Theory of vibrational anomalies in glasses

Abstract The theory of elasticity with spatially fluctuating elastic constants (heterogeneous-elasticity theory) is reviewed. It is shown that the vibrational anomalies associated with the boson peak can be qualitatively and quantitatively explained in terms of this theory. Two versions of a mean-field theory for solving the stochastic equation of motion are presented: the coherent-potential approximation (CPA) and the self-consistent Born approximation (SCBA). It is shown that the latter is included in the former in the Gaussian and weak-disorder limit. We are able to discuss and explain cases in which the change of the vibrational spectrum by varying an external parameter can be accounted…

Sound attenuation and anharmonic damping in solids with correlated disorder

We study via self-consistent Born approximation a model for sound waves in a disordered environment, in which the local fluctuations of the shear modulus G are spatially correlated with a certain correlation length The theory predicts an enhancement of the density of states over Debye's omega(2) law (boson peak) whose intensity increases for increasing correlation length, and whose frequency position is shifted downwards as lg. Moreover, the predicted disorder-induced sound attenuation coefficient r(k) obeys a universal scaling law F(k) = f (ke) for a given variance of G. Finally, the inclusion of the lowest-order contribution to the anharmonic sound damping into the theory allows us to rec…

Acoustic dynamics of network-forming glasses at mesoscopic wavelengths

The lack of long-range structural order in amorphous solids induces well known thermodynamic anomalies, which are the manifestation of distinct peculiarities in the vibrational spectrum. Although the impact of such anomalies vanishes in the long wavelength, elastic continuum limit, it dominates at length scales comparable to interatomic distances, implying an intermediate transition regime still poorly understood. Here we report a study of such mesoscopic domains by means of a broadband version of picosecond photo-acoustics, developed to coherently generate and detect hypersonic sound waves in the sub-THz region with unprecedented sampling efficiency. We identify a temperature-dependent fra…

What is the Right Theory for Anderson Localization of Light? An Experimental Test

Anderson localization of light is traditionally described in analogy to electrons in a random potential. Within this description, the random potential depends on the wavelength of the incident light. For transverse Anderson localization, this leads to the prediction that the distribution of localization lengths---and, hence, its average---strongly depends on the wavelength. In an alternative description, in terms of a spatially fluctuating electric modulus, this is not the case. Here, we report on an experimentum crucis in order to investigate the validity of the two conflicting theories using optical samples exhibiting transverse Anderson localization. We do not find any dependence of the …

Theory of heterogeneous viscoelasticity

We review a new theory of viscoelasticity of a glass-forming viscous liquid near and below the glass transition. In our model we assume that each point in the material has a specific viscosity, which varies randomly in space according to a fluctuating activation free energy. We include a Maxwellian elastic term and assume that the corresponding shear modulus fluctuates as well with the same distribution as that of the activation barriers. The model is solved in coherent-potential approximation (CPA), for which a derivation is given. The theory predicts an Arrhenius-type temperature dependence of the viscosity in the vanishing-frequency limit, independent of the distribution of the activatio…

Analytical description of the transverse Anderson localization of light

Disorder-induced single-mode transmission.

Localized states trap waves propagating in a disordered potential and play a crucial role in Anderson localization, which is the absence of diffusion due to disorder. Some localized states are barely coupled with neighbours because of differences in wavelength or small spatial overlap, thus preventing energy leakage to the surroundings. This is the same degree of isolation found in the homogeneous core of a single-mode optical fibre. Here we show that localized states of a disordered optical fibre are single mode: the transmission channels possess a high degree of resilience to perturbation and invariance with respect to the launch conditions. Our experimental approach allows identification…

Do we understand the solid-like elastic properties of confined liquids?

Recently, in polymeric liquids, unexpected solid-like shear elasticity has been discovered, which gave rise to a controversial discussion about its origin (1⇓–3). The observed solid-like shear modulus G depends strongly on the distance L between the plates of the rheometer according to a power law G ∝ L − p with a nonuniversal exponent ranging between p = 2 and p = 3 . Zaccone and Trachenko (4) have published an article in which they claim to explain these findings by a nonaffine contribution to the liquid shear modulus. The latter is represented as Δ G ∝ − ∑ λ = L , T 1 V … [↵][1]1To whom correspondence may be addressed. Email: giancarlo.ruocco{at}roma1.infn.it. [1]: #xref-corresp-1-1

Self-consistent Euclidean-random-matrix theory

Coherent potential approximation for diffusion and wave propagation in topologically disordered systems

Using Gaussian integral transform techniques borrowed from functional-integral field theory and the replica trick we derive a version of the coherent-potential approximation (CPA) suited for describing ($i$) the diffusive (hopping) motion of classical particles in a random environment and ($ii$) the vibrational properties of materials with spatially fluctuating elastic coefficients in topologically disordered materials. The effective medium in the present version of the CPA is not a lattice but a homogeneous and isotropic medium, representing an amorphous material on a mesoscopic scale. The transition from a frequency-independent to a frequency-dependent diffusivity (conductivity) is shown …

Heterogeneous shear elasticity of glasses: the origin of the boson peak

The local elasticity of glasses is known to be inhomogeneous on a microscopic scale compared to that of crystalline materials. Their vibrational spectrum strongly deviates from that expected from Debye's elasticity theory: The density of states deviates from Debye's law, the sound velocity shows a negative dispersion in the boson-peak frequency regime and there is a strong increase of the sound attenuation near the boson-peak frequency. By comparing a mean-field theory of shear-elastic heterogeneity with a large-scale simulation of a soft-sphere glass we demonstrate that the observed anomalies in glasses are caused by elastic heterogeneity. By observing that the macroscopic bulk modulus is …

Comment on "Universal Origin of Boson Peak Vibrational Anomalies in Ordered Crystals and in Amorphous Materials".

Vibrational excitations in systems with correlated disorder

We investigate a $d$-dimensional model ($d$ = 2,3) for sound waves in a disordered environment, in which the local fluctuations of the elastic modulus are spatially correlated with a certain correlation length. The model is solved analytically by means of a field-theoretical effective-medium theory (self-consistent Born approximation) and numerically on a square lattice. As in the uncorrelated case the theory predicts an enhancement of the density of states over Debye's $\omega^{d-1}$ law (``boson peak'') as a result of disorder. This anomay becomes reinforced for increasing correlation length $\xi$. The theory predicts that $\xi$ times the width of the Brillouin line should be a universal …