Where Be9 is being produced

Anna asked:

Could we graph the 9Be and 9Be/Al as a function of r/r0, where r and r0 is a) l outer radius of the cold fuel and then b)the outer radius of the hotspot; r-0 being the initial value.

Be9 production doesn't really have a radial profile in the same sense as e.g. the pressure or the temperature, because it is only produced at the location of the outer ablator material.  This ablator radius does shift with time, however, so I assume what she wanted was the rate of Be9 production plotted at the radius where it was produced.  Since the plots are multivalued (the ablator moves in and then out later on), I've color coded the passage of time, to clarify what is going on.

Here is a plot of the instantaneous rate of Be9 production versus the radius at which it was produced:

The radius is normalized by the initial Hot Spot radius (R0=0.012 cm) at the beginning of the simulation.  The pink is the inward traveling shell, and you can see Be9 production shut off dramatically as the shell turns around (and the dots tend to yellow).

Here is a plot of the ratio of the rates:

If the control were perfect, we'd expect the pink portion on the inward journey to be perfectly flat, and we see that it is indeed approximately flat, though not perfectly so.

It's also fun to look at the evolution of the integrated signal.  Here's the Be9 by itself:

Which as we expect, is monatonic until the production shuts off.  Here is the integrated ration of Be9/Al:

Here you clearly see the shutting off at the innermost radius, and you also get a feeling for how the overall rates die off very shortly after the ablator turns around (i.e. the burn stops and there are no more 14 neutrons incident on the outward moving shell).

Normalizing by the initial ablator radius (R0=0.02 cm) gives the exact same plots of course save the x axis, but these were part of what you asked for, so here they are (Note: The initial ablator radius is not the same as the initial size of the capsule, because our simulation does not start until the shock crosses the hotspot radius, well into the implosion process):

You'll notice this was not a particularly powerful implosion, the initial velocity I used in this simulation was 2.9e7 cm/sec (with a total yield of 2.47848e-03 (MJ) and 4.43250e+15 reactions).  I also tried a much higher implosion velocity of 3.5e7 cm/sec (yield of 7.93379e-03 (MJ) and 1.41888e+16 reactions) but it didn't really collapse much smaller to be honest.   But anyway here are the plots, in case you're interested (I have even higher implosion velocities to test, but they may be unrealistic??):

The points appear sparser than in the earlier plots because the time step decreased drastically during the simulation, and I had to thin the data by a factor of 10 to get it to plot in a reasonable time.

That's it for now, let me know if you've got any questions.

Integrated neutron scalar flux

Here are plots of the integrated neutron scalar flux as a function of radius.  Understand that this is the flux coming from spherical shells, not a cylindrical projection through a spherical capsule, as might be seen by an exterior camera.

These are very dirty:  The abscissae for the radial profiles change from time step to time step.  What I have done is picked some fixed radial bins, and used a spline to interpolate the radial profiles of the neutron flux at each time step to the fixed radial positions.  The "integration" is not a true numerical integral, but rather a sum of the values in the bins for each time step.  If anyone would like a modestly more accurate answer I can integrate numerically, but this seemed good enough for now.

For four sample implosion velocities, here are the integrated neutron fluxes, integrated for the entire length of the simulation (CLICK THE PLOT TO ENLARGE):

Here is a slightly zoomed in version of this same plot:

I also thought it might be interesting to see if the shape of this curve changes significantly as the state of the capsule changes (imploding versus exploding etc).  Here are the integrated neutron fluxes from the beginning of the simulation up until the time specified:

Not sure that I gained any particular insight from this... I'm sort of surprised by it, since hot spot radius changes by so much.  That's all for now.

Relative timing

Anna asked for a plot of the integrated neutron flux as a function of radius, for two cases, one in which all of the burn happens before the hot spot edge turns around and moves outward, and the other where most of the burn happens after the hot spot turns around.  My hunch was, we only have one of these cases appearing in our simulations.  This plot shows the hunch was correct:

I wondered if perhaps when we turned on the feedback (i.e. additional energy deposited into fuel from reactions, and energy leaving due to escaping radiation) this story could change.  Recall, however, that our code blows up when we add the feedback.  We do have the old mode (in which the feedback is a factor of 10 too small, and the run can finish before blowing up).  We refer to this as "Classic mode" (i.e. the mode with the fortuitous bug).   Using this I looked at the impact of feedback on these curves and found that the trend is entirely reversed:

I wanted to check if this might be an unexpected side effect of the bug, so I ran classic mode with both reactions and radiation off, just to make sure I captured the same behavior as the bug-free version.  Thankfully yes:

So it looks like the feedback causes the hot spot to turn around much earlier, in fact before peak burn is achieved.  I will make the integrated neutron flux plot with the case I trust, namely approx mode with reactions and radiation feedback off.

