Excitations of/on Fractal Networks

Abstract

Fractal symmetry (dilation invariance), as opposed to Euclidean symmetry (translation invariance), requires three dimensionalities to contain a physical description of the excitations of a fractal network: d, the Euclidean (or embedding) dimension; D, the Hausdorff (or fractal) dimension; and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaary % aaraaaaa!370D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar \bar d$$, the fracton (or spectral) dimension. The dynamical properties of percolarting networks are examined in this context. The vibrational density of states and the vigrational excitation dispersion law are calculated. The former is shown to be proportional to ω^d−1 in the phonon or long length scale reggime. A crossover is found at frequency ω_c, proportional to (p−p_c)^v[1+(θ/2)], where p is the bond occupancy probability and p_c the critical percolation bond occupancy. Here, v is the correlation fength exponent, and θ is the exponent appropriate to the range dependence of the diffusion constant [D(r) ∝ r^−θ]. At ω_c, a scaling argument shows that a reasonably rapid rise occurs in the vibrational density of states, to which we shall refer as a “fracton edge.” For frequencies higher than ω_c, the vibrational density of states continues to rise, but at a slower rate, proportional to EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaW % baaSqabeaaceWGKbGbaeHbaebacqGHsislcaaIXaaaaaaa!3AAF!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\omega ^{\bar \bar d - 1}}$$. The excitations in this regime are termed “fractons.” It is shown that the electrical conductance on a fractal network also depends on D and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaary % aaraaaaa!370D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar \bar d$$ . Use is made of Anderson localization scaling theory to show that, for EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaary % aaraaaaa!370D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar \bar d$$ < 2, the fracton eigenstates are localized. Very recent inelastic neutron diffraction measurements of the vibrational excitations of fused silica are shown to exhibit a density of states which agrees in form remarkably closely with a recent scaling model of phonon to fracton crossover.

The vibrational dispersion law also exhibits the effects of crossover. At long length scales, the dispersion law is linear, with a sound velocity proportional to (p−p_c)^vθ/2. Near crossover, the dispersion curve reduces its slope, exhibits an inflexion point, and then rises as the EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaac+ % caceWGKbGbaeHbaebaaaa!3889!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$D/\bar \bar d$$ power of the inverse length scale.

These results suggest that conventional electron-phonon interactions may be strongly modified on a fractal network. We have analyzed the two cases of localized electron-one fracton interaction, and a localized electron-two fracton interaction. To do so, we have had to take explicit account of the spatial extent of the vibrational wave function. This has necessitated the introduction of a fourth dimensionality, d_Φ, which determines the pythagorean range of the localized fracton wavefunction. The electronic relaxation rate differs from electronic site to electronic site. We are able to calculate the probability density for the electronic relaxation rate. The lack of a single relaxation rate results in a significant departure from an exponential time decay of the electronic departure from equilibrium. We find a time dependence which is faster than power law, but slower than exponential or stretched exponential. We suggest that these theoretical results call into question interpretations of phenomena involving electron-vibrational interactions in glassy materials which have been based on extended phonon states. We recalculate the energy and temperature dependence for the inelastic scattering time for extended electron states. Our results exhibit marked departure from those obtained with extended (and localized) phonon states.

We suggest a variety of calculations and experiments which can further elucidate the affect of fracton excitations on physical properties. We believe that fracton excitations may be relevant to some glassy materials above a crossover frequency. These excitations can have a profound influence on the vibrational density of states and the dynamics associated with electron-vibration interactions.