The various methods to measure the density of states by neutron inelastic scattering are discussed. Purely incoherent scattering data on silica aerogels have been obtained by taking the difference spectra between protonated and identical deuterated samples. The results so far cover the particles modes and the upper part of the fracton regime. Complemented with Brillouin scattering information, these measurements provide an overall picture of the absolute density of states, from which low temperature thermal properties can be calculated, in remarkable agreement with experiments. 1 INTRODUCTION The theory of vibrational excitations in fractal structures predicts the existence of "fractons" 111. Those vibrational modes are localized. Also, in contrast to long-wavelength acoustic phonons, which obey a linear dispersion law, o = v q , where w is the frequency, q is the wavevector, -and v is the constant sound velocity, fractons are expected to exhibit dispersion according to o C C ~ ~ I ~ 11.21. Here D is the fractal (Hausdorff) dimension, and k is a characteristic length for the fracton. Finally, the density of states (DOS) of fractons Z(w) was predicted to be proportional to wa -1 /I/. The dimensions obey the inequality d 2 D > z, with d the Euclidean dimension. As an example, for 3-d percolation clusters and scalar waves the values are D = 2.5 and a = 413. The above predictions are at strong variance with the well-known behavior of acoustic phonons, a@ the differences should be observable experimentally. Silica aerogels are ideally suited for the experimental study of fracton dynamics. They are porous solid materials. Large pieces can be prepated, so that their elastic properties can be measured by static as well as by ultrasonic techniques. The optical transparency of aerogels in a broad range of densities and preparation conditions allows the study of their vibrational dynamics by Brillouin and Raman light scattering. (I) Supported by Neutron Scattering Project IIKW, Natuurkunde, University of Antwerp, 8-2610 Wilrijk, Belgium (2) Laboratoire mixte CNRS-Commissariat a I'Energie Atomique. (3) Laboratory associated to the Centre National de la Recherche Scientifique (CNRS). No 1129. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989424 Structural studies /3,4/ have demonstrated that silica aerogels are made of elementary particles, of typical size a, with more or less defined surfaces. Those particles assemble to form fractal networks, up to an average cluster size 5. At length scales above (, the material is an homogeneous assembly of such clusters. Corresponding to these different length scales, three distinct vibrational regimes are expected, namely, with increasing frequency : phonons, fractons, and particle modes. A crossover from phonons to fractons at oco,12n -1 GHz has been demonstrated by Brillouin scattering experiments 151. This technique also allowed the determination of the dispersion curve for fractons 16,7/, and gave indication for the occurrence of localization at the phonon-fracton crossover. Evidence of an extended fracton domain, ranging approximately from 40 peV to 1 meV, was obtained from very-low frequency Raman scattering IS/. It is obviously of interest to measure directly the DOS of these excitations. 2 EXPERIMENTAL APPROACHES There are several ways to measure a vibrational DOS by inelastic neutron scattering. One basic idea is that the DOS is reflected in the incoherently scattered neutrons. Taking only the one phonon terms in a multiphonon expansion, the expression for the double differential incoherent cross section of a Bravais lattice