Equation For Distribution Function Research Articles

Past and present aspects of Italian plasma chemistry

The linking between the initial activities of the Rome–Bari group to their present activities is discussed in different fields of plasma chemical-physics. Several topics are presented including (1) plasma catalysis, (2) the interaction of alumina particles with thermal plasmas and its linking with the thermodynamics and transport properties of plasmas, and (3) non-equilibrium vibrational kinetics coupled with the Boltzmann equation for the electron energy distribution functions (eedf) in aerospace and cold plasma applications. The old activities in plasma catalysis pointed out the importance of atomic species affecting catalytic reactions opening also to the possibility of non-equilibrium vibrational distributions on the chemisorbed ad-atoms. This last aspect is discussed also taking into account recent developments. The study of the interaction of alumina particles with thermal plasmas was rationalized assuming that the step controlling the reaction could be the heat transfer plasma-particle. The model used a simplified fluid dynamic approach with the insertion of accurate values of thermodynamic and transport properties of thermal plasmas. This kind of activity can be considered precursor on the intense work in the field with particular attention on the importance of electronic excited states in affecting thermodynamics and transport of plasmas. Moreover, the plasma-particle interactions can be recovered to a given extent in the modern aspects of hypersonics in the boundary layer of re-entering vehicles as well as in the nozzle flow expansion. In both cases, we underline the existence of vibrational distributions (vdf) far from the Boltzmann behaviour and rates presenting an anti-Arrhenius trend as a function of the inverse of gas temperature. Concerning the non-equilibrium vibrational kinetics, we discuss two case studies. The first one deals with the behaviour of cold nitrogen plasmas pointing out the role of superelastic vibrational and electronic collisions on vdf and eedf. In this case, we discuss also the same case study comparing the results obtained by a complete set of electron molecule cross-sections acting on the whole vibrational ladder with the corresponding ones which consider transitions starting and arriving to the vibrational ground state of the molecule. The last case study concerns the dissociation kinetics of CO2 in cold plasmas, a topic largely studied in the past to emphasize the role of vibrational excitation in CO2 destruction. Recent results obtained using a sophisticated model based on the coupling of Boltzmann equation for the eedf and a state-to-state vibrational kinetics of the asymmetric ladder of CO2 are reported. The interplay between dissociation process by direct electron impact collision and by heavy particle collisions with vibrationally excited CO2 molecules is discussed.

Surface-Kinetics-Limited Ostwald Ripening of Spherical Precipitates at Grain Boundaries

Ostwald ripening of sufficiently large (usually macroscopic) precipitates is the late stage of the diffusion decomposition of a supersaturated solid solution, occurring through the formation of fluctuations and subsequent growth of centers (nuclei) of a new phase. The paper describes a theoretical study of the Ostwald ripening of spherical precipitates of a newly formed phase at the grain boundary of finite thickness with the diffusion of impurity atoms from the grain interior to the grain boundary considered. The precipitate growth is assumed to be limited by the kinetics of impurity atom imbedding into the precipitate rather than by the impurity atom diffusion inside the grain boundary. The speed of diffusion growth of spherical precipitate located on the grain boundary is found. A system of equations which describes surface-kinetics-limited growth of Oswald ripening of spherical precipitates on the grain boundary is formulated. This system consists of the equation of growth rate of the precipitate, the kinetic equation for the precipitates size distribution function which is normalized by the precipitates density, and the equation of the balance of matter in the system (the law of conservation of matter). The law of conservation of matter takes into account the atoms of impurities which are in solid solutions of the grain boundary and the body of the grain as well as in the precipitates which is the specifics of our problem. The asymptotic time dependences are found for the average and critical precipitate radius, supersaturation of solid solution of impurity atoms in the grain boundary, precipitate size distribution function, precipitate density, and for the factor of grain boundary filling with precipitates (the area covered by the precipitates per unit area of the grain boundary) and the total number of impurity atoms in precipitates. The factor of grain boundary filling with precipitates is a characteristic of the two-dimensional Ostwald ripening problem. A discussion of the limits of validity of obtained results is given.

