Excess Vibrational Modes and the Boson Peak in Model Glasses

The excess low-frequency normal modes for two widely used models of glasses are studied at zero temperature. The onset frequencies for the anomalous modes for both systems agree well with predictions of a variational argument, which is based on analyzing the vibrational energy originating from the excess contacts per particle over the minimum number needed for mechanical stability. Even though both glasses studied have a high coordination number, most of the additional contacts can be considered to be weak.

Figure 1

Density of vibrational states of 3D glasses of $N=1000$ particles, averaged over 100 configurations. We plot $D(\omega )$ for stressed systems (heavy curves) and $D^{(0)}(\omega )$ for unstressed systems (light curves) interacting via (a) the repulsive Lennard-Jones potential at $\varphi -{\varphi }_c={10}^{-6}$ (solid line), ${10}^{-2}$ (dotted line), 0.1 (dot-dashed line), and 0.2 (dashed line); and (b) the Lennard-Jones potential at $\rho =1.2$ (solid line), 1.4 (dotted line), and 1.6 (dashed line).

Figure 2

Left axis: The characteristic frequency ${\omega }^{†}$ (open symbols) calculated from numerical simulations and the corresponding theoretical predictions ${\omega }^{*}$ (crosses). Lines are to guide the eye and connect the theoretical points. Right axis: the fractional deviation from isostaticity $\delta Z_1\equiv (z_1-z_c)/z_c$ (closed symbols), as functions of (a) distance from the unjamming transition, $\Delta \varphi =\varphi -{\varphi }_c$, for repulsive Lennard-Jones mixtures, and (b) density $\rho$ for Lennard-Jones mixtures.

Figure 3

The energy cost of the lowest-frequency anomalous mode for an unstressed system, $\delta E^{(0)}$, as a function of $\mathcal{S}\equiv 1/N\sum _{ij}{V}^{\prime }(r_{ij})/r_{ij}$ for repulsive Lennard-Jones mixtures (solid circles) and Lennard-Jones mixtures (open triangles). The contribution of the stress term to the energy cost of a mode is $\delta E_3=A_3\mathcal{S}$, where $A_3$ is of order unity. The straight line fit corresponds to $\delta E^{(0)}=-1.6\mathcal{S}$.