Atomic Dynamics in Simple Liquid: de Gennes Narrowing Revisited

The de Gennes narrowing phenomenon is frequently observed by neutron or $x$-ray scattering measurements of the dynamics of complex systems, such as liquids, proteins, colloids, and polymers. The characteristic slowing down of dynamics in the vicinity of the maximum of the total scattering intensity is commonly attributed to enhanced cooperativity. In this Letter, we present an alternative view on its origin through the examination of the time-dependent pair correlation function, the van Hove correlation function, for a model liquid in two, three, and four dimensions. We find that the relaxation time increases monotonically with distance and the dependence on distance varies with dimension. We propose a heuristic explanation of this dependence based on a simple geometrical model. This finding sheds new light on the interpretation of the de Gennes narrowing phenomenon and the $\alpha$-relaxation time.

Figure 1

The distinct part of the van Hove correlation function, $G_d(r,t)-1$, for 3D model liquid iron at 2500 K: (a) constant $t$ plot and (b) constant $r$ plot. In (b), the results are normalized using respective $t=0$ values and the normalized correlation functions are fit with $\exp [-{(t/\tau )}^{\beta }]$ shown as short dashed lines.

Figure 2

The relaxation time determined from $G_d(r,t)$ in 3D liquid iron at a series of distances at temperatures of 2000, 2500, and 3000 K. The short-dashed straight lines serve as guides to the eyes.

Figure 3

The $r$-dependent relaxation time $\tau (r)$ determined from the normalized distinct part of the van Hove correlation function of model liquid iron in two dimensions at 2500 K (red triangle), three dimensions at 2500 K (black circle), and four dimensions at 3500 K (blue square) beyond the first peak position. The data points are shown in the form of $\log \{[\tau (r)-{\tau }_0]/{\tau }_r\}$ versus $\log (r/r_1)$ to highlight $\chi$ from the expected power law dependence, where ${\tau }_0$ and ${\tau }_r$ are taken from Table 1. The short dashed lines serve as guides to the eye.

Figure 4

Temperature dependence of ${\tau }_r$ and ${\tau }_0$ from linear fitting to the $r$-dependent relaxation time of the normalized distinct part of the van Hove correlation function of model liquid iron in three dimensions. The results are shown in the Arrhenius plot in (a) and ratios in (b). ${\tau }_M$ is Maxwell relaxation time. Short dashed lines serve as guides to the eyes.

Figure 5

Illustration of the de Gennes narrowing phenomenon: The wave-vector dependent relaxation time $\tau (Q)$ determined from the collective part of the intermediate scattering function at 2500 K in 3D liquid iron shown in (a) clearly demonstrates a characteristic slowing down in the vicinity of the peak position of $S(Q)$ shown in (b).