Results

7-23-08: In light of last week's new estimates for the nitrogen concentration in SSX, I ran a new set of simulations to see what the effect of a 0.003333% nitrogen impurity would be on the model spectra and model SXR filter signals. A number of N lines can be seen in the spectra, but the SXR model filter ratios only change by a few percent (except at T = 5 eV, when the change is more drastic). Therefore, while including N makes the simulations more accurate, it hasn't led to a breakthrough in getting the models to match the SXR data. Shown below are the model filter ratios (with data from a single time step of an SSX shot overlaid) for simulations without N (top) and with N (bottom).

7-16-08: I've created some new plots looking more closely at the impurity emission line ratios involving nitrogen. In summer 2006 we used the VUV monochromator to measure the intensity of the two N lines expected to be strongest in SSX: N V 124 nm and N IV 76.5 nm. These lines are produced by electron transitions analogous to those that produce the C IV 155 nm line and C III 97.7 nm line, respectively. We observed a typical signal strength of 100 μA for the N V 124 nm line, while the N IV 76.5 nm line could not be distinguished from the background noise. At the time, we assumed that this pair of results (a strong signal from one of the nitrogen lines and no signal from the other) was a contradiction, and we concluded that perhaps the 124 nm measurement had been contaminated by emission from the H Lyman alpha line. However, it appears that I should have investigated the expected ratios in more detail. Here is the 124 nm / 76.5 nm line ratio from steady state PrismSPECT simulations at a range of temperatures:

At T = 30 eV, the expected 124 nm / 76.5 nm line ratio is 56/1. In our observations of the 76.5 nm line in SSX, the average signal measured by the monochromator during the 40-60 μs interval was 1.75 μA, while the standard deviation of the measurements was 1.60 μA. The data was very noisy, thus it was impossible to conclusively state that the line had been detected. However, a signal strength of 2-3 μA is well within the uncertainty range implied by the data. This would imply a 124 nm / 76.5 nm line ratio of ~100 / 2 = 50, which is in perfect agreement with the models shown above given that the electron temperature in SSX is known to be between 20 and 35 eV. Therefore, it seems that we were misguided in questioning the 124 nm line measurement. I feel much better about this: we always saw separate peak signals for the H I 121.6 nm line and N V 124 nm line when scanning across wavelengths with the monochromator, and the lines shouldn't be wide enough to overlap.

Assuming that the 124 nm line measurement is indeed accurate, we should be able to improve upon our estimate of the relative concentration of nitrogen impurity in the plasma by comparing this line to a carbon line. The C IV 155 nm line is ideal because the expected line ratio is relatively flat between T = 20 eV and T = 100 eV. The following plot shows the N V 124 nm / C IV 155 nm ratio for a set of PrismSPECT simulations with equal concentrations of C and N:

The model line ratio is > 8/1 for all plausible SSX temperatures, and approximately 20/1 for our best temperature estimate. The measured line ratio is 100 μA / 1.5 mA = 1/15, so we can conclude that the carbon concentration is roughly 300 times greater than the nitrogen concentration in SSX. This does not contradict the previous result (featured in my thesis and the most recent version of our paper) obtained from analysis of the N IV 76.5 nm / C III 97.7 nm ratio, which placed an upper limit on the N / C abundance ratio (~ 1/100). However, all of my SXR simulations up to this point have assumed that there is no N in the plasma, while it now appears that nitrogen is more abundant in SSX than oxygen (we have estimated that there is 1000 times more C than O), so these simulations will need to be re-run. Perhaps this will help us make progress toward better fits between the SXR models and data. Incidentally, it is certainly suggestive that the N / O abundance ratio implied by the measurements outlined above is comparable to the ratio of their concentrations present in air.

7-2-08: The following plot shows the H 121.6 nm (Lyman α) / C III 97.7 nm line ratio calculated for a series of steady state simulations with 99% H and 1 % C.

By comparing these models with measured line strengths in SSX, we can estimate the absolute carbon impurity concentration in the plasma. For 9 shots on 8/2/06 and 8/3/06, the average Lyman α signal measured with the VUV monochromator was 15-20 μA. Meanwhile, we measure average signals of around 10 μA for the C III 97.7 nm line. Looking at the above plot, if the electron temperature in SSX is ~25-30 eV and there is 1% carbon in the plasma, we would expect the 121.6 nm / 97.7 nm ratio to be about 2/1. 25-30 eV is the mean T_e that we measure with the monochromator, and 2/1 is the line ratio that we measure, so it appears that 1% carbon impurity is a pretty good estimate for SSX.

It is desirable to place a lower limit on the carbon concentration, so that we can conclusively state that line emission dominates over the continuum at SSX impurity concentrations and we can calculate temperatures from SXR filter ratios without exact knowledge of absolute impurity concentrations. Based on our temperature measurements with the monochromator, it is safe to say that T_e > 15 eV at all times in SSX. At 15 eV, the above models predict that I_121 / I_97 = 1/10. Comparing this result with the 2/1 ratio we observe yields a carbon concentration 20 times lower than 1%.. Therefore, we can quite comfortably state that the concentration of C in the plasma is at least 0.05%. Several years ago, Victoria Swisher found that line emission dominates over the continuum as long as the C concentration is at least 0.01%

5-13-08: I recently ran several new time-dependent simulations to test whether equilibrium would still be reached quickly if the temperature of the plasma in the gun were significantly higher or lower than the final temperature after expansion into the flux conserver. I also ran a new steady-state simulation (T_e = 30 eV) using my latest atomic models (C200each11_6.atm and Hcomplete3_23_07.atm) as a basis for comparison. All of these simulations used my standard parameters (500 logarithmically-spaced time steps, 40 cm plasma thickness, 99% H / 1% C, etc.). For the time-dependent simulations, the initial atomic level populations were calculated based on LTE at room temperature (.025 eV); the temperature was instantaneously increased to its in-gun value at t = 0. All time-dependent simulations had T = 30 eV from t = 30 μs onwards. Each used the same ion density as a function of time, shown below for an example with T_gun = 60 eV.

T and n for the time-dependent and steady state simulations converge at t = 30 μs. In order for us to safely use steady state simulations for VUV monochromator and SXR temperature calculations, it must be the case that the two types of simulation become virtually identical within 10 μs after this (by t = 40 μs).

Here are the model spectra for the steady state simulation and for time dependent simulations with a low gun temperature (10 eV) and high gun temperature (60 eV) at t = 30 μs:

The spectra at t = 30 μs look quite similar, although differences in strengths of a given line in the three plots are greater than they appear due to the log-scale y-axes. At t = 40 μs, the spectra still aren't quite identical:

To actually determine whether steady state simulations are sufficient, a quantitative analysis would be more appropriate. I used PrismSPECT's Line Intensities tool to calculate the C III 97.7 nm / C IV 155 nm line strength ratio at a number of times during the simulations.

The "T_gun = 10 eV" simulation matches the steady state simulation as early as t = 32 μs. The "T_gun = 60 eV" simulation takes longer to reach equilibrium, but by t = 40 μs the 97.7 / 155 ratio still agrees with the steady state result to within less than 1%. Given the level of precision we are hoping to achieve in our temperature measurements, a discrepancy of this magnitude is of no concern, as the steady state line ratio changes by several orders of magnitude over the plausible temperature range (as shown here, for example). Therefore, it is safe for us to use steady state simulations for these calculations even if the actual electron temperature in the guns in SSX is as high as 60 eV.

7-13-07: I just finished running a full set of non-Maxwellian simulations with 1%, 5%, and 10% of the electrons distributed in a flat tail up to 500 eV. Here is an example of the temperature fitting for the 10% non-Maxwellian case (as before, the lines are the model filter ratios as a function of temperature and the points are the data for the time step):

This plot doesn't look very good: the model ratios barely differ over the 5-100 eV temperature range. We can see why by looking at some low-resolution spectra. Below are spectra from simulations at 10 eV (left) and 40 eV (right). Most the photons are between 300 and 400 eV, and the spectra at different temperatures don't really look all that different. Therefore, it's not surprising that the filter ratios don't change much in this case (they are basically equal to the ratio of the filter responsivities between 300 and 400 eV, although the Al filter captures an appreciable quantitiy of low-energy emission as well). This result shows that if 10% of the electrons in SSX were non-Maxwellian, SXR would not be a very useful diagnostic with it's current filters. However, 10% is probably too high, so an analysis of the simulations with lower non-Maxwellian fractions is more relevant.

Unfortunately, it turns out the model ratios for a 5% electron tail aren't any more useful. Once again, the filter ratios are pretty much constant, as shown below:

The model ratios for the simulations with 99% Maxwellian and 1% non-Maxwellian electrons vary a little bit more, but the data still don't match the models well at all:

Here's an example of the temperature derived for a shot using the 1% non-Maxwellian model:

Obviously this plot bears no resemblance to what is actually going on in the machine. I don't think these results necessary rule out a non-Maxwellian electron distribution, but this particular model (a flat distribution with peak energy 500 eV superimposed on a Maxwellian) doesn't fit the data very well. Part of the problem is that when there is a lot of high energy (300+ eV) emission at a range of temperatures, our SXR filters don't let us distinguish between the different temperatures very well. I definitely haven't exhausted the possibilities for models, and I would like to look into this further, but unfortunately I don't have time. Hopefully someone else will be able to continue the analysis. As a final note, I also looked into whether things look any better with the tin filter, but like the other ratios, the model ratios with Sn don't vary much from T = 5-100 eV (this is for the 1% non-Maxwellian model), and the data still don't fit the models:

7-10-07: Here are some spectra from the simulation that I discussed in my previous entry (same impurity fraction, density, atomic models, etc. that I have been using since December for SXR calculations, but with bigger spectra calculated). It's clear that once the plasma temperature reaches 40 or 50 eV, the 300+ eV spectral features are of comparable importance to those at E < 150 eV. There are no strong features at E > 500 eV, so I didn't include these energies in the plots even though they were used for the temperature calculation.

