Dynamic structure factor for classical motion in one-dimensional potentials

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Abstract
The dynamical correlation functions are considered for classical particles moving in general one-dimensional unbounded potentials. In particular, the dynamic structure factor S(k,ω), which predicts the intensity measured in a neutron-scattering experiment, is calculated in detail. According to the magnitude of the energy transfer, the analysis of S(k,ω) is usually divided into three regions: inelastic, quasielastic, and elastic. New results are obtained for each region: (a) The presence of a band structure in the inelastic zone, which often exhibits a rich intraband oscillatory behavior. The origin of these oscillations is discussed in detail. (b) The impossibility of having a quasielastic peak if the motion of the scatterer is Hamiltonian, with the important exception of particles moving in potentials that contain horizontal segments. (c) From a comparison between the elastic intensity and the Debye-Waller factor, which gives the elastically scattered intensity when weak frictional forces are added, it is argued that the difference between both functions gives the total quasielastic intensity in the presence of weak friction. The effect of a directional average and of the average over a parameter distribution are also investigated. The general results are illustrated with representative examples.