Unresolved resonance probability table sampling

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Jaakko Leppänen
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Unresolved resonance probability table sampling

Post by Jaakko Leppänen » Sat Apr 03, 2010 8:28 pm

The main challenge in validating the methods used for unresolved resonance probability table sampling is that the differences are generally quite small. It would be safer to always use this data, but unfortunately there is some significant penalty in the overall running time when the number of nuclides and materials becomes large. This is because Serpent relies heavily on several pre-calculated summation cross sections, which cannot be used as effectively when values are sampled during tracking.

Probability table sampling is still (in version 1.1.10) currently switched off by default, and it can be activated by setting the "ures" parameter, described in Section 5.5 of the User's Manual. The format of the input card is:

set ures <use> [<mode>] [<dilu>] [<iso 1> <iso 2> ...]

The first parameter ("<use>") gets value 1 or 0 to toggle the method on and off. The optional "<mode>" parameter determines how the sampled values are used with pre-calculated cross sections. There are three different options, as described in the manual. The mode should have no impact in the results, only in the overall calculation time. The only exception are cross sections that are used as detector response functions (Sec. 7.1.1 in the Manual), for which only mode 1 produces the correct result (infinite-dilute data is used with the other two modes). This is the mode used by default, although mode 2 generally runs faster in most geometries.

When switched on, probability table sampling is used by default for all nuclides for which the data exits. This means that in a typical burnup calculation the method is used for some 100 to 250 nuclides, most of which are present at very low concentrations. Since the difference to infinite-dilute cross sections comes from self-shielding effects, probability table sampling could be disabled for most of the nuclides without impacting the results. On the other hand, if the method is disabled for a nuclide with high concentration, the differences in transmutation cross sections would most likely cause the error to build up as the fuel is burnt.

The remaining parameters in the ures card can be used for enabling or disabling the method for single nuclides. The first option is to use the infnite dilution cut-off ("<dilu>"). This parameter defines a limit for atomic fractions, and probability table sampling is used only for nuclides with concentration above this limit. The criterion is checked before each burnup step and the treatment is swiched on once the concentration of a nuclide reaches the limit. The second alternative is to give an explicit list of nuclides for which the sampling is used. If the method is switched on, the treament is used only for the listed nuclides. If the method is switched off, it is used for all but the listed. The list remains the same throughout burnup calculation.

So the task is to find suitable criteria by which unnecessary probability table sampling can be dropped in order to speed up the calculation without affecting the results. Before getting into the actual criteria, it is necessary to find a test case where the effects of the method can be clearly seen. This has turned out to be more difficult than I expected.

The first test case that I run was a PWR pin-cell model with irradiated fuel. The actinide composition consists of 96.4% U-238, 1.4% U-235, 0.6% Pu-239, 0.2% Pu-240 and 0.1% Pu-241. Other uranium and plutonium isotopes, Am-241 and fission products are present at lower concentrations. To see the effects of probability table sampling, I calculated flux-weighted fission and capture cross sections as function of energy for the most important actinide isotopes. The differences between calculations run with and without probability table sampling should become visible when these cross sections are compared. If the concentration of a nuclide is high enough to have impact in the flux spectrum, the cross sections should be lowered in the unresolved region due to self-shielding. In the opposite case the sampled values simply reduce to average, infinite-dilute cross sections and the result should match the calculation without probability tables.

The figure below shows such comparison for U-238 capture. The unresolved resonance region for this nuclide is located between the two vertical lines, and the impact of probability table sampling is clearly visible. However, this was the only cross section where the difference was seen, and since the majorty of reactions takes place below the unresolved resonance region, the overall impact in integral parameters (k-eff, two-group constants, one-group transmutation cross sections) was completely negligible.

Image
Figure 1. Relative differences in flux-weighted capture cross sections of U-238 as function of energy (MeV).

