Fracton Dimension Research Articles

Detection of boson peak and fractal dynamics of disordered systems using terahertz spectroscopy

The boson peak is a largely unexplained excitation found universally in the terahertz vibrational spectra of disordered systems; the so-called fracton is a vibrational excitation associated with the self-similar structure of monomers in polymeric glasses. We demonstrate that such excitations can be detected using terahertz spectroscopy. In the case of fractal structures, we determine the infrared light-vibration coupling coefficient for the fracton region and show that information concerning the fractal and fracton dimensions appears in the exponent of the absorption coefficient. Finally, using terahertz time-domain spectroscopy and low-frequency Raman scattering, we experimentally observe these universal excitations in a protein (lysozyme) system that has an intrinsically disordered and fractal structure and argue that the system should be considered a single supramolecule. These findings are applicable to amorphous and fractal objects in general and will be valuable for understanding universal dynamics of disordered systems via terahertz light.

Is the Alexander–Orbach Conjecture Suitable for Treating Diffusion in Correlated Percolation Clusters?

How does a particle diffuse inside a percolation cluster? This question is of both scientific and practical importance, e.g. in drug-controlled release and vapour adsorption. Diffusion in fractal media is characterized by the fracton dimension, ds. The Alexander and Orbach conjecture indicates that ds = 4/3 for diffusion in classical percolation clusters and, after much research on the subject, it is still provides a very good approximation for ds in the case of uncorrelated percolation cluster structures. However, what happens to the value of ds when a particle is moving inside a correlated percolation cluster? In this work, this problem is studied via Monte Carlo computer simulation. Our results show that the Alexander and Orbach conjecture is not always fulfilled.

Observing the twinkling fractal nature of the glass transition

A fundamental understanding of the nature and structure of the glass transition in amorphous materials is currently seen as a major unsolved problem in solid-state physics. A new conceptual approach to understanding the glass transition temperature ( T g ) of glass-forming liquids called the twinkling fractal theory (TFT) has been proposed in order to solve this problem. The main idea underlying the TFT is the development of dynamic rigid percolating solid fractal structures near T g , which are said to be in dynamic equilibrium with the surrounding liquid. This idea is coupled with the concept of the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid and liquid clusters interchange or “twinkle” at a cluster size dependent frequency ω TF , which is controlled by the population of intermolecular oscillators in excited energy levels. The solid-to-liquid cluster transitions are in accord with the Orbach vibrational density of states for a particular fractal cluster g( ω) ~ ω df − 1 , where the fracton dimension d f = 4/3. To an observer, these clusters would appear to be “ twinkling.” In this paper, experimental evidence supporting the TFT is presented. The twinkling fractal characteristics of amorphous, atactic polystyrene have been captured via atomic force microscopy (AFM). Successive two-dimensional height AFM images reveal that the percolated solid fractal clusters exist for longer time scales at lower temperatures and have lifetimes that are cluster size dependent. The computed fractal dimensions, ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of percolating clusters in accord with the TFT. The twinkling dynamics of polystyrene within its glass transition region are captured with time-lapse one-dimensional AFM phase images. The autocorrelation cluster relaxation function was found to behave as C( t) ~ t − 4/3 and the cluster lifetimes τ versus width R were found to be in excellent agreement with the TFT via τ ~ R 1.42. This paper provides compelling new experimental evidence for the twinkling fractal nature of the glass transition.

Thermodynamic properties of V3Te4

The heat capacity of V3Te4 has been measured as a function of temperature using an adiabatic calorimeter, and its thermodynamic properties were evaluated. We have determined the Debye characteristic temperature and fracton dimension of crystalline V3Te4.

Fractal properties of lysozyme: A neutron scattering study

The spatial structure and dynamics of hen egg white lysozyme have been investigated by small-angle and inelastic neutron scattering. Analysis of the results was carried using the fractal approach, which allowed determination of the fractal and fracton dimensions of lysozyme, i.e., consideration of the protein structure and dynamics by using a unified approach. Small-angle neutron scattering studies of thermal denaturation of lysozyme have revealed changes in the fractal dimension in the vicinity of the thermal denaturation temperature that reflect changes in the spatial organization of protein.

Charging dynamics of the electric double layer in porous media

Electric double layer in porous media is studied with direct numerical simulations of the Nernst-Planck-Poisson equation. The time evolution of the charging process of the electric double-layer along a straight pore is first studied, and confirm that the time evolution obeys a power law of the exponent 1/2. We find that the diffusion constant increases effectively by the effect of the width of the pore. Next it is found that the time evolution of the charging process in fractal porous media obeys a power law, and the exponent alpha is related to the fracton dimension. Finally, we propose a coupled map lattice model for the creation of pore structures by gas activation processes, and perform numerical simulation of the charging dynamics of the electric double layer.

