Electron Impact Reactions
For each electron impact reaction, the reaction formula is the first piece of information that is required. Most published papers involving the numerical modeling of plasmas include a table of the reactions which make up the plasma chemistry.
Once the formula is entered the dependency of the reaction rate on the electron energy needs to be supplied. The most common way of accomplishing this is to specify cross-section data for each of the electron impact reactions. The cross-section data can be hard to find or not even exist. Cross-section data is available for most of the common gases. The LXCat Plasma Data Exchange Project (www.lxcat.net) is an excellent source to obtain electron impact cross-sections. When using data from the LXCat database it is encouraged to refer to the data as suggested in the LXCat website. Many models in COMSOL use data from LXCat. Any reutilization of these data should also be reference following LXCat norms. It is also encouraged to download the data directly from the LXCat database.
Cross-section data must be in a specific format in order to be imported into COMSOL Multiphysics. See Importing Collision Cross-Section Data.
If cross-section data is not available then the rate coefficient can be specified with a constant value, an Arrhenius expression, or by a lookup table. In the case of an Arrhenius expression, the Arrhenius coefficients must be specified. If it is preferred to specify the rate coefficient using a lookup table the table of rate or Townsend coefficient versus mean electron energy need to be loaded.
Depending on the type of electron impact collision, the following data is also required:
•
The mass ratio of the electron to the target species for elastic collisions.
•
The energy loss (in eV) for inelastic collisions.
•
The ratio of statistical weights between the upper and lower levels of the heavy species if a detailed balance is required.
detailed balance
If a detailed balance is requested the cross–section data for the reverse reaction is created automatically using the principle of the detailed balance (Ref. 1)
where σ is the cross section for the direct reaction, ε is electron energy, Δε is the excitation energy, and glow and gup are the statistical weights of the lower and upper levels of the heavy species.
In practice, if an electron impact reaction created from a cross-section import file is defined as
e+Ar=>e+Ars
and the reverse reaction is requested the following reaction is created automatically
e+Ars=>e+Ar
with the cross section being defined by the principle of detail balance.
When a reverse reaction is requested the ratio of the statics weights needs to be provided. For atomic species the static weight of a level is defined as
where J is the total electronic angular momentum. The values for J can be obtained from Ref. 2. As an example, if the first four 4s levels of argon are to be used in a plasma model as a lumped level, the statistical weight of the excited lumped state is 12, which results from the sum of the statistical weights of the individual levels Table 2-1.
Table 2-1: Atomic data for the 4s levels of argon Ref. 2.
Term
J
Level (eV)
g
2
11.548
5
1
11.624
3
0
11.723
1
1
11.828
3
For more details on the application of the principled of detail balancing and on the computation of statistical weights for molecules the reader is referred to Ref. 3 (page 267).