5.2 Plasma Density Peak in the Afterglow
6.1.5 High Resolution Spectroscopy
6.1.5.1 Line Broadening
An electron collision with an excited atom or ion temporarily produces a local electric field that shifts its energy levels through the Stark effect. The cumulative effect of many such collisions is to produce “Stark broadening” of the line profile, along with additional wavelength shifts and profile distortions [137, Chapter 5]. The characteristic shape of a Stark broadened line is a Lorentzian profile [138, Section 10.6]:
φL(λ) = 1 π
wS/2
(λ−λ0)2+ (wS/2)2, (6.2) whereλ0 is the line center wavelength andwS is the full-width at half-maximum (FHWM).
For non-hydrogenic atoms and ions, the quadratic Stark effect applies3, andwS is directly proportional to the electron density:
wS ≈wm(Te)ne (6.3)
where the temperature dependence of the Stark parameter wm is usually relatively weak.
Emission lines can also be broadened by the Doppler effect due to the random thermal motions of the emitting species, which leads to a Gaussian line profile:
φG(λ) = s
4 ln 2
πwD2 exp −4 ln 2 (λ−λ0)2 w2D
!
, (6.4)
where the Doppler FWHM is
wD =λ0
r8 ln 2kBTi
mic2 . (6.5)
Additional effective line broadening, known as instrumental broadening, is introduced by the optical system. The instrumental function is often approximately Gaussian, although it is not guaranteed to be. A final source of measurable line profile distortions in the Caltech experiments is Zeeman splitting, which is proportional to the magnetic field strength [139]. This effect was expected to be smaller than the others discussed here and will not
3Here the label “quadratic” refers to the fact that the term linear in the applied electric field vanishes when calculating the energy level shift using perturbation theory—it should not be confused with the dependence of the Stark width on density, which is in fact linear (see Eq. 6.3).
be considered further, but it could have possibly become important if the magnetic field strength was amplified toB 0.1 T by the jet dynamics.
When multiple sources of broadening are present, the overall line profile is a convolu- tion of the constituent profile functions. The convolution of two Lorentzians is another Lorentzian with FWHMwL=wL1+wL2, and two Gaussians combine to yield a Gaussian with FWHM wG =
q
wG12 +wG22 . The convolution of a Gaussian profile and a Lorentzian profile is called a Voigt profile [140, p. 302]:
φV(λ) = Z ∞
−∞
φG λ0
φL λ−λ0 dλ0 ∝
Z ∞
−∞
exp−4 ln 2(λ0−λ0)2 w2G
(λ−λ0−λ0)2+ w2L2dλ0. (6.6) By fitting this profile function to the measured shapes of spectral lines from MHD-driven jets, the Lorentzian and Gaussian FWHMswLand wG were determined. The contribution to these FWHMs from instrument broadening was then subtracted off, leaving only the Stark and Doppler widthswS and wD, from which the electron density and ion temperature were calculated using Eqs. 6.3 and 6.4, respectively.
Voigt profiles were fit to the shapes of argon emission lines using IDL routines written by T. R. Metcalf [141] and D. M. Zarro [142]. The analysis focused on the Ar II lines at 372.931 nm and 434.806 nm; neither line had a particular large Stark broadening parameter wm4, but they were bright enough in our jets and sufficiently isolated from other lines to enable clean fitting to the line profiles (however, the 434.806 nm line exhibited asymmetric profile distortions when the plasma density was high that made the determination of ne and Ti from the line shapes unreliable, so the 372.931 nm line data was ultimately used to derive the results presented in Sec. 6.5.1).
The instrument function was measured by observing emission lines from weakly ionized, low density spectral lamps, which were expected to have minimal Stark and Doppler broad- ening. The line shapes were nearly Gaussian, but Voigt profiles with a small Lorentzian component gave a superior fit away from the line center (in the line “wings”). With a 50µm entrance slit width on the spectrometer, analysis of the Ar I lines at 394.90 nm and 416.42
4Averaging over measurements from a number of authors tabulated by Konjevi´c et al. [143], we adopt the valueswm= 2.0 pm/1022m−3 for the Ar II 372.931 nm line andwm= 2.7 pm/1022m−3 for the Ar II 434.806 nm line.
372.8 372.85 372.9 372.95 373 373.05 102
103 104 105
Wavelength (nm)
Intensity (arb. units)
Voigt Lorentzian Gaussian Instrument
372.8 372.85 372.9 372.95 373 373.05
102 103 104
Wavelength (nm)
Intensity (arb. units)
Voigt Lorentzian Gaussian Instrument
Figure 6.6: Examples of Voigt profile fits to the Ar II 372.931 nm line (data shown as dots).
Lorentzian and Gaussian fits with the same amplitude, continuum level, and total FWHM (approximated from the Gaussian and Lorentzian components of the Voigt fit usingwtotal≈ q
wG2 +wL2) are shown for comparison, and the Voigt profile fit to the instrument function is also shown. Both spectra were from pre-ionized jets (Vgas,RF = 700 V, Vgas,inner = 0 V, Vgas,outer = 750 V) at z = 2.0 cm. The left panel shows data taken at t = 1.3–2.3 µs, when the plasma density was low and the line shape was between Gaussian and Lorentzian (the Voigt profile fit gave wL = 4.4 pm and wG = 15.3 pm), while the right panel shows the approximately Lorentzian line shape (wL = 30.5 pm, wG = 17.9 pm) obtained with a high plasma density at t= 4.3–5.3 µs. The ICCD camera had a constant, non-uniform background level (typically 250-265 counts when the camera was cooled to−10◦ C), which was subtracted off before making all profile fits.
nm gave the following Gaussian and Lorentzian instrumental FWHMs:
wIG ≈2.1 pixels ; wIL ≈0.7 pixels. (6.7)
Examples of Voigt profile fits to the 372.931 nm emission line from pre-ionized jets are shown in Fig. 6.6, along with pure Lorentzian and pure Gaussian fits, which may be easily distinguished from one another because the Lorentzian function decays much more slowly away from the line center. Plotting the line profiles with a logarithmic y-axis was valuable for assessing the quality of the fit in the wings. At times when the electron density was low along the chosen line of sight, the measured line shape was between Gaussian and Lorentzian, as shown in the left panel of Fig. 6.6, while for the highest densities measured, the Stark broadening contribution dominated and the line shape was approximately Lorentzian (see the right panel of Fig. 6.6).
Ar II 294.29 nm Ar IV 291.30 nm
Ar III 351.14 nm (two lines)
Ar II 349.13 nm (two lines)
Ar III 350.36 nm Ar II 347.67 nm
Figure 6.7: Sample argon jet spectra from shots with Vgas,RF = 700 V, Vgas,inner = 0 V, and Vgas,outer = 750 V. The emission lines used for line ratio calculations are labeled. The optical fiber was pointed directly at the base of the jet with no lens used, the ICCD was exposed from t= 3.3–4.3µs, and the spectrometer entrance slit width was 300µm.