technique for the measurement of the time-varying optical absorption caused by the various metastables. The measured loss of helium singlet metastable atoms indicates destruction after diffusion to the wall, upon collisions with single atoms, and by conversion into triplet metastable atoms in collisions with thermal electrons. The triplet metastable atoms are destroyed upon diffusion and by collisions with two neutral atoms which result in the formation of molecules in the $2{^{3}\ensuremath{\Sigma}_{u}}^{+}$ state. These molecules have a natural lifetime of at least 0.05 second and are destroyed upon diffusion to the wall and by collisions with slow electrons or other metastables. The product of the diffusion coefficient and the helium atom density at 300\ifmmode^\circ\else\textdegree\fi{}K is 1.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ (${\mathrm{cm}}^{2}$/sec) (atom/cc) for the atomic metastables and 1.0\ifmmode\times\else\texttimes\fi{}${10}^{19}$ (${\mathrm{cm}}^{2}$/sec) (atom/cc) for the molecular metastable. The cross sections for the destruction of the singlet atomic metastables are 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$ and 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}20}$ ${\mathrm{cm}}^{2}$ for collisions with thermal (300\ifmmode^\circ\else\textdegree\fi{}K) electrons and neutral atoms, respectively. The three-body combination coefficient for the conversion of triplet atomic metastables into molecular metastables is 2.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}34}$ ${\mathrm{cc}}^{2}$ ${\mathrm{atom}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ at 300\ifmmode^\circ\else\textdegree\fi{}K.

A. V. Phelps

Papers

Momentum transfer cross sections for slow electrons in he, ar, kr, and xe from transport coefficients,

Determination of momentum transfer and inelastic collision cross sections for electrons in nitrogen using transport coefficients.

Momentum-Transfer and Inelastic-Collision Cross Sections for Electrons in O-2, CO, and C O-2

Rotational Excitation and Momentum Transfer Cross Sections for Electrons in H 2 and N 2 from Transport Coefficients

Absorption Studies of Helium Metastable Atoms and Molecules