Inelastic Incoherent Neutron Scattering in Some Proteins

Inelastic incoherent neutron scattering in hen egg white lysozyme, alpha-chymotrypsin from bovine pancreas, and human serum albumin has been investigated. The effect of the isotopic hydrogen-deuterium exchange on the vibrational spectrum of lysozyme has been studied by analyzing the experimentally obtained density of states (DOS). Phonon and fracton regimes have been identified in the low-frequency region of the DOS of deuterated lysozyme. It has been shown that deuteration of lysozyme does not lead to significant changes in the phonon and fracton dimensions. At the same time, the most pronounced differences in G(ω) of deuterated and protonated lysozyme were identified at frequencies above 100 cm − 1 . The DOS of alpha-chymotrypsin from bovine pancreas and human serum albumin at 285 K have been determined.

Subdiffusion-assisted reaction kinetics in disordered media

We present a theory for describing the reaction process occurring in disordered media withenergetically disordered trapping sites and spatial constraints. The theory is based on ageneralized fractional reaction–diffusion equation, which describes the time evolution of themean distribution of a particle performing a continuous time random walk on a fractalnetwork. The motion of a particle is subdiffusive because of the spatial constraints and/orthe random detrapping times described by a waiting time distribution given byψ(t)∼t−(1+α) with 0<α<1. Assuming that the reaction occurs at a separation of contact, the reactionand transport processes are decoupled and the kinetic information for thereaction is expressed in terms of the reaction-free Green’s function obtainedwith the reflecting boundary condition at the separation of contact. Thesurvival probability of a reactant pair is shown to decay asymptotically asτ−α|ds/2−1|,where ds is the fracton dimension of the fractal network under consideration. We also check thevalidity of the analytical results by comparison with Monte Carlo simulation results.

Measurement of fracton dimensionality in liquids by four‐photon Rayleigh‐wing spectroscopy

AbstractFour‐photon spectroscopy of the Rayleigh wing was applied to study distilled and Milli‐Q water, liquid carbon tetrachloride (CCl4), 30% aqueous solution of hydrogen peroxide (H2O2 ) and 0.4 mM aqueous solution of α‐chymotrypsin protein, in the spectral range from − 10 to 10 cm−1. The observed spectra allowed the measurement of the fracton dimensionality as d = 2.87 ± 0.01 in Milli‐Q water, 2.97 ± 0.02 in the protein solution, 3.00 ± 0.02 in CCl4 and H2O2 and 3.15 ± 0.05 in distilled water. The variation of fractal dimensionality can be interpreted as indication of the structural differences in the investigated liquids. In the presence of protein and hydrogen peroxide molecules, this difference indicates that the hydrogen‐bond network of Milli‐Q water is restructured in the solution. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Defect attributed variations of the photoconductivity and photoluminescence in the HVPE and MOCVD as-grown and irradiated GaN structures

The effect of native and radiation induced defects on the photoconductivity transients and photoluminescence spectra have been examined in GaN epitaxial layers of 2.5 and 12 μm thickness grown on bulk n-GaN/sapphire substrates by metal-organic chemical vapor deposition (MOCVD). For comparison, free-standing GaN as-grown samples of 500 μm thickness, fabricated by hydride vapor phase epitaxy (HVPE), were investigated. Manifestation of defects induced by 10-keV X-ray irradiation with the dose of 600 Mrad and 100-keV neutrons with the fluences of 5 × 1 0 14 and 10 16 cm −2 as well as of 24 GeV/ c protons with fluence 10 16 cm −2 have been revealed through contact photoconductivity and microwave absorption transients. The amplitude of the initial photoconductivity decay is significantly reduced by the native and radiation defects density. Synchronous decrease of the steady-state PL intensity of yellow, blue and ultraviolet bands peaked at 2.18, 2.85, and 3.42 eV, respectively, with density of radiation-induced defects is observed. The decrease of the PL intensity is accompanied by an increase of asymptotic decay lifetime in the photoconductivity transients, which is due to excess-carrier multi-trapping. The decay fits the stretched exponent approximation exp[ - ( t / τ ) α ] with the different factors α in as-grown material ( α ≈ 0.7 ) and irradiated samples ( α ≈ 0.3 ). The fracton dimension d s of disordered structure changes from 4.7 to 0.86 for as-grown and irradiated material, respectively, and it implies the percolative carrier motion on an infinite cluster of dislocations net in the as-grown material and cluster fragmentation into finite fractons after irradiations.

