Derivation of Kinetic Equations for Slow-Neutron Scattering

It is shown that for moderately dense classical gases Van Hove's correlation functions $G(r, t)$ and Gₛ(r, t) can be calculated from kinetic equations. The kinetic equations are derived from the cluster expansion of a one-particle distribution function which describes the propagation of a density-momentum impulse. For times short compared with the time between collisions, the density series can be used directly. But for longer times a different treatment is necessary because of the increasing importance of multiple collisions. Considering only the dominant terms, the series is shown to be the iterated solution of an integral equation. In lowest order, this equation has the form of a linearized Boltzmann equation in which the collision kernel is nonlocal because of the finite extent of the particles, and is non-Markoffian if the duration of a collision is finite. The present approach permits a systematic treatment of coherent as well as incoherent scattering. In particular, we demonstrate an earlier assertion that the appropriate equations for studying the two kinds of scattering are, respectively, the linearized Boltzmann equation and the neutron transport equation. The effects of statistical correlation are introduced approximately in the kinetic equations in lowest order.