Diffusive Motions in Water and Cold Neutron Scattering

TLDR

The authors derive a formula for the quasi‑elastic peak broadening that reduces to simple diffusion in the long‑time limit and approaches a constant in the short‑time limit, and they compute inelastic scattering using gas and Debye models of water to compare with experiment. The calculated cold‑neutron scattering cross section shows a non‑Lorentzian quasi‑elastic peak, with broadening consistent with a τ₀≈4×10⁻¹² s, and the results support a quasi‑crystalline model of water.

Abstract

Using a model of liquid water in which a molecule, in its equilibrium position, performs an oscillatory motion for a mean time ${\ensuremath{\tau}}_{0}$, and then diffuses by continuous motion for a mean time ${\ensuremath{\tau}}_{1}$, and repeats this sort of motion, the differential scattering cross section for cold neutrons has been calculated. It is found that the shape of the "quasi-elastic" scattering is, in general, not Lorentzian. The formula for the broadening of the quasi-elastic peak assumes a simple form in two limiting cases: In case (i) ${\ensuremath{\tau}}_{1}\ensuremath{\gg}{\ensuremath{\tau}}_{0}$, it reduces to the formula derived on the simple diffusion theory; and in case (ii) ${\ensuremath{\tau}}_{1}\ensuremath{\ll}{\ensuremath{\tau}}_{0}$, the broadening is the same as in case (i) if ${\ensuremath{\kappa}}^{2}D{\ensuremath{\tau}}_{0}\ensuremath{\ll}1$, and it approaches the asymptotic value $\frac{2\ensuremath{\hbar}}{{\ensuremath{\tau}}_{0}}$, if ${\ensuremath{\kappa}}^{2}D{\ensuremath{\tau}}_{0}\ensuremath{\gg}1$, where $\ensuremath{\hbar}\ensuremath{\kappa}$ is the momentum transferred to the system and $D$ is the diffusion coefficient of water. The observed value of the broadening can be explained for a value of ${\ensuremath{\tau}}_{0}=4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ sec. Besides, the theoretical quasi-elastic scattering in case (ii) has certain interesting features which are in general agreement with experiment. In part II of this paper, inelastic scattering (hindered translations only) of cold neutrons has been calculated using two different models of water: (a) a gas model and (b) a Debye model; and the results have been compared with experiment.The general shape of both the quasi-elastic and inelastic scattering of cold neutrons and the magnitude of the diffusive broadening seem to support a quasi-crystalline model of water.