Observation of High-Frequency Transverse Phonons in Metallic Glasses

Using inelastic neutron scattering and molecular dynamics simulations on a model Zr-Cu-Al metallic glass, we show that transverse phonons persist well into the high-frequency regime, and can be detected at large momentum transfer. Furthermore, the apparent peak width of the transverse phonons was found to follow the static structure factor. The one-to-one correspondence, which was demonstrated for both Zr-Cu-Al metallic glass and a three-dimensional Lennard-Jones model glass, suggests a universal correlation between the phonon dynamics and the underlying disordered structure. This remarkable correlation, not found for longitudinal phonons, underscores the key role that transverse phonons hold for understanding the structure-dynamics relationship in disordered materials.

Figure 1

The dynamic and static structure factors in ${\mathrm{Zr}}_{46}{\mathrm{Cu}}_{46}{\mathrm{Al}}_8$ MG. (a) Dynamic structure factor $S(\mathit{Q},E)$, measured by INS with $E_i=50\mathrm{meV}$ on ARCS at RT. The dashed curve is calculated by the function, $E_{0^{*}}|\sin [\pi (Q/Q_{\max })]|$, which defines the quasi Brillouin zone (QBZ), a term analogous to the Brillouin zone in crystalline materials in order to facilitate discussions of the phonon dynamics over different $Q$ regions. Here, $Q_{\max }=2.74{\AA }^{-1}$ is the position of the first sharp diffraction peak and $E_0=19.02\mathrm{meV}$ is an estimated Einstein frequency for vibration. (b) Static structure factor $S(Q)$, measured by synchrotron x-ray and neutron total scattering at RT, demonstrated the fully amorphous nature of the sample. The inset shows a schematic configuration of the cluster-centered structure in ${\mathrm{Zr}}_{46}{\mathrm{Cu}}_{46}{\mathrm{Al}}_8$ MG.

Figure 2

Phononlike dispersion relationship in ${\mathrm{Zr}}_{46}{\mathrm{Cu}}_{46}{\mathrm{Al}}_8$ MG. (a),(c) Phonon $\mathit{Q}$-dependent GDOS reduced from INS data and the corresponding neutron-weighted MD simulation results based on the VHF method. (b),(d) Maps of the second derivatives of (a) and (c) in the vicinity of the second QBZ, respectively. Only the absolute values are plotted to trace the peaks, following the practice in Refs. [35, 36]. (e),(f) The longitudinal and transverse $\mathit{Q}$-dependent VDOS obtained from MD simulations by the DM method. The results clearly show that the transverse branch has an intensity maximum around $Q=3.8{\AA }^{-1}$. The olive and magenta curves in (a),(c),(e),(f) are calculated longitudinal and transverse phonon dispersions based on the analytical theory for a disordered system [4, 6, 16, 17, 27]. The green and yellow lines in (a),(c),(e),(f) are calculated dispersions based on longitudinal and transverse sound velocities measured by the RUS method. These data are consistent with each other in the low $Q$ regime. The blue stars in (c),(e),(f) marks the boundary of the second QBZ, where the transverse phonons manifest. For the purpose of comparison, the color bars are plotted in relative intensities.

Figure 3

MD simulation results based on the DM method. (a) The total VDOS, and partial VDOS for each element, and (b) the participation ratio (PR) for each mode. For comparison, the experimental GDOS is superimposed in (a), which was obtained by averaging a $Q$ range of $2.5–9.0{\AA }^{-1}$ for the INS data. (c) $Q$ dependence of the transverse and longitudinal VDOS $D_{\alpha }(Q,E)(\alpha \in L,T)$ at $E=17\mathrm{meV}$. These peaks indicate the plane-wave characteristics in this mode. (d) Comparison of $D_T(Q,E)$ at $E=10$, 17, and 23 meV. The mode of $E=17\mathrm{meV}$ shows stronger transverse wave characters around $Q=3.8{\AA }^{-1}$.

Figure 4

Peak widths of the longitudinal and transverse phonon spectra vs $S(Q)$ obtained by MD simulations based on the DM method. The widths were obtained by fitting the MD spectra with a Gaussian function [20, 27]. (a) Empirical potential for ${\mathrm{Zr}}_{46}{\mathrm{Cu}}_{46}{\mathrm{Al}}_8$ MG. (b) Lennard-Jones (LJ) potential for a model glass. A one-to-one correlation can be seen between the apparent peak width and $S(Q)$ for the transverse phonon mode. These results suggest a universal correlation between the transverse phonon dynamics and the underlying disordered structure. No correlation was found for the longitudinal phonons. These LJ results are in scaled units.