DESCRIPTION OF THE MODEL

The aims of the model introduced in this thesis are as follows:

(1) To simulate both the discharge pulse and afterglow processes in the plasma.

(2) The model must be able to treat processes described by second and

higher-order differential equations in a state-variable form.

(3) The ratio of helium to neon in the buffer gas should be variable.

(4) The code should be applicable to other recombination lasers by

alteration of the relevant atomic, optical and thermodynamic parameters.

The model is based on an infinitely long plasma column of radius R confined by insulating walls (Figure 3.1). The plasma has five (seven) components consisting of electrons and helium (neon) and strontium neutrals and ions. Electrons and ions are accelerated in the axial direction z under the influence of a longitudinal electric field E^. The column is axially homogeneous. A

uniform longitudinal electric field is assumed. Mathematically, the plasma is

completely described by electrical, atomic, optical and thermodynamic parameters. These are then cast as the state variables x^. This approach has

two advantages over conventional methods of plasma modelling:

(1) The equations can always be written as a coupled set of first-order differential equations. By solving these equations numerically, the state matrices may be updated at each successive time step. This allows non-linear and time-varying characteristics to be included. Furthermore, state variable analysis provides a powerful method of manipulating the results obtained by analysis of the A matrix.

(2) The plasma processes can be directly linked to the generalised circuit

analysis program (GCAP) described in Chapter Two.

If all particle temperatures are considered to be constant, then the coefficients of the particle balance equation are constant. However, on dropping the unrealistic assumption of constant particle temperatures across the discharge tube cross-section to include explicit radial dependence of each of the state variables, the situation becomes considerably more complicated. The coefficients become radially and temporally dependent, and the electron temperature becomes a function of the neutral gas concentration and the electric field. Furthermore, constant pressure and a low degree of ionisation cannot be assumed in a pulsed gas discharge at high peak currents. As the degree of ionisation of a pulsed plasma varies with time, so too do its properties such as

thermal conductivity which is a function of electron temperature, its spatial

and temporal gradients, the degree of ionization and the discharge geometry.

Calculations described in this chapter extend the two-fluid model of Ecker

and Zoller^^ for a DC discharge to apply to pulsed systems. This necessarily

introduces non-linearities of second and higher orders. All state variables

describing an inhomogeneous plasma are allowed to be spatially and temporally dependent. The state variables are governed by a system of coupled differential

equations with variable coefficients whose solution is not obtainable by conventional methods. The state equations describing the atomic, optical, thermodynamic and electrical processes are derived in turn below.