7-8-07: Based on my earlier results showing that carbon lines in the 300-400 eV range have a significant effect on SXR signals for simulations with standard Maxwellian electron distributions, it is clear that we should be considering these energies in all of our SXR calculations. I ran a new set of simulations with spectra calculated all the way up to 1000 eV (all other parameters were identical to the simulations used for my thesis, which only considered spectral features at hν < 150 eV). Here are the resulting average temperatures for counter-helicity (left) and single spheromak (right) shots:

It would be hard to convince anyone based on these plots that the electron temperature in SSX shows evidence for reconnection heating. There is also little apparent difference between counter-helicity and single-spheromak shots, besides the fact that single spheromaks produce weaker SXR signals and therefore show higher variance at early and late times.

I wanted to delve further into this result and make sure it is really more accurate than my past results, so I made some plots comparing the model and data filter ratios at individual time steps (I did something similar last July to make sure my code was working). In the plots below, model filter ratios as a function of temperature are shown by the solid lines, and data ratios are shown as individual points, plotted at the x-value corresponding to the best-fit temperature. Error bars on the data points assume that there is a 30% uncertainty in each data ratio (this number is entirely arbitrary, but not unrealistic I don't think). The following plots show the temperature fits for three time steps during a counter-helicity merging shot.

A couple of things are immediately apparent from these plots. 1. Even though the code doing a fine job of determining the best-fit temperature given the model and data at hand, the best-fits aren't very good--in the top two plots, the models can't match the data to within 30% no matter what temperature is chosen. 2. We have essentially no ability to differentiate between different temperatures above 50 eV, because the model spectra produce the same filter ratios at all the high temperatures.

Problem #1 gets much worse when we include data from the tin filter, so this clearly isn't the answer:

In order to get a handle on where the differences lie between these calculations and the old ones, I re-made the above plots for the same shot and time steps as shown above, but this time I used the model spectra that cut off at 150 eV (the parameters for this calculation are the ones that were used for my thesis plots). Problem #1 is slightly improved for these models (see the three plots below), although the fits still sometimes aren't good to within 30%, and problem #2 seems to be somewhat less severe as well. However, just because the data appears to fit these models better doesn't mean they are more correct. I can see no way to justify ignoring spectral features above 150 eV, especially since we can readily identify them and expect that they will contribute to SXR signals.

7-5-07: In the interest of only changing one thing at a time, I went ahead and re-calculated the average SXR temperature profiles using my modified code (now I'm minimizing the sum of the relative differences between simulated and measured filter ratios rather than the absolute differences). The results are shown below (counter helicity merging on the left, single spheromaks on the right). Both temperature profiles look similar to the old ones, although both the size of the peak in the counter helicity temperature profile at 35-40 μs and the magnitude of the uncertainties for the counter helicity temperatures have increased slightly. I'm going to calculate these temperatures again using the results of a new simulation which includes spectral features at energies up to 1000 eV--I'll post the results soon.

7-4-07: I ran a 30 eV Maxwellian simulation yesterday to assess the importance of lines above 150 eV for SXR. The spectrum is shown on the left with a logarithmic y-axis, and a smoothed spectrum (Resolution setting in PrismSPECT = 10) with a linear y-axis is shown on the right.

Although there are a number of lines visible between 300 and 400 eV, they are several orders of magnitude weaker than the high energy lines produced in simulations with a 10% non-Maxwellain electron tail. In order to quantify the effect that these lines will have on SXR signals, I've run the spectrum through my TfromSXR code, first considering only energies up to 150 eV, and then considering energies all the way up to 1000 eV.

So it turns out that the effects of including spectral features at E > 150 eV are non-trivial: all filter signals increase by at least 1e-7, representing a relative increase of well over 10% for the Sn, Ti, and Zr filters. I should definitely calculate spectra out to at least 500 eV in the future.

The next step is to calculate SXR signals for the non-Maxwellian spectra that I have produced. I think the fact that the C V and C VI lines at >300 eV are substantially weaker in Maxwellian simulations than in non-Maxwellian simulations is good news, because it means that SXR signals will be different for the two conditions--this gives me hope that we'll be able to use these simulations to confirm or rule out the presence of non-Maxwellian electrons in SSX.

7-3-07: Here is the spectrum for a simulation with 10% of the electrons in a flat distribution out to 500 eV and the rest distributed in a 30 eV Maxwellian. This time I had PrismSPECT calculate the spectrum all the way out to 500 eV, rather than stopping at 150 eV as usual. There aren't a whole lot of lines at higher energies, but those that are present dominate the spectrum. I wasn't really expecting this, but I probably should have, since many of the dominant ionization stages (C V, C VI, O VII) have resonance lines at E > 150 eV. To highlight the relative intensities of different lines, I've included both log (left) and linear (right) plots of the spectrum.

C V 304.4 eV (4.1 nm) 1s(1)2p(1) 3P[1] to 1s(2) 1S[0]
C V 307.9 eV (4.0 nm) 1s(1)2p(1) 1P[1] to 1s(2) 1S[0]
C V 354.5 (eV (3.5 nm) 1s(1)3p(1) 1P[1] to 1s(2) 1S[0]
C VI 367.5 eV (3.4 nm) 2p(1) 2P[1/2] and 2P[3/2] to 1s(1) 2S[1/2]

To get a feel for how much influence these lines would have on SXR signals, I used PrismSPECT to produce a low-resolution (10) version of the spectrum:

As I mentioned above, it turns out that the high energy features dominate the spectrum, so they will be incredibly important for SXR. I have a feeling this effect may not be specific to plasmas in which there is a high energy electron tail: the C V lines will probably be just as strong in a normal Maxwellian simulation, because the C V fraction is higher for a simple 30 eV Maxwellian than for an electron distribution with a high energy tail. I feel kind of silly for only calculating spectra out to 150 eV in the past--I knew that the C V lines at 3-4 nm would be strong, but I never actually did the conversion from nm to eV to realize that those wavelengths were not included in my previous calculations. I'm running a single Maxwellian simulation right now to look into the importance of high energy spectral features.

I've recalculated the SXR filter responsivities out to 1000 eV--I think I'll start calculating PrismSPECT spectra over this entire energy range because there may be a few strong lines even above 500 eV. This time I used an interpolation tool in Origin to match the photodiode data to the filter data, and the results look more reasonable than my previous calculation.

7-1-07: I ran a simulation with 1% carbon, .001% oxygen, and 1% copper with temperatures ranging from 5 eV to 100 eV to see how the SXR electron temperature calculation would be affected by having a large copper impurity in the plasma. To my initial dismay, I got temperature profiles that looked like this:

Closer inspection confirmed that there was not in fact a bug in the calculation. Rather, 15 eV really was the best fit temperature at almost every time step based on the fitting procedure used in my code. Here's a sample of the filter ratios produced by the model spectra:

And for comparison, the measured filter ratios during the 35-39 μs interval for the shot shown above on the right:

It is apparent that even though the measured filter ratios change considerably as a function of time, the sum of (measured-model)^2 will always be minimized by choosing T = 15 eV (note: all three model ratios continue to decrease at higher temperatures not shown in the table). The fact that the ratios involving the aluminum filter are consistently too low in the models provides pretty good evidence that there is not a whole lot of copper in SSX (recall that the C and Cu concentrations in this simulation were equal).

However, I've become a bit concerned about one aspect of my temperature fitting code. The code minimizes the sum of the chi-squared statistics for each filter ratio, defined as:

I didn't have a good way to estimate the error in each ratio, so I just set all the errors equal to 1 for the calculation. I realize that this means large filter ratios are weighted more heavily than small ratios in picking which model best fits the data. For instance, in the above example, the Ti / Zr ratio doesn't contribute much to the overall sum of the chi-squared since it is the smallest in magnitude (see the discussion on my journal page). Rather than setting all the errors to be equal, it seems to me that it would be better to assume that the errors in each ratio are proportional to the size of the ratio in the data. This will ensure that all ratios are given appropriate weight, and consequently that the calculation is internally consistent (using the current method, a best-fit temperature calculated by considering the filter ratio Al / Ti might be different from the temperature calculated using the ratio Ti / Al).

I made the change to my code and recalculated temperature profiles for the two shots shown above. The results didn't change much:

Error estimation issues aside, I think it's safe to conclude that there is much less copper than carbon in SSX. I'd like to place further constraints on the amount of copper present using the VUV monochromator, but it will be difficult given the lack of strong copper lines and our current signal-to-noise ratio. The best copper lines to observe seems to be the Cu VIII lines at 90.0 and 93.8 nm. Here are the ratios of the strengths of these lines compared to our old friend C III 97.7 nm as a function of temperature:

Back to the temperature calculations from SXR. I went back to using the old simulation results with only carbon and oxygen impurities and recalculated the temperatures for the two shots shown above (both are from August 1). The plots on the left were produced by my old code which assigns equal errors to all ratios, and the plots on the right were produced by assuming the error in each ratio is proportional to its magnitude. There is a noticeable difference in the temperatures calculated for the first shot, but not much change for the second shot. I'll recalculate the average temperatures reported in my thesis and post the results soon.

6-29-07: I used the Lawrence Berkeley Lab site to re-calculate response functions for our SXR filters up to 500 eV. We have responsivity data for the photodiodes up to 1000 eV, but the data points are sparse, so I had to interpolate between them before I could multiply the photodiode and filter response functions together. Here are the results:

I should note that some aspects of this plot are almost certainly not accurate. I used IDL's spline interpolation tool to fill in the missing data points for the diode responsivity, but I think there were times when trying to fit a polynomial or other similar function to the data wasn't the best approach. In particular, consider the following data:

The interpolated response function from IDL features a valley at ~280 eV and a peak at ~320 eV, but the actual responsivity is probably flat in the 260-340 eV range. I'm not sure what to do about this--one option would be to just assume that all the data points we have are connected by straight lines--but it would be nice to have some more detailed responsivity data from the photodiode manufacturer (having no points between 340 and 650 eV isn't ideal...).

