Time-resolved and multiple-angle Thomson scattering on gas-puff Z-Pinch plasmas at pinch time

The development of a time-resolved Thomson scattering diagnostic that views the scattered light at two angles while duplexing the spectrometer system could allow the measurement of electron density in addition to electron temperature in gas-puff z-pinches at pinch time using a single spectrometer and streak camera.¹ For α ≡ 1/kλ_De ≳ 1 (where k = k_s − k_l is defined in Fig. 1 and λ_De is the electron Debye length), Thomson scattering is in the collective regime, and the spectrum is dominated by the ion acoustic feature, which is generally used to make measurements of fluid flow velocity, relative electron-ion drift velocity, and electron and ion temperatures.^2,3 The intensity of the ion acoustic feature spectrum is dependent on electron density, but absolute calibration has not been performed on this system. Consequently, ion acoustic feature Thomson scattering is not generally used to determine electron density. However, due to the relation of peak separation in the ion feature to the collection angle and density as well as temperature, a comparison of Thomson scattering spectra collected at two different angles to the laser allows in principle a determination of the electron density.³ 

For this paper, we used a streak camera coupled to a spectrometer and collection optics at 30° and 90° to the laser to measure the evolution of the scattering spectrum in a neon gas-puff z-pinch plasma near pinch time. Based on whether the collection angle of the scattered light is large or small, the spectral feature dependence on the electron density is more or less sensitive to variations in density.⁴ Using this differing sensitivity, we developed and tested a method for extracting density data.

Given plasma parameters such that kλ_De ≈ 0.5 − 1.5, the peak separation is sensitive to electron density, and this measurement technique may be able to supplement other electron density diagnostics, such as laser interferometry, which becomes ambiguous when large refraction effects occur due to steep density gradients, such as at stagnation in a z-pinch.⁵ The setup of the spectrometer and streak camera that here allows simultaneous observation of two angles on a single camera could also be used to observe two different scattering volumes.

The rest of this paper is organized as follows: Sec. II reviews relevant Thomson scattering theory, and Sec. III describes the experimental arrangement. Section IV presents the data and analysis, and Sec. V concludes with a summary.

The spectral profile obtained here is known as the ion acoustic feature; its two peaks are located at plus and minus the ion acoustic frequency of the plasma relative to the center of the spectrum. Plasma parameters are measured by comparing the experimental data to the theoretical spectral density function, S(k, ω), where ω = ω_s − ω_l. (More about S(k, ω) can be found in the literature.⁴)

We focus on the separation between the two peaks, which is related to the electron density and the electron and ion temperatures by the following expression:

where λ_l is the laser wavelength, θ is the angle between k_s and k_l (see Fig. 1), T_e is the electron temperature, Z is the average ionization state, T_i is the ion temperature, and m_i is the ion mass. Equation (1) is simplified in the range where ZT_e ≳ 3T_i; the ion acoustic peaks become resolvable, and their separation is dominated by electron temperature. The dependence on electron density comes from the electron Debye length,⁶ λ_De = 7.43 × 10²T_e/n_e.

Consider the kλ_De in the denominator of Eq. (1). The wave number yielded by the scattering setup can be approximated by k ≅ 2k_l sin θ/2 (Fig. 1). Through the dependence of kλ_De on k and therefore θ, it is possible to use two different collection angles to obtain spectra with differing values of kλ_De and consequently differing sensitivities to electron density (see Fig. 1). In the experiments performed here, with electron temperatures on the order of 500 eV and densities on the order of 1 × 10²⁰ cm⁻³, a 30° collection angle yields kλ_De = 0.1. At a 90° collection angle, kλ_De = 0.28. The 30° collection angle has a smaller kλ_De, and therefore the separation of the ion acoustic peaks in the 30° spectra is less dependent on electron density than the peak separation at 90°. By comparing the peak separations obtained from 30° to 90°, given sufficient resolution and signal-to-noise ratio, it is possible to infer electron density.

