2.3 Atomic Relaxation

The ionization of an atomic electron in shell $k$ by an incoming particle leaves behind a vacancy in the atom's electronic structure $i$. The inner-shell vacancy in an atom is filled by an outer-shell electron, leading to the emission of a fluorescence photon or the ejection of an Auger electron. This process leaves additional vacancies in outer shells, triggering the production of more photons or electrons, resulting in a recursive process. These intricate relaxation processes are often referred to as relaxation cascades, for which a simplified visual diagram can be found in Lorence et al. [3] and Naceur et al. [65]. Because such calculation can be rather intensive for high-Z atoms, only particle productions with a probability of occurrence following an initial ionization event greater than 0.1 % are included in this work. Finally, only the $N_{t}$ specific electron cascade transitions which result in electron or photon with energy greater than the cutoff energy $E_{G+1/2}$ are kept for the following calculations.

The differential cross-sections corresponding to the production of either fluorescence ($p'=\gamma$) or Auger electron ($p'=\texttt{e-}$) following a specific electron cascade transition $j$, where $1 \le j \le N_{t}$, with the produced particle energy, $\Delta E_{i,k,j}$, and the probability of occurrence of the $j$ electron cascade, $\eta_{i,k,j}^{p'}$, is given by

\[\sigma_{s}^{i,k,j,p\rightarrow p'}(E \rightarrow E') = \eta_{i,k,j}^{p'} \delta(E'-\Delta E_{i,k,j}^{p'}) \sigma^{i,k,p}_{t}(E) \,,\]

where $\sigma^{i,k,p}_{t}(E)$ is the $k$-shell cross-sections of either inelastic electron ($p=\texttt{e-}$), inelastic positron ($p=\texttt{e+}$) or photoelectric interaction ($p=\gamma$) given in previous sections. The values of $\Delta E_{i,k,j}^{p'}$ and $\eta_{i,k,j}^{p'}$ are computed using the relaxation data from the JENDL-5 library [16], based on the EADL library [36], as proposed by Hébert and Naceur [66]. The data is available for $Z \in \left\{6,100\right\}$ for subshells K, L1 to L3, M1 to M5, N1 to N7, O1 to O7, P1 to P3 and Q1. For $Z \le 5$, the produced relaxation radiation energy is very low and can be ignored in transport calculations. The production of fluorescence photon and Auger electron is assumed to be isotropic.