The Boltzmann equation describes the evolution of a distribution function, f, in six-dimensional phase space:

To be able to solve the Boltzmann equation and thus compute the EEDF, drastic simplifications are necessary. A common approach is to expand the distribution function in spherical harmonics. For high precision, six or more terms might be needed. If the EEDF can be assumed almost spherically symmetric, the series can be truncated after the second term. The EEDF becomes symmetric because in elastic collisions with neutral atoms, the direction of motion of the electrons is changed and their loss of energy is small due to large mass difference. The two-term approximation is valid for all values of reduced electric field (the ratio of the electric field to the number density of the background gas) if inelastic scattering can be neglected. There is no minimum in the cross section, and the cross section for momentum transfer is related to the electron energy by a power-law dependence (Ref. 3).

When the Boltzmann equation option is selected, it becomes necessary to solve a 1D reaction/convection/diffusion equation to compute the EEDF. The equation is somewhat special because the source term is nonlocal and the convection and diffusion coefficients depend on the integral of the solution. The stationary two-term Boltzmann equation, including the effects of electron-electron collisions is:

The source term, S represents energy loss due to inelastic collisions. Because the energy loss due to an inelastic collision is quantized, the source term is nonlocal in energy space. The source term can be decomposed into four parts where the following definitions apply:

where xk is the mole fraction of the target species for reaction k, σk is the collision cross section for reaction k, Δεk is the energy loss from collision k, and δ is the delta function at ε = 0. The term changes slightly when equal energy sharing is used:

Note the factor of 4 differs from the factor of 2 used in Ref. 1, as was later corrected in Ref. 4. The term, λ is a scalar valued renormalization factor which ensures that the EEDF has the following property:

An ODE is implemented to solve for the value of λ such that Equation 3-2 is satisfied. The rate coefficients are computed from the EEDF by way of the following integral

In order to set the mean electron energy to a specific setpoint, a second Lagrange multiplier is introduced to solve for the reduced electric field, such that Equation 3-3 is satisfied. The weak form of the constraint is:

The drift velocity is an important quantity for two reasons. Firstly, it provides a convenient way of comparing the results of the Boltzmann equation to experimental data. The second reason is that it allows the Townsend coefficients to be computed using:

and Δεk is the energy loss for collision k. If the collision is elastic, the energy loss is defined as:

A plot of the drift velocity for different distribution functions versus the reduced electric field for oxygen is shown in Figure 3-1. Experimental data is also included in the plot.

The reduced transport properties are computed using the following integrals

The computed transport coefficients have little dependence on the type of EEDF. However, the rate coefficients for excitation and ionization processes are highly dependent on the shape of the EEDF, due to the exponential drop off in the population of electrons at energies exceeding the activation threshold. Figure 3-2 plots the ionization rate coefficient for oxygen for the types of distribution function.