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

Abstract

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 $\ensuremath{\hbar}\mathbf{q}$ and of the energy transfer $\ensuremath{\hbar}\mathrm{\ensuremath{\omega}}$ the dynamical form factor $S(\mathbf{q}\phantom{\rule{0.25em}{0ex}}\mathrm{\ensuremath{\omega}})$ for neutron scattering is controlled by this effect, and is independent of the vibrational spectrum. In this regime, we show that the frequency width $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\omega}}}_{q}$ of $S(\mathbf{q}\phantom{\rule{0.25em}{0ex}}\mathrm{\ensuremath{\omega}})$ 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 $d\ensuremath{\sim}\sqrt{t}$ 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 $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\omega}}}_{q}$ for coherent scattering is predicted to increase.