Comparative analysis of pork tenderness prediction using different optical scattering parameters

This study aimed at the assessment of pork tenderness using hyperspectral backscattering imaging technique. Backscattering images for 54 pork samples in the spectral range of 480–900 nm were characterized by 3-parameter Lorentzian distribution function (LD) and diffusion equation (DE), which represent a statistical model and a theoretical model respectively. Characteristic parameters spectra were obtained and employed to establish partial least squares regression (PLSR) models for the Warner-Bratzler Shear Force (WBSF) values of pork samples. By contrast, the results showed that LD function interpreted pork scattering profiles more accurately than DE model between 480 and 600 nm. The PLSR model based on two-LD parameter combinations (the asymptotic value a and the peak value b of the backscattering profile) gained the best prediction results, with correlation coefficient in the prediction set (Rp) of 0.902, root mean square error of prediction (RMSEP) of 5.218 N, indicating that hyperspectral backscattering imaging technique coupled with LD model is preferable for rapid and nondestructive evaluation of pork tenderness.

Optical characteristics of overstressed nanosecond discharge in atmospheric pressure air between chalcopyrite electrodes

The results of the investigation of the optical characteristics of a high-current nanosecond discharge in atmospheric pressure air between chalcopyrite electrodes (CuInSe2 and CuSbSe2 compounds) in conditions of overstressed discharge gap are presented. When using electrodes with a relatively large radius of curvature of the working part (3 mm) and a special system for isolating a part of the surface of the electrodes, a relatively uniform nanosecond discharge was obtained in millimeter intervals. Electrical and optical (spectral and temporal) discharge characteristics were investigated. Based on the solution of the Boltzmann equation for the electron energy distribution function, the parameters of the air plasma with small additions of the vapor of easily ionized impurity, in the form of copper vapor, were simulated.

Dusty plasma of a dc glow discharge in different noble gases

A self-consistent non-local model of the positive column of a dc glow discharge in helium, neon and argon with dust particles is presented. A kinetic Boltzmann equation for the electron energy distribution function, drift-diffusion equations for ions and dust particles, the Poisson equation for electric field were calculated self-consistently for different discharge conditions. The radial distributions of the discharge plasma and the dust particles parameters were obtained for different discharge gases. The influence of the buffer gas sort is revealed and discussed.

Experimental and numerical study of a low-frequency ferromagnetic enhanced inductively coupled plasma

Low-frequency ferromagnetic-enhanced inductively coupled plasma (FMICP) has been investigated under conditions typical for large-scale plasma processing. Radial profiles of ion density and plasma floating potential were determined with a Langmuir probe. A self-consistent radial kinetic model of argon FMICP was developed, based on the simultaneous solution of a non-local Boltzmann equation for the electron energy distribution function, balance equations for the ion and metastable argon atom densities, the thermal balance equation and the Poisson equation for a self-consistent radial electric field. A satisfactory agreement between the numerical and experimental results was found that confirms the validity of the presented approach to the description of the FMICP.

Derivation of equations of multimoment hydrodynamics for a gas of particles with internal structure

The equations for pair distribution functions are used to derive the equations of multimoment hydrodynamics for a gas of particles with internal structure. The equations for pair functions are derived in terms of semi-classical approximation. The basic property of the pair functions is established. In conformity with basic property, these functions remain unchanged in time along the trajectory of the inertia center of pair. The basic property of the pair distribution functions reveals the existence of infinite number of principle hydrodynamic values. The equations of multimoment hydrodynamics are constructed using limited number of principle hydrodynamic values. Selected principle values specify measurable moments. The measurable moments are represented by linear combination of principle and non-principle hydrodynamic values. The general structure of constructed multimoment conservation laws is identical to the structure of appropriate multimoment conservation laws in a gas of structureless particles. Each of the multimoment conservation laws is divided into two separate equations. The first group of conservation equations corresponds to convective phenomena. The second group of conservation equations corresponds to dissipative phenomena. Derived equations of multimoment hydrodynamics are designed for interpreting the behavior of medium states, which are far removed from the state of statistical equilibrium. Classic hydrodynamics encountered the problems when interpreting the unstable phenomena. The possibility of improvement of classic hydrodynamics equations for a gas of particles with internal structure is sought on the way toward an increase in the number of principle hydrodynamic values.

