Self-Shielded Thermal Reactor Problems

Thermal neutrons are usually important when light materials (for example, water or graphite) are present. Such problems are characterized by the presence of upscatter. In addition, thermal assemblies are often strongly heterogeneous because of high absorption cross sections. Thermal upscatter cross sections are available on some of the MATXS libraries, and the heterogeneity is usually treated by fine-group structures and/or heterogeneous self-shielding methods.

  TEST 9 -- ORNL-1 BENCHMARK
  0  1  0  1  1  1  0  3  0  0
  69  2  93  20  40  1  1  7  1  0
  GLOP
  GLOP  300.  1.  1  34.595/
  1  1  H1  .066228  H20/
  1  1  016  .033736  FREE/
  1  1  N14  1.869E-4  FREE/
  1  1  U234  5.28E-7  FREE/
  1  1  U235  4.8066E-5  FREE/
  1  1  U236  1.38E-7  FREE/
  1  1  U238  2.807E-6  FREE/
  CHI
  STOP

This benchmark is modeled as an homogeneous sphere of uranyl nitrate in water without a container (Ref. 17). It is important to use a MATXS library like one of the 69-group libraries that has both upscatter cross sections and self-shielding factors. Note that 20 upscatter groups are allowed, for a total table length of 4+69+20 = 93 groups. Using 40 thermal groups gives a thermal cutoff of 2.60 eV; that is, there is no upscatter to groups above this energy. The self-shielding treatment accounts for the mixture effects and the possibility of escape from the sphere (the mean chord length is 34.595 cm). Free-gas thermal scattering is used for all materials except hydrogen, which is bound in water.

Most ENDF/B-VI thermal evaluations give the scattering from an atom as bound in a molecule or crystal lattice, and the effects of the different atoms in a molecule must be combined in TRANSX to obtain the net thermal scattering cross sections. As an example, water in the sample problem above is a combination of H bound in H$_2$O and free-gas oxygen. However, some ENDF/B evaluations give the scattering for the entire molecule in the tabulation for the principal scatterer. In these cases, the thermal cross sections are renormalized to be used with the principal scatter. The TRANSX user must be careful not to include free-gas scattering for the secondary atoms. The following table lists the thermal evaluations available in ENDF/B-VI and indicates which treatment of secondary atoms was used. This table also indicates which evaluations require both inelastic and elastic parts. The elastic names (whether coherent or incoherent) are obtained by appending a dollar sign (\$) to the names given. The names in parentheses give the binding state for the principal scatterer. For example, C in graphite, or H in water. The MATXS names are based on this binding condition. Elastic scattering can be either coherent (coh) or incoherent (iel). In either case, the MATXS name is constructed by appending a dollar sign (\$). The last column gives the appropriate treatment to use for the secondary atom, if any. -------------------------------------------------------- Material MATXS Elastic Secondary Name Name Treatment Treatment -------------------------------------------------------- Be BE coh BeO BEO coh none C(graphite) GRAPH coh C(polyethylene) POLY iel free C C$_6$H$_6$ C6H6 none D(D$_2$O) D2O free O H(H$_2$O) H2O free O Zr(ZrH$_n$) ZRH iel H(ZrH$_n$) H(ZrH$_n$) ZRH iel Zr(ZrH$_n$) -------------------------------------------------------- As examples, the TRANSX material specifications for graphite, beryllium oxide, methane, and ZrH might take the forms

   1  1  CNAT  1.  GRAPH  GRAPH$/
   2  2  BE9   1.  BEO  BEO$/
   2  2  O16   1.  NONE/
   3  3  H1    6.  C6H6/
   3  3  CNAT  6.  NONE/
   4  4  H1    1.  ZRH  ZRH$/
   4  4  ZR    1.  ZRH  ZRH$/

For self-shielding problems like this one, it is necessary to define a mixture in order to calculate the $\sigma_0$ values. However, the user often needs the ``constituent'' self-shielded microscopic cross sections for such applications as concentration searches or depletion calculations. These constituents can be requested by defining additional output mixes and by using mix specifications containing the key words ``CC'' (for cell constituent) or ``RC'' (for region consituent). Sample problem 9 would become

  .
  .
  .
  69  2  93  20  40  8  1  14  1  0
  GLOP  H1  016  N14  U234
  U235  U236  U238
  .
  .
  .
  1  1  U238  2.807E-6  FREE/
  2  1  H1  1.  CC/
  3  1  016  1.  CC/
  .
  .
  .
  8  1  U238  1.  CC/
  STOP

The entry ``CC'' tells the code to use the microscopic cross section that was added into the mix by the normal mix specification for this material and region. (The ``CC'' and ``RC'' commands are equivalent for a homogeneous case like this one.) The transport code now has the choice of using the macroscopic cross section or remixing using the micros. If the densities in a concentration search were to vary enough to change the self-shielding, it might be necessary to repeat the TRANSX calculation for the new mix.

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