These experiments were carried out on COBRA, the pulsed power generator at Cornell University.⁷ It was operated in long pulse mode, with a current rise time of approximately 200 ns and a peak current of 0.85-1 MA. The load was a neon gas-puff z-pinch, initiated by a solenoid-actuated gas-puff valve mounted on the cathode of COBRA. The gas-puff valve, shown in Fig. 2, consists of a central jet coaxial to two concentric annuli. The valve is opened about 1.4 ms before the start of the current, and the gas is pre-ionized by a gas discharge electron source located 1.5 cm above the anode plane, which is in turn 2 cm above the cathode.⁸ The neutral gas density profile used for these experiments was determined using planar laser induced fluorescence (PLIF):⁹ the center jet and inner annulus produced number densities of about 2 × 10¹⁵ cm⁻³, and the density in the outer annulus was about 5 × 10¹⁴ cm⁻³. All times are given with respect to pinch time (t = 0 ns), which is defined as the rise of the x-ray burst measured on a photoconducting diamond detector (PCD) filtered with a 15 μm beryllium foil, i.e. when the z-pinch stagnates on the axis.

The probe beam used is a 10 J Nd:YLF frequency-doubled laser at 526.5 nm, with a full-width half maximum (FWHM) of 2.3 ns. The laser is split into two pulses (each with 2.5 J) that are delayed by 4 ns relative to one another along the same beam line so as to cover a longer timespan within the duration of a z-pinch. For more details on the splitting of the beam, see the work of Banasek et al.¹⁰ The laser is focused to a 350 μm spot size at the pinch axis, 12 mm above the cathode. A linear array of fiber optics is used to couple the scattered light to the spectrometer, a 750 mm Czerny-Turner, with a 2400 ℓ/mm diffraction grating. At the collection end, the fiber bundle is split into two separate bundles, one mounted at 30° to the laser and the other at 90° (see Fig. 1). The two bundles join into a single linear array, which couples to the spectrometer. Only one fiber (100 μm diameter) from each side is used; the others are masked to prevent the spectra from overlapping on the streak camera. Using a sweep speed of 11 ns, the resulting temporal resolution from the Hamamatsu model C7700 high dynamic range streak camera is 250 ps. The 125 mm focal length collection lenses, with diameters of two inches, were placed 56 cm from the pinch axis, collected light from an approximately 0.05 mm³ scattering volume radially spanning 400 μm across r = 0 and fed it to the fiber bundle. In order to obtain time-resolved Thomson scattering spectra from the same scattering volume, at two scattering angles, the spectrometer slit was left wide open and the fiber bundle was mounted perpendicular to the slit, as done by Montgomery and Johnson. Therefore, the 100 μm diameter of each fiber, respectively, constituted the effective slit width. In this manner, we observed time-resolved spectra from two angles on the streak camera, as shown in Fig. 3(a), where the 90° fiber is on the left and the 30° fiber is on the right. It is clear that ion acoustic feature peak separation scales with the collection angle.

Given the effective slit width of 100 μm and 2400 ℓ/mm diffraction grating, the spectral FWHM is 0.6 Å. In Fig. 3(b), the theoretical spectral density curve, S(k, ω), that gives the best fit to the data is given by the red profile.

Because ion acoustic peak separation depends on both T_e and n_e, it is not useful to fit each spectrum to both parameters at once. To perform the fit, the experimental data are compared to the theoretical spectral density function by varying four parameters: bulk flow velocity, relative electron-ion drift velocity, ion temperature, and either electron temperature or electron density. Based on the differing sensitivities of collection angles to electron density, we developed an iterative fitting method to extract electron density. First, the less sensitive 30° spectrum was fit in order to find T_e, using an input electron density of 2 × 10¹⁹ cm⁻³. This guess at electron density yields the temperatures shown in Fig. 4(a). Second, the more sensitive 90° spectrum is fit with density as a variable parameter, using the electron temperatures determined from the 30° spectrum [see Fig. 4(b)].

The best fit density values are calculated by using a least-squares method that minimizes the χ² value of the curve fit to the spectrum (see Fig. 3). Finally, using these best fit values of density determined from the 90° spectrum [the location of the triangles in Fig. 4(b)], the 30° spectrum is fit again to obtain more accurate electron temperatures, as shown in Fig. 4(c). The errors on these measurements were determined via a Monte Carlo method.^11,12 Further iterations did not provide significantly improved best fit spectra.

