Dynamical structure factor of topologically disordered systems: high-frequency sound attenuation with realistic particle correlations and vector displacements

The dynamical structure factor $S(p,E)$ of a glass, probed by inelastic x-ray or neutron scattering, displays an acoustic (Brillouin) peak whose linewidth $\Gamma(p)$ measures the attenuation of high-frequency sound. In 2001, Martin-Mayor, Mézard, Parisi and Verrocchio computed $S(p,E)$ of a topologically disordered solid within Euclidean random matrix (ERM) theory — particles at uncorrelated random positions, scalar displacements, and an expansion in the inverse number density $1/\rho$ — and left a list of open problems: inclusion of the strong particle correlations of real glasses beyond the superposition approximation, the sensitivity of the linewidth coefficients to the chosen interaction, the vectorial (longitudinal/transverse) structure of the vibrations, experimental test of the predicted linewidth scaling regimes over a wide momentum range, comparison with mode-coupling theory, and a resummation for the density of states near its maximum. Later literature resolved several of these — the resummation (Grigera et al., 2001), the perturbative vector extension (Ciliberti et al., 2003), and the experimental clarification of the scaling structure, which established a Rayleigh $\omega^4$ law at low frequency crossing to $\omega^2$ at the boson peak (Baldi et al., 2010) — and corrected Martin-Mayor et al.'s (2001) asymptotic claim of a $\Gamma\propto p^2$ regime at the lowest momenta (Ganter and Schirmacher, 2010/2011). The surviving open core is a first-principles theory of $S(p,E)$ and $\Gamma(p)$ for topologically disordered systems with correlated particle positions beyond the leading-order substitution $f(r)\to f(r)g(r)$, a self-consistent vector (longitudinal/transverse) treatment, and quantitative, parameter-free prediction for real glass formers.

Background

In an inelastic scattering experiment on a solid, one measures the dynamical structure factor $S(p,E)$: the probability that the probe transfers momentum $p$ and energy $E$ to the sample. In the gigahertz-to-terahertz range a glass shows a sound-like peak in $S(p,E)$ (the Brillouin, or high-frequency sound, peak) whose width $\Gamma(p)$ measures how fast this acoustic excitation is damped. In the 1990s, inelastic x-ray scattering (IXS) established two robust facts across many glass formers: the peak position disperses linearly in frequency to momenta of the order of the inverse interparticle distance, and the linewidth follows an approximately quadratic law, $\Gamma\propto p^2$, with remarkably weak temperature dependence. Both facts were puzzling. Hydrodynamic sound attenuation also gives $\Gamma\propto p^2$, but with a viscosity prefactor that varies strongly with temperature, which is not observed. Conversely, wave scattering off a dilute ensemble of defects in an otherwise ordered medium obeys Rayleigh scattering, $\Gamma\propto p^{4}$, which is what any perturbative treatment of a crystal with weak disorder yields.

The difficulty is specific to what is called topological disorder: in a glass there is no reference lattice, so the medium that carries the wave and the objects that scatter it are the same particles, and the usual separation between propagating background and scatterers is meaningless. To treat this, Mézard, Parisi and Zee (1999) introduced Euclidean random matrices (ERM): the Hessian (the matrix of second derivatives of the potential energy, whose eigenvalues squared are the squared vibrational frequencies) is built by placing particles at random positions and coupling them through a distance-dependent spring function $f(r)$. For uncorrelated (Poisson) positions, the disorder average can be performed and the spectrum computed in an expansion in the inverse number density $1/\rho$, where $\rho$ is the particle density.

Martin-Mayor, Mézard, Parisi and Verrocchio (2001) carried this program through for the dynamical structure factor, to two loops in $1/\rho$ (for scalar, collinear displacements). In the low-momentum regime the imaginary part of the self-energy takes the form

$\mathrm{Im}\,\Sigma(p,E)=\rho\tilde f(0)\left[\frac{(p\sigma)^{d+2}A_1}{\rho\sigma^d}+\frac{(p\sigma)^{d}A_2}{(\rho\sigma^d)^2}\right],$

