Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

51the secondary neutron can appear with either of two energies. This region is called thedouble-valued region. If the condition (3.22) is not satisfied only the positive sign in theRHS of Equation (3.21) gives a physically realistic solution.The angles for the inelastic scattering are selected nearly exclusively from differentia!cross-sections presented in terms of the center-of-mass angles. The transformation into thelaboratory angle is given by:whereg + (cos-Q)v mCOS-O == — 7 = = —=r (3.Z.5)Vl + 2g(cosi c »,, mF g 21A(A + I)QFor the actual angle (or cosine of angle) selections, few data are available on the angulardistribution.In the center-of-mass system scattering is nearly isotropic for the low- and intermediateenergy (Sl MeV) neutrons, thus the differential cross-section is frequently represented bya low-order (N £ 6) Legendre polynomial expansion:tT cm(E 0,p. cm) = ^ 2 (2n + Df n(E)P n(JO (3.24)where a(E) is the average cross section of inelastic scattering and f n(E) is the expansioncoefficient to be recorded in the cross-section library (f~, = 1).Recipes for selecting scattering angles from tables of Legendre polynomials are giver,in Appendix 3D.The scattering may be highly anisotropic if only one excitation level is considered. NoIall energy levels can be treated individually. In the statistical gas model, the excitation levelstructure is replaced by a continuum. The energy of the secondary neutron is selected froma density function:P(E) = C 1EeXPt-C 2(E 11)E], O *, E < E 11,where C 1is a normalization constant, C 2(E 0) is an empirical function, depending on thetarget nucleus, and E mis the maximum energy of the secondary neutron.An actual selection scheme is given in Appendix 3D.4. Scattering of Thermal NeutronsAt the end of the slowing down process the energy of the neutron becomes comparableto the thermal motion energy of the atoms (or molecules) of the target material. Then theneutrons can either lose or gain energy during collisions with the target nuclei. In a completetreatment, thermal scattering is influenced by such factors as interference between the targetatoms or chemical binding. Generally, the phenomenon is treated with the free gas model'"where the two minor effects mentioned above are neglected.