Translational/Diffusive Behaviour

The neutron scattering from many liquids (e.g., water and liquid methane) can be represented using a solid-type scattering law \(S_s(\alpha,\beta)\) and convolving it with a translational scattering law \(S_t(\alpha,\beta)\). There are two options for the translational term: the effective width model, and the free gas model.

Effective Width Model

Egelstaff and Schofield have proposed an especially simple model for diffusion called the “effective width model”, which has analytical forms for both \(S_t(\alpha,\beta)\) and the associated frequency distribution function \(\rho_t(\beta)\).

\[S_t(\alpha,\beta)=\frac{2c\omega_t\alpha}{\pi}~\mathrm{exp}\Big[2c^2\omega_t\alpha-\beta/2\Big]~\sqrt{\frac{c^2+0.25}{\beta^2+4c^2\omega_t^2\alpha^2}}~K_1\Big[\sqrt{c^2+0.25}\sqrt{\beta^2+4c^2\omega_t^2\alpha^2}\Big]\]

\[\rho_t(\beta)=\frac{4c\omega_t}{\pi\beta}\sqrt{c^2+0.25}~\sinh(\beta/2)~K_1\left[\beta\sqrt{c^2+0.25}\right]\]

where \(K_1(x)\) is a modified Bessel function of the second kind, \(\omega_t\) is the translational weight, and \(c\) is the diffusion constant.

Free Gas Model

Alternatively, the translational term can be represented using a free gas model. While a free gas model is obviously most applicable for cloud of non-interacting gas atoms, it has been used to represent the translation component for liquid moderators.

\[S_t(\alpha,\beta)=\frac{1}{\sqrt{4\pi\omega_t\alpha}}~\mathrm{exp}\left[\frac{-(\omega_t\alpha+\beta)^2}{4\omega_t\alpha}\right]\]