Figure 4(a) shows that the initial guess for electron density (2 × 10¹⁹ cm⁻³) when fitting to obtain temperature does not yield agreement between the 30° and 90° spectra for t > 1 ns, indicating that the density used is incorrect during the second pulse. Figure 4(b) shows a best fit electron density near 2 × 10¹⁹ cm⁻³ in the 3 ns period before pinch time, increasing to 1.5 × 10²⁰ cm⁻³ in the 3 ns period after pinch time. The electron temperature rises from around 375 eV to 475 eV in the 3 ns before pinch time and falls from around 450 eV to 350 eV after pinch time. In Shot 4869 (Fig. 5), from 6 to 9 ns post-pinch, the best fit density falls to around 8 × 10¹⁹ cm⁻³, and the electron temperature has decreased to the 200-300 eV range. When the best fit density values in Fig. 4(b) are used to recalculate T_e rather than assuming that n_e = 2 × 10¹⁹ cm⁻³, the temperatures determined from 30° to 90° are in much better agreement. The 30° T_e does not change appreciably, but the 90° T_e falls to match the 30° T_e when the new best fit density is used: 90° is more sensitive to changes in density, so this was expected.

The 3 ns before pinch time exhibits best fit density values that bounce perhaps nonphysically within the range given by the error bars, in contrast to the 3 ns after pinch time, where the best fit values sit consistently around 1.5 × 10²⁰ cm⁻³ [Fig. 4(b)]. This can be better understood by recalling Eq. (1). Earlier in time, the electron temperature is slightly lower, meaning that α is higher and the peak separation is less sensitive to changes in density. In this regime, there is a larger range of densities that will satisfy a spectral profile for a given temperature.

The large density range spanned by the error bars yielded by this measurement technique makes it clear that in order to use the ion acoustic feature to determine precise electron densities, the plasma being examined must be in a certain parameter range, namely, lower α, while still remaining in the collective Thomson scattering regime. The gas-puff z-pinch plasmas investigated here have α values on the order of 4 (at 90°) to 9 (at 30°). Rewriting the pertinent term of Eq. (1) in terms of α gives

The larger the α value of the plasma, the less effect the 1/α² term has on the separation between the peaks. For a plasma with higher electron temperatures and/or lower electron densities, 1/α² is larger, making more of an impact on peak separation. The size of the error bars on the density measurements reflects the insensitivity of peak separation to electron density in this high-α regime.

In these experiments, the range of densities that will enable a good fit to the observed spectrum at a particular temperature is large. Although this does not yield precise densities, it does provide us with the knowledge that choosing any density within the calculated range [(0.01–1.0) × 10²¹ cm⁻³ before pinch time, decreasing to a smaller range after] will have a nominal effect on the peak separation and therefore the inferred electron temperature. Figure 6 shows the range of temperatures obtained by using four densities within the range shown in Fig. 4(b). All of the temperatures plotted in Fig. 6 are within error bars of one another, confirming that the gas-puff z-pinch plasma at pinch time is not in an experimental regime where the peak separation is very sensitive to changes in electron density.

To determine regimes where this technique of density measurement could be more accurate, we examine how the separation of the ion acoustic peaks is affected by α over a range of plasma parameters and experimental configurations. Figure 7 displays the electron temperature and density regions where a comparison of the ion-acoustic features measured from two pairs of angles should yield a more precise electron density measurement with a probe laser at 526.5 nm. Both wire array z-pinches^13,14 and some laser-produced plasmas^15,16 are in this range. The parameters in the gas-puff z-pinch plasmas described in this paper are marked by the red X.

In summary, a duplexed coupling method was used to display two angles of Thomson scattering spectra on the same streak camera. This method of coupling the optical fibers to the spectrometer can additionally be used in the future to provide time resolution at two locations within a plasma, rather than to observe the same scattering volume from two angles, as was performed here. As the scattering was in the collective regime, α ≳ 1, electron temperatures were determined at the two angles, providing a consistency check. Additionally, the differing sensitivities to electron density at the two angles were used to attempt to measure that plasma parameter, but the spectra were not adequately sensitive to density at our values of α.

Although the calculated range of densities that will satisfy a profile with a given temperature is large, it provides a window of appropriate density inputs for analysis of future data in similar regimes. The results obtained suggest that increased sensitivity to the procedure used here is necessary to obtain a useful estimate of electron density. Useful estimates would require a regime where ∼0.5 < α < ∼2, which is achieved in some laser-produced and wire array plasmas.

Thanks to the technical team, Harry Wilhelm, Dan Hawkes, and Todd Blanchard, for machine operation and maintenance. This work is supported by NNSA SSAP under DOE Cooperative Agreement Nos. DE-NA0001836 and DE-NA0003764 and the Lawrence Livermore National Laboratory Subcontract No. B619181.