"Dynamics of Field-Reversed Configuration in SSX"
Abram Falk

Summary:

Abram's thesis focuses on two diagnostics that he designed and built for SSX--the soft x-ray detector (SXR) and the Mach probe. His intro describes several motivations for laboratory plasma research, including trying to understand astrophysical phenomena such as coronal heating, and the pursuit of practical fusion power. A steady-state fusion reactor requires that an extremely hot plasma (350 million K or 30 keV) be confined by gravity, inertia, or magnetic fields. The spheromak is a magnetic confinement scheme that relies on fields produced entirely by currents within the plasma itself. Spheromaks have a toroidal magnetic structure but relax to force-free equilibrium within a simple container (unlike a Tokamak, which resides in a toroidal container), and mathematically are solutions to the Grad-Shafranov equation. Abram includes a detailed description of the formation of spheromaks and Field Reversed Configurations (FRCs) in SSX. He notes that while MHD and kinetic simulations predict that FRCs should be highly unstable to the m=1 tilt mode, actual experimental FRCs are anomalously stable. He also includes a chapter on basic plasma theory, covering topics such as Debye shielding, MHD, Alfvén waves, frozen-in-flux, and Sweet-Parker magnetic reconnection.

"Soft x-ray" refers to extreme ultraviolet light at energies between 10 and 150 eV. X-ray spectrometers are very costly and therefore impractical for SSX, so the low-resolution SXR is a good alternative for obtaining spectral information at these energies. The detector consists of broadly sensitive photodiodes each filtered by different thin metallic foils. The original foils used were 100 nm thick Al, Zr, and In, and 50 nm thick Ti. Although the low spectral resolution of the detector prohibits the observation of individual spectral lines, the diodes have extremely fast time resolution (~ 700 ps when properly biased). Similar diagnostics have been built at several other experiments, including the CDX-U spherical torus at the Princeton Plasma Physics Laboratory and the Madison symmetric torus reversed-field-pinch. Another common approach is the Ross filter technique, in which the K-edges (e.g. the sudden drop at 72 eV in the responsivity of the Al filter used in SSX) of two filters of the same material but different thicknesses are used to take measurements in a small passband.

The SXR photodiodes have approximately linear quantum efficiency at energies above 10 eV. Abram also includes additional details about the photodiodes and how they are connected to the detector circuits. The photodiodes are sensitive to impacts by charged particles, so the holes leading to each filter were surrounded by strong magnets to deflect ions and electrons. Abram calculates that ions would have to have energies greater than 440 eV to make it past the filters. These could only exist in large numbers if reconnection produced extremely energetic ion beams. A camera flash test was used to ensure that the filters were not passing significant amounts of visible light. The Al, In, and Zr filters all generated a current at least three orders of magnitude weaker than the current on the unfiltered diode, but the Ti current was only 2 orders of magnitude less than the unfiltered current. Abram suggests that the Ti filter may have developed a pinhole, or it may simply pass more visible light than the others because it is thinner.

Soft x-rays can be produced by bremsstrahlung radiation or free-free emission (radiation emitted by accelerating charged particles), line emission (radiation emitted as a result of bound electrons undergoing atomic transitions), and the recombination continuum. Line radiation in laboratory plasmas tends to overwhelm the bremsstrahlung continuum, even at relatively low impurity concentrations. Residual gas traces taken between SSX shots found negligible impurity gas pressures, but during shots concentrations of carbon and oxygen in SSX may be as high as a few percent each due to particles being released from the flux conserver walls.

Spect3D, a more complex version of PrismSPECT, was used to produce model spectra for a variety of plasma sizes, temperatures, densities, and atomic makeups. Spect3D allow density and temperature to vary spatially, but this feature was not used. Models used assumed coronal equilibrium. Before the experimental data was fit to the models, the measured SXR signals were divided by the time-dependent density of the plasma squared, as measured by a He-Ne interferometer. This removes the effect of variations in the density, because both bremsstrahlung and line radiation depend on n^2. Model spectra were smoothed using Spect3D, and then the filter signals that each model would have produced were calculated. The Levenberg-Marquardt Method was used to find best fits for both a normalization constant (corresponding to the density and volume of plasma that SXR sees) and the electron temperature as a function of time. The fitting method used gave more weight to data points with smaller uncertainties (uncertainties were calculated by taking the standard deviation of the detector noise in the neighborhood of a point; each point represented the average signal over a 1 us interval).

The results of initial SXR measurements show that the diode signals rise as the plasma gets cleaner and its density falls (for example, over the course of consecutive shots after the chamber was vented and re-sealed). This occurs because high concentrations of impurities impede heating and activity in the plasma. Interpretation of SXR data is aided by comparison with data from magnetic probes. Three main peaks are seen in the SXR signals, around 38 μs, 53 μs, and 65 μs. The first peak corresponds to the time when poloidal magnetic reconnection peaks, while the second peak occurs as the FRC tilts from the m=0 mode to the lower energy m=1 mode. The cause of the third peak is still a mystery.

Temperature fits were calculated using four categories of model spectra as hypotheses: bremsstrahlung radiation only, recombinations lines and continuum plus bremsstrahlung but no impurities, 0.5% C and O impurities, and 2% C and O impurities. Using the pure bremsstrahlung model, it was determined that counter-helicity merging shots have higher temperatures than co-helicity shots because of greater magnetic reconnection and also have longer lifetimes, presumably because FRCs are more stable than spheromaks. Finding reliable temperatures from the models that include impurities is difficult, because the best-fit temperatures show a strong dependence on the precise impurity concentrations used. However, all models suggest a mean temperature in SSX of 30 ± 10 eV.

In addition to working with SXR, Abram used a Mach probe to calculate ion drift velocities. I will skip these sections since they are not directly relevant to my research.

Applications to my research:

Abram was involved in the design and implementation of SXR, one of the two main SSX diagnostics that I am focusing on for my project, so his thesis will be an extremely useful reference for information about how the detector works and why it was chosen for SSX. His results are also of great interest to me, especially since we have been thus far unsuccessful in using SXR to make meaningful inferences about the SSX electron temperature. Although he produces several reasonable-looking temperature profiles, his most plausible results come from calculations using model spectra with no impurities, which is clearly not an accurate representation of the SSX plasma. Calculations using models with carbon and oxygen included lead to temperature profiles with essentially random variations much like the ones I have obtained. Therefore, I am not convinced that his results are any more reliable than the temperatures I have calculated so far; it remains to be seen if useful information can be garnered from SXR.