Inelastic neutron scattering from damped collective vibrations of macromolecules

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The present article describes the theory of neutron scattering from harmonically vibrating molecules in the framework of classical Langevin and Smoluchowski dynamics. For both levels of description, the input quantities are a positive definite matrix of force constants and a positive definite friction matrix. Starting from the Fokker–Planck equation of the Ornstein–Uhlenbeck process, the intermediate scattering functions for coherent and incoherent scattering as well as related static correlation functions are derived. It is demonstrated that the Langevin description on a coarse-grained time scale is identical with the Smoluchowski description if friction dominates. Inelastic neutron spectra are discussed in terms of the coherent and incoherent dynamic structure factors and the vibrational density of states. It is shown that incorporating friction on the atomic level is not equivalent to broadening the lines of the corresponding normal mode spectrum. The theory is applicable to low frequency and large amplitude motions of macromolecules as proteins and DNA. Vibrations at higher frequencies can be treated as long as they can be described in the classical approximation.