Line intensity ratios

For comparing with experiments (as well as for testing the model), integrated line intensity ratios are a convenient tool. In general, the ratios depend not only on ne and Te, but also on the quenching rate kQ. This is demonstrated in figure 4.9, for at pair of lines at 616.2 nm and at 643.9 nm, which are lines that are relatively sensitive to changes in the electron temperature and density, and have sufficient intensity for measurements. This figure shows the ratio of the wavelength integrated intensity as a function of n_e and T_e, without quenching (left) and with a quenching rate k_Q = 3.3 · 10⁵ s⁻¹. It is clear that quenching has an important effect on the line ratio in the displayed parameter ranges. Only at high electron densities n_e> 10²² m⁻³ the ratio

CHAPTER 4 A CRM for calcium 4.5. Results

is unaffected by the collisional quenching. It should be noted that line escape factors were not used, i.e. the transitions are assumed to be optically thin.

Figure 4.9: Temperature and density sensitive line ratio I(λ=616.2 nm)/I(λ=643.9 nm) ∝ n6/n₈ with (right) and without quenching (n1 = 1 .0 · 10²¹ s⁻¹, xCa+= 1 .0 ). Only at high electron densities ne> 10²² m⁻³ the ratio is unaffected by the quenching at a rate of 3 .3 · 10⁵ s⁻¹. An optically thin plasma is assumed (ΘL= 1 ).

A comparison with measurements is made in figure 4.10, where again Θ_L = 1 is used, as well as a quenching rate of k_Q = 8 · 10⁵ s⁻¹. The figure contains the calculated line intensity ratio for the previous line pair (a), as well as for 616.2 nm and 527.0 nm (b). The measured data is determined from spectra recorded using the ´echelle (top view, r = 0, at various times) for a series of shots with 0.4 g/l CaCl2in tap water. For the first four data points, the intensity was measured from the broadening of Hβ in the same spectra (see figure 3.23, so the electron temperature can be directly read from the graph. It decreases from just over Te= 5500 K (not shown) at t = 5 ms to 4500 K at t = 45 ms, determined from the 616.2 nm / 643.9 nm line pair (a) and about 500 K higher from line pair (b). At later times, the measured line ratios are consistent with an electron electron density decreasing at a rate of roughly one decade per 60 ms. The electron temperature is also found to decrease. At t = 155 ms, Te is estimated to be close to 3000 K, at later times the ratios are relatively insensitive to the temperature. It should be noted that the consistency of the line ratios depends on the value assumed for the quenching rate. To get the best agreement, the value of k_Qwas adjusted to 8 · 10⁵s⁻¹, as compared to 3.3 · 10⁵s⁻¹used for other results presented here. For completeness, the same comparison is shown using k_Q = 3.3 · 10⁵ s⁻¹ in appendix E.

It should be mentioned that the assumption xCa+= 1 is likely not realistic initially. At temper-atures above 5000 K, other species (such as atomic hydrogen) will reach a considerable ionization degree and are expected to provide the bulk of the electrons [Fussmann, private communication].

Nevertheless, the same agreement between measurement and simulation is not obtained when the ion-stage contribution (i.e. three-particle- and radiative recombination) to the ASDF is ne-glected. Line ratios obtained in the latter case are shown in figure 4.11, using the same set of parameters as in figure 4.10.

As stated before, line escape factors were were not used for the calculation of the line ratios so far. When values of ΘL, calculated using lp = 8 cm and the estimated line widths according to equation (4.16) are included, the ratios become somewhat different, as is visible in figure 4.12.

The line ratios are higher at intermediate n_e and high T_e, due to stronger absorption of the line at 616.2 nm (also see figure caption). In the limit of high n_e an increase of the line-ratio occurs that is a consequence of the assumption x_Ca+= 1. The lower state of the lines at 527.0 nm and

4.5. Results CHAPTER 4 A CRM for calcium

Figure 4.10: This graph shows the calculated line ratios of two line pairs, with a relatively large difference in upper level energy (see the top-left legend and the Grotrian diagram in figure 4.1). As a consequence, the line ratios are relatively temperature sensitive. Also included in the graphs are measured line ratios, determined from spectra recorded using the ´echelle spectrometer (top view) at different times t after triggering the discharge (indicated by the text labels near the data points, the exposure time is between 1 and 6 ms.). For the first four points (squares) the density has been measured, using Stark broadening of the H_β line in the same spectra. The horizontal error bars indicate the measurement error in the Stark broadening measurements. For the other shots (triangles) the density could not be measured and the horizontal placement is estimated from the calculated and measured line ratios. The measured points clearly indicate that quenching is important (compare with figure 4.9). Although overall agreement is good, some small inconsistencies can be seen between the temperatures determined using ratios labeled (a) and (b).

Line escape factors were not used and the other simulation parameter are shown in the top-left corner.

Figure 4.11: Same line ratio as in figure 4.10, but with no contribution from the ion stage to the ASDF. Ra-tio (b) is higher by a factor 1 to 3, with respect to figure 4.10. This can be understood by the fact that the upper level of the line at 527.0 nm (q = 16 ) has a higher energy than that of the line at 643.9 nm (q = 8 ).

The ion stage gives the highest con-tribution to the population of the states that are closest to the contin-uum (also see the Boltzmann plots).

CHAPTER 4 A CRM for calcium 4.5. Results

643.9 nm (p = 3) shows an increase of its population relative to p = 2 (the lower state of the line at 616.2 nm) due to recombination. That such an increase is not shown in the measurements is likely explained by xCa+ 1 initially, as was discussed before. Also, the line widths due to Stark broadening, estimated using equation (4.16) may be too low, resulting in overestimating of the optical thickness at high ne.

Figure 4.12: Same line ratio as in figure 4.10, but with line es-cape factors included. The lines at 527.0 nm and 643.9 nm share the same lower state (p = 3 ) whereas 616.2 is a transition to the lower state p = 2 . At most (intermediate) values of ne and Te, the line ratios decrease, due to the larger popula-tion of the lower state p = 2 . In the limit of low density and tem-perature, the effect of absorption be-comes negligible, whereas for low T_e and high density (n_e& 10²¹ m⁻³), the line ratios increase as the state with p = 3 is more strongly popu-lated due to recombination. The lat-ter is a consequence of the assump-tion ne= nCa+ and most likely not realistic for the plasmoid in it’s

ini-tial phase. ¹⁰