Integral Thinning

The criteria used for deciding when to stop halving intervals during resonance reconstruction are complex. Modern resonance evaluations can be very detailed, yielding hundreds of thousands of energy points when fully expanded. These very large tabulations are expensive to produce, to store, and to Doppler broaden or average. However, almost all of the uses of RECONR output result in integrating over the cross sections; this is true for both Doppler broadening and multigroup averaging, and it is also true for all ultimate applications, even continuous-energy Monte Carlo transport calculations. Therefore, if the reconstructed resonance cross sections contain features that cause negligible effect on these integrals, these features can be removed without affecting the ultimate results significantly. This is why RECONR includes an option for carrying out "integral thinning" while it is generating its resonance cross sections.

The main user input to RECONR is the reconstruction tolerance, ERR. It is given as a fractional quantity, e.g., 0.001, to represent 0.1%. The default behavior is to set the auxiliary thinning parameters to standard values ERRMAX=20*ERR and ERRINT=ERR/10000. RECONR will then always halve intervals until the fractional difference between the computed cross sections and the interpolates are less than ERRMAX for all reactions. Once this has been achieved for an interval, it will check to see how much the panel will change the resonance intergal. If the change is smaller than ERRINT, the subdivision process will be terminated. When the subdivision terminates, it is possible to make a rough estimate of the effect of the termination on the resonance integral. As an example of this process, here is an example of the output provided by RECONR when processing U-238 from ENDF/B-VI:

         number of user and resonance nodes = 3834
         points in initial unionized grid = 4230
         points added by linearization = 215

                       estimated maximum error due to
                       resonance integral check (errmax,errint)
                       and significant figure truncation

            upper      elastic    percent error    capture    percent error
            energy     integral  res-int sig-fig   integral  res-int sig-fig
          1.00E-04     2.15E+01   0.000   0.000    1.86E+02   0.000   0.000
          1.00E-03     2.15E+01   0.000   0.000    5.88E+01   0.000   0.000
          1.00E-02     2.15E+01   0.000   0.000    1.86E+01   0.000   0.000
          1.00E-01     2.15E+01   0.000   0.000    5.91E+00   0.000   0.000
          1.00E+00     2.14E+01   0.000   0.000    1.96E+00   0.000   0.000
          2.00E+00     6.32E+00   0.000   0.000    3.32E-01   0.000   0.000
          5.00E+00     7.90E+00   0.000   0.000    6.29E-01   0.000   0.000
          1.00E+01     1.47E+01   0.000   0.000    1.30E+02   0.000   0.000
          2.00E+01     5.27E+00   0.000   0.000    6.59E-01   0.000   0.000
          5.00E+01     1.02E+02   0.000   0.000    1.09E+02   0.000   0.000
          1.00E+02     1.89E+01   0.000   0.000    1.26E+01   0.000   0.000
          2.00E+02     5.05E+01   0.000   0.000    1.34E+01   0.000   0.000
          5.00E+02     1.93E+01   0.000   0.000    5.14E+00   0.004   0.000
          1.00E+03     1.50E+01   0.001   0.000    2.45E+00   0.014   0.000
          2.00E+03     1.47E+01   0.002   0.000    1.33E+00   0.042   0.000
          5.00E+03     1.77E+01   0.006   0.000    1.09E+00   0.113   0.000

         points added by resonance reconstruction =140468
         points affected by resonance integral check =103592
         points affected by significant figure reduction =     0
         final number of resonance points =144493
         number of points in final unionized grid =144913

Note that the output shows the elastic and capture integrals over roughly logarithmic energy bands running through the resolved range (1e-5 eV to 5 keV). The "res-int" column shows an estimate of the error made by integral thinning, e.g., a maximum of 0.1% for U-238 capture at high energies. About 100000 points were removed from the final energy grid by the integral check. This is a major savings for a fairly small impact on the least known part of the U-238 cross section.

If this increased error is unacceptable, it can be reduced in two ways: either set ERRMAX closer to ERR, or make ERRINT smaller than ERR/10000.


23 January 2013 T-2 Nuclear Information Service