Phonon-Fracton Crossover on Fractal Lattices
Recently there has developed a growing interest in the dynamical properties of structures which have a fractal geometryl. Alexander and Orbach2 were the first to point out that three dimensionalities are required to describe excitations on fractals: the embedding (Euclidean) dimension d; the Hausdorf (fractal) dimension D; and the fracton (spectral) dimension \( \tilde d\). The fractal dimension D describes how the mass of the geometrical object depends on length scale whereas the spectral dimension \( \tilde d\) characterizes the low frequency behaviour of the density of states. It is well known3 that percolating networks near the critical threshold p have a self-similar(fractal) geometry on small length scales and a homogeneous(Euclidean) structure on larger scales. The length scale which separates these two regions is the percolation correlation length εp which diverges as pc is approached from above. Orbach et al4,5,6 have recently used scaling arguments and effective medium approximation (EMA) calculations to predict that there is a sharp increase in the density of vibrational states at a crossover frequency ωc which corresponds to excitations with wavelengths of the order of εp. They have conjectured4,5,7 that glasses and perhaps all amorphous materials may exhibit similar behaviour which can be understood in terms of a crossover from phonons to fractons.
KeywordsFractal Dimension Lattice Spacing Small Length Scale Fractal Lattice Crossover Frequency
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- 1.B. B. Mandelbrot, “The Fractal Geometry of Nature,” Freeman, San Francisco (1979).Google Scholar
- 6.A. Aharony, S. Alexander, O. Entin-Wohlman, and R. Orbach, Phys. Rev. 831, 2565 (1985).Google Scholar
- 7.R. Orbach, Geilo Lectures (1985).Google Scholar
- 9.R.W. Southern and P.D. Loly, J. Phys. A18, 525 (1985).Google Scholar
- 10.R. Stinchcombe, Geilo Lectures (1985).Google Scholar
- 12.A. R. Douchant, M.Sc. Thesis, University of Manitoba, (1985).Google Scholar
- 18.P. Bak, Geilo Lectures (1985).Google Scholar