A quantum kinetic approach for calculating low-field mobility in black phosphorus crystals and multilayer phosphorene

Analytic expressions for the low-field mobility have been obtained in black phosphorus crystals and multilayer phosphorene. Acoustic and optical phonons, charged impurities and surface roughness are adopted as the scattering system. Theoretical considerations are based on a quantum kinetic equation and special form of the non-equilibrium distribution function (shifted Fermi distribution). Our calculations reveal that the hole mobility in black phosphorus crystals is limited by scattering with both acoustic and optical phonons over a wide temperature range of 10–400 K. The hole mobility in multilayer phosphorene is thus limited by impurity and optical phonon scattering in this temperature range.

Non-equilibrium electron and vibrational distributions under nanosecond repetitively pulsed CO discharges and afterglows: II. the role of radiative and quenching processes

The non-equilibrium vibrational distributions and electron energy distributions of CO in nanosecond repetitively pulsed discharges and afterglows have been determined from a coupled solution of the time-dependent Boltzmann equation for the electron energy distribution function (eedf) of free electrons, the master equations for vibrational levels of CO and the electronic excited states of CO, O and C atoms. The optically thick plasma conditions have been investigated in a companion paper (part I), while in the present paper we also show the results obtained by allowing radiative emission processes (optically thin plasma) as well as electronic excited state collisional quenching processes. Two case studies, which differ for the duration of the afterglow following each pulse (1 μs and 25 μs case studies) are discussed, and each pulse is characterized by a time-dependent electric field profile in the range 0–20 ns. The results, which depend on the number of pulses considered in the discharge and the corresponding afterglow duration, show several peaks in the eedf due to super-elastic electronic collisions. On the other hand, the quenching process of the a3Π electronic state of CO determines the pumping of vibrational quanta in the v = 27 level, which in turn largely modifies the vibrational distribution function (vdf) of CO. As a consequence, the quenching of the a3Π state increases the reactivity of CO through the Boudouard reaction, and under given conditions, this channel can become more important than the dissociation rates by electron impact collisions.

Experimental and numerical study of a dust cloud formation in the stratified positive column of a dc glow discharge in helium

The experimental and theoretical investigations of the formation of dust particle clouds in the stratified positive column of a dc glow discharge in helium were performed. The size and shape of a dust cloud that levitated in the strong electric field of the striations in a vertically oriented discharge tube were measured under different helium pressures. Axial electric field strength was also experimentally estimated. A model for radial distributions of all dusty plasma parameters of the positive column of the dc glow discharge was developed to describe the obtained experimental results. The model is based on the solution of a non-local Boltzmann equation for an electron energy distribution function, drift-diffusion equations for ions and dust particles, and a Poisson equation for a self-consistent radial electric field. The experimental and calculated results show that the size of the dust cloud decreases with the increase in the gas pressure, and the axial electric field strength pressure dependencies have minima. At low gas pressures, the dust particle charge number density increases and exceeds the electron density that strongly influences the plasma parameters.

Structural properties of β-[formula omitted] nanorods under compression and torsion: Molecular dynamics simulations

In recent years, one-dimensional (1D) magnetic nanostructures, such as magnetic nanorods and chains of magnetic nanoparticles have received great attentions due to the breadth of applications. Especially, magnetic nanorods has been opened an area of active research and applications in medicine, sensors, optofluidics, magnetic swimming, and microrheology since they possess the unique magnetic and geometric features. This study focuses on the molecular dynamics (MD) simulations of an infinitely long crystal β-Fe2O3 nanorod. To elucidate the structural properties and dynamics behavior of β-Fe2O3 nanorods, MD simulation is a powerful technique. The structural properties such as equation of state and radial distribution function of bulk β-Fe2O3 are performed by lattice dynamics (LD) simulations. In this work, we consider three main mechanisms affecting on deformation characteristics of a β-Fe2O3 nanorod: 1) temperature, 2) the rate of mechanical compression, and 3) the rate of mechanical torsion.

