Quasi-elastic scattering of neutrons by dilute polymer solutions: I. Free-draining limit

In the so-called "free-draining limit", successive units of a long molecule equalise their average orientation by a diffusion process along the chain. For low values of the momentum transfer and of the energy transfer the dynamical form factor S (q0. 25em{0ex}) for neutron scattering is controlled by this effect, and is independent of the vibrational spectrum. In this regime, we show that the frequency width ₐ of S (q0. 25em{0ex}) is small and proportional to q^4. The unusual q^4 law is related to the fact that, in a time t, a signal travels a distance d but the corresponding distance in space is only of order d^1/2 for a coiled polymer. On the other hand, if the chain is stretched this argument breaks down and the width ₐ for coherent scattering is predicted to increase.