Fractals

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Pascal Monceau - One of the best experts on this subject based on the ideXlab platform.

  • Effects of deterministic and random discrete scale invariance on spin wave spectra
    Physica E-low-dimensional Systems & Nanostructures, 2012
    Co-Authors: Pascal Monceau, Jean-claude Serge Lévy
    Abstract:

    The properties of magnon spectra in low dimensional random Sierpinski Fractals, deterministic Fractals, and percolation clusters are calculated and compared. While deterministic scale invariance leads to singular continuous spectra with gaps and degenerated levels, random scale invariance leads to continuous density of states which enables to define low frequency exponents; the exponents associated to random percolation clusters and to Sierpinski Fractals are significantly different, even if their fractal dimensions are very close. The study of spacing levels shows quantitatively that the degeneracy is linked to geometrical symmetry in deterministic Fractals while it is strongly reduced by random discrete scale invariance.

  • Effects of deterministic and random discrete scale invariance on spin wave spectra
    Physica E: Low-dimensional Systems and Nanostructures, 2012
    Co-Authors: Pascal Monceau, Jean-claude S. Levy
    Abstract:

    International audienceThe properties of magnon spectra in low dimensional random Sierpinski Fractals, deterministic Fractals, and percolation clusters are calculated and compared. While deterministic scale invariance leads to singular continuous spectra with gaps and degenerated levels, random scale invariance leads to continuous density of states which enables to define low frequency exponents; the exponents associated to random percolation clusters and to Sierpinski Fractals are significantly different, even if their fractal dimensions are very close. The study of spacing levels shows quantitatively that the degeneracy is linked to geometrical symmetry in deterministic Fractals while it is strongly reduced by random discrete scale invariance

  • Spin waves in deterministic Fractals
    Physics Letters A, 2010
    Co-Authors: Pascal Monceau, Jean-claude Levy
    Abstract:

    We calculate spin wave spectra of low-dimensional self-similar nanostructures, namely deterministic Sierpinski carpets, in the framework of short ranged ferromagnetic exchange. The integrated density of states of magnetic excitations are shown to exhibit singularities resulting in devil's staircase spectra which remain at all iteration steps; the spectra are found to be singular continuous functions of the frequency with numerous gaps and plateaux linked to symmetry and degeneracy of the associated eigen-modes. These spin wave spectra are shown to be sensitive not only to the fractal dimension, but to additional connectivity properties of the structures as already pointed out in the study of critical properties of Fractals. This result is closely linked to the fractal subdimensions arising from the set of eigenvalues of the connectivity matrix describing the construction of the fractal structure; our set of results strongly suggests that the link between the integrated density of states and fractal subdimensions is a much more general feature of deterministic Fractals; connectivity marks mode localization. Lastly, magnetic Fractals are shown to be able to filter radiofrequencies.

  • Spin waves in Fractals
    2009
    Co-Authors: Pascal Monceau, Jean-claude Serge Lévy
    Abstract:

    Spin wave frequencies and profiles of several Sierpinski Carpets taken at different iteration levels are calculated. Spin wave spectra of these Fractals with short ranged ferromagnetic exchange are found to be singular continuous functions of the frequency with quite numerous steps and cliffs, i.e. energy gaps and degenerate modes, resulting in devil's staircase spectra. The study of connectivity reveals the existence of several different connectivity areas within each fractal. It marks mode localization and symmetry. Spin wave modes are found to be quite sensitive to fractal topology and connectivity as already observed about critical properties of Fractals. The extension of the properties of magnetic excitations, first to random Fractals, then to elastic waves as well as to electronic states in Fractals is introduced. Applications to spin wave resonance in dilute magnetic semiconductors are discussed.

Jean-claude Serge Lévy - One of the best experts on this subject based on the ideXlab platform.

  • Effects of deterministic and random discrete scale invariance on spin wave spectra
    Physica E-low-dimensional Systems & Nanostructures, 2012
    Co-Authors: Pascal Monceau, Jean-claude Serge Lévy
    Abstract:

    The properties of magnon spectra in low dimensional random Sierpinski Fractals, deterministic Fractals, and percolation clusters are calculated and compared. While deterministic scale invariance leads to singular continuous spectra with gaps and degenerated levels, random scale invariance leads to continuous density of states which enables to define low frequency exponents; the exponents associated to random percolation clusters and to Sierpinski Fractals are significantly different, even if their fractal dimensions are very close. The study of spacing levels shows quantitatively that the degeneracy is linked to geometrical symmetry in deterministic Fractals while it is strongly reduced by random discrete scale invariance.

  • Spin waves in Fractals
    2009
    Co-Authors: Pascal Monceau, Jean-claude Serge Lévy
    Abstract:

    Spin wave frequencies and profiles of several Sierpinski Carpets taken at different iteration levels are calculated. Spin wave spectra of these Fractals with short ranged ferromagnetic exchange are found to be singular continuous functions of the frequency with quite numerous steps and cliffs, i.e. energy gaps and degenerate modes, resulting in devil's staircase spectra. The study of connectivity reveals the existence of several different connectivity areas within each fractal. It marks mode localization and symmetry. Spin wave modes are found to be quite sensitive to fractal topology and connectivity as already observed about critical properties of Fractals. The extension of the properties of magnetic excitations, first to random Fractals, then to elastic waves as well as to electronic states in Fractals is introduced. Applications to spin wave resonance in dilute magnetic semiconductors are discussed.

Jean-claude S. Levy - One of the best experts on this subject based on the ideXlab platform.

  • Effects of deterministic and random discrete scale invariance on spin wave spectra
    Physica E: Low-dimensional Systems and Nanostructures, 2012
    Co-Authors: Pascal Monceau, Jean-claude S. Levy
    Abstract:

    International audienceThe properties of magnon spectra in low dimensional random Sierpinski Fractals, deterministic Fractals, and percolation clusters are calculated and compared. While deterministic scale invariance leads to singular continuous spectra with gaps and degenerated levels, random scale invariance leads to continuous density of states which enables to define low frequency exponents; the exponents associated to random percolation clusters and to Sierpinski Fractals are significantly different, even if their fractal dimensions are very close. The study of spacing levels shows quantitatively that the degeneracy is linked to geometrical symmetry in deterministic Fractals while it is strongly reduced by random discrete scale invariance

Jean-françois Colonna - One of the best experts on this subject based on the ideXlab platform.

Alexander I. Kuklin - One of the best experts on this subject based on the ideXlab platform.

  • Scattering from surface Fractals in terms of composing mass Fractals
    Journal of Applied Crystallography, 2017
    Co-Authors: A. Yu. Cherny, Eugen Mircea Anitas, Vladimir A. Osipov, Alexander I. Kuklin
    Abstract:

    It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass Fractals. Various approximations for the scattering intensity of surface Fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensity I(q) ∝ q^{D_{\rm s}-6}, where 2 < Ds < 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝ r−τdr, with Ds = τ − 1. The distribution is continuous for random Fractals and discrete for deterministic Fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface Fractals, such as the fractal iteration number and the scaling factor.

  • small angle scattering from the cantor surface fractal on the plane and the koch snowflake
    Physical Chemistry Chemical Physics, 2017
    Co-Authors: Alexander Yu Cherny, Vladimir A. Osipov, Alexander I. Kuklin, E. M. Anitas
    Abstract:

    The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface Fractals can be decomposed into a sum of surface mass Fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ qDs-4, with 1 < Ds < 2 being the fractal dimension of the perimeter. The curve I(q)q4-Ds is found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

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