Blake E. Cherrington

ELEMENTARY THEORY OF A GAS DISCHARGE

This chapter focuses on the elementary theory of a gas discharge. The simplest model that includes interactions is the Lorentz gas. In this model, it is assumed that the electrons are free to move in a stationary uniform background of ions and neutrals that provide a viscous damping force. The electrons are assumed to interact with each other only through the space charge fields. The chapter discusses the Langevin equation and its application. The solution of Langevin equation provides the directed motion of an average electron that is assumed to be representative of all of the electrons in the discharge. The results for one electron is multiplied by the density of electrons, and the result is assumed to represent the gas discharge as a whole. Moreover, the equations obtained for the conductivity and dielectric constant are useful for obtaining good approximations of the actual behavior of a gas discharge in the area of investigation involving the propagation of electromagnetic waves through ionized media.

DISTRIBUTION FUNCTIONS AND THE BOLTZMANN EQUATION

This chapter discusses the distribution functions and the Boltzmann equation. To obtain a realistic idea of the processes occurring within a plasma, the distribution of velocities of the electrons within the plasma should be determined. The equations for each and every particle should be solved separately. The properties of a plasma that are observed are because of the contributions of large numbers of particles; therefore, a statistical description is perfectly adequate. This statistical description is provided by the distribution function. To give a complete specification of the properties of a system of particles, it is necessary to know the position and velocity of each particle. The entire system at any instant of time is specified by the positions of the points in phase space, where each point represents a particle. As a necessity for the validity of the statistical approach, each elemental volume in phase space must contain a large number of particles. The number of particles in any elemental volume of the phase space is specified by a distribution function.

DC DISCHARGES — THE POSITIVE COLUMN

This chapter focuses on the case where the source and loss terms balance. This is the steady-state condition that applies to direct current (DC) discharges and high radio frequency (RF) discharges. The analysis is applied to the positive column. This is the common regime for gas discharges and gas lasers. Depending on the pressure, current, and gas medium that is under consideration, the dominant electron loss mechanism can be diffusion, recombination, or attachment. The most common regime is when the loss of electrons by diffusion balances the production of electrons by collisional ionization. The other situation is where attachment or recombination dominates the electron loss. In the analysis of diffusion-dominated discharges described in the chapter, the fluid equations were used. The source term was assumed as one-step ionization of a neutral by an electron. The chapter discusses longitudinal electric field in the diffusion-dominated discharge. The simplest approach to determine the electric field along the positive column is to balance the energy gain from the electric field with the energy loss due to elastic collisions. Inelastic energy losses should also be included if they constitute a significant energy loss.

ATOMIC NEUTRAL GAS LASERS

This chapter examines a number of gas laser systems on the basis of fundamental gas discharge processes. A laser is an optical oscillator, and it must consist of a medium with optical gain and a positive feedback system so that oscillation can be produced. The chapter discusses small signal gain, saturation intensity, and amplitude of oscillation. The concept of saturation plays an important role in laser amplifiers and oscillators. The product of the small signal gain and the saturation intensity represents the power that can be extracted per unit length of the amplifier. The chapter examines the Helium–Neon (He–Ne) laser and focuses on the collisions between the helium metastables and neon ground state atoms to produce neon atoms in excited states corresponding to the upper levels of conventional laser transitions. The important processes for pumping the He–Ne laser involve the resonant transfer of energy by inelastic atom–atom collisions between excited helium atoms and ground state neon atoms. The chapter also discusses rate equations, population inversion, and gain.

ELECTRON-DENSITY DECAY PROCESSES

This chapter discusses the application of fluid equations for the examination of the fundamental processes occurring in gas discharges. The first application deals with the electron density rate processes occurring within a discharge. These processes deal primarily with the continuity equation but now include important collisional processes such as ionization, recombination, and attachment. The chapter discusses the important electron loss processes—diffusion, recombination, and attachment—that determine the behavior of a plasma when there is no electron source. It also discusses the case of ambipolar diffusion. It is assumed that the electrons are free to diffuse through a uniform neutral background gas and that the ions have no substantial effect on this process. This free diffusion, however, can only occur in low density plasmas where the Coulomb forces can essentially be neglected. In higher density plasmas, such as commonly occur in gas discharges, the diffusion rates must be determined by simultaneously solving the two-fluid equations for electrons and ions, keeping in mind that the strong Coulomb forces will tend to maintain overall space charge neutrality.

