#### 2.1. Atomic Fundamental Parameters

In the context of the MCDF method, an atomic state is characterized by the electronic configuration

A, the total angular momentum

J, the total magnetic quantum number

M, and all other quantum numbers

$\alpha $ necessary to fully characterize that state. In the absence of external fields, the energies of these states are degenerate in the total magnetic moment, that is, there are

$2{J}_{i}+1$ states with identical energies. The set of these degenerate states is called an atomic level, which will then be characterized by

$(A,J,\alpha )$. The MCDFGME code developed by Desclaux and Indelicato [

23,

24] allows for the calculation of the level energies

${E}_{i}$ as well as radiative

${R}_{if}$, and radiationless

${R}_{if}^{NR}$ transition rates between levels

i and

f. From these quantities one can obtain the other atomic fundamental parameters.

The fluorescence yield of an one-hole atomic configuration

A is defined as the probability that a vacancy in the i

^{th} subshell is filled through a radiative transition

where the sum in

i is over all the initial levels belonging to the configuration

A and the sum in

f is over all allowed final atomic levels. The quantity

${g}_{i}=2{J}_{i}+1$ takes into account the degeneracy of the initial level.

#### 2.2. Relativistic Calculations

Level energies and transition probabilities, both radiative and radiationless, were obtained using the multiconfiguration Dirac–Fock method, fully implemented in the general relativistic MCDFGME code developed by Desclaux and Indelicato [

23,

24].

We will give here a brief description of the MCDF method. For a detailed description we refer the reader to Refs. [

23,

25,

26,

27]. The relativistic Hamiltonian is taken in the form

where

${h}_{a}^{\mathrm{D}}$ is the one-electron Dirac Hamiltonian, and

${V}_{ab}^{\mathrm{CB}}$ describes the sum of Coulomb repulsion and Breit interaction between the

ath and the

bth electron. Furthermore, the code also accounts for radiative corrections, namely, self-energy and vacuum polarization. For details on Quantum Electrodynamics (QED) corrections, we refer the reader to Ref. [

28]. Nuclear size effects were taken into account by using a uniformly charged sphere, and the atomic masses and the nuclear radii were taken from the tables by Audi et al [

29], and Angeli [

30], respectively.

For the determination of the radiative transition probabilities we followed the formalism proposed by Löwdin [

31] to treat the nonorthogonality effects, and consider independently the initial- and final-state wave functions obtained in the so-called optimized level scheme. The length gauge was used for all radiative transition probabilities.

For radiationless transitions, we assumed that the creation of the inner-shell hole is independent of the decay process. The continuum-electron wavefunctions are obtained by solving the Dirac-Fock equations with the same atomic potential of the initial state. To ensure orthogonality in these calculations, no orbital relaxation was allowed between the initial and final bound state wavefunctions.

The code was used in the single-configuration approach, with the Breit interaction and the vacuum polarization terms included in the self-consistent field process, and other QED effects included as perturbations.

#### 2.3. Line Shapes

The width of an atomic level

i is given by the sum of all the allowed transition probabilities from this level to all lower-energy levels, both through the emission of a photon (radiative)

${R}_{i{f}^{\prime}}^{R}$ and through the emission of an electron (radiationless)

${R}_{i{k}^{\prime}}^{NR}$, multiplied by

ℏ [

32],

where the index

${f}^{\prime}$ represents all one-hole levels that the initial level

i can decay to radiatively, and the index

${k}^{\prime}$ represents all two-hole levels that the level

i can decay to via a radiationless emission. The width of a particular transition is connected to its parent- and daughter- level’s lifetimes, thus depending not only on the transition rate from

$i\to f$, but also on all possible decay pathways that the atom can undergo, both from the initial and the final level. Thus, we can define the width of a given transition to be the sum of the widths of the initial and final levels, as

An x-ray “line” is composed, in general, of a multiplet of lines resulting from fine structure levels as shown in Refs. [

11,

13,

32], and thus when trying to compute the line width one has to sum Equation (

5) for all of the initial and final levels,

i and

f, respectively, in the x-ray line manifold,

where the sums run over the initial levels

i belonging to the one-hole configuration

A and over the final levels

f belonging to the one-hole configuration

B. This method of computing linewidths is henceforth labeled as “Method 1”.

For example, the transition width of the K${\alpha}_{1}$ multiplet (in Siegbahn notation) is usually obtained as the sum of the K$\left(1{s}^{-1}\right)$ and L${}_{3}\left(2{p}_{3/2}^{-1}\right)$ manifolds (in IUPAC notation), where K represents all of the fine structure levels that arise from a one-hole configuration with a hole in the K shell, while the same is valid for the L${}_{3}$ subshell.

