3 Data processing and plasma diagnostics

   
3.1 Spectroscopic data processing

Firstly, for each spectral interval at each instant where measurements were performed, the average spectra of the five runs taken with and without mirror M3 were obtained. Averaged spectra differed from the individual spectra by less than 5%, which gives a good idea of the reproducibility of the plasma source in different pulses. By comparing both averaged spectra and using the algorithms described by González (1999, 2000), it has been possible to detect and reconstruct spectral profiles when necessary. It is important to note that self-absorption was detected in less than 10% of the whole spectral profiles and, in less than 10% of these cases, the reconstructed profile differed from the measured one without mirror M3 by more than 20% in the peak intensity. These profiles have been rejected from further calculations.

Figure 2: An example of KrII lines recorded in this experiment.

After dividing the averaged spectra by the spectrometer transmittance functions, all of them were fitted to sums of Lorentzian functions plus a luminous background with a linear dependence (Gigosos et al. 1994). This is justified since the Stark effect is the dominant spectral line broadening mechanism at the electron densities achieved in this plasma. Differences between the experimental spectra and the fits were usually lower than 0.5%. The fitting algorithm allows us to determine simultaneously the center, asymmetry, line width and area of each profile. As it can be seen in Fig. 2, even the very overlapped weakest lines have been considered in the fit, not as an objective by themselves, but with the aim of obtaining an accurate measurement of the intensity of their closest isolated spectral profiles. The final uncertainty estimated for the intensity measurement is lower than 15%. This procedures have been apllied to all KrI and KrII lines.

   
3.2 Electron density

Concerning the 15 measured interferometric recordings, they have been processed according to the algorithms developed and described by Aparicio et al. (1998) and de la Rosa et al. (1990). They allow us to obtain for each wavelength an average curve of the phase evolution changes along the plasma life $\Delta\psi_{\lambda_i}(t)$ (i=1, 2) and from them, the electron density curve $N_{\rm e}(t)$, according to the expression:

$$n_{\rm e}(t)=\frac{4\pi\varepsilon_0m_{\rm e}c^2}{q_{\rm e}^2... ...lambda_1\Delta\psi_{\lambda_2}(t)} {\lambda_1^2-\lambda_2^2}$$ (1)

L being the plasma column length, which as has usually been demonstrated for this plasma source, is assumed to be the lamp length.

When comparing the $N_{\rm e}(t)$ curve measured with the two-wavelength method (Eq. (1)) with that obtained at a single wavelength, the differences were always lower than 5%, which indicate the negligible influence of the bound electrons to refractivity changes in this plasma. The electron density curve is shown in Fig. 3, where for each instant, an 10% error bar has been considered. This is the uncertainty estimated for the electron density in this work.

Figure 3: Electron density evolution curve. An error bar of 10% has been included to the value obtained at each instant of the plasma life.

   
3.3 Temperature

Relative to temperature measurements, it is a common hypothesis to assume that KrII excitation temperature $T^{\rm exc}_{\rm KrII}$, Saha temperature and kinetic electron temperature take similar values in collision-dominated plasmas like those generated in this experiment (van der Mullen 1990). The KrII excitation temperature was obtained from the Boltzmann-plot of some KrII lines, measured in this work, for which the transition probabilities were known. These A_ki-values were taken from Fuhr & Wiese (1998) and from Castro et al. (2001). In Table 1, both sets of data are shown and those employed here are labelled with an asterisk. The criterium to select the reference data was to use the data from Castro et al. in all cases except for those lines not measured by them, now measured, and for which Fuhr and Wiese provide data. It is important to note that the values from Castro et al. (2001) were also obtained in an emission experiment by using as a reference the data from Fuhr & Wiese (1998), so that the whole set of A_ki-values employed in this work corresponds to the same absolute scale.

 

 

Table 1: Comparison of the transition probabilities measured from Castro et al. (2001) with the reference employed in their work (Fuhr & Wiese 1990). The A_ki-values taken as reference in this work have been labelled with an asterisk.

$\lambda$ (nm)	Aki (108 s-1)	Aki (108 s-1)
	(Fuhr & Wiese 1990)	(Castro et al. 2001)
457.720	0.960	0.831*
458.285	0.760	0.812*
461.528	0.540	0.509*
461.915	0.810	0.748*
481.176	0.170*
482.518	0.190	0.208*
483.207	0.730	0.787*
484.660		0.762*
530.866	0.024*
533.341		0.500*

Once the A_ki reference data is selected, a Boltzmann-plot is made for each instant of the plasma life where spectra were taken. Those corresponding to instants 90 $\mu$s and 130 $\mu$s are plotted in Fig. 4. The linear behaviour detected demonstrates that the plasma can be well described by a partial local thermodynamic equilibrium model, at least in the energy level interval considered (16.60-20.86 eV). By using the KrII excitation temperature $T^{\rm exc}_{\rm KrII}$ obtained from the slope b of the linear fit ( $b=-1/kT^{\rm exc}_{\rm KrII}$) and the ordinate at null energy, it is possible to calculate the transition probabilities corresponding to all the measured lines, including those for which this value was previously known. In this method, the original set of data is assumed to be a good one as a whole, but the individual data may have uncertainties which sometimes are significant. This is the reason why both sets of data do not necessarily coincide. This explains the differences between the results from Castro et al. (2001) and those from Fuhr & Wiese (1990) or those found between our results and those of Castro et al. (2001). Temperature has also been obtained by the KrII/KrI intensities ratio in those instants where KrI and KrII lines with enough intensity could be recorded. Another method, based on the assumption that $N_{\rm KrII}/Z_{\rm KrII}={\rm cte}$ along the plasma life (Gigosos et al. 1994), was employed to calculate the temperature, $N_{\rm KrII}$ being the KrII density and $Z_{\rm KrII}$ its partition function. The resulting three evolution curves are shown in Fig. 5. The very good agreement among these three methods, specially between the first two ones, seems to confirm that the plasma is well described by a pLTE model and is very near to total LTE. The quality of the linear fits and the agreement with temperatures obtained from the other methods allows us to estimate the $T^{\rm exc}_{\rm KrII}$ uncertainty lower than 10%.

Figure 4: Two examples of a Boltzmann plot performed in different instants of the plasma life. Population of excited states is plotted against the corresponding energy level.

Figure 5: Temperature evolution measured from Boltzmann-plot, from consecutive krypton intensities ratios and from the assumption $N_{\rm KrII}/Z_{\rm KrII}={\rm cte}$.