2.3 Atomic Relaxation
The ionization of an atomic electron in shell $k$ by an incoming particle (photoelectric absorption, Møller or Bhabha scattering) leaves behind a vacancy in the atom's electronic structure of element $i$. The inner-shell vacancy is filled by an outer-shell electron, leading to the emission of either a fluorescence photon (radiative transition) or an Auger / Coster–Kronig electron (non-radiative transition). The transition leaves additional vacancies in outer shells, which themselves relax, leading to a cascade of secondary emissions. A simplified visual diagram of these cascades can be found in Lorence et al. [3] and Naceur et al. [73].
2.3.1 Cascade Selection
For high-Z atoms a complete relaxation cascade may comprise hundreds of distinct paths. Two filters are applied to retain only the cascades that contribute significantly to the transport calculation:
- Only cascades with a probability of occurrence following the initial ionization event greater than
set_minimum_probability(ηmin)(default $\eta_{\min} = 0.1\%$) are included. - Among these, only the $N_{t}$ transitions which produce an electron or photon with energy above the cutoff energy $E_{G+1/2}$ are kept. Cascades that deposit all of their energy below the cutoff contribute only to the local energy / charge deposition and do not enter the multigroup scattering matrix.
2.3.2 Differential Cross-Section per Cascade
The differential cross-section corresponding to the production of either a fluorescence photon ($p'=\gamma$) or an Auger electron ($p'=\texttt{e-}$) along a specific cascade transition $j$ ($1\le j \le N_{t}$), associated with a produced-particle energy $\Delta E_{i,k,j}^{p'}$ and an occurrence probability $\eta_{i,k,j}^{p'}$, is
\[\sigma_{s}^{i,k,j,p\rightarrow p'}(E \rightarrow E') = \eta_{i,k,j}^{p'}\,\delta(E'-\Delta E_{i,k,j}^{p'})\,\sigma^{i,k,p}_{t}(E) \,,\]
where $\sigma^{i,k,p}_{t}(E)$ is the $k$-shell ionization cross-section associated with one of the following incident channels:
\[p = \texttt{e-}\]
: inelastic-collision (Møller) cross-section on subshell $k$ (Section 2.2.1),\[p = \texttt{e+}\]
: inelastic-collision (Bhabha) cross-section on subshell $k$ (Section 2.2.1),\[p = \gamma\]
: photoelectric cross-section on subshell $k$ (Section 2.1.3).
The total cross-section for the production of relaxation radiation is therefore the sum over every cascade $j$ that survives the filtering of Section 2.3.1.
2.3.3 Data Source and Element Coverage
The values of $\Delta E_{i,k,j}^{p'}$ and $\eta_{i,k,j}^{p'}$ are precomputed for every element and every initial vacancy using the relaxation data of the JENDL-5 library [24], itself derived from the EADL library [44], following the approach of Hébert and Naceur [74]. The data is available for $Z\in\{6,\ldots,100\}$ and covers the subshells K, L1–L3, M1–M5, N1–N7, O1–O7, P1–P3 and Q1. For $Z\le 5$ the produced relaxation radiation has very low energy and is ignored: the corresponding energy is treated as locally deposited.
2.3.4 Angular Distribution
The production of fluorescence photons and Auger electrons is assumed to be isotropic in the laboratory frame. This corresponds to using only the $\ell = 0$ Legendre moment of the differential scattering cross-section: every higher Legendre moment of the relaxation contribution vanishes. The assumption is justified by the random orientation of the residual ion at the moment of relaxation.
2.3.5 Coupling with the Triggering Interactions
The relaxation cascade is triggered only when the underlying ionizing interaction is computed in a subshell-dependent way. Concretely:
Inelastic_Collisionwithset_is_subshells_dependant(true)triggers Møller/Bhabha-driven relaxation,Photoelectricwithmodel = "epdl97"triggers photoelectric-driven relaxation.
When Relaxation() is added to the interaction list, every consistent combination of triggering interaction and produced particle (Fluorescence or Auger) is activated. The convenience constructors Fluorescence() and Auger() restrict the activation to the radiative or non-radiative channels, respectively.