Collisional Radiative Model

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

• Spontaneous decay: $A_{m \rightarrow n} / A_{n \rightarrow m}$
• Electron-impact excitation/de-excitation: $q^e_{m \rightarrow n} / q^e_{n \rightarrow m}$
• Ion-impact excitation/de-excitation: $q^i_{m \rightarrow n} / q^i_{n \rightarrow m}$
• Impurity-impact excitation/de-excitation: $q^Z_{m \rightarrow n} / q^Z_{n \rightarrow m}$
• Electron-impact ionization: $I^e_n$
• Ion-impact ionization: $I^i_n$
• Impurity-impact ionization: $I^Z_n$
• Charge exchange with ions: $X^i_n$
• Charge exchange with impurities: $X^Z_n$

The above rate coefficients have units of $cm^3/s$ and are calculated by averaging the respective collisional cross sections with a Maxwellian of the relevent species. The population of the $n^{th}$ energy level of a neutral atom, $f_n$, can be described by the following time dependent differential equation $$\frac{df_n}{dt} = - \left ( \sum_{k=i,Z} f_n d_k X^k_n + \sum_{k=e,i,Z} f_n d_k I^k_n \right ) \\ + \sum_{m>n} \left (f_m A_{m \rightarrow n} + \sum_{k=e,i,Z} (f_m d_k q^k_{m \rightarrow n} - f_n d_k q^k_{n \rightarrow m}) \right ) \\ + \sum_{n>m} \left (-f_n A_{n \rightarrow m} + \sum_{k=e,i,Z} (f_m d_k q^k_{m \rightarrow n} - f_n d_k q^k_{n \rightarrow m}) \right )$$ where the $d_k$ are the respective target densities.

By rearranging the terms and letting $q^k_{n \rightarrow m}$ represent excitation/de-excitation depending on the order of the indices we can get the following equation $$\frac{df_n}{dt} = C_{n,n} f_n + \sum_{m \ne n} C_{n,m} f_m$$ where $$C_{n,n} = - \left [ \sum_{k=i,Z} d_k X^k_n + \sum_{k=e,i,Z} d_k I^k_n + \sum_{m \ne n} \left ( A_{n \rightarrow m} + \sum_{k=e,i,Z} d_k q^k_{n \rightarrow m} \right ) \right ]$$ and $$C_{n,m} = A_{m \rightarrow n} + \sum_{k=e,i,Z} d_k q^k_{m \rightarrow n}$$

The system of differential equations can be compactly represented as a matrix multiplication. $$\frac{d \mathbf{f}}{dt} = \mathbf{C} \cdot \mathbf{f}$$

The solution of this matrix differential equation takes the form of a matrix exponential

$$\mathbf{f}(t) = e^{\mathbf{C} t} \cdot \mathbf{f}(0) = \mathbf{S} \cdot e^{\mathbf{\Lambda} t} \cdot \mathbf{S}^{-1} \cdot \mathbf{f}(0)$$

where $\mathbf{f}(t)$ is a vector of the neutral population flux [1/s] for each energy state at time $t$, $\mathbf{S}$ is the matrix of the eigenvectors of $\mathbf{C}$ and $\mathbf{\Lambda}$ is a diagonal matrix containing the eigenvalues of $\mathbf{C}$. 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 $t$, $\mathbf{n}(t)$, is given by

$$\mathbf{n}(t) = \mathbf{S} \cdot ( \mathbf{\Lambda}^{-1} \cdot e^{\mathbf{\Lambda} t} - \mathbf{\Lambda}^{-1} ) \cdot \mathbf{S}^{-1} \cdot \mathbf{f}(0)$$

If $t$ represents the time spent inside a grid cell the neutral density can be calculated by dividing the above equation by $V_{cell}$. The total neutral density of a mc marker is shown below.

As you can see over time the total number of neutrals decreases exponentially.

Fortran References

• colrad: Fortran implementation
• get_rate_matrix: Constructs rate matrix $A$
• AtomicRates: Derived type that stores populating and de-populating transitions