Simple Binary Collision Model for Van Hove'sGs(r, t)

It is noted that the Van Hove self-correlation function $G_s(r, t)$ for a dilute fluid is determined by a linearized Boltzmann equation identical to that occuring in the theory of neutron diffusion. A simple model for the collision integral, in which a molecule emerges from a collision with a Maxwellian distribution, allows some interesting analytic results to be obtained. The double Fourier transform $S_s(\kappa , \omega )$, of $G_s(r, t)$ is expressible in terms of the probability integral for complex arguments. Since $S_s(\kappa , \omega )$ is directly proportional to the Mössbauer line shape, the transition of the line shape from Doppler broadening to simple diffusion broadening is explicitly exhibited as a function of momentum transfer. The spatial moments of $G_s(r, t)$ are calculated as a function of time. The model used gives the same mean-square displacement as does the Langevin equation. The resulting $G_s(r, t)$ is not, however, Gaussian as is shown by the mean fourth power of the displacement which is 25% greater at intermediate times than would be predicted by the Langevin equation. The non-Gaussian effects lead to an appreciable narrowing of $S_s(\kappa , \omega )$ for intermediate values of $\kappa$.