TwoBody Scattering 

Elastic scattering (ENDF MT=2) and discrete inelastic neutron
scattering (with MT=5190) are both examples of twobody kinematics
and are treated together by GROUPR. The feed function required for
the grouptogroup matrix calculation may be written
where f(E,E',ω) is the probability of scattering from E to E' through a centerofmass cosine ω and P_{ls} is a Legendre polynomial for laboratory cosine μ. The laboratory cosine corresponding to ω is given by and the cosine ω is related to secondary energy E' by where A' is the ratio of the emitted particle mass to the incident particle mass (A'=1 for neutron scattering). In these equations, R is the effective mass ratio where A is the ratio of target mass to neutron mass, and Q is the energy level of the excited nucleus (Q=0 for elastic scattering). Integrating the defining equation for the twobody feed function over secondary energy gives where ω_{1} and ω_{2} are evaluated using the kinematics equations with E' equal to the upper and lower bounds of g', respectively. The scattering probability is given by
where the Legendre coefficients are either retrieved directly from the
ENDF/B File 4 or computed from File 4 tabulated angular distributions
using subroutines from the GROUPR module called The integration over ω is performed simultaneously for all Legendre components using Gaussian quadrature. The quadrature order is selected based on an estimate of the polynomial order of the integrand using the known order of the CM angular distribution plus a factor depending on atomic weight ratio AWR. More terms are needed for light targets. The resulting twobody feed function for higher Legendre orders is a strongly oscillatory function of incident energy in some energy ranges. Such functions are very difficult to integrate with adaptive techniques. Gaussian methods, on the other hand, are capable of integrating such oscillatory functions exactly if they are polynomials. Since a polynomial representation of the feed function is fairly accurate, a Gaussian quadrature scheme was chosen for GROUPR. The scheme used is also well adapted to performing many integrals in parallel. In GROUPR, all Legendre components and all final groups are accumulated simultaneously. NJOY takes pains to locate the "critical points" in incident energy so that the Gaussian integrations are performed over energy ranges that do not contain discontinuities. Here is an example of a multigroup elastic scattering matrix for carbon computed using GROUPR for a 30group structure: 
group constants at t=3.000E+02 deg k 16.6s for mf 6 and mt 2 elastic initl final group constants vs legendre order group group 0 1 2 3 1 1 8.087E+00 4.532E01 1.144E02 4.205E05 2 1 7.572E01 2.232E01 1.690E02 4.363E04 2 2 4.003E+00 4.900E01 2.363E02 4.544E04 3 2 7.525E01 2.219E01 1.676E02 4.303E04 3 3 3.995E+00 4.879E01 2.348E02 4.484E04 4 3 7.574E01 2.234E01 1.686E02 4.318E04 4 4 3.985E+00 4.891E01 2.357E02 4.498E04 5 4 7.539E01 2.223E01 1.678E02 4.294E04 5 5 3.986E+00 4.880E01 2.348E02 4.474E04 ... 28 26 8.308E02 4.928E02 8.439E03 1.234E02 28 27 2.545E01 7.532E02 6.513E03 2.423E02 28 28 5.256E01 4.740E01 3.849E01 2.805E01 29 26 4.329E03 4.075E03 3.606E03 2.988E03 29 27 1.217E01 4.973E02 1.258E02 1.592E02 29 28 1.841E01 1.053E01 2.710E02 4.737E04 29 29 5.124E01 4.633E01 3.788E01 2.799E01 30 27 2.721E02 2.127E02 1.255E02 5.134E03 30 28 6.865E02 1.719E02 1.846E02 1.032E02 30 29 1.834E01 1.016E01 1.937E02 7.491E03 30 30 6.055E01 5.428E01 4.372E01 3.182E01
Note that the "ingroup" cross section is much larger than the outscattering cross section, which is normally limited to a fairly small energy loss by kinematics. Also note that the scattering tends to be fairly isotropic at low energies (low group numbers here), but it becomes very forward peaked for inscatter in group 30. This is demonstrated by the slow decrease for higher Legendre orders (a delta function of forward scattering would have the same cross section for each l order). 
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23 January 2013  T2 Nuclear Information Service  ryxm@lanl.gov 