Movies!!

Earlier I developed a generic tool that would take a dataset and generate a plot of any tagged quantity (e.g. 'TIME' or 'ELECTRON_DENSITY') versus any other tagged quantity.  The tags are in the header of the data files.  The tool is generic in the sense that the file can include any new data column or stream that it wants, as long as the header information is added.

Yesterday and this morning, I used this generic tool to create another generic tool, namely a movie generator.  This can be used to make movies of any pair of tags that are dumped out at each time step.  Here are a few examples, although it would take quite a long time to explore the parameter space completely:

Pressure profile movie:

Mass density profile movie:

The evolution of the RIF neutron energy spectrum:

I think it might be nice to add the locations of the shock, the edge of the hotspot, and the outer boundary (the Lagrangian coordinate marking the location of the "Ablator" in this simulation).  I also think I'll try and make a movie of the doppler shifted spectrum of the 14 MeV neutrons, and also plot our Be9 cross sections on the same plot, so as we can see the overlap region as a function of time.

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Addendum:  Jerry wanted to know if we can do webm files instead of flv files.  Here goes:

Sensitivity to cross section curves

GH, as previously mentioned, gave new cross section curves to Anna, and I have been examining the sensitivity to the "dipsy doodle" today.  Here is a plot of the three cross section curves I have tested:

The results are puzzling to me.  I expect the green curve to produce the most signal, the red an intermediate amount, and the blue the smallest.  This is not exactly what I am finding, oddly.  Here are the results for the ratio of Be9/Al:

And of course the control is the same for all three curves, which were all run at an input velocity of 2.9e7 cm/sec.  The signal curves alone look as follows:

The only thing I could think of was that although the cross section curves appear to be the same at low energies, in fact they are not.  This is born out, and explains the trends.  What we learn here is that the most important place to measure the cross sections with fidelity is at the lowest energies (which is not surprising given that this is where they will overlap with the doppler shifted 14 neutron spectrum).  Here's a plot to convince you:

This makes it clear that red should give less signal than blue, even though after 14.8, the cross section is larger.    Green wins because of the dispsy doodle.  Cross sections are apparently critical to this measurement.

Velocity Diagnostic with updated cross section tables

Gerry Hale sent Anna new cross section tables.  These are larger by factor of 2 since the cross section was measured using neutrons from only one decay channel of 9Li, and also includes a resonance that was previously thought to be wrong.  Here are the same plots as in the post from Mar 27th. You can click on any plot to make it larger.


This plot is the cumulative ratio.

This plot is the ratio of the rates.

I have also plotted the signal and the control separately.  Here is the rate of signal (generation of Be9):

And here is the rate of the control:

Finally the cumulative measures for Be9 (signal) and Al (control), individually:

The take home message, most easily seen in the Be9 rate plot, is that with the new cross sections there is more Be9 being produced, especially around peak burn time.  Incidentally, there is also a new density/pressure model in the cold fuel (see the previous blog post for details) which is affecting these results through minor differences in the burn.  However, it is likely that the change in the cross section is what is driving the more efficient production of Be9.  

Better Shell Density Profile

Inspection of this plot from the Sanz et al. paper shows that our simplified model of a constant density in the cold fuel as a function of radius is not what simulations of the acceleration phase would predict.

Indeed, their theoretical model is a quadratic function plus a cut off.  For this reason, I put a new shell density model into the code, and ran it.   Here is the $\rho_0(x)$ that I used.

The new density model affected the pressure model in only one of the possible configurations, namely, before the shock and the outer boundary cross, in the region between the shock and the outer boundary.  Currently the pressure model is isobaric behind the shock.  The approximation I made was that in the cold un-shocked region, between the shock position and the outer boundary defining the edge of the ablator shell, the material is isothermal and the shape of the pressure profile is affected by the changing density.  Unfortunately, for all the models I ran, the shock has crossed the outer boundary at peak burn time, therefore the profile plots were all qualitatively similar.  The only difference was that the shock had propagated further by peak burn time in the new model.  The initial conditions look similar to those used in the Sanz paper, here are examples of the initial mass and pressure profiles:

But in order to see the effects of this on other quantities like RIF production, which depend on the pressure, I'll have to find a simulation for which the shock has not yet crossed the outer boundary by the time of peak burn.  Possibly even more of a fizzler than this.   I'll try and get back to this soon. 

For the moment, however, I am going to go back to the velocity diagnostic, and re-run the simulation turning on the radiation losses and the feedback from reactions.  We shall see if we are getting the blow-up behavior seen in the garnier paper.