3.1.2. Electron Temperatures

As in the previous work (Moreno-Diaz et al. [24]), the electron temperature can be estimated by a Boltzmann plot under the assumption that the plasma is in local thermodynamic equilibrium (LTE) (Griem [27]; Griem [29]) using:

$$I\_{ij}^{\,\,\lambda} = (A\_{ij} \mathbb{S}i/\mathcal{U}(T)) \text{N} \cdot \exp(-E\_i/kT) \tag{2}$$
 
$$\ln \left( I\_{ij}^{\,\,\lambda}/A\_{i\,\,\widetilde{\text{J}}i} \right) = \ln(\text{N}/\mathcal{U}(T)) - (Ei\% T)$$

For a transition from a higher state *i* to a lower state *j*, *I*λ*ij* is the integrated measured integral line intensity in counts per second, <sup>A</sup>*ij* is the transition probability, λ is the wavelength of the transition, *Ei* is the excited level energy, *gi* is the energy and statistical weight of level *i*, *U*(*T*) is the atomic species partition function, *N* is the total density of emitting atoms, *k* is the Boltzmann constant, and *T* is the temperature in Kelvin. If the dependency of ln(*I*λ*ij*/*Aijgi* ) vs. *Ei* is plotted for lines of known transition probability (Boltzmann plot) the resulting straight line would have a slope of <sup>−</sup>1/*kT*, and, therefore, the temperature can be obtained without any previous knowledge of the total density of atoms or the atomic species partition function.

In several of the described cases, the temperature can be determined directly through this procedure because, as already mentioned, the emission of Mg II close to 279.5 nm, although weak, it was sufficient for the desired temperature estimation. Spectral lines used in the Boltzmann plot along with the transition probabilities and the energies corresponding to different starting levels were presented in Table 2 of the previous work (Moreno-Diaz et al. [24]). In addition, that table shows the ωref (impact broadening factors) measured by different authors that is necessary for the line self-absorption corrections.

In order to correct for the above mentioned self-absorption of Mg II lines, a procedure similar to that previously used and described in Moreno-Diaz et al. [24] was followed. Voigt profiles were adjusted to Mg II lines to determine their Lorentz broadening and apparent electron densities (*Ne*\*). These values were then compared to the real electron densities (*Ne*), estimated from the Hα-line, in order to obtain the corresponding self-absorption (SA) coefficients in every line. Finally, with these coefficients, the relative intensity in the limit of null self-absorption (negligible self-absorption) was estimated.

An example of the results obtained by means of this procedure is displayed in Figure 7. In this Figure, for times up to 2 μs after laser pulse (500 ns temporal gate), a temperature of 18,700 ± 2000 K can be determined before correcting the line's self-absorption. Once the self-absorption has been corrected, temperatures of 15,200 ± 2500 K are estimated. In the calculations, the uncertainty includes the statistical error from fitting of the temperature value and the statistical uncertainty in the intensities (~10%). These temperature values are maintained within the experimental range for temporal gates of 1000 ns and 2000 ns.

This process was also used for a 3 μs delay from the laser pulse and with those gate times where the intensity of the Mg II lines in the LSP conditions allowed for their discrimination from the background noise (500 ns and 1000 ns). The result obtained in this case was 14,000 ± 2500 K (using the self-absorption corrected values).

For delays equal to or greater than 3.5 ns, the estimation of the temperature by means of this method was been possible because of the lack of clear discrimination of the emission of the Mg II levels from the noise.

In a previous work of the authors (Moreno-Diaz et al. [24]) temperatures of about 10,900 K were estimated by means of an indirect procedure at a delay of 5 μs from the laser pulse. The values obtained here are in-line with that result and with the values that can be found in the bibliography without the presence of water (i.e., with ambient air).

The final results obtained in this work with this first procedure for electron densities and temperatures are shown in Figure 8. As it can be observed, the electron density and temperature decrease with an increase in the delay time. These values are similar to those obtained by other authors in similar experiments without water, as expected from the results obtained in previous work.

**Figure 7.** Boltzmann plot for Mg II spectral lines from Laser Induced Plasma (Al 2024 target) in LSP conditions. The spectrum was recorded at a 2 μs delay time from the laser pulse and a gate time of 500 ns.

**Figure 8.** Electron density and electronic temperature versus delay time obtained with a gate time of 500 ns.
