Abstract
Scattering is known to be a powerful method of analysing fractal systems. A classical heuristic result in small-angle X-ray or neutron scattering relates the scattered intensity I(q) in the high-frequency regime to some power-law of the momentum transfer q
where D is some fractal dimension of the scatterer. In the last decade, the case of one-dimensional systems has been studied more rigorously, in the framework of optical and quantum scattering ([1], [2],[13],[19],[14]). For deterministic fractals such as Cantor-like features, the heuristic argument needs to be refined. As was first pointed out in ([1]), some scaling law actually exists if one performs frequency averages on the scattering data. Indeed as figure 1.1 shows, the reflection coefficient for quantum scattering on a triadic Cantor measure gives a “wrong” scaling. Only after some suitable averages have been performed, the right scaling behavior is recovered (figure 1.2). Recently, a general method was proposed ([7]) to retrieve a fractal dimension, namely the correlation dimension, of a potential barrier from the reflection amplitude. In this paper, we wish to give further developments to this method. In particular, we show how the same kind of results can be extended to the scattering phase for half-line scattering. Moreover we shall show how to recover the self-similarity (if there is any) of the potential through a large scale renormalization of the scattering amplitude and scattering phase, respectively.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allain, C, Cloitre, M. (1985): Optical diffraction on fractals. Phys. Rev. B, 33(5), 3566–3569.
Allain, C, Cloitre, M. (1987): Spatial spectrum of a general family of self-similar arrays. Physical Review A, 36(12), 5751–5757.
Daubechies, I. (1992): Ten Lectures on Wavelets. SIAM.
Deift, P., Trubovitz, E. (1979): Inverse scattering on the line. Comm. Pure Appl. Math., 32, 121–251.
Grossmann, A., Morlet, J. (1984): Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal., 15(2).
Guerin, CA., Holschneider, M. (1995): l2-Fourier asymptotic of self-similar measures. Preprint CPT, (P.3266).
Guerin, CA., Holschneider, M. (1996): Scattering on fractal measures. J. Phys. A: Math. Gen., (29), 7651–7667.
Guerin, CA., Holschneider, M. (1997): On equivalent definitions of the correlation dimension for a probability measure. Joum. Stat. Phys.
Holschneider, M. (1994): Fractal wavelet dimension and localization. Comm. Math. Phys., 160, 457–473.
Holschneider, M. (1995): Wavelets, an Analytical Tool. Oxford University Press.
Holschneider, M. (1996): Large scale renormalisation of fourier transforms of self-similar measures and self-similarity of Riesz measures. J. of Math. Anal and Appl, (200), 307–314.
Hutchinson, J.E. (1981): Fractals and self similarity. Ind. Univ. Math. Jour., 30(5), 713–746.
Jaggard, D.L., Sun, X. (1990): Reflection from fractal multilayers. Opt. Lett., 15, 1428–1430.
Konotop, V.V., Yordanov, O.I., Yurkevitch, I.V. (1990): Wave transmission through a one-dimensional Cantor-like fractal medium. EuroPhys.Lett, 12(6), 481–485.
Lau, K.S., Wang, J. (1993): Mean quadratic variations and Fourier asymptotics of self-similar measures. Monatsh. Math., 115, 99–132.
Pesin, Y.B. (1993): On rigorous mathematical definition of the correlation dimension and generalized spectrum for dimension. Joum. Stat. Phys., 71(3/4), 529–547.
Strichartz, R.S. (1990): Self-similar measures and their Fourier transform 1. Indiana Univ. Math. Journ., 39(3), 797–817.
Strichartz, R.S. (1993): Self-similar measures and their Fourier transform 2. Trans. Am. Math. Soc, 336(1), 335–361.
Sun, X., Jaggard, D.L. (1991): Wave interactions with generalized Cantor bar fractal multilayers. J. Appl. Phys., 70, 2500–2507.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag London Limited
About this paper
Cite this paper
Guerin, CA., Holschneider, M. (1997). Potential Scattering on Fractals in One Dimension. In: Lévy Véhel, J., Lutton, E., Tricot, C. (eds) Fractals in Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0995-2_20
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0995-2_20
Publisher Name: Springer, London
Print ISBN: 978-1-4471-1253-2
Online ISBN: 978-1-4471-0995-2
eBook Packages: Springer Book Archive