It may well turn out that these issues are unimportant--if there is little emission by the plasma at 200+ eV, then calculating the SXR responsivities at high energy exactly shouldn't be necessary. I'll run some new simulations soon that produce model spectra out to these energies.

6-28-07: The figure in my thesis that I used to estimate the relative concentrations of carbon, nitrogen, and oxygen in SSX was based on simulations with old (possibly incorrect) atomic models and used other parameters that have since been optimized, so I thought it would be a good idea to run a new simulation for use in our paper. The plots below show the N IV 76.5 nm / C III 97.7 nm and O V 63.0 nm / 97.7 nm line ratios for the new simulation with equal concentrations (0.1%) of C, N, and O.

Compared to the old result, the N IV 76.5 nm / C III 97.7 nm ratio has decreased by up to a factor of two (the minimum value in the plausible SSX temperature range is now ~ 28), while the minimum value of the O V 63.0 nm / C III 97.7 nm ratio is ~ 970 (similar to the old result). Thus the conclusion that the O/C number ratio in the plasma is 1/1000 still seems reasonable, while the upper limit on the N/C ratio is weakened somewhat (based on our failure to detect the 76.5 nm line in the lab, we will continue to assume a negligible nitrogen concentration for the purpose of SXR simulations).

Yesterday I ran my first successful simulation using an analytic expression for the electron distribution in PrismSPECT. The simulation had 1% carbon and .001% oxygen impurities, 90% of the electrons distributed in a 30 eV Maxwellian, and 10% of the electrons in a flat distribution running from .0001 eV to 500 eV. The following spectrum was produced (compare to the dual Maxwellian and single Maxwellian spectra below):

As in the case of the dual Maxwellian simulation, the addition of high energy electrons doesn't seem to lead to a lot of high-energy emission lines; rather, the low-energy spectrum gets denser as the ionization balance and levels populations change. The following tables summarize the results from the three simulations:

As we would expect, the addition of a population of high-energy electrons causes the plasma to become more ionized. The effect is slightly greater for the slideaway distribution than for the dual Maxwellian. The following table summarizes the effect on SXR signals of adding non-Maxwellian electrons.

6-22-07: The results of my first simulation with a dual Maxwellian electron energy distribution are in. The simulation had 1% carbon and .001% oxygen impurities, just like my standard simulations for SXR, 90% of the electrons were distributed in a Maxwellian at 30 eV and 10% were described by a hot 100 eV Maxwellian. The spectrum on the left was produced by the dual Maxwellian simulation, while the spectrum on the right was produced by a simulation that was identical but had electrons distributed in a single Maxwellian at T = 30 eV.

The two spectra are noticeably different: there are significantly more lines in the spectrum from the hot-electron simulation, particularly at relatively low energies. This seems surprising, but can probably be explained by the changes in the impurity ionization balances caused by the 100 eV electrons. For the single Maxwellian simulation, the dominant carbon ions are C V (fraction = .998), C IV (.001), and C VI (.001). For the dual Maxwellian simulation, on the other hand, the mean charge of carbon has increased: the dominant ions are C V (.557), C VI (.431), and C VII (.011). Similarly, the dominant oxygen ions in the single Maxwellian simulation are O VII (.791), O VI (.143), O V (.064), and O IV (.002), while for the dual Maxwellian simulation they are O VII (.950), O VI (.042), O V (.007), and O VIII (.001). Given the non-negligible quantity of O VIII present in this simulation, it seems that I may want to switch to a more complex oxygen atomic model (the model I've been using has only the ground state for O VIII).

So compared to the single 30 eV Maxwellian simulation, the dual Maxwellian simulation led to a higher predicted signal through the aluminum, titanium, and zirconium filters and a lower signal through the tin filter. This result seems reasonable based on visual comparison of the above spectra and the filter responsivities.

On another note, Joe MacFarlane suggested at one point last year that a non-Maxwellian electron distribution could potentially help explain the anomalously strong C III 229.7 nm line observed in SSX (i.e. by increasing the population of excited levels in C III). However, the CIII 97.7 nm / CIII 229.7 nm line ratio from this dual Maxwellian simulation was ~22, in precise agreement with the ratio calculated from single Maxwellian simulations at T > 20 eV (recall that we observe a line ratio of approximately 1:1 in the lab). Therefore, it appears unlikely that a population of hot electrons could explain the measured 229.7 nm line intensity.

Back to my standard single Maxwellian simulations: I realized today that I never looked into what the optical depths are for the carbon lines we observe in SSX, and PrismSPECT has a simple tool for doing so. If the 97.7 nm line were optically thick, for example, it could explain why it doesn't appear as bright relative to the 229.7 nm line as we would expect. However, as the plots below show, no carbon lines are optically thick at SSX temperatures and densities. The plots are for model plasmas with thickness (40 cm) and density (5e14 ions/cm^3) appropriate for SSX, with a 1% carbon impurity. The electron temperatures are 20 eV (left) and 30 eV (right). At T = 5 eV, several lines are optically thick; however, overall line strengths in the spectra tend to decrease as the plasma temperature increases, and for a 20 eV plasma the strongest line (C IV 155 nm) has an optical depth well below 0.1. 20 eV is the lower limit for our temperature estimates in SSX, so it appears that optically thick lines are not something that we need to worry about.

6-21-07: To investigate the equilibrization time for copper and see if it's still safe to use steady state simulations, I ran a time-dependent simulation with 99% H and 1% Cu. The basic simulation parameters were the same as for the time-dependent simulations I ran last year (500 logarithmically-spaced time steps, 40 cm thickness, n = 8e15 ions/cm^3 for 0 μs < t < 10 μs, decreasing linearly to reach n = 5e14 ions/cm^3 for t > 30 μs), with an electron temperature of 30 eV. The simulation produced the following warning:

"Potential problem detected in time-dependent population solver. Possible solution: decreasing time step size."

However, the results mostly appear to be reasonable, although the fractions of very high ionization stages present don't seem right (they are small, so they won't effect the spectrum produced, but they should be zero). Using 500 logarithmically-spaced time steps has always been sufficient before, and I'm not eager to increase the number since the simulation took 35-40 hours to run as it was. However, I'll consider it if we decide it's necessary to make these results more exact.

The model spectrum at t = 50 μs is shown below (left), with the steady state spectrum produced last week also included for comparison (right).

The spectra are similar, although the time-dependent spectrum is missing a number of lines between 40-60 eV that are present in the steady-state spectrum. The following tables show the ionization balance at t = 50 μs from the time-dependent simulation (left) compared to the ionization balance in the steady-state simulation (right):

The ionization balances are qualitatively similar; the dominant ions in both simulations are Cu VIII through Cu X, so it appears that 50 μs is indeed enough time for the copper in the plasma to have a large number of electrons stripped off and reach a near equilibrium state. However, the fractions present of individual subdominant ions differ in the two tables, sometimes by large factors. In both cases, it appears that there are intricacies of modeling the ionization of a complex atom that PrismSPECT is not handling correctly (i.e. I'm not confident that the calculated fractions of ions higher than Cu XV are realistic).

The time evolution of the copper ionization balance is shown in the two plots below. These plots confirm that the plasma quickly (< 10 μs) burns through the C I through C VII ionization stages. Therefore, my conclusion from the steady state simulations still holds: if there is copper in the SSX plasma, we will have difficulty detecting any lines with the VUV monochromator, but the higher energy lines will still affect our SXR measurements.

In the wavelength range accessible with the VUV monochromator, the strongest copper lines at t > 30 μs in the time-dependent simulation were:

Cu XI 77.7 nm 3s(2)3p(6)4p(1) 2P [15] --> 3s(2)3p(6){2S}4s(1) 2S
Cu XI 82.4 nm 3s(2)3p(6)4p(1) 2P [5] --> 3s(2)3p(6){2S}4s(1) 2S

These three appear as broad bumps on the spectrum, but PrismSPECT's Line Intensity Viewer gives their strengths as 10-100 times stronger than the Cu XI lines above:

Cu VIII 85.7 nm 3s(2)3p(6)3d(3){4F}4p1 3G --> 3s(2)3p(6)3d(3){4F}4s(1) 5F
Cu VIII 90.0 nm 3s(2)3p(6)3d(3){4F}4p1 5F --> 3s(2)3p(6)3d(3){4F}4s(1) 5F
Cu VIII 93.8 nm 3s(2)3p(6)3d(3){4F}4p1 3G --> 3s(2)3p(6)3d(3){4F}4s(1) 3F

There are no copper lines of even moderate intensity at wavelengths between 100 nm and 300 nm. Looking for the 5 lines listed above could allow us to place an upper limit on the amount of copper in the plasma, although with our current signal/noise ratio, the limit won't be very stringent.

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6-15-07: Here is the spectrum from a simulation with 99% hydrogen and 1% copper (T = 30 eV and n = 5e14), plotted as a function of wavelength (left) and energy (right):

There are a ton of lines between about 40 and 100 eV, as well as an apparent 100-fold increase in the continuum (which David says it is actually just a dense forest of lines). As I mentioned yesterday, most of the strongest lines are from the dominant ionization stages present: C VII, C VIII, C IX, C X, etc. These spectral features will obviously have an effect on SXR measurements if Cu is present in the plasma. Unfortunately, it looks like we won't be able to use the VUV monochromator to find out.