where $\sigma$ is the range of the interaction, $d$ the spatial dimension, $\tilde f(0)$ the zero-momentum value of the Fourier transform of $f$, and $A_1$, $A_2$ are dimensionless numbers of order one that depend on the shape of $f$. In $d=3$ the two terms generate a three-regime picture for the linewidth: a $\Gamma\propto p^2$ regime at sufficiently small $p$ (where the two-loop term dominates), a crossover at a momentum $p_c$ with $(\sigma p_c)^2=\frac{A_2}{A_1}\frac{1}{\rho\sigma^3}\propto 1/N_R$ (Martin-Mayor et al. 2001, Eq. (60)) to the Rayleigh-like $\Gamma\propto p^4$ law, and finally, at momenta of order the inverse interparticle spacing, the collapse of the peak onto the density of vibrational states, where the width necessarily grows more slowly and can mimic $p^2$. The authors suggested that the experimentally observed $p^2$ might be this last transient behavior, and proposed an experimental test over a wider momentum range. In a final section they showed that particle correlations (the pair correlation function $g(r)$, which gives the probability of finding a particle at distance $r$ from another) can be included at leading order through the substitution $f(r)\to f(r)g(r)$, which amounts to the superposition approximation of liquid-state theory (factorizing higher-order correlation functions into products of pair functions); they also showed this substitution fails at order $1/\rho$ for diagrams containing strings of diagonal terms, which instead generate an exponentially suppressed tail of the density of states.

The paper closed with the open issues that define the problem: real glasses and supercooled liquids have strong particle correlations, and a full inclusion beyond the superposition approximation was still missing; the coefficients $A_1$ and $A_2$ depend strongly on the choice of $f(r)$, so quantitative predictions for realistic systems (silica, fragile glass formers) were out of reach; the computation used scalar displacements, while the true vibrations are vectorial and split into longitudinal and transverse components with respect to the momentum $p$; the predicted scaling regimes awaited experimental verification; a comparison with the mode-coupling theory computation for hard spheres (Götze and Mayr, 2000), which found $p^2$ only at very low peak frequencies, was desirable; and the density of states near its maximum was not correctly captured at one loop because the zeroth-order delta function splits off from the continuous part, requiring a partial resummation of the $1/\rho$ expansion. The subsequent literature resolved several of these items, corrected the central low-momentum asymptotics, and reformulated what remains.

Problem Statement

Surviving open core. The later literature resolved the resummation of the density of states at realistic densities (Grigera et al., 2001), the perturbative vector extension (Ciliberti et al., 2003), and the experimental determination of the linewidth scaling structure (Rayleigh $\omega^4$ at low frequency crossing to $\omega^2$ at the boson peak; Baldi et al., 2010), and it corrected the low-momentum asymptotics of Martin-Mayor et al. (2001) (Ganter and Schirmacher, 2010/2011). What remains open, and constitutes the problem to be solved, is: construct an analytically controlled (perturbative or self-consistent) theory of the dynamical structure factor $S(p,E)$ and the high-frequency sound attenuation $\Gamma(p)$ of a topologically disordered system in which (a) the particle positions are correlated according to a realistic pair structure — going beyond the leading-order substitution $f(r)\to f(r)g(r)$ and specifying which classes of correlation-induced diagrams are included, resummed, or neglected — and (b) the displacement field is vectorial, with correctly separated longitudinal and transverse branches treated at self-consistent (not merely perturbative) level; and use this theory to predict $\Gamma(p)$ and $S(p,E)$ quantitatively, without adjustable parameters, for a realistic glass former (for example vitreous silica or a named fragile glass former), in a form directly comparable with inelastic x-ray or neutron scattering data. In the target theory, (a), (b) and the parameter-free prediction hold jointly. The theory is posed for a single named glass former. The predicted $S(p,E)$ and $\Gamma(p)$ exhibit the experimentally established scaling structure: a Brillouin-peak dispersion linear in $p$ and a linewidth crossover from a Rayleigh-like $\omega^4$ at low frequency to approximately $\omega^2$ at the boson peak. Here 'without adjustable parameters' means $f(r)$ and $g(r)$ are fixed by an independent microscopic model of the chosen former rather than fitted to the compared $S(p,E)$ or $\Gamma(p)$ data, which are taken from a reference inelastic x-ray or neutron scattering dataset for that former. The theory reduces, in the appropriate limiting regimes, to the uncorrelated-scalar two-loop self-energy of the 2001 computation — its computed $\mathrm{Im}\,\Sigma(p,E)$ expression given in the Background, not its superseded $\Gamma\propto p^2$ low-momentum claim — and to the corrected Rayleigh lowest-frequency asymptotics of the Euclidean random matrix model (self-energy $\Sigma(k,z)\propto k^2 z^{d/2}$); these two limits occupy disjoint regimes, so the $\Gamma\propto p^2$ claim is not a limiting case of the target theory.

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