Examples

LoKI-B: Find the Ionization Rate of Argon

This example shows how to use LoKI-B to find the ionization rate of argon using the cross sections from the LXCat database.

The steps to find the ionization rate coefficient are as follows:

  1. Download the electron-impact cross-sections of interest from a database (e.g. LXCat).

  2. Use LoKI-B to solve the electron Boltzmann equation and obtain the ionization rate coefficient.

  3. Tabulate the ionization rate coefficient in a CRANE-acceptable format.

LoKI-B solves the Electron Boltzmann Equation (EBE) and obtains the ionization rate coefficient \(k_i\) as a function of the reduced electric field \(E/N\). The EBE is solved for a range of \(E/N\) values from 0.001 to 1000 Td. The resulting rate coefficients are then tabulated by LoKI-B, and we pre-process these into a CRANE-acceptable format.

Our computational tool of choice is LoKI-B, which requires MATLAB. If you do not have access to MATLAB, you can use BOLSIG+. The proper usage of LoKI-B or BOLSIG+ is not discussed here.

Obtain Cross Sections from LxCat

The first step is to save the electron-impact cross-sections of interest in a tabulated format. The cross-sections can be obtained from the online LXCat database. The LXCat database contains a large number of electron-impact cross-sections for a wide range of species. The cross-sections are tabulated in a .txt file, which can be downloaded from the LXCat website.

The .txt LXCat file contains a header with information about the cross-sections, and the cross-sections of each process are tabulated separately. Each table contains the cross-section \(\sigma\) (\(\text{m}^2\)) in the second column as a function of electron energy \(\varepsilon\) (\(eV\)) in the first column for the given process. For our simple problem, we nominally need just the ionization cross-section. However, we will also include the metastable excitation cross-section, and importantly, the elastic momentum transfer cross-section. The elastic momentum transfer cross-section is needed to calculate the electron energy loss rate, and therfore, necessary to obtain an accurate EEDF.

LxCat has multiple databases for Ar. Here, we use the Morgan database. From the Morgan database, we select cross-sections for the following processes:

  • e + Ar -> e + Ar: elastic momentum transfer

  • e + Ar -> e + Ar*: metastable excitation

  • e + Ar -> e + e + Ar+: ionization

The cross-sections are tabulated in a .txt file, which can be downloaded from the LxCat website. We have saved the .txt file in the directory crane/tutorials/TwoReactionArgon/data as Ar_Morgan.txt. The file is modified such that: (a) only the metastable excitation pathway is included (i.e. we exclude the “total excitation” process), (b) Ar* is renamed to Ar(eff), and (c) the first comment of each process describes the reaction from the ground state Ar(1S0), which is parsed by LoKI-B.

Warning

In this example, we are only interested in obtaining the ionization rate coefficient from LoKI-B. However, this DOES NOT mean the ionization cross section should be the only one tabulated. The other cross sections, especially elastic scattering, are absolutely necessary to accurately reflect the relaxation of the electrons.

Using LoKI-B to Calculate the Ionization Rate Coefficient

In this section, we set up a LoKI-B input file named Ar_lumped.in that provides the necessary information of operating conditions and cross-sections to solve the EBE and obtain the ionization rate coefficient using LoKI-B. This LoKI-B input file Ar_lumped.in that was used for this tutorial can be found in crane/tutorials/TwoReactionArgon/data/Ar_lumped.in. Please note that CRANE does not read or write this file in any way; Ar_lumped.in must be copied into an appropriate location with LoKI-B, an external software to CRANE.

The input file is reported below. For an explanation on the functionality and use of each of the working conditions, electron kinetics configurations, outputs, etc., please refer to the LoKI-B documentation. However, a brief explanation can be provided. As shown, we have set the range of reduced electric field values to 0.001 to 1000, the excitation frequency is set to 0 for a direct-current field, the gas pressure is 101325 Pa = 1 atm, the gas temperature is 295 K, the electron kinetics calculation is set up to solve the electron Boltzmann equation, the input file points to the Ar_Morgan.txt file that contains the desired cross-sections, the population of the gases and states is listed, and after execution, the program outputs various data files, including the rate coefficients for each process.

Place both Ar_lumped.in and the cross section file Ar_Morgan.txt in the directory LoKI/Code/Input/Argon, and run LoKI-B in MATLAB with the command

>> lokibcl('Argon/Argon_lumped.in')

while in LoKI/Code to execute the input file.

After the input file is executed, a new directory LoKI/Output/ArLumped is generated which includes the output lookup table lookUpTableRateCoeff.txt. This file contains the rate coefficients in units of \(\text{m}^3/\text{s}\) for each process versus the reduced electric field in Td.

Tabulate the Ionization Rate Coefficient in CRANE Format

To tabulate the ionization rate coefficient in a CRANE-acceptable format, the following Python script is a suitable example:

This writes a new two-column file ionization.txt, parsable by CRANE, where the first column is the reduced electric field in Td, and the second column is the ionization rate coefficient in \(\text{cm}^3/\text{s}\).

Now that we have found the ionization rate coefficient and saved it in a file, we are ready to build the CRANE input file for our problem.