[from A Primer on Modeling and Simulation of Plasma Chemistry by W.L. Morgan © 2003]

Chapter 3: Electron Swarm Calculations and Analysis

1. Introduction

I noted in Chapter 2 that our three sources of electron impact cross section data are electron beam measurements, ab initio quantum theoretical calculations, and the deduction of cross sections from electron transport or swarm measurements.

Deducing collision cross sections and intermolecular potentials from measurements of transport coefficients in dilute gases dates from the early days of kinetic theory [1, 2]. The viscosity, diffusion coefficient, and thermal conductivity are functions of the collision cross sections, which are, in turn, functions of the intermolecular potentials.

This approach was carried over to the deduction of cross sections for electron collisions in gases by analysis of measured electron transport coefficients. With the advent and general availability of electronic computers in the 1960s and 1970s A. V. Phelps [3-5] and R. W. Compton [6] and their collaborators were able to develop and refine the methodology and numerical algorithms for inverting transport data.

In electron swarm measurements a burst or swarm of electrons is produced, which then drifts along an electric field applied to a low pressure gas. Various transport coefficients, such as the drift velocity, transverse and longitudinal diffusion coefficients, attachment and ionization coefficients, and excitation coefficients can be measured as functions of the applied electric field. The collision cross sections, which are related to the transport coefficients through Boltzmann’s transport equation [7], can be extracted by a process of inversion.

Whereas the beam and ab initio approaches directly yield cross sections from which we can compute the transport coefficients used in modeling:

Cross sections Þ calculated transport coefficients

in the swarm approach, cross sections themselves are derived from measured transport coefficients, as described by the sequence below:

Measured transport coefficients Þ Cross sections

Cross sections Þ calculated transport coefficients

On the one hand, the cross sections that are derived from swarm data cannot be expected to possess the accuracy and detailed structure of good beam measurements or ab initio calculations. On the other hand, they naturally produce cross section sets that accurately reproduce the macroscopic observables that are relevant to real plasmas. For example, drift velocities or mobilities are directly connected with the plasma conductivity and, hence, with the power deposition in a discharge. Electron diffusion coefficients and ionization and attachment coefficients are intimately related to the ionization balance of a plasma. These are the quantities that are used directly in most plasma models and that are measured in laboratory plasmas.

Modern high-vacuum beam measurement techniques and modern ab initio multichannel quantum calculations performed on super computers can provide very accurate cross sections for low-energy elastic and inelastic collisions. Often, however, when such data are assembled into a model for a molecule and transport calculations are performed, the agreement with measured transport or swarm data is poor. This leaves us somewhat dubious about the value of modeling a processing reactor using very detailed and correct cross sections if the model does not reproduce very accurately the plasma measurables in a well-defined, well-controlled, swarm experiment. An easily conceivable example of this is one where we assemble a detailed model using what we consider to be accurate cross sections from disparate sources and find that the computed Townsend ionization coefficient differs from that measured in a drift tube by an order of magnitude or more.

An example of this can be seen in Figure 1 [8], which is a graph of the ionization coefficient in argon versus the quantity E/N, the electric field divided by the gas number density. As we will see below, E/N is an extremely important parameter in electron transport theory. We see that choice of cross section set has a profound effect upon the ionization coefficient. I will discuss these data further in a later chapter.

Figure 1 Measured and calculated ionization coefficients in argon

The consequences of not calculating accurate electron transport coefficients can be seen in Figure 2.

Figure 2 Two-dimensional modeling results at three power levels in a plasma reactor showing the differences in calculated electron density due to differences in electron collision cross sections

The Peterson and Allen cross section set is a collection of measured and calculated cross sections for argon from the published literature that have commonly been used in upper atmospheric chemistry calculations. That labeled “Kinema” is the cross section set labeled “Morgan” in Figure 1. These discrepancies come about due to inaccuracy in the calculated electron drift velocity and ionization coefficient in argon.

There are two reasons for the disagreements between the measured and calculated quantities, even for simple gases such as argon or helium. First, the individual and independent errors, in both magnitude and energy dependence, in the separate cross sections from different sources conspire to produce a possibly sizable overall error. Second, including all known cross sections does not necessarily mean that all possible or important collision processes have been included in the model. This is somewhat analogous to the missing matter problem in cosmology: All that we know may be only a fraction of what there is. This is where swarm analysis can make a very important contribution. By their nature, swarm-derived cross sections include all possible processes, either explicitly within other cross sections. This is another reason why swarm-derived cross sections often differ from beam measurements and calculations.

The best procedure for dealing with the potential problem of a collection of cross sections producing erroneous plasma transport coefficients is to

1. Assemble the most complete models that we can, using data from beam measurements and theory.

2. Perform swarm calculations for conditions appropriate to transport measurements when such data are available.

3. Systematically renormalize the cross sections in order to reproduce the measured transport coefficients.

References

1. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, NewYork, 1954).

2. D. A. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1976).

3. L. S. Frost and A. V. Phelps, Phys. Rev. 127, 1621 (1962).

4. A. G. Engelhardt, A. V. Phelps, and C. G. Risk, Phys. Rev. 135, A1566 (1964).

5. R. D. Hake, Jr. and A. V. Phelps, Phys. Rev. 158, 70 (1967).

6. L. G. H. Huxley and R. W. Crompton, The Diffusion and Drift of Electrons in Gases (Wiley, New York, 1974).

7. W. L. Morgan and B. M. Penetrante, Comp. Phys. Comm. 109, 432 (1990).

8. A. V. Phelps, personal communication; see jilawww.Colorado.edu/www/research/colldata.html