Physica 136B (1986) 285-290 North-Holland, Amsterdam Chapter 10 Disordered systems NEUTRON SCATTERING FROM FRACTALS J.K. KJEMS 1, T. FRELTOFT l, D. RICHTER 2 and S.K. SINHA 3 tRis¢ National Laboratory, DK-4000, Roskilde, Denmark 2Kernforschungsanlage Jiilich, D-5170 Jiilieh, Fed. Rep. Germany 3Exxon Research and Engineering Company, Annandale, N.J., USA Invited paper The scattering formalism for fractal structures is presented. Volume fractals are exemplified by silica particle clusters formed either from colloidal suspensions or by flame hydrolysis. The determination of the fractional dimensionality through scattering experiments is reviewed, and recent small-angle neutron scattering studies of the variaion with aggregation rate are presented. These results allow a very detailed comparison to be made with the theoretical scattering curves. Preliminary incoherent inelastic data on the low-frequency dynamics of hydroxylated silica particle aggregates show a clear deviation from a Debye spectrum and give the first experimental estimates for the spectral dimensionality. 1. Introduction Condensed matter systems are most often con- sidered to have a uniform density perturbed by fluctuations on a characteristic length scale, i.e. the atomic scale for crystals, liquids and amorph- ous materials, and the correlation length for systems near a critical point. However, as pointed out by Mandelbrot [1], there are numerous exam- ples of naturally occurring structures that appear self-similar after magnification (also called dil- ution symmetry or scale invariance), and where one cannot define a unique characteristic length scale. Instead, these structures can be character- ized by an exponent called the fractal dimension, df, which is similar to the more rigorously defined Hausdorff-Besikowitch dimension, and which describes the decay of the density-density correl- ation function with distance. There are in general two different classes of fractal structures, namely (i) aggregated clusters of small prticles which represent volume fractals; and (ii) systems with irregular but sharply defined interfaces and grain boundaries which may represent surface fractals. Here we shall concern ourselves only with bulk fractals. It is not surprising that scattering expriments with neutrons, X-rays, and light have proven valuable in the efforts to characterize fractal structures because, as discussed in the following section, there is a direct relation between the exponent for the scattering law and the exponent, de, for the ideal systems. In real systems the fractal character extends only over a limited range of length scales, and the observed scattering profiles will be affected by these bounds. An explicit formalism for the scattering from volume fractals that takes into account both the upper and lower cut-offs has recently been worked out [2], and the results are discussed below. The expected scattering law for surface fractals can be found elsewhere [3, 4]. So far, most of the experimental studies have focused on the determination of the fractal di- mension and a range of values, 1.8 < df <2.6, have been found for the different particle aggre- gate clusters [2, 5, 6]. This range coincides with the values found in numerical simulations of the clustering process based on the diffusion-limited aggregation model (DLA) [7, 8] and on more complex cluster-cluster aggregation models [9]; the results indicate that df could be the signature of a particular aggregation mechanism. In the third section some recent small-angle neutron scattering (SANS) experiments are de- scribed which aim at a more detailed understand- 0378-4363/86/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)