The collisions that a neutral particle experiences as it travels through a plasma changes the distribution of its energy level population. The types of collisions that FIDASIM takes into account is as follows
The above rate coefficients have units of and are calculated by averaging the respective collisional cross sections with a Maxwellian of the relevent species. The population of the energy level of a neutral atom, , can be described by the following time dependent differential equation where the are the respective target densities.
By rearranging the terms and letting represent excitation/de-excitation depending on the order of the indices we can get the following equation where and
The system of differential equations can be compactly represented as a matrix multiplication.
The solution of this matrix differential equation takes the form of a matrix exponential
where is a vector of the neutral population flux [1/s] for each energy state at time , is the matrix of the eigenvectors of and is a diagonal matrix containing the eigenvalues of . The fractional flux of a neutral traveling through a uniform plasma is shown below.
As you can see the relative populations between states converges fairly quickly.
The number of neutrals in a given state after a time , , is given by
If represents the time spent inside a grid cell the neutral density can be calculated by dividing the above equation by . The total neutral density of a mc marker is shown below.
As you can see over time the total number of neutrals decreases exponentially.