# Influence of Pre-Ionized Plasma on the Dynamics of a Tin Laser-Triggered Discharge-Plasma
## 3 Theoretical Calculations  
To simulate the emission from tin ions around 13.5 nm, the transition probability (\(gA\)) and oscillator strengths (\(gf\)) versus wavelength (\(\lambda\)) are calculated for a range of tin ion stages by Hartree-Fock configuration interaction (HFCI) Cowan codes [27]. Here configuration interaction (CI) is taken into account between the 4\(d^{m}\), 4\(d^{m-1}\)_mf_, 4\(d^{m-1}\)_t_p_, and 4\(p^{5}\)4\(d^{m+1}\) (_m_ = 4-6, \(t\) = 5-7, \(n\) = 1-8) configurations, as it will considerably influence the \(gA\), the wavelength [28], and narrow the bandwidth. The Slater Condon parameters are set according to [PERSON] [29]. There are tens of thousands of transition lines in the EUV band for highly charged tin ions, and the lines from different charge states and  
Figure 2: Typical waveforms for the voltage (a, black line), current (b, red line), IRD diode X-ray signal (c, blue line), and the laser signal (d, pink line) under two laser-current delays. (**A**) The high voltage was maintained until the pre-ionized plasma was ignited, and the electrical breaks under the laser-current delay 0.68 \(\upmu\)s. (**C**) The tin pre-ionized plasma was generated by laser long before the high voltage was loaded between the anode and the tin cathode. (**B,D**) The X-ray signals, which are enlarged by 20 times, have been magnified corresponding to conditions (**A**) and (**C**) respectively. different transition arrays overlap with each other, forming an unresolved transition array (UTA) or quasi-continuous band. Therefore, it is necessary to broaden the spectral lines when simulating the spectral lines. Statistical methods are used to calculate the averaged wavelength versus ion stages according to Equation (1) [30]. As shown in Figure 3A, the 13.5 nm emission in 2% bandwidth (13.365-13.635 nm) is mainly generated by the Sn\({}^{8+}\)-Sn\({}^{13+}\) 4_d_-4\(f\) and 4_p_-4\(d\) transitions. \[\overline{\lambda}=\sum_{i=1}^{N}\left(\lambda_{i}\cdot gA_{i}\right)/\sum_{i=1 }^{N}gA_{i}\ \ \ \Delta\lambda=\sqrt{\sum_{i=1}^{N}\left[(\lambda_{i}-\overline{\lambda})^{2} \cdot gA_{i}\right]/\sum_{i=1}^{N}gA_{i}} \tag{1}\]  
A collision-radiation (CR) model of Colombant and [PERSON] [31] is used to approximately estimate the ion stage distribution. The ion fraction \(f_{z}\) of charge \(z\) is given in Equation (2) for rate coefficients of collisional ionization \(S\), radiative recombination \(\alpha_{\rm r}\) and three-body recombination \(n_{\rm e}\alpha_{3b}\). Figure 3B shows the ion fractions and average ions stages versus electron temperature for tin to the 15 th ion stage during the electron temperature range from 1 to 70 eV. The electron density is choosing as \(10^{19}\) cm\({}^{-3}\), which is typical for LTD source [13, 14]. Moreover, the electron density has almost no effect on the ion stage distributions when the electron density is under \(10^{21}\) cm\({}^{-3}\)[32]. It is seen that the optimal temperature for Sn\({}^{8+}\)-Sn\({}^{13+}\) ions, which can emit the emission around 13.5 nm, mainly lies in the range of 15-50 eV. The dominant fractional abundance of Sn\({}^{8+}\)-Sn\({}^{13+}\) ions should be less than 60% of the total ions for a given \(T_{e}\). \[f_{z}=\frac{n_{z+1}}{n_{z}}=\frac{S(z,T_{e})}{\alpha_{r}(z+1,T_{e})+n_{e} \alpha_{3b}(z+1,T_{e})} \tag{2}\]  
The LTD plasma is optically thin when comparing it with the LPP plasma [7]. In this way, the opacity effect is ignored. A Boltzmann distribution appropriate to the temperature is used to weight the population of the upper states for each stage [33]. Figure 4 shows the simulated tin spectra as a function of temperature and wavelength. It is clear from these simulated spectra that the optimum electron temperature is around 30 eV, where the emission around 13.5 nm is highest. The spectra in the range 14-16 nm, which are mainly generated by the 4_d_-4\(f\) and 4_p_-4\(d\) transitions of Sn\({}^{6+}\)-Sn\({}^{8+}\) ions, can be seen when the electron temperature is lower than 20 eV.