One thing that's clear is that the presence of copper cannot explain the SXR tin filter anomaly. Rather, the effect would go in the opposite direction: if the plasma had copper in it, we would expect to see even stronger signals in the Al, Ti, and Zr filters relative to the Sn filter signal, as we can see by comparing the copper spectrum to a plot of SXR filter responsivities:

Incidentally, there are some strange things going on with the Cu ionization balance my most recent simulation: ions as high as Cu XX are present with fractions greater than 1e-4. In the previous simulation (which was identical to this one except that it included 1% carbon as well as 1% copper), the concentration of ions above Cu XIII was negligible. It makes sense that removing carbon would cause the copper mean charge to increase (less free electrons floating around to recombine), but I'm surprised that the effect would be this strong.

This afternoon, I created a new copper atomic model that included 50 excited states for all ions and then re-ran the simulation discussed above. The ionization balance changed considerably, as shown in the following screenshots (The ionization balance shown on the left corresponds to the copper atomic model with only ground states included for ions higher than C XV, and the ionization balance shown on the right corresponds to the more complete copper atomic model):

Both ionization balances are non-monatonic at times (e.g. there is more Cu XX than Cu XIX for the ionization balance shown on the left). It's apparent that something is not quite right. Adding more levels to the atomic model seems to produce changes in the right direction, but in any case, PrismSPECT doesn't seem to be correctly accounting for the levels that are left out like it usually does for simpler atoms. I probably won't worry about this any more for now, but if we ever want to do precision modeling using copper it will be an issue.

6-14-07: I recently created a copper atomic model and ran a simulation to try to identify Cu lines that we might look for with the VUV monochromator. To keep the simulation time reasonable, I just did a single run at 30 eV, with a 1% copper impurity. I also included a 1% carbon impurity for the purpose of line strength comparisons. The following spectrum was produced:

Surprisingly, none of the strongest lines are copper. The usual carbon lines at 97.7 nm, 117.6 nm, and 155 nm are visible, as are Lyman α and β, but the copper lines are also weaker than a number of carbon lines that we have never been able to detect with the VUV monochromator, including C IV 77.0 nm, C IV 80.0 nm, C IV 110.8 nm, C IV 116.8 nm, C V 227.2 / 227.8 nm, and C IV 252.5 nm. These results suggest that the previous failures to detect copper in the SSX plasma may not mean that the impurity is not present but rather that there are no strong copper lines in the UV at SSX temperatures and densities. At 30 eV, there are appreciable quantities of Cu VI through Cu XII, and Cu IX is the dominant ion, so we would expect the resonance lines of these ions to be the strongest copper lines.

3-25-07: Here are some results from a simulation using a new hydrogen atomic model with all energy levels included (previously we had been using a 2-level hydrogen atom). All other parameters for the simulation were the same as those used for the final SXR temperature calculations in my thesis (n = 5e14 ions/cm^3, 1% C, .001% O, etc.). For a preliminary analysis of the importance of hydrogen emission lines for SXR, I'll focus on simulation results from three temperatures: 10 eV, 30 eV, and 50 eV. Here is the simulated spectrum for T = 10 eV:

At 10 eV, the strongest UV hydrogen line, Lyman α (1216 Angstroms), makes a noticeable contribution to the spectrum, although it is several orders of magnitude weaker than the C III 1175 Angstrom line. Lyman β (1025 Angstroms) can also be seen on the above spectrum. Since hydrogen is the dominant element in the plasma, we might expect its lines to be stronger than the strongest carbon and oxygen lines. However, we see that this is not the case, because even at the relatively low (by SSX standards) temperature of 10 eV, the vast majority of hydrogen atoms are ionized.

At 30 eV, Lyman α is still relatively strong, but Lyman β no longer reaches the threshold intensity necessary to be displayed by PrismSPECT. The strongest lines in the 30 eV spectrum are from the O VI 1035 Angstrom transition.

At 50 eV, Lyman α and β are both visible, but they still appear less intense than the strongest C and O lines (I should note that the H lines tend to display significant thermal broadening due to hydrogen's low atomic mass, so eyeballing the plots and noting that the C and O lines are taller does not serve as conclusive proof that there are more photons being produced by the carbon and oxygen transitions than by Lyman α).

For greater quantitative insight into the impact of the new hydrogen atomic model, I ran the model spectra through my IDL code and compared the model filter signals for the 2-level and complete H models. As we might expect from the plotted spectra, the inclusion of hydrogen emission lines had a small but non-negligible effect on the filter signals. The net effect on the filter ratios will be quite small, since the relative signal increases seem to be similar across filters for a given temperature. Nevertheless, I plan to use the full hydrogen atomic model for all my calculations from this point forward. Even with the complete model, hydrogen doesn't have very many energy levels, so the increase in simulation times will be minor.

Taken as a whole, these results are good news. I had been worried that the oversimplified hydrogen model used in our simulations might have led to significant inaccuracies in the SXR temperature calculations in my thesis, but it appears that this was not the case. Of course, the mysteries surrounding the Sn filter remain. It's interesting to note that the relative weakness of the Lyman series lines (compared to C and O lines) in simulations probably explains why we have been having difficulty observing Lyman α with the VUV monochromator (see August 4, 2006).

3-5-07: Here's the powerpoint presentation that I used to give my thesis talk on February 26.

12-20-06: I made some plots of the average temperature derived from SXR measurements for 23 single spheromak and 23 counter-helicity shots from August 1. Here they are (counter-helicity merging is on the left).

The temperatures are higher overall than those derived from UV line ratios, but the relative temperatures of counter-helicity and single spheromak shots during the middle of the shot are in approximate agreement (both measurements suggest that counter-helicity shots are 5-10 eV hotter).

12-15-06: Here's a comparison of the temperatures I derive using the two SXR simulations I did recently. The images on the left correspond to the simulation that used an oxygen atomic model with only ground states for OI, OII, OVIII, and OIX, and the images on the right are calculated from the simulation that used an oxygen atomic model with 100 energy levels for each ion.

The temperatures derived using the two different oxygen models are almost identical. Given the other sources of uncertainty in our SXR calculations, it seems that the simpler atomic model is sufficient.

12-14-06: I ran a simulation for SXR with 1% carbon and 0.001% oxygen impurity number fractions. We observe a CIII 97.7 nm / OV 63.0 nm line ratio of approximately 1:1 in SSX; the new simulated ratio is within an order of magnitude of this for temperatures between 15 eV and 100 eV, so it appears that the 1000/1 carbon to oxygen ratio is in the right ballpark.

12-10-06: I ran some time-dependent simulations using our final carbon model (C200each11_6) in order to make sure we're still okay using steady-state simulations. All time-dependent simulations had T = 30 eV and n = 8e15 ions/cm^3 for the first 10 μs, decreasing linearly to 5e14 ions/cm^3 at t = 30 μs. The ionization balances for simulations with different starting levels populations converged quickly, just like before:

Furthermore, the emission spectrum at t = 50 μs for the simulation with starting level populations calculated from LTE at 0.025 eV [left] looks identical to the spectrum produced by a steady-state simulation with T = 30 eV and n = 5e14 ions/cm^3 [right]:

12-3-06: I made a new carbon atomic model with the number of CV energy levels increased from 200 to 296 (the maximum possible) and re-ran the simulation whose results were shown on November 29 with T = 60 eV and varying density. The resulting plots of ionization fraction vs. density were indistinguishable from the previous simulation, so we can conclude that the unexpected behavior of the CV and CVI fractions was not a result of having too few CV levels.

11-29-06: Based on our examinations of the atomic model files, it seems that we've done everything correctly when constructing the 200-level per ion carbon atomic model. Our hypothesis is that the unexpected density dependence in the CIII 97.7 nm / CIV 155 nm line ratio because they are subdominant ionizations stages present in very low concentrations, so small absolute changes in the CIII and CIV fractions correspond to large relative changes which can dramatically effect the line ratio. In an effort to bring some closure to this issue, I've made several plots summarizing the density and temperature dependence of the carbon ionization balance for simulations using the 200-level per ion atomic model. The following plot shows the temperature dependence of the ionization balance for a typical SSX density:

As shown above, the CIII and CIV abundances are several orders of magnitude less than the CV abundance at most temperatures. The following plots show the density dependence of the ionization balance for three different temperatures. The CV fraction appears to be relatively constant at low densities, suggesting that the density dependence in the CIII and CIV fractions at low densities indeed arises because they are trace ions.

One thing that slightly worries me is the change in the CV and CVI fractions at around n=10^9 ions/cm^3 in the 60 eV plot. Both ions are present in fairly high concentrations, and coronal equilibrium should still hold at densities this low, so I'm not sure what's going on. I made a plot that zooms in on the upper portion of the above graph:

11-19-06: As outlined on my journal page, we're once again struggling with density-dependent emission line ratios that we didn't expect and don't really understand.. Here's a plot of the CIII 97.7 nm / CIV 155 nm line ratio from a simulation I ran with the 200-level per carbon ion atomic model [the .atm file can be downloaded here] that I had settled on using for all my simulations:

There are essentially no regimes in which the line ratio DOESN'T depend on density. This is surprising--we don't expect to observe a density dependence at low densities, where the coronal approximation holds. It may be that we observe strange behavior because CIII and CIV are present in extremely low concentrations at the temperatures of interest (CV is by far the dominant ion). However, non-physical explanations must be investigated as well. One possibility was that I somehow made carbon atomic model file wrong; to test this, I added levels to the 304-level atomic model that David and I made on October 25 [the original .atm file can be found here] to make it have 200 levels per ion. The results were the same as shown above, suggesting that either there was nothing wrong with my other 200-level-per-ion atomic model, or that by modifying the October 25 atomic model, I somehow messed it up too. As a follow-up test, I again modified the October 25 model, this time by adding a single energy level to the CIV ion. The line ratios produced look the same as those derived earlier from the October 25 model, so the simple act of modifying that model did not lead to any glitches. Rather, it was the addition of many more energy levels that effected the relevant level populations and changed the values of the 97 nm / 155 nm line ratio.

