Introduction

This page describes how to obtain electron transport data for use in plasma fluid simulations. Typical transport parameters are:

The electron mobility \( \mu_e \)
The electron diffusion coefficient \( D_e \). Simulations are usually not very sensitive to this coefficient, so for simplicity a scalar diffusion coefficient is typically used.
Ionization coefficient(s) \( \alpha \), which describe the mean number of ionization events per electron per unit distance
Attachment coefficient(s) \( \eta \), which describe the mean number of attachment events per electron per unit distance
Electron-neutral rate coefficients

Cross sections

Transport coefficients are typically computed from electron-neutral cross sections using a so-called Boltzmann solver, see Boltzmann solvers. The first step in computing transport coefficients is therefore to select a set of cross sections for a given gas mixture. The most convenient source for such cross sections is lxcat, which aims to

address, at least in part, the well-recognized needs for the community to organize the means of collecting, evaluating and sharing data both for modeling and for interpretation of experiments.

Electron-neutral cross sections are usually given in tabulated form for a range of electron energies.

To download electron-neutral cross sections from lxcat, take the following steps:

Go to data center, browse and download
Highlight scattering cross sections and electrons
Next highlight all databases (unless you already know what you are looking for)
Highlight the ground states present in your gas mixture, for example N2, O2 and Ar
Next, the types of cross sections has to be selected. Highlight all physical processes: attachment, ionization, rotational and excitation. Furthermore, a choice has to be made for the elastic collision cross sections, choosing between elastic and effective. This choice is discussed below in Elastic collision cross sections, but you can simply select both here.
A list of cross section databases is now shown. It is important to use a so-called complete set of cross sections, which means that all relevant electron energy loss processes are accounted for. For each gas component, select such a set using the + button.
Next, you can see a picture of the included cross sections and download the data in TXT format. The beginning of the text file describes the format that is used. Furthermore, each database has a description with often important information, and instructions on how to cite the database, which is also important.

Next, the downloaded cross sections can be used in a Boltzmann solver, see Boltzmann solvers.

Elastic collision cross sections

There exist various types of elastic collision cross sections on lxcat, categorized under elastic and effective. Before, there were also momentum cross sections, but a post from Sergey Pancheshnyi indicates that these have been renamed to effective. According to lxcat documentation:

"elastic" is used to denote the elastic momentum transfer cross section and where "effective" denotes the total momentum transfer cross section (sum of elastic momentum transfer and total inelastic cross sections). The latter is useful for solving the Boltzmann equation in the 2-term approximation.

Some further documentation is present in a post from Leanne Pitchford:

The cross sections for each species include EITHER elastic cross sections OR momentum transfer cross sections.

Elastic cross sections : these correspond to elastic momentum transfer cross sections.

Momentum transfer cross sections : these correspond to a total momentum transfer or "effective momentum transfer" cross sections. These include the effects of inelastic collisions as is appropriate for use in the two-term spherical harmonic expansion. See, for example, Baraff and Buchsbaum, Phys. Rev. 130, 1007 (1963) and Sec. IIB of Pitchford and Phelps, Phys. Rev. A 25, 540 (1982). Where data is available, the effective Qm is set equal to the sum of the inelastic cross sections plus the elastic momentum transfer cross section. This is an approximate relation.

In other words, the effective cross sections include the momentum loss of all types of electron-neutral collisions, not only elastic processes. It depends on the type of Boltzmann solver (see Boltzmann solvers) which type of cross section can be used. Effective cross sections should probably only be used with a two-term Boltzmann solver.

A further complication is that different types of elastic cross sections can be found, for example the elastic momentum transfer cross section and the elastic total cross section. The main idea is the following: when an electron elastically collides with a gas molecule, the scattering angle for the electron is generally anisotropic. In principle, such collisions should be described with differential cross sections, that give the probability of scattering per unit angle \( \frac{d\sigma}{d\Omega} \). These differential cross sections can be integrated to give the total cross section

\[ \sigma_{\mathrm{total}} = \int_{0}^{2\pi} \int_{0}^{\pi} \frac{d\sigma}{d\Omega} \, \sin(\theta) d\theta d\phi \]

which describes the probability of an elastic collision, regardless of scattering angle. Note that there is usually no dependence on the \( \phi \) direction, so that this just gives a factor \( 2 \pi \). In practice measuring or computing differential cross sections is difficult. The main effect these collisions is to reduce the momentum of electrons. How much momentum is lost is described by the momentum transfer cross section, given by

\[ \sigma_{\mathrm{mt}} = \int_{0}^{2\pi} \int_{0}^{\pi} [1- \cos(\theta)] \frac{d\sigma}{d\Omega} \, \sin(\theta) d\theta d\phi \]

The factor \( 1- \cos(\theta) \) here weighs the cross section by the fraction of momentum that is lost. In a model it is convenient to assume that electrons scatter isotropically, meaning that all angles are equally probably. The differential cross section then simplifies to

\[ \frac{d\sigma}{d\Omega} = \frac{\sigma_{\mathrm{isotropic}}}{4 \pi} \]

With this assumption, the momentum transfer cross section is simply given by

\[ \sigma_{\mathrm{mt}} = \sigma_{\mathrm{isotropic}} \]

In other words, one can directly use the momentum transfer cross section in an isotropic scattering model, and have the correct amount of momentum loss.

