# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy
## 3 X-ray Pulse Transmission in Saturable Absorption Conditions  
The impact of saturable X-ray absorption on time distribution of photons in X-ray pulses penetrating elemental iron medium was examined, starting from the time-dependent Beer-Lambert law:  
\[\mathrm{d}I(x,t)=-I(x,t)\;\sigma\;n\;\mathrm{d}x, \tag{2}\]  
with \(I(x,t)\) denoting photon flux (number of photons per unit area and per unit time) at depth \(x\) and moment \(t\) (in cm\({}^{-2}\)s\({}^{-1}\)), \(\sigma\) is the interaction cross section (in cm\({}^{2}\)), dominated by the photoabsorption component at photon energy close to the K-shell binding energy, and \(n\) is the concentration of the target atoms (in cm\({}^{-3}\)). At a low intensity of the incident X-ray beam (low \(I(x,t)\)), atoms have enough time to return to the ground state before arrival of another photon; thus, the atom concentration \(n\), being effectively the concentration of capable absorbing centers, remains constant across the target depth and over the course of the pulse. At a higher intensity, atoms are excited with a higher rate, which makes a significant part of them unable to interact with many arriving photons. This phenomenon is referred to as saturable absorption. It necessitates consideration of a variable concentration of active absorbers \(n(x,t)\), dependent on both \(x\) and \(t\) through the \((x,t)\)-dependent photon flux and the dynamics of ground state reestablishment. To account for the variable concentration of active absorbers \(n(x,t)\), we employed a two-state picture of atom, with the ground state \(N_{1}\) and the excited state \(N_{2}\). In this simple model, an atom in the ground state \(N_{1}\), having absorbed a photon, is excited to the state \(N_{2}\), and during the excited state lifetime \(\tau\), it is unable to interact with photons. Once time \(\tau\) passes, the atom returns to the ground state \(N_{1}\) and is again capable of absorbing a photon. With this model, we describe the concentration of active absorbers \(n(x,t)\) as  
\[n(x,t)=n_{a}\;N_{1}(x,t),\]where \(n_{a}\) stands for the concentration of atoms in the target (in cm\({}^{-3}\)) and \(N_{1}(x,t)\), a dimensionless value in the range (0,1), is the average ground state population per atom at depth \(x\) and moment \(t\). With this relation, Equation (2) takes the following form  
\[\mathrm{d}I(x,t)=-I(x,t)\;\sigma\;n_{a}\;N_{1}(x,t)\;\mathrm{d}x. \tag{4}\]  
The population \(N_{1}(x,t)\) meets the relation  
\[\mathrm{d}N_{1}(x,t)=\left[-I(x,t)\;\sigma\;N_{1}(x,t)+N_{2}(x,t)\;\frac{1}{ \tau}\right]\;\mathrm{d}t, \tag{5}\]  
where the introduced \(N_{2}(x,t)\), similar to \(N_{1}(x,t)\), is a dimensionless value in the range (0,1) and denotes the average excited state population per atom at depth \(x\) and moment \(t\). One can read from Equation (5) that \(N_{1}(x,t)\) decreases with the average number of photoabsorptions per atom (first summand) and grows with the average number of deexcitations per atom (second summand). With the condition \(N_{1}(x,t)+N_{2}(x,t)=1\), which ensures a constant number of available ground and excited states, Equation (5) can be rephrased in the following way  
\[\mathrm{d}N_{1}(x,t)=\left[-\left(I(x,t)\;\sigma+\frac{1}{\tau}\right)\;N_{1 }(x,t)+\frac{1}{\tau}\right]\;\mathrm{d}t. \tag{6}\]  
We simulated the influence of saturable X-ray absorption on time distribution of photons in an X-ray pulse using the system of Equations (4) and (6):  
\[\left\{ \begin{array}{c}\frac{\mathrm{d}I(x,t)}{\mathrm{d}x}=-I(x,t)\;\sigma \;n_{a}\;N_{1}(x,t)\\ \frac{\mathrm{d}N_{1}(x,t)}{\mathrm{d}t}=-\left(I(x,t)\;\sigma+\frac{1}{\tau} \right)\;N_{1}(x,t)+\frac{1}{\tau}\end{array} \right.. \tag{7}\]  
The above system of differential equations for long lifetime \(\tau\) (\(1/\tau\to 0\)) takes the form described in Ref. [43] and can be solved analytically for functions \(I(x,t)\) and \(N_{1}(x,t)\), although the solutions are rather complicated.