Returning to the possibility that the density-dependence is a real effect caused by the fact that CIII and CIV are not the dominant ion stage: I noticed some interesting behavior in the ionization stage fractions for the simulations using the 200-level-per-ion carbon atomic model. At T = 20 eV, the CIV fraction is constant for n < 10^13 ions/cm^3, while the CIII fraction varies (so the density dependence in the line ratio at low n is caused by CIII):

At T = 60 eV, on the other hand, when the fraction of CIV is lower, both the CIII and CIV fractions vary with density at low n.

11-12-06: I ran a new simulation with 400 energy levels per carbon ion (1957 levels total) and added the calculated 97.7 nm / 155 nm line ratio to the plot from November 8. The results are shown below--the ratio is almost identical for the 200-level and 400-level simulations, suggesting that using 200 levels per ion is sufficient to model the important atomic processes.

11-8-06: After figuring out how to create atomic models with fine structure, we also needed to make sure the model used for the October 29 results had enough atomic levels to capture all the key spectral properties. I tested this by creating a new carbon atomic model with 200 energy levels for each ion (1061 levels total, since CVI only has 60 levels and CVII only has 1) and then re-calculating the 97.7 nm / 155 nm line ratio for n = 5e14 ions/cm^3 and T between 5 eV and 100 eV. I found that the line ratio changed by a factor of ~1.5 at most temperatures, so it looks like we should use the more complex atomic model.

This finding agrees with my previous results from SXR modeling suggesting that at least 200 atomic levels per ion are necessary to produce accurate spectra. To be safe, I'm going to run one more simulation with 400 atomic levels per carbon ion to make sure that the line ratio of interest doesn't continue to change as I increase the complexity of the carbon atomic model further.

10-29-06: Today was one of those rare days in this project when things got simpler instead of more complex. Now that we have finally finished re-running steady-state PrismSPECT simulations with our new, more detailed atomic models, the density-dependence of the 97.7 nm / 155 nm line ratio has essentially disappeared. Compare the plot on the left, which shows the 97.7 nm / 155 nm ratio for the three densities we used in our most recent round of simulations, with the plot on the right, derived from the simulations run back in July, in which changing the density by an order of magnitude altered the line ratio by a factor of 5 or more.

Here are the new calculated average temperature profiles for counter-helicity shots from August 2. I made separate plots for each ion density used in simulations, but in light of the result presented above, the plots should and do look quite similar.

The densities shown are, from top to bottom, 1e14 ions/cm^3, 5e14 ions/cm^3, and 2e15 ions/cm^3:

Here are the results for single-spheromak shots from August 3. These have extremely large uncertainty ranges, so it seems like they're not quite ready for prime time. This may be because a Mach probe radial scan was in progress during the shots used to calculate the temperature profiles. Although the probe was at least 6 cm out of the center of the machine for all shots and therefore should not have been in the VUV monochromator line of sight, its presence may have led to large shot-to-shot variations in plasma structure. Once again the temperatures shown are calculated from simulations with n = 1e14 ions/cm^3, n = 5e14 ions /cm^3, and n = 2e15 ions/cm^3, respectively.

9-14-06: Joe MacFarlane pointed out that the atomic models that I was using didn't use the most detailed calculations possible for determining level populations, so David and I have been working on making new carbon models that include fine structure and term splitting for the calculation of populations and of spectra. However, we were confused about the effects of a couple of changes that we tried to make in Atomic Model Builder, so I put together a presentation for Joe MacFarlane outlining our questions.

8-17-06: It turns out that my earlier conclusion that the density-dependent ionization balance in simulations was caused by spontaneous emission was incorrect. After running simulations over a wider range of densities, it became clear that there is a threshold density where the ioniziation state fractions begin to change. Turning off spontaneous emission simply moves this threshold to a somewhat higher density, so we didn't see the ionization balance changing when we only range simulations between 5e13 and 5e15 ions/cm^3. So the good news is that nothing was wrong with PrismSPECT. However, it appears that the density-dependent ionization state fractions are real--the coronal equilibrium approximation seems to break down at lower densities than we previously thought. This means that we will have to know the density in SSX precisely if we want to derive accurate temperatures from emission line ratios. I put together a presentation summarizing our investigations into the causes of the density dependence.

8-10-06: I've discovered that the density dependence of my simulated ionization state fractions and line ratios that I noticed earlier is somehow caused by spontaneous emission. Here are some plots of CIII and CIV ionization state fractions from planar, steady-state PrismSPECT simulations at T = 20 eV and densities of 5e13, 5e14, and 5e15. The carbon atomic model used had 200 levels for each ion.

The two sets of simulations shown are identical except for the spontaneous emission multiplier. With spontaneous emission turned off, the ionization state fractions are essentially independent of density, as they should be. However, when it is turned on, they vary by roughly a factor of two when the density increases by two orders of magnitude. Since spontaneous emission is an internal atomic process, I can't conceive of any physical reason that it would depend on density. I'll post another update when I have more insight into this issue.

8-9-06: I've finished a first draft of a significant portion of the theory section of my thesis: here's the latest version in post script format.

8-7-06: Since the VUV monochromator wavelength calibration has been an important issue for my project this summer, I've put together a new calibration curve based on the lines that I've observed. The slope and intercept are different than those determined by Slava during his thesis project several years ago, perhaps because the monochromator has moved, thereby changing the incident angle of light on the diffraction grating.

The slope on the linear fit is extremely well-constrained, so we can apply the calibration equation above with a fair degree of certainty as we look for more spectral lines in the future.

I've finished processing the data from our second SXR lollipop experiment, on August 1. The results are posted here. The Sn filter signal was about 3-4 μA when the line of sight was observed with both UV fused silica and sapphire, while the other three filters produced signals of less than 1 μA. These signals are less than 1.5% of the unblocked signals for each filter. The sapphire lens passes light at slightly lower wavelengths than the UVFS lens, and previously I calculated that the Sn signal was higher with sapphire obscuring SXR than with UVFS in place. However, this time I was more careful to only consider shots in which the Sn filter was definitely blocked completely (the sapphire lens has a smaller diameter than the UFVS lens, so it is difficult to position it so that it completely blocks the line of sight of all four SXR diodes), and this difference essentially disappeared. This makes the result somewhat less interesting, because it probably rules out the possibly that the Sn filter responsivity increases rapidly at energies between 5 and 10 eV.

8-4-06: We ran over 400 SSX shots this week, and during most of these we took VUV monochromator data. This gave me a chance to look for some of the oxygen and nitrogen lines that are strong in my PrismSPECT simulations. I looked for the NIV 76.5 nm, NV 123.9 nm, OIV 55.4 nm, OV 63.0 nm, OIV 79 nm, and OVI 103.5 nm lines. The results are tabulated in an Excel file (we tried many monochromator wavelength settings to find the setting that produced the strongest signal for each line, but only the results from the best settings are shown in the spreadsheet). Only the OV 63.0 nm and NV 123.9 nm lines produced measurable signals. The 63.0 nm line was over 1000 times stronger than the CIII 97.7 nm line in simulations that I ran with equal concentrations of carbon and oxygen impurities. As shown in the spreadsheet, the average signal we observed for the 63.0 nm line was about 7 μA. Signals for the 97.7 nm line from the same run day were in the 5-10 μA range, so the two lines are of comparable intensity in SSX. This implies that the oxygen concentration in SSX is probably two or more orders of magnitude less than the carbon concentration. This conclusion is supported by the lack of observable signals from the OIV and OVI resonance lines.

Strong signals in the 100 μA range were seen for the NV 123.9 nm line during counter-helicity merging shots on August 2. However, these results are somewhat questionable because the NIV 76.5 nm line, which is of comparable intensity in simulations, was not detected. It is possible that our observations of the 123.9 nm line were contaminated by hydrogen Lyman α line emission at 121.6 nm. However, as the following diagrams show, it appears that we were able to isolate and observe two separate lines, one of which was probably Lyman α and the other of which was the NV 123.9 nm line.

VUV monochromator signal from shot #108 on August 2 (the data has been smoothed in 1 μs bins and the baseline signal has been subtracted out). The monochromator was set at 121.4 nm--based on the calibration curve we have inferred from observing other lines, this signal should be from the NV 123.9 nm line.

VUV monochromator data from shot #143 on during single-spheromak runs on August 3. The monochromator was set at 119.0 nm, so this line should be Lyman α. Observations of Lyman α were also made on August 2, but the signals were corrupted by feedback from the Mach probe signals.

VUV monochromator data from shot #114 on August 2. The monochromator was set at 119.8 nm, so this signal comes from a segment of the spectrum between the two lines shown above. The signal is very noisy, but the average signal appears to be near zero, implying that we were indeed observing a region with little or no line emission. This result suggests that the NV 123.9 nm and Lyman α lines have been correctly identified, although the observed Lyman α emission is weaker than expected. More observations are needed to conclusively decide the issue.

If the 123.9 nm line measurements can be trusted, there is a significant nitrogen impurity concentration in the plasma, although carbon remains the dominant impurity element. This conclusion would be more robust if we could observe the NIV 76.5 nm line, but so far monochromator signals at this wavelength have been indistinguishable from background noise.

