| Dynamic measurements on fragile supercooled liquids strongly suggest that a development of clusters of modest sizes, i.e., of several molecules across, underlies the anomalous dynamics observed at low temperatures. However, examinations of the structure factor, S( k), obtained from scattering experiments have failed to reveal any corresponding growth in the correlation length, ξ, of the density autocorrelation function, H(R). In the belief that clusters do form in these liquids, we first show that the conventional mode of analysis based on the Fourier transform S(k) of H(R) is reliable only for detecting a diverging ξ, and then introduce a new method, called the “periscope method,” tailored for deducing ξ of only modest size.;The method, represented by the “periscoped function,” D( k; n), of H(R), is designed to determine for each k the length, lk, over which H(R) follows the functional form of sin( kR)/(kR), which we establish to be the only relevant pattern of behavior for H(R) of isotropic three-dimensional systems, and the longest lk is identified as ξ. The two-parameter function D(k; n) is an adaptation of S(k) in that it is the Fourier transform of the density autocorrelation function evaluated within a finite volume, vn, demarcated by a “periscope window” whose size is controlled by n, the “range index;” physically, D(k; n) corresponds to the scattering signal that would be obtained from vn situated inside a system. The form of the periscope windows is chosen so that the resulting expression for D(k; n) is an integral of the product H(R)·sin(kR)/( kR) with a finite, n-dependent upper limit on the R-range as opposed to ∞ in S(k). The behavior of H(R) is referred to the pattern sin(kR)/( kR) over increasing ranges of R, and ξ eventually determined, by analyzing the n-dependence of D(k ; n) at fixed k's. The formulation of D(k; n) as a physical observable and not as an ad hoc integral of H(R)·sin( kR)/(kR) requires that special conditions be placed on the periscope windows but ensures that the ξ thus extracted characterize cooperative effects that are physically meaningful. The periscope method is also applicable to correlations of variables other than density, but to date only for isotropic three-dimensional systems. |