Parameters of plasma of high-current nanosecond discharge in air with small admixtures of copper vapor

Introduction . The use of a discharge in air of atmospheric pressure opens the possibility of developing windowless UV lamps that are promising for applications with significant mechanical overloads under various extreme conditions, since they do not contain any quartz elements (windows or bulbs). The output characteristics of such point lamps strongly depend on the plasma parameters of the nanosecond discharge with small admixtures of the vapor metal from which the electrodes are made. At present, there are no data on the parameters of air plasma with small admixtures of copper vapor under the operating conditions of a point lamp, which inhibits the establishment of mechanisms for populating the excited levels of atoms and copper ions in such a plasma. This in turn inhibits the long-range optimization of the output characteristics of such point sources of UV radiation. Purpose . The purpose of the study was to determine the emission characteristics in the spectral range of 200-250 nm of the plasma of high-current nanosecond discharge, between copper electrodes in air at atmospheric pressure under conditions of strong overstress of the discharge gap and to carry out a numerical simulation of the parameters of the plasma of air with small copper impurities under conditions close to those realized in the experiment. Methods . The nanosecond discharge in air of atmospheric pressure between copper electrodes was ignited. The distance between the copper electrodes was 1 mm, which led to the ignition at atmospheric pressure air of a point diffuse discharge of a spherical shape with a diameter of 1 mm. The standard program was used to solve the Boltzmann kinetic equation for the electron energy distribution function (EEDF) for modeling air plasma atmospheric pressure parameters with small copper vapor admixtures, since under the experimental conditions the maximum value of the E/P parameter reached about 530 V cm -1 torr -1 , which is less than the critical value of E/P for nitrogen according to the local electron runaway criterion – 590 V cm -1 torr -1 . Results . Numerical simulation of the plasma parameters established that for the range of reduced electric field strength 610 Td-810 Td, at which experimental studies of the characteristics of a point nanosecond discharge were carried out, mean electron energies varied within the range of 12.2 -15.6 eV; the value of the electron concentration was 10.7·10 21 m -3 – 8.7 ·10 21 m -3 at a current density of 765 · 10 6 A / m 2 on the surface of the copper electrode. The excitation constants of the 2 P 3/2, 1/2 , 2 D 3/2 , 2 D 5/2 , 4 F 7/2 , - states of the copper atom and its ionization were in the range (7.3 – 8.5) 10 -13 m 3 /c, ( 0.29 - 0.28) 10 -13 m 3 /c, ( 0.22 - 0.22) 10 -13 m 3 /c, (0.19 – 0.22) 10 -15 m 3 /c and (0.33 -0.45) 10 -13 m 3 /c,, respectively. The maximum excitation rate constant for nitrogen molecules was for the sum of the singlet states: they were (1.4 10 -14 ) m 3 / s for an electric field strength of 810 Td. The specific losses of the discharge power in the plasma on inelastic collisions of electrons with the components of the gas mixture were maximal for nitrogen molecules and reached 32% for excitation of the sum of singlet nitrogen states. Conclusion . Radiation of the plasma is concentrated in a narrow spectral interval of 200-250 nm, is promising for use as a point UV lamp; the spectral lines 248.9 and 254.4 nm Cu II are promising for estimating the electron temperature in a plasma. Mean electron energy, electron mobility, specific electric power losses and the rate constants of elastic and inelastic scattering of electrons by copper, argon and carbon dioxide, oxygen and nitrogen molecules, the ratio of which was 3: 7.098: 0.266: 159.157: 593.438, depending on the magnitude of the reduced electric field (ratio of the electric field strength (E) to the total concentration of atoms and gas molecules (N)). The range of changes in the parameter E / N = 1-1300 Td (1 ∙10 -17 - 1 ∙ 10 -15 V ∙ cm 2 ) and included the values of the parameter E / N, which were realized in the experiment (610-810) Td.