7-31-06: On Friday we tested the SXR filter responses at energies lower than 10 eV using the device that Chris and I built. Two "lollipop-like" structures composed of a screw with a circular piece of sapphire or UV fused silica glued to the top were attached at 90 degree angles to a rod inserted below SXR. By rotating the rod, we could block the lines of sight to SXR with either of the two pieces of glass, or they could be arranged so that no light was blocked. Both glass lenses do not transmit high energy x-ray or UV light. The transmission of UVFS cuts off below ~170 nm (above ~7.3 eV), and the transmission of sapphire cuts off below ~150 nm (above ~8.3 eV). We took data during counter-helicity merging shots. Here are the results tabulated in an Excel file. The sheet entitled "Data" has average filter signals from t = 40-60 μs for each shot, and the sheet entitled "Averages" has the mean signals for each position of the lollipop. There were a number of shots in which we weren't sure whether the sapphire window was completely blocking all four SXR diodes--I left these out when calculating the averages. When one of the two windows was blocking the SXR line of sight to the plasma, the filter signals decreased by about two orders of magnitude. The Sn filter consistently had the highest signals when the windows were in place, but the magnitudes of the currents registered were less than 10 microamperes, or about 1-2% of the normal unblocked currents. Both windows have transmission probabilities of above 80% for photons throughout their transmission range, so it is unlikely that we missed a substantial number of low energy photons in these measurements. In conclusion, there is evidence for some transmission below 10 eV, especially through the Sn filter, but it seems unlikely that this effect can fully explain the unexpectedly high Sn signals that we have been seeing all summer.

7-26-06: I ran a set of steady-state PrismSPECT simulations with T ranging from 4 to 80 eV and used the resulting spectra to recalculate the SSX plasma temperature from SXR data. Here are the results for the counter-helicity shot shown in the first two plots from 7-25-06:

All three model filter ratios decrease monotonically above 40 eV, so the addition of higher model temperatures had no effect on the calculated plasma temperature--it was still found to be 36 eV.

I've modified my IDL code to calculate and display the temperature derived from SXR filter ratios at 1 μs intervals. For the same counter-helicity shot analyzed above (the above plot was an average temperature over the interval [40 μs, 60 μs]), here are the results:

Since magnetic reconnection occurs during this shot, we would expect the temperature to rise at around 50 μs, but the plot shows just the opposite. This suggests that the calculated temperatures may bear little resemblence to the actual conditions in SSX during the shot. The time-resolved temperature appears to agree with the average temperature of 36 eV derived above, providing some evidence that my code is still working correctly. I only used data from the Al, Ti, and Zr filters for this calculation, so it appears that the Sn filter is not the sole cause of our difficulties. Here's the temperature profile for another counter-helicity shot from June 20. Once again, the temperature fluctuations appear to be more or less random:

7-25-06: The following plots detail the latest results in my SXR temperature calculations. The colored lines show filter ratios calculated from PrismSPECT steady-state simulations with 0.1% carbon, oxygen, and nitrogen impurities. The individual colored symbols are the corresponding ratios from the SXR data, plotted at the temperature where the models best fit the data. The data pictured are the average filter ratios from a June 20 high-density shot with counter-helicity merging. I will eventually add error bars to the plot, but I haven't decided on the best way to calculate the errors in the ratios. Currently my code assumes that the errors on every data point are equal in magnitude when calculating the best-fit T by minimizing the chi-squared statistic.

For this shot, my code found the best-fit temperature to be 28 eV, but none of the temperatures provided a good fit for every ratio. This was to be expected, since we already knew that the measured filter ratios from SSX cannot be explained by any realistic spectra given the know filter responsivities (one possible explanation is that the filters are not opaque in the UV or visible--we will test this experimentally in the near future). The experimental signals from the tin filter are particularly anomalous; here are the results from the same shot with the tin ratios left out of the calculation:

Now it is possible to find a reasonably good match between the model and data filter ratios. It is apparent that I will need to run simulations that include higher temperatures, since the calculated best-fit temperature corresponds to the second hottest possible model. Finally, I repeated the calculation for a low density (400 μs gas valve delay) shot with counter-helicity merging on June 23. The plasma should have been hotter for this shot because glow-discharge cleaning was run on the gun and flux conserver beforehand. However, my code again finds that the temperature was 36 eV.

This time the model does not fit the data as well, even at the best-fit temperature. However, there could be a better fit at higher temperatures, or the impurity concentrations in the model used may not be correct (this seems quite likely). In any case, when we leave out the data from the Sn filter, the results of my temperature modeling are promising. More experimental checks (e.g. switching out the filters for their duplicates and measuring the filter signals when we insert a piece of glass in front of them) are necessary before we can be confident that the results presented here are correct. Computationally, the next step is to use SXR data to calculate the time evolution of the electron temperature over the course of a shot and see how these results compare to the temperatures calculated from VUV line ratios.

7-24-06: My discovery that the CIII 97 nm / CIV 155 nm line ratio depends on the plasma density was surprising, because both collisional ionization and radiative recombination should scale with the square of the density. To further investigate this issue, I made plots of the simulated CIII and CIV ionization fractions as a function of temperature and density:

So the ionization balance depends on density. For some reason collisional ionization is increasing at a faster rate than radiative recombination as the density increases, so the plasma on average becomes more ionized at higher densities. The above plots can at least partially explain the dependence of the CIII 97 nm / CIV 155 nm line ratio on density; the CIII fraction decreases by a larger factor than the CIV fraction does when the density increases by a factor of 10, so the net effect is that the line ratio of interest gets smaller at higher densities.

I ran a set of zero-width, steady-state simulations with 0.1 % nitrogen included along with 0.1% carbon and 0.1% oxygen as impurities. The SXR filter signals I calculated are tabulated in an Excel file. As in previous cases, the units of the signals are meaningless, but their magnitudes can be used for comparison. At low temperatures, the addition of nitrogen causes the filter signals to increase by a factor of two or more, but at T = 20 eV and above the signals are nearly unaffected. In fact, the signals with C+N+O are sometimes slightly smaller than the signals for C+O alone. This concerns me slightly, because I didn't expect the different impurities to have any effect on one another, but the differences are small, and the signals at these temperatures for a model with N impurity alone are extremely weak. We can approximately derive the signals for the C+N+O model by adding the signals from the N model and the C+O model together, but this doesn't work exactly.

On another note, the simulations imply that for equal concentrations of the three elements, the strongest nitrogen and oxygen lines are many times more intense than the carbon lines we have observed, so if there is an appreciable amount of N or O in the plasma we should easily be able to observe these lines with the VUV monochromator. Two of the strongest lines in the 10-150 eV range for the temperatures considered are OV 63.0 nm and NIV 76.5 nm. The following plots show the ratios of the strengths of these two lines to the strength of the CIII 97.7 nm line that we have been observing with the VUV monochromator:

We're confident that the electron temperature in SSX is greater than 10 eV, so if we try to observe the 63.0 nm or 76.5 nm lines with the VUV monochromator, we should find that the signal is at least as strong (and possibly several orders of magnitude stronger) than the signal we saw for the 97.7 nm line. If we do not see these lines, then amounts of nitrogen and oxygen in the plasma must be negligible compared to the amount of carbon present.

7-21-06: Today I ran two steady-state simulations to get an idea of how the 97.7 nm / 155 nm line ratio depends on the plasma density. It turns out there is a fairly strong dependence, as shown in the following plot:

From this result it is clear that misestimating the plasma density by a factor of 10 would have a large effect on the temperatures that we calculate. Fortunately, we can measure the time-resolved density for individual shots using interferometry, so this shouldn't create too much additional uncertainty in the calculated temperature profile.

Here are some spectra from steady-state, zero-width PrismSPECT simulations with .1% nitrogen impurity and no carbon or oxygen in the plasma. The strongest lines are at 76.5 nm (NIV) and 123.9 nm (NV). We'll be able to anaylze whether there is actually any nitrogen in the SSX plasma by looking for these with the VUV monochromator.

I'll post results of simulations with carbon, nitrogen, and oxygen together within the next few days.

I went back to my plot of T vs. t derived from the 97.7 nm / 155 nm line ratio and calculated the uncertainty by finding the standard deviation of the signal for each line over the set of 4 shots and then calculating the maximum and minimum possible values that the ratio could take on at each time step. These maximum and minimum ratios were then plugged into my temperature fitting code. I've added uncertainties to the temperature plot I posted yesterday--the bounds on T are shown with dotted lines.

The uncertainties are quite large, especially after t = 70 μs. However, we should be able to decrease the uncertainties by taking more shots at each wavelength and increasing the sample size.

7-20-06: Since our efforts to use SXR signals to derive plasma temperatures have yet to prove fruitful, I've spent a couple of days focusing on whether VUV monochromator measurements of the CIII 97.7 nm and CIV 155 nm lines can be used to calculate the time evolution of the electron temperature over a shot. We need to measure the line ratio in order to calculate temperature, but the monochromator cannot take data on two lines simultaneously, so I had to average signals for each line from several different shots and then find the ratio of the averages. The following temperature profile was calculated from data taken during counter-helicity shots on June 20. The electron temperature rises from 16 eV to a peak of about 25 eV during reconnection and then decreases steadily. The monochromator signals are very weak before 35 μs and after 90 μs, so data from these times cannot be trusted to accurately indicate the temperature.

The above plot was derived from 4 shots with the monochromator tuned to observe the 97.7 nm line and 4 shots with the monochromator tuned to observe the 155 nm line. The first plot below shows the calculated average line strength ratio [white] that led to this temperature profile, and the line ratios for individual pairs of shots [colors]. The second plot below shows how the 97.7 nm / 155 nm ratio falls as the temperature increases in PrismSPECT simulations; comparing these plots, we can see how the measured average line ratio led to the temperature profile shown above.

7-18-06: Following yesterday's result showing that the atomic models I've been using may not be detailed enough to give correct simulated SXR filter signals, I did a more systematic comparison between models today. I ran steady-state zero-width simulations for T from 4 eV to 40 eV using atomic models of carbon and oxygen with 50, 100, and 200 energy levels per ionization state and compared the filter signals derived from these simulations to the signals for a set of identical simulations using the most complicated models available in Atomic Model Builder. The results are posted here. Signals that differ by more than 10% from the signal for the run with complete models are shown in red. This happens often for the 50-level and even 100-level models (the Zr filter signal is effected most strongly by making the atomic models more complex), but only once for the 200-level model (for the Sn filter at 8 eV). Using the complete atomic models is highly impractical due to excessive computation time, so some sacrifices will have to be made. With this in mind, I've decided that the 200-level models are good enough, especially since at that point it seems unlikely that we'll be able to match experimental and simulated filter signals with a confidence interval of less than 10% anyway.

