## 3 Result and Discussion
### Electron Density of FEGLD  
The hydrogen line broadening effects of electron density on Stark broadening is one of the most widely used techniques for electron density measurement [45]. Both H\({}_{\alpha}\) and H\({}_{\beta}\) could be used for the calculation. In this work, the H\({}_{\alpha}\) is preferred because of its high line shape sensitivity and adequate emission intensity. The emission line of H\({}_{\alpha}\) is the  
Figure 4: Typical discharge voltage and current waveforms for (**a**) positive pulse, and (**b**) negative pulse. convolution of the Gaussian and Lorentzian lines. The Gaussian profile was determined by both the Doppler and instrument broadening, in which the fullwidth at half maximum (FWHM) of the Gaussian is given by:  
\[\Delta\lambda_{\mathrm{G}}=(\Delta\lambda_{\mathrm{D}}^{2}+\Delta\lambda_{ \mathrm{I}}^{2})^{\frac{1}{2}} \tag{1}\]  
where \(\Delta\lambda_{\mathrm{D}}\) and \(\Delta\lambda_{\mathrm{I}}\) are the FWHM of the Doppler broadening and instrumental broadening, respectively. Doppler broadening mainly depends on the temperature of the emitter [46]. The FWHM of the Doppler broadening is given by the expression [47]:  
\[\Delta\lambda_{\mathrm{D}}=7.16\times 10^{-7}\lambda(\frac{\mathrm{T}_{ \mathrm{g}}}{\mathrm{M}})^{\frac{1}{2}} \tag{2}\]  
where \(\lambda\) is the emission wavelength, \(\mathrm{T}_{\mathrm{g}}\) is the gas temperature of plasma in K and M is the atomic weight of hydrogen atoms in g\(\cdot\)mol\({}^{-1}\). The broadening induced by the detection system is called apparatus function or instrumental broadening [41]. The instrument broadening in this work is determined by measuring the FWHM of the mercury lamp characteristic spectrum at 435.8 nm. Its value is given as:  
\[\Delta\lambda_{\mathrm{I}}=0.078\;\mathrm{nm} \tag{3}\]  
while grating 1200 lines\(\cdot\)mm\({}^{-1}\) is used. Lorentzian profiles are mainly determined by the Stark, van der Waals, resonance, and natural broadening, and the expression could be given by:  
\[\Delta\lambda_{\mathrm{L}}=\Delta\lambda_{\mathrm{V}}+\Delta\lambda_{\mathrm{ S}}\;+\Delta\lambda_{\mathrm{R}}+\Delta\lambda_{\mathrm{N}} \tag{4}\]  
Natural broadening is caused by the uncertainty principle of atomic stable state energy, and the expression could be given by:  
\[\Delta\mathrm{E}=\frac{\mathrm{h}}{\Delta\lambda_{\mathrm{N}}}=\frac{\mathrm{h }}{2\pi\mathrm{r}} \tag{5}\]  
where \(\Delta\mathrm{E}\) is uncertainty of the energy level, \(\mathrm{h}\) is Planck constant, \(\tau\) is energy level lifetime in the order of \(10^{-9}\). The FWHM of natural broadening could be expressed as:  
\[\Delta\lambda_{\mathrm{N}}=2\pi\mathrm{r} \tag{6}\]  
The value of natural line broadening is very small by calculation, in the orders of magnitude of \(10^{-5}\) nm, which can be negligible in this study [48]. Resonance broadening occurs by electric dipole of the upper- or the lower-level transit to the ground state. The formula for the FWHM of resonance broadening could be written as follows [47; 49]:  
\[\Delta\lambda_{\mathrm{R}}=8.61\times 10^{-30}\lambda^{2}\lambda_{\mathrm{r}} \mathrm{f}_{\mathrm{r}}\mathrm{N}_{\mathrm{i}}(\frac{\mathrm{g}_{\mathrm{i}}} {\mathrm{g}_{\mathrm{k}}})^{\frac{1}{2}} \tag{7}\]  
where \(\lambda\) is the wavelength of the spectral line, \(\mathrm{f}_{\mathrm{r}}\) is the absorption oscillator strength, which is 0.6407 when the transition is 2s-3p for H\({}_{\alpha}\) line [50], \(\lambda_{\mathrm{r}}\) is the wavelength of the resonance line; \(\mathrm{g}_{\mathrm{k}}\) and \(\mathrm{g}_{\mathrm{i}}\) are the statistical weights of its upper and lower levels, which are 8 and 18, respectively [49]. \(\mathrm{N}_{\mathrm{i}}\) is the ground state number density. Since FEGLD operated in the open air with low H\({}_{2}\)O concentration, the discharge happened on the liquid surface with low temperature, which is not enough to boil the water to improve the H\({}_{2}\)O concentration highly. The resonance broadening should be out of consideration in this work.