Characterization of as-grown and heavily irradiated GaN epitaxial structures by photoconductivity and photoluminescence

The influence of radiation defects on photoconductivity transients and photoluminescence (PL) spectra have been examined in semi-insulating GaN epitaxial layers grown on bulk n-GaN/sapphire substrates. Defects induced by 10-keV X-ray irradiation with a dose of 600 Mrad and 100-keV neutrons with fluences of 5×10 14 and 10 16 cm −2 have been revealed through contact photoconductivity and microwave absorption transients. The amplitude of the initial photoconductivity decay is significantly reduced by the radiation defect density. A simultaneous decrease with radiation-induced defect density is also observed in the steady-state PL intensity of yellow, blue and ultraviolet bands peaked at 2.18, 2.85, and 3.42 eV, respectively. The decrease of the PL intensity is accompanied by an increase of asymptotic decay lifetime, which is due to excess carrier multi-trapping. The decay can be described by the stretched exponential approximation exp[ - ( t / τ ) α ] with different values of α in as-grown material ( α ≈ 0.7 ) and irradiated samples ( α ≈ 0.3 ). The value of the fracton dimension d s of the disordered structure, evaluated as d s = 2 α / ( 1 - α ) , changes from 4.7 to 0.86 for as-grown and irradiated material, respectively, implying percolative carrier motion on an infinite cluster of dislocations net in the as-grown material and cluster fragmentation into finite fractons after irradiation.

Magnetic properties of the weak itinerant-electron ferromagnetNi75Al25: II. The effect of compositional disorder

The results of a detailed study of the magnetic properties of well-characterized polycrystallineNipAl100−p () alloys are presented and discussed in the light of the existing theories. Extreme care hasbeen exercised in the sample preparation to ensure that the site disorder (invariablypresent in any alloy system) does not interfere with the compositional disorderbrought about by the reduction in the concentration of the magnetic (Ni) atoms.Thus, the observed variation in the magnetic properties with Ni concentration(p) is solely controlled by the compositional disorder. Like site disorder, compositional disordersmears out the sharp features in the density of states (DOS) curve near the Fermi level,EF, and reducesthe DOS at EF, N(EF), andthereby causes a fall (an enhancement) in the values (value) of the spontaneous magnetization at 0 K,M0, the spin-wavestiffness at 0 K, D0, andthe Curie temperature, TC (zero-field differential susceptibility at 0 K,χ0). However, compositional disorder, unlike site disorder, gives rise to smooth variations inN(EF), the inverse Stoner enhancement factor , M0,D0,TC,D0/TC andχ0 withp. These variationsin the case of M0(p), D0(p) and TC(p) are very well described by the power laws , and with p>pc (pc = the percolation threshold for the appearance of long-range ferromagnetic order) predictedby the percolation theories for these quantities on a regular three-dimensional(d = 3) percolating network. The alloys in question exhibit a crossover in the spin dynamics fromthe hydrodynamic (magnon) to critical (fracton) regime at a well-defined temperatureTco*(p). An elaborate analysis of the magnetization data in terms of the percolation modelspermits a reasonably accurate determination of the magnon-to-fracton crossover line in themagnetic phase diagram, the percolation-to-thermal crossover exponent, fractal dimension,fracton dimensionality, the percolation critical exponents for spontaneous magnetization,spin-wave stiffness, correlation length and conductivity. The results of this analysisalso vindicate the Alexander–Orbach conjecture and the Golden inequality ford = 3 percolating ferromagnetic networks.

Anomalous diffusion exponents in continuous two-dimensional multifractal media.

We study diffusion in heterogeneous multifractal continuous media that are characterized by the second-order dimension of the multifractal spectrum D2, while the fractal dimension of order 0, D0, is equal to the embedding Euclidean dimension 2. We find that the mean anomalous and fracton dimensions, d(w) and d(s), are equal to those of homogeneous media showing that, on average, the key parameter is the fractal dimension of order 0 D0, equal to the Euclidean dimension and not to the correlation dimension D2. Beyond their average, the anomalous diffusion and fracton exponents, d(w) and d(s), are highly variable and consistently range in the interval [1,4]. d(w) can be consistently either larger or lower than 2, indicating possible subdiffusive and superdiffusive regimes. On a realization basis, we show that the exponent variability is related to the local conductivity at the medium inlet through the conductivity scaling.

