# Electron Plasma

July 12, 2014  The electron energy distribution function (EEDF) plays an important role in plasma modeling. Various approaches can be used to describe the EEDF, such as Maxwellian, Druyvesteyn, or using a solution of the Boltzmann equation. Today, we will demonstrate the influence the EEDF has on a plasma model’s results. Additionally, we present a way to compute the EEDF with the Boltzmann Equation, Two-Term Approximation interface.

The electron energy distribution function (EEDF) is essential in plasma modeling because it is needed to compute reaction rates for electron collision reactions. Because electron transport properties can also be derived from the EEDF, the choice of the EEDF you use influences the results of the plasma model. If the plasma is in thermodynamic equilibrium, the EEDF has a Maxwellian shape. In most plasmas, for technical purposes, deviations from the Maxwellian form occur.

### Describing the EEDF

To describe the EEDF, several possibilities are available, such as a Maxwell or Druyvesteyn function. In addition, a generalized form is available, which is an intermediate between the Maxwell and the Druyvesteyn function.

#### Functions That Describe the EEDF

 Maxwell \beta_1=\Gamma(5/2)^{3/2}\Gamma(3/2)^{-5/2}, \ \beta_2=\Gamma(5/2)\Gamma(3/2)^{-1} Druyvesteyn \beta_1=\Gamma(5/4)^{3/2}\Gamma(3/4)^{-5/2}, \ \beta_2=\Gamma(5/4)\Gamma(3/4)^{-1} Generalized \beta_1=\Gamma(5/2g)^{3/2}\Gamma(3/2g)^{-5/2}, \ \beta_2=\Gamma(5/2g)\Gamma(3/2g)^{-1}

Here, ϵ is the electron energy, (eV); \varphi is the mean electron energy, (eV); and g is a factor between 1 and 2. For a Maxwell distribution function, g is equal to 1, while g equals 2 for a Druyvesteyn distribution. Lastly, \Gamma is the incomplete Gamma function.

#### Druyvesteyn

As the Druyvesteyn EEDF is based on a constant (electron energy independent) cross section, the Maxwellian EEDF is based on constant collision frequency. The distribution functions assume that elastic collisions dominate, thus the effect of inelastic collisions (e.g., excitation or ionization) on the distribution function is insignificant. In such a case, the distribution function becomes spherically symmetric. In elastic collisions with neutral atoms, the electrons’ direction of motion is changed, but not their energies (due to the large mass difference).

#### Maxwellian

If the electrons are in thermodynamic equilibrium among each other, the distribution function is Maxwellian. However, this is only true if the ionization degree is high. Here, electron-electron collisions drive the distribution towards a Maxwellian shape. Inelastic collisions of electrons with heavy particles lead to a drop of the EEDF at higher electron energies. Therefore, a Druyvesteyn distribution function often gives more accurate results for a lower ionization degree.       