Nuclear Scatter of Neutron Spin States

Abstract This chapter derives the partial differential cross sections for neutron scatter from a nucleus, which accounts for the neutron spin and the nuclear spin. First covered are the preliminary background topics of angular momentum vectors, spin vectors, and vector operators, the Heisenberg uncertainty principle and commutation of operators, the neutron spin operator, and the neutron spin–lowering and –raising operators. First, the partial differential cross section for nuclear scatter of the neutron spin-up and spin-down states is dervied. Next derived for polarized neutron scatter is the partial differential cross section, which includes both the neutron spin state and nuclear spin state, via the combined neutron spin operator and nuclear spin operators. Covered next are the neutron nuclear scatter length, which accounts for the neutron spin states. Thermal averaging is then taken into account, and the total partial differential cross section for neutron spin state scatter is derived, as well as the neutron spin state scatter lengths for an ensemble of nuclear spins and isotopes. Finally, the partial, differential, and total cross section for neutron coherent and incoherent scatter are derived from an ensemble of atoms of varying nuclear spins and isotopes, which accounts for neutron spin states.