Thermal Quantities and Resonance Integrals

In the thermal reactor business, quite a bit of insight into the behavior of a particular evaluation can be obtained by examining certain thermal cross sections and resonance integrals. In a reactor, most fission neutrons are born from fission reactions at thermal energies. The quantities of interest there are the fission cross section itself, the competing capture cross section, and the number of fission neutrons produced per reaction, nu-bar. The deviation of the shape of the cross sections from a 1/v shape is also of interest. The fission neutrons are produced at energies up around 1 MeV, and they have to slow down to thermal energies before producing more neutrons to continue the chain reaction. The quantities of interest here are the fission and capture resonance integrals, which help determine the probability that a neutron will disappear before reaching thermal energies. For these reasons, the following simple parameters have come into wide use:
  • fission cross section at thermal
  • fission nu-bar at thermal
  • capture cross section at thermal
  • thermal Maxwellian fission integral
  • thermal Maxwellian capture integral
  • fission g-factor (deviation from 1/v)
  • capture g-factor
  • thermal alpha integral
  • thermal eta integral
  • thermal K1 integral
  • fission resonance integral
  • capture resonance integral

The conventional thermal energy is 0.0253 eV. If the cross section has a 1/v shape, its integral weighted against a Maxwellian spectrum for 0.0253 eV is given by

Maxwellian Average

where the factor G provides a measure of the deviation from a 1/v shape. It will be unity for a pure 1/v. Similarly, the alpha, eta, and K1 integrals are Maxwellian averages of the following quantities:

Thermal Alpha
Thermal Eta
Thermal K1

The quantity K1 is known to be an especially good indicator for keff. The following is an example for U-235 from Release 3 of ENDF/B-VI:

         thermal quantities at 300.0 K = 0.0259 eV
              fission xsec at 0.0253:  5.8472E+02
             fission nubar at 0.0253:  2.4338E+00
              capture xsec at 0.0253:  9.8575E+01
                 fission xsec at tev:  5.7830E+02
                fission nubar at tev:  2.4338E+00
                 capture xsec at tev:  9.7349E+01
            thermal capture integral:  8.5429E+01
            thermal capture g-factor:  9.9022E-01
          capture resonance integral:  1.4337E+02
            thermal fission integral:  4.9323E+02
            thermal fission g-factor:  9.6239E-01
              thermal alpha integral:  1.6117E-01
                thermal eta integral:  2.0059E+00
                 thermal k1 integral:  6.2301E+02
          fission resonance integral:  2.7750E+02

A good reference for exerimental values of these quantities is S. F. Mughabghab, Neutron Cross Sections, Volume I, Neutron Resonance Parameters and Thermal Cross Sections, Part B, Z=61-100, Academic Press, 1984. Its values for the quantities in the table above are
  • thermal capture cross section = 98.3 +/- 0.8 b
  • thermal fission cross section = 582.6 +/- 1.1 b
  • thermal nu-bar = 2.4251 +/- .0034
  • thermal alpha = 0.1687 +/- .0015
  • tehrmal eta = 2.0751 +/- .0033
  • fission g factor = 0.9761 +/- .0012
  • capture resonance integral = 144 +/- 6 b
  • fission resonance integral = 275 +/- 5 b
which compare fairly well with the numbers from BROADR.


23 January 2013 T-2 Nuclear Information Service