A comprehensive scheme for the analysis and use of thermal neutron scattering data is described here. Experimental work has been carried out at Chalk River with a 4-rotor high speed chopper system operated in conjunction with a multicounter multichannel neutron time-of-flight system. This apparatus allows many measurements to be made of the probability of an incident neutron of given energy to be scattered through a given angle with a given energy change. The results of such experiments are reduced to a “Scattering Law,” which is a function S of two variables (α and β) representing the momentum and energy transferred in the scattering process. Scattering Laws have been measured for many moderators at many temperatures. The scattering law can be divided into interference and self-terms; for the application to neutron spectrum calculations it is sufficient to consider the interference term as a small correction to the self-term, [in practice the interference term is calculated and subtracted from the measured S(α, β)]. The self-term is the double Fourier transformation of the self-correlation function of Van Hove, which represent the motion of an individual atom in the system. This function can be represented as a gaussian distribution of the atomic position plus correction terms. One objective of the analysis of the measured scattering law is to show that these correction terms contribute little to the scattering. The width of this gaussian distribution is a function of time and its double differential with respect to time is the velocity correlation function for an atom in the system. Thus the velocity correlation function is the function which makes the major contribution to the scattering cross sections. It can be shown that the fourier transform [p(β)] of the velocity correlation may be derived from measurements of the scattering law. Once p(β) has been obtained the self-part of the scattering law for all values of α and β may be calculated. These calculations serve to smooth, interpolate, and extrapolate the measurements of the scattering law and ensure that the scattering law conservation rules are satisfied, and therefore form a reliable basis for calculating cross sections to be used in any neutron spectrum calculation. In estimating the accuracy to which measurements, analysis, and calculations of the scattering law need be made, it is necessary to evaluate a “sensitivity function” for each neutron spectrum problem, and thus for each scattering law a variety of sensitivity functions need to be evaluated. Two simple examples are discussed which show, separately, the effect of absorption and temperature gradients.