The K

${\alpha}_{1,2}$ and K

${\beta}_{1,3}$ line shapes were obtained, by means of a simulation, from the atomic structure data computed in this work. Similarly to what has been done in previous works (see Refs. [

11,

13]), the emission line shape (without Doppler broadening) is obtained from the individual transitions between fine structure levels. This is accomplished by summing the contribution of every Lorentzian distribution whose centroid is the transition energy and a natural width given by Equation (

4). For comparison with experimental results one has to convolute the simulated line shape with the spectrometer instrumental function in order to get the proper broadening. The intensity of each individual transition is obtained through the expression:

where

${N}_{i}$ is a scaling factor that represents the rate of formation of ions in level

i per unit volume (in units of s

${}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$cm

${}^{-3}$), assumed here to be equal for all levels. The factor

g is the sum of the multiplicities of all the possible levels in the subshell that contains level

i.

${R}_{if}^{R}$ is the radiative rate of a transition from the one-hole level

i to the one-hole level

f, while

${R}_{ik}^{NR}$ corresponds to a radiationless-transition rate between the one-hole level

i to the two-hole level

k. Once the spectra are synthesized, we can in principle fit it with multiple Lorentzians in a similar fashion to the analysis of experimental spectra to obtain natural linewidths [

12,

13,

33]. If this method is applied to the simulated spectra including only the natural widths of the transitions in a given x-ray line, we will obtain the theoretical natural widths of the lines in question. This will be labeled in this work as “Method 2”. The difference between Method 1 and Method 2 is that in the latter, the influence of the broadening of the line shape due to the energy spread of the multiplet, is already taken into account, whereas in the first method we are just summing the weighted widths of each transition as if all of them had the same energy.

The ratio of the two line intensities belonging to the same initial configuration, such as the K

$\beta $/K

$\alpha $ ratio, depends only on the radiative transition rates, and can be written as:

where the sum in

f is over all levels belonging to the one-hole configuration

B and the sum in

${f}^{\prime}$ is over all levels belonging to the one-hole configuration

C. In Equation (

6), the term

${g}_{i}/g$ represents the probability that a given ion is left in level

i after ionization, assuming that all the states are equally probable. This approximation is only valid if the cross sections for the creation of the holes are similar. Note that if we are comparing peaks that originate from the same levels, this factor is the same for both. The second term corresponds to a transition fluorescence yield, that is basically the probability that a radiative transition from level

$i\to f$ occurs taking into account all of the other decay possibilities.

If the simulated spectra were to be compared with experimental results, some broadening mechanisms would have to be included in the simulation. One of the most common ways of including the instrumental broadening or line broadening due to the thermal motion of the atoms that are emitting the x rays (in a gas or plasma), is through the convolution of the natural line shape with a Gaussian profile [

34]. This convolution of a Lorentzian and a Gaussian distributions is usually known as a Voigt profile and can be computed by the direct calculation of

where

and

where

E is the energy,

${E}_{0}$ is the energy centroid of a particular transition,

$\mathsf{\Gamma}$ is the full-width at half-maximum (FWHM) of the Lorentzian profile and

$\sigma $ is the standard deviation of the Gaussian distribution, related to the FWHM

$\alpha $, by

$\alpha =2\sigma \sqrt{2ln2}$.

${I}_{i,f}$ is the intensity of the transition, given by Equation (

6).

Although there is no closed analytical form for the Voigt profile, it can be obtained from the real part of the Faddeeva function,

$w\left(z\right)$ by

where

$z=\frac{E+i\mathsf{\Gamma}}{\sigma \sqrt{2}}$. As with the Fadeeva function, which can be computed from different algorithms [

35], there are some other representations of the Voigt function that are usually given in terms of special functions. Some examples are the confluent hypergeometric function [

36], the Whittaker function [

37], and the Complex Error function [

38]. In any simulation, the instrumental broadening can be simulated by the use of Equation (

11) or, if the instrumental function is closer to a Lorentzian than to a Gaussian shape as is the case of some crystal spectrometers [

11,

13], a Lorentzian shape can also be used. This Lorentzian line shape will have a FWHM obtained by summing the natural width to the spectrometer energy resolution, because the convolution of two centered Lorentzian functions is still a Lorentzian function whose width is just the sum of the two. These procedures have been recently applied to compare high-resolution measurements with simulated spectra [

12,

13,

15].