5.2 Plasma Density Peak in the Afterglow
6.1.5 High Resolution Spectroscopy
6.1.5.3 Effect of Re-Absorption
Emission line ratio measurements will yield misleading results if an appreciable number of photons of either wavelength are re-absorbed before leaving the plasma, since the measured intensity of each line will no longer be proportional to the upper level population density.
The absorption cross section is highest at the line center wavelength and lower in the wings, meaning that re-absorption distorts the line profile and thus corrupts Stark and Doppler broadening measurements as well. In particular, the electron density or ion temperature may be overestimated if the measured intensity at the line center is attenuated by re- absorption because the apparent FWHM will actually correspond to the true line width at
5Note, however, that the Ar III 351.14 nm and Ar II 349.13 nm lines were actually each composed of two nearby lines, at 351.11 nm and 351.17 nm (both Ar III) and at 349.12 nm and 349.15 nm (both Ar II), respectively. When the spectrometer was operated with a relatively wide entrance slit (300µm), the separate lines could not be resolved, and the line pairs appeared as single lines with a Gaussian profile due to the dominant instrumental broadening. On the other hand, when a narrow 50µm entrance slit was used, the individual line profiles could be distinguished, but they overlapped on the spectrum, making reliable fitting to the line shapes difficult. Therefore, using the [Ar III 350.36 nm] / [Ar II 347.67 nm] ratio instead of the [Ar III 351.14 nm] / [Ar II 349.13 nm] ratio was preferred in this case.
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
1 cm < lmfp< 5 cm
lmfp> 5 cm 434.81 nm nAr II/ ne= 0.01
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
1 cm < lmfp< 5 cm
lmfp> 5 cm 434.81 nm nAr II/ ne= 0.1
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
lmfp> 5 cm
1 cm < lmfp< 5 cm 434.81 nm nAr II/ ne= 0.9
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp> 5 cm
372.93 nm nAr II/ ne= 0.01
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
1 cm < lmfp< 5 cm lmfp< 1 cm
lmfp> 5 cm 372.93 nm nAr II/ ne= 0.1
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
1 cm < lmfp< 5 cm lmfp> 5 cm 372.93 nm nAr II/ ne= 0.9
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp> 5 cm
347.67 nm nAr II/ ne= 0.01
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
1 cm < lmfp< 5 cm
lmfp> 5 cm 347.67 nm nAr II/ ne= 0.1
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
1 cm < lmfp< 5 cm lmfp< 1 cm
lmfp> 5 cm
347.67 nm nAr II/ ne= 0.9
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp> 5 cm
350.36 nm nAr III/ ne= 0.005
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
1 cm < lmfp< 5 cm
lmfp> 5 cm
350.36 nm nAr III/ ne= 0.05
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
lmfp> 5 cm
1 cm < lmfp< 5 cm 350.36 nm nAr III/ ne= 0.45
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp> 5 cm
291.30 nm nAr IV/ ne= 0.003
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
lmfp> 5 cm
1 cm < lmfp< 5 cm 291.30 nm nAr IV/ ne= 0.03
Te(eV) ne(m−3)
2 4 6 8 10 12 14 16 18 20
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 1023
lmfp< 1 cm
lmfp> 5 cm
1 cm < lmfp< 5 cm
291.30 nm nAr IV/ ne= 0.3
Figure 6.8: Absorption mean free paths for the Ar II 434.81 nm, Ar II 372.93 nm, Ar II 347.67 nm, Ar III 350.36 nm, and Ar IV 291.30 nm lines as a function of ne and Te, with various assumed values of the ratio of the relevant ion’s population density to the electron density. The populations of excited states within a given ionization stage were assumed to follow a Boltzmann distribution, and Ti =Te was assumed.
an intensity below the half maximum.
The MHD-driven jets were transparent in camera images that were not saturated, so they were optically thin (meaning that the absorption mean free path was much longer than the plasma radius) in the continuum. However, it is still possible that there was non-negligible re-absorption at the wavelengths of the strongest emission lines. Indeed, given that re-absorption of line emission was important in the RF discharge, as discussed in Appendix E, it was reasonable to expect that it might play a significant role in the jet plasma as well.
Using φ(ν) = λ2/c
φ(λ) to convert from frequency to wavelength, Eq. E.4 for the absorption cross section becomes
σλ = λ4 8πc
gα gβ
Aαβ
1− gβnα
gαnβ
φ(λ). (6.8)
As an approximation of the Voigt profile [137, Section 6.2], assume that the line profile is roughly Lorentzian, but with a total FHWM wtotal =
q
w2S+w2D. Then according to Eq.
6.2, the profile function at the line center is φ(λ0) = 2/π q
w2S+w2D, and the line center cross section is
σλ0 = λ40 4π2c
gα gβAαβ
1− gβnα
gαnβ
1 q
wS2 +w2D
. (6.9)
For lack of a better simple alternative, we will assume that the level populations follow a Boltzmann distribution,nα/nβ = (gα/gβ) exp (−(Eα−Eβ)/kBTe), as in the “partial LTE”
model discussed in Sec. 6.1.5.2. However, since the jet plasma may be far from ionization equilibrium,lmf p will be evaluated for several different possible ionization balances instead of using the Saha equation to calculate nArII,nArIII, andnArIV as a function of Te. This is a reasonable approach because the relaxation time for the excited state populations in a time-varying plasma is much shorter than the timescale for the ionization balance to reach a steady state [74, p. 225].
Contour plots of the absorption mean free pathlmf p= (nβσλ0)−1as a function ofneand Te are shown in Fig. 6.8 for several lines of interest. Since the spectroscopy measurements presented in this chapter will focus on determining the properties of the bright narrow
section at the base of the pre-ionized jets, we take lmf p = 1 cm as the cutoff below which re-absorption would start to have a deleterious effect on the analysis. It is clear that all of the lines studied were likely to be re-absorbed if the ionization stage from which they arose was dominant; on the other hand, lines associated with ions whose population densities were
≤1% of the electron density had negligible re-absorption. Since the ionization balance was not well known, the actual effect on the data was uncertain, but re-absorption should be kept in mind as a possible confounding factor. A rough estimate from the measured line ratios shown in Figs. 6.12 and 6.15 suggests that Ar III or Ar IV was the dominant ion present.