Quantum kinetics approach to calculation of the low field mobility in the hole inversion layers of silicon MOSFET’s

Analytic expressions for low field mobility have been obtained in the quantized p-type inversion layers. The confining potential is approximated by a triangular quantum well. Main attention is paid to study the dependence of the hole mobility on transverse effective field at different temperatures and concentrations of the ionized impurities. Acoustic and optical phonons, charged impurities, and surface roughness have been adopted as scattering system. Theoretical considerations are based on the quantum kinetic equation and special form of the non-equilibrium distribution function (shifted Fermi distribution). Calculations show that the acoustic phonon limited mobility does not depend on the transverse effective electrical field $$E_\mathrm{eff} $$ and has a temperature dependence closer to experiment than known expression for the universal mobility. At the same time, the mobility limited by scattering with optical phonons and surface roughness is proportional to $$E_\mathrm{eff} ^{-1/3}$$ and $$E_\mathrm{eff} ^{-2}$$ , respectively. The mobility limited by scattering by ionized impurities is a weak function of the transverse effective field. Results of the calculations are compared with known experimental data.

Hot electrons in metal films at low temperatures (Review)

This review discusses research completed as a result of the author's cooperation with the Department of Igor Mikhailovich Dimitrenko, from 1975 to 1992. Based on kinetic equations for electron and phonon distribution functions, we theoretically considered hot electrons at low temperatures in metal films deposited on dielectric substrates with high thermal conductivity. We present a brief analysis of the major results related to hot electrons in superconducting films, the problem of heat removal from a sample, the form of a hot electron distribution function when thermalization criterion is not satisfied, the size effect in heat transfer from metal films, the tunneling of hot electrons, as well as the generalization of these results for metals with arbitrary dispersion of electrons and phonons in terms of Eliashberg spectral functions.

Gluon quasidistribution function at one loop

We study the unpolarized gluon quasidistribution function in the nucleon at one loop level in the large momentum effective theory. For the quark quasidistribution, power law ultraviolet divergences arise in the cut-off scheme and an important observation is that they all are subjected to Wilson lines. However for the gluon quasidistribution function, we first point out that the linear ultraviolet divergences also exist in the real diagram which is not connected to any Wilson line. We then study the one loop corrections to parton distribution functions in both cut-off scheme and dimensional regularization to deal with the ultraviolet divergences. In addition to the ordinary quark and gluon distributions, we also include the quark to gluon and gluon to quark splitting diagrams. The complete one-loop matching factors between the quasi and light cone parton distribution functions are presented in the cut-off scheme. We derive the P^z evolution equation for quasi parton distribution functions, and find that the P^z evolution kernels are identical to the DGLAP evolution kernels.

Isotropic-Nematic Transition and Demixing Behavior in Binary Mixtures of Hard Spheres and Hard Spherocylinders Confined in a Disordered Porous Medium: Scaled Particle Theory.

We develop the scaled particle theory to describe the thermodynamic properties and orientation ordering of a binary mixture of hard spheres (HS) and hard spherocylinders (HSC) confined in a disordered porous medium. Using this theory, the analytical expressions of the free energy, the pressure, and the chemical potentials of HS and HSC have been derived. The improvement of obtained results is considered by introducing the Carnahan-Starling-like and Parsons-Lee-like corrections. Phase diagrams for the isotropic-nematic transition are calculated from the bifurcation analysis of the integral equation for the orientation singlet distribution function and from the conditions of thermodynamic equilibrium. Both the approaches correctly predict the isotropic-nematic transition at low concentrations of hard spheres. However, the thermodynamic approach provides more accurate results and is able to describe the demixing phenomena in the isotropic and nematic phases. The effects of porous medium on the isotropic-nematic phase transition and demixing behavior in a binary HS/HSC mixture are discussed.