7-17-06: My simulation times have been increasing since I started doing planar PrismSPECT simulations (instead of zero-width) and including oxygen as an impurity, so I decided to test if I could get away with using simpler atomic models. I did a steady-state zero-width scan from T = 4 eV to 40 eV using a carbon model with only CIII, CIV, and CV included and an oxygen model with OII-OVII included but with only 50 levels for each ionization state (my previous oxygen model had 100). I compared the signals through each of the SXR filters to the signals for a previous run that used more complete atomic models. The results are detailed here. The units of the filter signals are meaningless, since there remains some confusion about the units of the spectral output for zero-width simulations, but the numbers can still be used to compare the two simulations. Unfortunately, it appears that I simplified the atomic models too much; using the simpler models causes some of the filter signals to decrease by as much as 40%, and the decrease is not uniform across different filters (the Zr filter is disproportionally effected, implying that we are losing lots of lines at E > 50 eV), so the filter ratios will be altered if I use the simpler models. These results underscore the importance of having the right impurities in my simulations--I expect that adding nitrogen may have a significant effect on the filter ratios, as will changing the relative concentrations of oxygen and carbon.

Today I processed a new set of SXR data from a large set of single-spheromak shots taken last week. Single-spheromak shots have the simplest plasma dynamics, so they should produce SXR signals that we can easily understand. However, I'm still finding that the Sn filter has the strongest signal in all shots--a result that seems highly unlikely based on the known filter responsivities. Averaged over 30 shots that I looked at, the filter ratios were:
Sn : Al : Ti : Zr = 19.8 : 9.8 : 2.5 : 1.0
The complete results are posted here. While it is perhaps comforting to know that our anomalous filter signals were not caused by high-energy jets or other strange processes initiated by magnetic reconnection, we have yet to come up with a plausible explanation for our data. The next step is switching out the SXR filters for their duplicates to make sure that we get consistent results.

7-13-06: I put together a powerpoint presentation summarizing the interesting results we have been getting concerning the CIII 229.7 nm emission line. It can be accessed here.

7-12-06: I ran a PrismSPECT simulation with 0.1% oxygen impurity in addition to the standard 0.1% carbon impurity, and the result was a much more dense spectrum. The following spectra are from 1 cm planar simulations with T = 15 eV, starting level populations calculated from LTE at 0.025 eV, and the standard SSX density profile (8 x 10^15 cm^-3 in the gun and 5 x 10^14 cm^-3 in the flux conserver).

The plasma with oxygen included produced simulated SXR filter signals that were more than 100 times stronger than those produced by the plasma with only carbon. The signals calculated by my IDL code were:

Carbon only:
Al filter = 2.7 x 10^-6 Amperes
Sn filter = 3.3 x 10^-7 Amperes
Ti filter = 8.0 x 10^-7 Amperes
Zr filter = 9.8 x 10^-8 Amperes

Carbon and Oxygen:
Al filter = 1.1 x 10^-3 Amperes
Sn filter = 5.9 x 10^-4 Amperes
Ti filter = 2.1 x 10^-4 Amperes
Zr filter = 1.9 x 10^-5 Amperes

These currents are of the same order of magnitude as the currents experimentally observed in SSX (usually hundreds of microamperes).

7-10-06: In order to check whether my IDL code that calculates the SXR signals that would be produced by a given PrismSPECT model spectrum is working correctly, I did a sample PrismSPECT run at 15 eV. The filter ratios calculated by my code were:
Al / Sn = 8.59
Al / Ti = 3.45
Al / Zr = 29.13
PrismSPECT can calculate low-resolution versions of its spectra, allowing one to make a rough visual estimation of the SXR filter ratios we should expect. In the following plots, the left-side y-axis has units of erg/cm^3/ster/s/eV, corresponding to the emissivity (shown in black) from t = 50 μs in the simulation. The right-side y-axis corresponds to the SXR filter responsivities.

The intensity of the spectrum declines significantly above 50 eV, so we can zoom in for a clearer view:

From the above diagram, it is apparent that the aluminum filter should have by far the strongest signal, followed by the titanium filter (which its peak responsivity in the vicinity of 30 eV, where the spectrum is also strong), then the tin filter, and finally the zirconium filter should have the weakest signal, since it has almost no response at E < 50 eV. This ordering agrees with the filter ratios I calculated above. The numerical values of the ratios also seem reasonable, although making an accurate estimate is difficult.

I've begun to get to the source of the unit-conversion problem I was having going from simulated spectra to SXR signals. It seems that the issue is with the PrismSPECT "zero-width" simulations that I've been using. When I run "planar" simulations with a finite plasma thickness (e.g. 1 cm), the spectral intensities are much more reasonable. There also may be an inconsistency between the PrismSPECT spectra and the line strengths calculated by the "line intensities" tool. Here is a powerpoint presentation I made detailing these findings.

7-7-06: To close out the week I've got several new results to share. Yesterday I ran some time-dependent PrismSPECT simulations to determine how the time taken to reach equilibrium depends on the plasma density. My analytical calculations suggest that the two quantities should be inversely proportional, and the simulation results appear to agree with this hypothesis. Here are some plots of ionization state fractions for plasma densities of 1 x 10^14 ions/cm^3, 2 x 10^14 ions/cm^3, 5 x 10^14 ions/cm^3, and 8 x 10^15 ions/cm^3. All five simulations had starting atomic level populations calculated from LTE at .025 eV and plasma temperatures of 15 eV.

The times at which the fractions of CIII and CIV in the plasma peak are inversely proportional to the density. For example, CIII peaks near t = 1 μs for 8 x 10^15 ions/cm^3, around t = 6 μs for 1 x 10^15 ions/cm^3, around t = 12μs for 5 x 10^14 ions/cm^3, around t = 30 μs for 2 x 10^14 ions/cm^3, and around t = 60 μs for 1 x 10^14 ions/cm^3. Likewise, the time taken for CV to become the dominant state is inversely proportional to the density. Since CV is the dominant ionization state in equilibrium, this result confirms that the equilibrization time depends on 1/density. In my time-dependent simulations for SSX up to this point, I have started with a density of 8 x 10^15 ions/cm^3 for the first 10 μs, when the plasma is in the gun. 10 μs is plenty of time for the plasma to reach equilibrium at this density, so I have seen very fast equilibrization. However, the results presented here suggest that if the density in the gun was somewhat less than previously assumed, or if the ionization state fractions actually cannot come to equilibrium within the gun because of other factors, the effect on the simulation results could be significant.

Today I also made some more progress on my SXR code, and I wrote up a description of the unit conversion between emissivity (erg/cm^3/ster/s/eV) and amperes (the units of the SXR signals) in Latex. I still need to add a diagram to the document, but here's a first draft.

Today I ran a steady-state PrismSPECT simulation to determine how the CIII 97.7 nm / CIV 155 nm line ratio varies with changing plasma temperature. The simulation used a plasma density of 1 x 10^15 ions/cm^3, and spectra were calculated for temperatures between 1 eV and 30 eV at intervals of 1 eV. Since I have found that time-dependent simulations converge to equilibrium fairly quickly, the steady-state results should be a good approximation for the results I would get if I ran time dependent simulations at each temperature. In addition to calculating the 97.7 nm / 155 nm line strength ratio, I recalculated the CIII 229.7 nm / CIV 155 nm ratio, since this simulation used a finer temperature grid than the early steady-state simulations I did:

The CIII 97.7 nm / CIV 155 nm line strength ratio falls as plasma temperature increases--this makes sense because the CIII ionization state abundance decreases more than the CIV abundance at high temperatures. The ratio is about 1 at 10 eV, 1/4 at 15 eV, and 1/10 at 20 eV. Observations of this line ratio in SSX found it to be between 1/6 and 1/24, suggesting an average plasma temperature between approximately 15 and 25 eV. As I have discussed extensively in previous entries, the CIII 229.7 nm / CIV 155 nm line ratio is quite low throughout the range of likely plasma temperatures, in marked contrast to the experimental results. If we can attribute this discrepency to an anamolously strong 229.7 nm line in experiments, then we can still use the 97.7 nm / 155 nm line ratio to derive a useful estimate for the plasma temperature, which can then be compared to results from soft x-ray modeling.

On June 23 we ran a set of SSX shots after performing helium glow discharge cleaning on the flux conservers and the guns. After cleaning, the plasma should contain fewer impurity ions, but because of some sort of resonance process that we do not fully understand, the emission lines always become stronger. These shots were no exception; relative to the shots on June 20, all the carbon lines we looked at were stronger, but the intensity of the CIII 229.7 nm line increased disproportionally. I used my IDL code to process the VUV monochromator data and re-calculate the 229.7 nm / 155 nm line ratio. Over the course of the shots the plasma density was varied by altering the time delay between the opening of the gas valve to inject hydrogen into the guns and the time when the capacitor was discharged to ionize the gas.

For a time delay of 400 μs (lowest density):
Comparing peak currents: 229 nm / 155 nm ~ 1/4
Comparing average currents: 229 nm / 155 nm ~ 1/7

For a time delay of 500 μs:
Comparing peak currents: 229 nm / 155 nm ~ 1/3
Comparing average currents: 229 nm / 155 nm ~ 1/8

For a time delay of 600 μs:
Comparing peak currents 229 nm / 155 nm ~ 1/4
Comparing average currents: 229 nm / 155 nm ~ 1/8

Combining data for all densities:
Comparing peak currents: 229 nm / 155 nm ~ 1/4
Comparing average currents: 229 nm / 155 nm ~ 1/7

The standard gas valve time delay for high-density SSX shots is 730 μs. However, there was not enough data at this density from this set of shots to calculate line ratios. More detailed run-by-run results, including uncertainty estimates, can be found in this excel file.

