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 $\hslash \mathbf{q}$ and of the energy transfer $\hslash \mathrm{\omega }$ the dynamical form factor $S(\mathbf{q}\mathrm{\omega })$ for neutron scattering is controlled by this effect, and is independent of the vibrational spectrum. In this regime, we show that the frequency width $\mathrm{\Delta }{\mathrm{\omega }}_q$ of $S(\mathbf{q}\mathrm{\omega })$ 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 $d\sim \sqrt{t}$ 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 $\mathrm{\Delta }{\mathrm{\omega }}_q$ for coherent scattering is predicted to increase.