Hopping‐randomwalk of electrons at localized band tail states in random fractal fluctuation

Power law decays in dispersive transient current and photoluminescence in amorphous semiconductors are explained from hopping-randomwalk of carriers at localized band tail states in random self-affine fluctuation of the 3.9 fractal dimension. The hopping-randomwalk is reported. The mean square distance is given by 〈r(t)2〉 ∼ tκ with κ < 1, which shows anomalous diffusion. The hopping area where electrons pass through is complicated and has a fractal dimension. The fractal and fracton dimensions of the randomwalk are calculated at different temperatures. The fractal and fracton dimensions of the hopping-randomwalk in amorphous semiconductors is shown to be able to be obtained from the dispersive transient current and photoluminescence decay. It is shown that the temperature behavior of both dimensions in a-Si:H agrees with that in the simulation of the hopping-randomwalk. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Green's function approach to nonclassical reaction kinetics in fractal media.

We present an accurate description for the mean field kinetics of the reversible recombination reaction occurring in fractal media, which is in good agreement with Monte Carlo simulation results from an intermediate to a long time region. The central dynamic quantity is the Green's function for the generalized diffusion equation that is defined in a hypothetical space whose dimensionality is given by the fracton dimension d. An exact expression for the Green's function that is valid for arbitrary values of d is presented.

Raman study of fractal voids in hot-pressed SiO 2

The sensitivity of the dynamic (vibrational) properties of fractal nanocracks in hot-pressed silica to the thermal and mechanical prehistory of the material was studied by low-frequency Raman scattering. Samples were prepared from gel-derived and fused powdery products. The Raman data demonstrate the fractality of elementary continuity defects. From the frequency dependence of the Raman intensity, the values of the fracton (spectral) dimension that reflects the mid-range order in glass, and the dimensions of fractals in real space were calculated.

Magnon-fracton crossover in quenched random site-diluted ferromagnets

The strong departure from the Bloch ${T}^{3/2}$ law behavior of magnetization in dilute amorphous ${(T}_{p}{\mathrm{Ni}}_{1\ensuremath{-}p}{)}_{80}{\mathrm{B}}_{16}{\mathrm{Si}}_{4}$ $(T=\mathrm{Fe},\mathrm{Co})$ alloys is shown to be a manifestation of the crossover from hydrodynamic to critical spin wave dynamics induced by the diverging correlation length near percolation threshold. An appropriate choice of the density of states for magnetic excitations on self-similar (fractal) percolation network permits an accurate determination of the magnon-to-fracton crossover line in the magnetic phase diagram of quenched random site-diluted ferromagnets. By unambiguously demonstrating that the fracton dimensionality ${d}_{f}\ensuremath{\simeq}4/3$ and the conductivity percolation critical exponent ${\ensuremath{\sigma}}_{p}&lt;2,$ the present results vindicate the Alexander-Orbach conjecture and the Golden inequality for percolating network with Euclidean dimension $d=3.$

Kinetics of N 2O catalytic decomposition over three-dimensional fractals

A Monte Carlo (MC) random walk algorithm is employed to investigate the kinetics of N 2O catalytic decomposition over three-dimensional fractals. The simulation results show that usual kinetic equations based on the mean field theory are not applicable to the reaction over fractals, and a different rate law containing a fractional reaction order is more suitable for describing the reaction kinetics over fractals. For the annihilation reaction involved in N 2O decomposition over fractals, the effective reaction order is found to range from 2 to 1+2/ d s depending on the annihilation constant, where d s is the fracton dimension.

Fractal approach to the exciton-phonon coupling and structure of J aggregates

We present a study on exciton-phonon coupling in molecular aggregates at low temperature probed by photon-echo and resonance Raman experiments. A large peak in the low-frequency Raman spectrum is interpreted as a boson peak. Our results for the J aggregates of the dye 5,${5}^{|\mathrm{I}\mathrm{H}}$, 6,${6}^{|\mathrm{I}\mathrm{H}}$-tetrachloro-1,${1}^{|\mathrm{I}\mathrm{H}}$-diethyl-3,${3}^{|\mathrm{I}\mathrm{H}}$-di(4-sulfobutyl)-benzimidazolocarbocyanine indicate an anomalous vibrational density of states with a spectral (or fracton) dimension of 1.44 at 3.9 K. The spectral dimension increases with temperature, suggesting that the topological dimension of the aggregates in solution is 1 and that an increase in temperature leads to an increase in effective branching of the aggregate.

Fracton excitation and Lévy flight dynamics in alkali silicate glasses

We have examined the relaxation behavior of alkali metal ions in lithium metasilicate glasses by means of molecular dynamics simulation. We have observed a change of slope of the mean squared displacement at \ensuremath{\sim}300fps. In shorter time regions, localized motion of lithium ions within neighboring sites is observed, which is caused by the small fracton dimension (fracton excitation). On the other hand, an accelerated motion of particles due to cooperative jumps is found, which characterizes the diffusion and conduction mechanisms of the alkali metal ions in longer time regions. The dynamics of accelerated motion is discussed in relation to L\'evy flight dynamics.