Here is a link to an order-of-magnitude calculation I did to determine the average time for an electron to collisionally excite an ion in the SSX plasma. I wrote up the calculation in Latex, both to make it easily readable and for practice for my thesis.
Download ps

The collision rate can also be calculated from fundamental plasma parameters. This approach yields a similar result for the SSX plasma but does not tell us what fraction collisions between electrons and ions are inelastic and therefore lead to excitation.

This week we've taken some new SSX data using the VUV monochromator and soft x-ray detector, and I've written some IDL code to process the monochromator data so I can calculate line strength ratios to compare to my simulation results. My code starts with data from the oscilloscope:

Then it smoothes the data over 1 μs bins and converts the voltage to a photocurrent:

Then it subtracts out a baseline current calculated from a shot in which we closed the entrance slit to the VUV monochromator.

Finally the program finds the peak value of the current and the average current over an interval in the middle of the shot (40 μs to 60 μs).

Since the VUV monochromator has not been accurately calibrated, we took data at several wavelengths near each spectral line in order to make sure that we were centered on the line. Once we were confident that we had the wavelength right, we did three or four shots each at 97.7 nm, 155 nm, and 229.7 nm, both with the PMT output plugged directly into an oscilloscope and with a current amplifier used to get a stronger signal.

Comparing peak currents for shots with the amplifier, I found the line ratios to be:
229 nm / 155 nm ~ 1 / 17
97 nm / 155 nm ~ 1 / 22

Comparing average currents for shots with the amplifier, I found the line ratios to be:
229 nm / 155 nm ~ 1 / 15
97 nm / 155 nm ~ 1 / 24

Comparing average currents for shots without the amplifier, I found the line ratios to be:
229 nm / 155 nm ~ 1 / 6
97 nm / 155 nm ~ 1 / 6

The 229 nm / 155 nm ratio is still nowhere near the value expected from simulations at T = 15 eV (~ 1 / 2000), but the 97 nm / 155 nm ratio is at least of the right order of magnitude in the simulations. More detailed run-by-run results, including uncertainty estimates, can be found in this excel file.

Since the CIII 229 nm to CIV 155 nm line strength ratio has been very different in my simulations and the experimental data, I've also used PrismSPECT to calculate the ratio of the CIII 97 nm line to the same CIV 155 nm line. This CIII line is much stronger in the simulations, and it will give us another reference point for comparison when we take more data with the VUV spectrometer.

As shown in the plot, once the plasma has finished expanding, the line ratio calculated by PrismSPECT is about .27.

We're still stumped by the large difference between the experimental CIII 229.7 nm to CIV 155 nm line strength ratio (about 1/4) and the simulation results (around 5 x 10^-4). I ran a simulation today in which the plasma electron temperature was 30 eV in the gun and then cooled to 15 eV over the next 20 μs, while the density decreased over the same time scale as it had in my previous simulations. Here is a comparison of these results to the simulation results for T constant at 15 eV:

After 50 μs, the differences in the spectra for the two runs that were present at early times have largely disappeared.

The fractions of carbon in the CIII, CIV, and CV ionization states begin to converge once the temperature in the two simulations becomes equal (this happens at 30 μs), but the fractions still differ somewhat around 50 μs.

Once the temperatures in the two simulations become equal, the line strength ratios between the CIII 229.7 nm line and the CIV 155 nm line are very similar. At 50 μs, the numerical value of the ratio for the simulation with varying temperature was .0004837, while for the simulation with constant temperature it was .0005095.

The results of these simulations show that changing the plasma temperature in the gun has a noticable but small effect on parameters such as line intensity ratio later in the run. As in previous simulations I have done, it is apparent that changes in plasma properties such as temperature and density are reflected in changes in the ionization balances and spectra on very short time scales, 10 μs or less. Therefore, it appears nearly impossible to contrive a set of plasma parameters for the first 30 μs of the shot that will lead to results at late times that are drastically different from those I have shown.

I also ran a simulation with CV energy levels included (previous runs only included CIII and CIV). In order to see more lines I increased the range of energies to display in the spectra to 4-20 eV (previous spectra only went from 4-10 eV), corresponding to a wavelength range of 62 nm to 310 nm. This produced the following spectrum:

Interestingly, even though CV is by far the dominant ionization state at 15 eV, there are no strong CV emission lines in the energy range shown in the above spectrum. In other words, the differences between this spectrum and those for previous simulations that didn't include CV arise only because the spectra was calculated for a wider range of wavelengths, and not because we were missing prominent CV transitions before. The six strong lines visible in the figure above are:

CIII 976.99 A
CIII ~1175 A (actually 6 separate lines)
CIII 1247.34 A
CIV 1548.13 and 1550.72 A
CIII ~1923 A (actually 5 separate lines)
CIII 2297.5 A

Calculating the intensity ratio of the CIII 97 nm line to the CIV 155 nm doublet might be a good alternative to the ratio we have been considering so far (between the CIII 229.7 nm line and the CIV 155 nm doublet), since this ratio appears from the above spectrum to be much closer to one.

As one would expect barring some unforseen glitch, the addition of CV energy levels to the model had no effect on the CIII 229.7 nm to CIV 155 nm line ratio--its value at 50 μs remained equal to .0005095.

Now that I've settled on using 500 logarithmically-spaced time steps for my simulations, I've been investigating the effect of changing the initial atomic level populations. Here of the results of three simulations with initial level populations calculated from LTE at .025 eV, 1 eV, and 5 eV:

By 50 μs, the spectra for runs with the three different sets of starting conditions appear identical.

The fractions of carbon in the CIII, CIV, and CV ionization states for the three simulations differ early on but are identical for times later than approximately 10 μs.

The ratio of the intensity of the CIII 229.7 nm line to the CIV 155 nm line is almost exactly the same all three simulations, especially for t > 10 μs. These results demonstrate that the simulation output for times greater than 10 μs (after the SSX plasma has left the gun) has little dependence on the initial atomic level populations. This means that we can safely use room temperature (.025 eV) LTE populations to start without worrying about whether this precisely describes the physical reality.

My next goal was to compare the results of time-dependent and time-independent PrismSPECT simulations. I used a time-dependent simulation with 500 logarithmically-spaced time steps, starting levels populations corresponding to LTE at .025 eV, constant electron temperature of 15 eV, and ion density starting at 8 x 10^15 cm^-3 for 10 μs then decreasing linearly to reach 5 x 10^14 cm^-3 by 30 μs. The steady-state simulation had the same temperature and density as the time-dependent simulation at t > 30 μs (T = 15 eV, n = 5 x 10^14 cm^-3). The following spectra were produced:

The spectra appear very similar, although not identical. Other results are as follows (all results from the time-dependent simulation comes from t = 50 μs):

Line strengths: CIII 229.7 nm / CIV 155 nm
Steady state = .0005279
Time dependent = .0005095

The results from the two simulations differ, but not to a large degree. For example, the line strength ratio, one of the primary parameters that can be experimentally observed, differs by less than 5% between the two simulations. This example suggests that even time-independent simulations can be expected to produce relatively reasonable results, although the divergence of steady-state and time-dependent results would certainly grow as greater complexity was added to the simulation temperature and density profiles.

The powerpoint presentation Timesteps.ppt has been updated. A plot of the CV ionization state fractions has been added, along with results from t = 50 μs, line strength ratios, and additional analysis.

Although I previously showed the simulation results at early times (< 10 μs) to be vary if the number of time steps was increased from 100 to 500 or 2500, what we really care about are the results at later times (around 50 μs), because this is when we will experimentally observe the SSX plasma. As shown above, the spectra at 50 μs for the three simulations appear practically identical.

Likewise, the ionization state fractions between 40 and 60 μs are identical for the three simulations.

One of ways that we'll compare simulation results to experimental data is by looking at line strength ratios. The above plot shows the ratio of the intensity of the CIII 229.7 nm line to the CIV 155 nm line for each of the simulations. The 100-time-step run gives somewhat different results at times up to about 40 μs, but all three are nearly the same after that. The numerical values of the ratio at 50 μs are:
100 time steps: CIII / CIV = .0005092
500 time steps: CIII / CIV = .0005095
2500 time steps: CIII / CIV = .0005095

In summary, increasing the number of simulation time steps from 100 to 500 has a significant effect on output during the first 5-10 μs of the shot, and further increasing the number of steps to 2500 also slightly alters the results at early times. However, by 50 μs, the three simulations give virtually identical spectra, ionization state fractions, and line strength ratios. We only care about the simulation results at later times—the plasma can only be observed after it has left the gun, and reconnection occurs at about 50 μs. Therefore, 100 time steps are probably sufficient, but given the large discrepancy between the 100-time-step and 500-time-step runs at t < 5 μs, it wouldn’t be a bad idea to use 500 steps to be safe. Simulation time appears to scale roughly linearly with the number of time steps used: the 500-step simulation took only about 6 minutes to run, while the 2500-step simulation took approximately 30 minutes. Little efficiency will be sacrificed, then, by using 500 steps per run instead of 100.

Here are the results of a set of simulations I did to determine how many time steps we need to use to get consistent and physically reasonable results from our time-dependent PrismSPECT simulations.
Timesteps.ppt

Here are the results of some steady-state Prism-SPECT simulations I did to determine the temperature dependence of the ratio between the strength of the CIII 229 nm emission line and the CIV 155 nm line that Doc and Chris have observed:
CIIIoverCIVvsT.ppt