THE FLUID EQUATIONS

The large majority of the basic processes occurring within gas discharges can be explained on the basis of the fluid equations. The use of the fluid equations suggests that the plasma is considered to act as a whole or as a conducting fluid rather than as individual particles. The condition for validity of this approximation is that the distance between particles be small with respect to the interparticle forces and the mean free path << scale of change of macroscopic quantities. This condition is usually well satisfied in most gas discharges and the fluid equations are found to be useful. There are three main fluid equations: (1) the equations of continuity, (2) momentum conservation, and (3) energy conservation. In a gas discharge, the electron gas and the neutral gas are the main focus. For the neutral gas, the main case of interest is the determination of the spatial temperature distribution when the source of energy is from collisions with the electron gas. Moreover, the Langevin approach to the determination of important discharge parameters gives good results for the electron gas. For the neutral gas, the heat flow terms represent the total heat loss.

Chapter 5 – TRANSPORT COEFFICIENTS

Publisher Summary This chapter focuses on the anisotropic part of the distribution function from which the appropriate net motion or drift of electrons and hence, the transport coefficients can be derived. The Boltzmann equation is used to solve the electrical conductivity of an ionized medium under the influence of an externally applied alternating current (AC) electric field. There is a direct conversion possible between conductivity and mobility; therefore, the mobility can be directly determined from the conductivity. The chapter presents some equations to calculate ion mobilities. The ion–molecule interaction is usually simpler than that for electrons because at low energies the polarization interaction potential applies, and the collision frequency and mobility are independent of electron energy. At higher energies, the ion–molecule interaction becomes a hard sphere interaction and the collision frequency increases with ion velocity; therefore, the mobility decreases. If a plasma is nonuniform in space, then there will be a diffusion of particles from regions of high density to areas of low density. This process can be characterized by a diffusion coefficient, and the diffusion coefficient can be determined using the Boltzmann equation expansion.

Chapter 3 – COLLISIONS

Publisher Summary This chapter discusses the fundamental aspects of collisions in plasmas by relating it to the collision frequency to the collision cross section and then to the differential scattering cross section, which contains the essential features of the collisional interaction between two particles. The collision cross section depends on electron velocity or energy. The dependence is because of the fact that electron–molecule collisions represent an interaction between the incoming electrons and the electrons, plus the nucleus of the molecule. This gives rise to a spatially varying interaction potential between the electron and the molecule that causes the cross section to vary with the relative velocities of the two collision partners. The chapter discusses the scattering of a particle by a central force, that is, the potential because of the scattering partner is purely radial. The scattering theory helps derive the differential and total scattering cross sections. The chapter also describes electron–molecule hard-sphere collisions.

Chapter 9 – EXCITED SPECIES

Publisher Summary This chapter discusses the excited atomic and molecular species within the discharge because it is these species that are critical to the determination of light output for lamps or optical gain and available power for lasers. The description of the electrons is a necessary prerequisite to the study of excited species because electron impact excitation is either the primary or secondary source of excitation of the excited species. To determine the density of the excited species of interest, either on a steady state or time varying basis, it is necessary to evaluate the production and loss mechanisms. The excited species can be produced by electron impact excitation from ground or lower excited species, by collisional energy exchange with excited atoms or molecules, or by radiative cascade from higher excited species. The excited species density can be determined by evaluating the loss processes. There are two distinct cases: (1) when radiative decay dominates and (2) when collisional decay dominates.

Chapter 11 – ION LASERS

Publisher Summary This chapter discusses ion lasers. There are a number of gaseous media in which laser action has been obtained in the ionized species. These can be broken down into the rare gas—argon and krypton—and the metal vapor—lead, calcium, germanium, indium, tin, mercury, cadmium, and zinc systems. The chapter describes helium–cadmium (He–Cd) laser as an example of metal vapor lasers. The He–Cd laser is simple to construct and operates at high efficiency. This laser uses inelastic atom–atom collisions as an excitation method. The rare-gas ion lasers includes argon and krypton ion lasers and is of considerable interest because high powers (∼ 100 W) are available on a continuous wave (CW) basis in the visible through ultraviolet spectrum. These lasers are of quite low efficiency; therefore, the power supplied must be high to obtain appreciable power out. The chapter also discusses argon ion laser. For the CW argon ion laser, there is a two-step excitation process leading to a quadratic dependence of the upper laser level population on discharge current.