Principles of Considering the Effect of the Limited Volume of a System on Its Thermodynamic State

The features of a system with a finite volume that affect its thermodynamic state are considered in comparison to describing small bodies in macroscopic phases. Equations for unary and pair distribution functions are obtained using difference derivatives of a discrete statistical sum. The structure of the equation for the free energy of a system consisting of an ensemble of volume-limited regions with different sizes and a full set of equations describing a macroscopic polydisperse system are discussed. It is found that the equations can be applied to molecular adsorption on small faces of microcrystals, to bound and isolated pores of a polydisperse material, and to describe the spinodal decomposition of a fluid in brief periods of time and high supersaturations of the bulk phase when each local region functions the same on average. It is shown that as the size of a system diminishes, corrections must be introduced for the finiteness of the system volume and fluctuations of the unary and pair distribution functions.

Effect of Channel Geometry and Properties of a Vapor–Gas Mixture on Volume Condensation in a Flow through a Nozzle

A method of direct numerical solution of the kinetic equation for the droplet size distribution function was used for the numerical investigation of volume condensation in a supersonic vapor–gas flow. Distributions of temperature for the gas phase and droplets, degree of supersaturation, pressure, fraction of droplets by weight, the number of droplets per unit mass, and of the nucleation rate along the channel were determined. The influence of nozzle geometry, mixture composition, and temperature dependence of the mixture properties on the investigated process was evaluated. It has been found that the nozzle divergence angle determines the vapor–gas mixture expansion rate: an increase in the divergence angle enhances the temperature decrease rate and the supersaturation degree raise rate. With an increase or decrease in the partial pressure of incondensable gas, the droplet temperature approaches the gas phase temperature or the saturation temperature at the partial gas pressure, respectively. A considerable effect of the temperature dependence of the liquid surface tension and properties on gas phase parameters and the integral characteristics of condensation aerosol was revealed. However, the difference in results obtained with or without considering the temperature dependence of evaporation heat is negligible. The predictions are compared with experimental data of other investigations for two mixtures: a mixture of heavy water vapor with nitrogen (incondensable gas) or n-nonane vapor with nitrogen. The predictions agree quite well qualitatively and quantitatively with the experiment. The comparison of the predictions with numerical results from other publications obtained using the method of moments demonstrates the usefulness of the direct numerical solution method and the method of moments in a wide range of input data.

Impact of Initial Electron Development on Sodium Plasma Generation Using LIBORS Technique: Numerical Modeling

The present work reports an investigation on the role played by Na3+ ions formed through triatomic associative ionization collision of Na(4d) atoms with Na2 ground state molecules during the early phase of sodium plasma generation by laser ionization based on resonance saturation (LIBORS). Such ionization mechanism is observed experimentally for the first time by Tapalian and Smith (1993) [1]. In their experiment, stepwise atomic excitations are created using two CW dye lasers; one laser is tuned to 589 nm to excite the Na(3s) to Na(3p) D2 transition of sodium and the other laser is tuned 569 nm to excite the Na(3p) to Na(4d) transition. The analysis is grounded on a numerical study of the role of seed electron processes on the temporal evolution of sodium plasma formation by laser irradiation. A previously developed numerical model based on LIBORS technique is modified and adopted. In the present study, the sodium atom is treated as an atom comprises 22 levels namely: a ground state, 18 excited states and three ionic states (atomic, molecular and tri-atomic). The model tackled various collisional and radiative processes that act to enhance and deplete the free electrons generated in the interaction region. The contribution of these processes is signified by solving numerically a system of time-dependent rate equations, which couple the generated atomic and ionic species with the laser fields. Meanwhile, it solves the time-dependent Boltzmann equation for the electron energy distribution function (EEDF) of the generated electrons. The computed values of the EEDF, time evolution of both excited states population and the formed ionic species considering the individual effect of associative ionization, Penning, and photo-ionization and triatomic associative ionization justified the important effect of each of these ionizing processes in creating the early stage electrons. These seed electrons are assumed to rapidly gain energy through superelastic collisions leading eventually to plasma development.

Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.