So in my opinion, it is pretty safe to say that omitting the probability table sampling for other nuclides than U-238 for the sake of speed does not compromise the results of typical LWR transport or burnup calculations. Even for U-238 the effect is very small, although you may see statistically significant differences in homogenized group constants calculated using other than the conventional two-group structure.

I will try to find a fast reactor case where the importance of the unresolved resonance region is more pronouced to see how the cut-offs and other ures options actually work in practice. There is also another issue, related to the "spectrum method" used for calculating transmutation cross sections (see item 16 in the topic about known issues). I will get to that later. Also note that these are all Serpent vs. Serpent calculations. I am pretty confident that the methodology works in principle, and I have validated it by comparing to MCNP results. I'll run some more comprehensive validation calculations later, once I find a good test case.
- Jaakko

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Re: Unresolved resonance probability table sampling

Post by Jaakko Leppänen » Thu Apr 08, 2010 12:54 am

I got some good fast reactor input files from KTH (thanks!) and ran similar comparison as discussed in the previous message for a PWR fuel. The first case is a lead-cooled fast reactor assembly and the second case a pin-cell model of sodium-cooled fast reactor fuel. Figures 1 and 2 show the differences for capture and fission in the SFR case.

Image
Figure 1: Relative differences (in %) in flux-weighted capture cross sections as function of energy (MeV). Sodium-cooled fast reactor fuel.

Image
Figure 2: Relative differences (in %) in flux-weighted fission cross sections as function of energy (MeV). Sodium-cooled fast reactor fuel.

The total difference in the one-group capture cross section of U-238 is almost 1%, and there are several reactions for which the difference exceeds 0.2%. It is noticeable that the effect is clear even for nuclides with relatively low concentration. The atomic fraction of Pu-238 is only 0.7%, and the error in one-group cross section is 0.6%. Since self-shielding is obviously not dependent only on the concentration, but also on the magnitude of the cross sections, a cut-off criterion based on atomic fraction alone may not be the best solution. I also noticed that the effect is strongly dependent on temperature. The example case was run using cross sections generated at 300K. The differences become smaller as the temperature increases.

I started running burnup calculations using the two fast reactor models and I'll get back to the topic once the calculations are completed.
- Jaakko

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Re: Unresolved resonance probability table sampling

Post by Jaakko Leppänen » Thu Apr 22, 2010 3:45 pm

I have completed the burnup calculations. It took a while because I decided to fix the methodological flaw related to the spectrum method (xscalc 2) when used with probability table sampling (see item 16 in the topic about known issues).

The SFR and LFR fuels were burnt to 100 MWd/kgU burnup at a constant power. Infinite multiplication factor as function of burnup for the LFR case is plotted below.

Image
Figure 1: Infinite multiplication factor as function of fuel burnup in the LFR case

The differences are clearly seen when the results are compared to a reference case with probability sampling on and cross sections calculated by directly tallying the reaction rates (xscalc 1):

Image
Figure 2: Relative differences (in pcm) in the infinite multiplication factor as function of fuel burnup in the LFR case

The error from the use of infinite-dilute cross sections is several hundred pcm, which is much higher compared to thermal reactors. The old implementation of the spectrum method doesn't give much better results either, since the errors in the microscopic transmutation cross sections are reflected in the isotopic composition as the fuel is burned. The problem was fixed by calculating the cross sections in two parts: 1) direct tallying in the unresolved region and 2) spectrum method outside the unresolved region. The corrected methodology will be available in update 1.1.11.

Similar results as for k-eff are observed in material compositions. The isotope most affected by probability table sampling Pu-239:

Image
Figure 3: Concentration of Pu-239 as function of fuel burnup in the LFR case

The errors in the infinite-dilute calculation and the old spectrum method are about 0.3-0.4%, while the corrected spectrum mehtod curve practically overlaps the reference case.

The next stage is to look for ways to speed up the calculation by leaving out the probability table treatment for isotopes at insignificant concentrations. It may be a while before I get to run the tests, as I am currently in the process of moving back to Finland and the practical arrangements are taking most of my time.
- Jaakko

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