Three-body processes

One has to be careful with three-body processes, such as three-body attachment to oxygen: e + O2 + O2 -> O2- + O2. There are two options. The first is to multiply the corresponding cross section with the density of the third body (in this case O2) in the correct units before applying a Boltzmann solver. The second is to essentially have a negligible cross section for this process in the Boltzmann solver, but to scale it back to the desired amplitude afterwards in the simulation model. This second option is convenient not fully consistent. It can lead to problems when the three-body process has a significant effect on the shape of the electron distribution function, for example at high pressure.

Boltzmann solvers

Transport data can be computed from cross sections with a so-called Boltzmann solver. The idea is to solve Boltzmann's equation for electrons under simplified conditions, for example by assuming that the electron distribution function \( f(\vec{x}, \vec{v}, t) \) is homogeneous in space, and that the electric field is constant and homogeneous. A few Boltzmann solvers are listed below.

Bolsig+ (Popular two-term Boltzmann solver, closed source)
Bolos (Open-source two-term Boltzmann solver, implemented in Python)
Magboltz (Particle / Monte Carlo method implemented in Matlab, open source)
Particle_swarm (Particle / Monte Carlo method, open source, implemented in Fortran)

Bolsig+

An easy way to use Bolsig+ is through the online interface at lxcat via the online calculations button. (The online version has some downside however, see below!) After selecting cross sections from the desired database(s), you could for example use the following options

option	value	description
EEDF	non-Maxwellian	The EEDF is generally non-Maxwellian
E/N range	1-1200 Td	A reasonable range for streamer simulations
`# of points`	100	This seems to be the limit for online calculations
E/N profile	exponential	To have more points at lower fields, quadratic could also work
`Tgas`	300 K	Gas temperature
Super-elastic collisions	include	Probably not significant at low gas temperature and high E/N
Growth model	temporal	Temporal is recommended, see this paper

After pressing run calculations, you can download the swarm parameters in a text file. This file can be converted to input for the afivo-streamer code with the script tools/bolsig_convert.py, for example like:

./bolsig_convert.py swarm.txt my_new_file.txt

where swarm.txt is the output of Bolsig+. Execute bolsig_convert.py with the -h options to see its full usage. The output should now contain the following:

Mean energy (eV)
Mobility *N (1/m/V/s)
Diffusion coefficient *N (1/m/s)
Townsend ioniz. coef. alpha/N (m2)
Townsend attach. coef. eta/N (m2)
Data for specific collisions, either selected by type or by name. By default ionization and attachment processes are included.

One still has to define the reactions before the file can be used as input for the afivo-streamer code, see Transport data format

A more advanced way to compute transport data is to download an offline version of a Boltzmann solver and compute the data locally, for example using BOLSIG+ (Windows), BOLSIG- (linux), or particle_swarm. When input data is linearly interpolated, it is best to compute a large number of data points to reduce interpolation errors, in particular in the rate coefficients, which are often far from linear. Another option is to enable cubic interpolation of the input data, see Transport data interpolation.

Transport data format

Finally, a list of reactions has to be added by hand, since it is hard to deduce the species appearing on the right-hand side from the output of a Boltzmann solver. The format of these reactions is illustrated in the files in the transport_data folder. A standard file for O2 could look like this

reaction_list
-----------------------
# Ionization
e + O2 -> e + e + O2+,field_table,C43 O2 Ionization 12.06 eV
# Attachment
e + O2 + O2 -> O2-,field_table,C27 O2 Attachment
e + O2 -> O-,field_table,C28 O2 Attachment
-----------------------

In this example, the reactions are defined by a field_table, which tabulates the reaction rates versus reduced electric field. The strings C43 O2 Ionization 12.06 eV etc. refer to the names of the corresponding tables. Note that the three-body reaction rate constant here will be multiplied by the density of O2 squared. The cross section for this process should therefore not have been pre-multiplied by the O2 density when it was used in the Boltzmann solver.

Further details about the format of chemical reactions can be found in the chemistry documentation.

Transport data interpolation

To speed up computations, transport and reaction coefficients are stored in lookup tables, see lookup_table_fortran. These tables have a fixed size and spacing, so that values can quickly be obtained without searching through a list. These tables are constructed and used as follows:

Step 1 The input data is read in from a text file

Step 2 The input data is interpolated to have a regular spacing in E/N. This interpolation is controlled by the following parameters:

table_data%min_townsend = 0. (minimum E/N)
table_data%max_townsend = 1000. (maximum E/N)
table_data%size = 1000 (number of points in the table)
table_data%input_interpolation = linear (or cubic_spline)

Note there are two options for the interpolation of input data: linear (the default) and cubic spline interpolation. The interpolated data is then stored in a lookup table.

Step 3 When a value is required at a certain E/N, the values in the lookup table are linearly interpolated.

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