Boson peak in the dynamical structure factor of network- and packing-type glasses

TL;DR

The paper establishes a unified framework connecting the boson peak to a dispersionless excitation band in the dynamical structure factor $S(q,\omega)$, showing that the BP arises from non-phononic excitations superposed on a Debye-like phonon background. Using three glass models (silica, LJ, SS), it compares two routes to obtain the vibrational density of states $g(\omega)$ from $S(q,\omega)$: (i) an incoherent high-$q$ route that reconstructs $g(\omega)$ from $S_L(q,\omega)$, and (ii) a wavenumber-resolved low-$q$ route that integrates over $0<q<q_D$ to decompose $g(\omega)$ into transverse and longitudinal, and further into phononic and non-phononic parts. The results show that BP is predominantly carried by transverse modes, with a dispersionless band responsible for the excess $g(\omega)$ atop the Debye background; these findings are corroborated by an effective-medium theory for random spring networks near isostaticity. Together, the MD and EMT analyses provide a coherent, experimentally accessible picture of BP in both network- and packing-type glasses and connect BP phenomenology to the underlying spatial structure of vibrational excitations.

Abstract

Glasses are structurally disordered solids that host, in addition to crystalline-like phonons, vibrational excitations with no direct phononic counterpart. A long-standing universal signature is the excess vibrational density of states~(vDOS) over the Debye prediction, known as the boson peak~(BP), which has been extensively reported via inelastic neutron and X-ray scattering measurements of the dynamical structure factor $S(q,ω)$. Here we quantify the vDOS directly from dynamical-structure-factor data and clarify the microscopic origin of the BP. We contrast two routes to extract the vDOS from $S(q,ω)$: (i) using high-wavenumber $q$ data beyond the Debye wavenumber $q_D$ to access predominantly incoherent scattering and recover the vDOS in a manner analogous to velocity-autocorrelation-based approaches; and (ii) integrating $S(q,ω)$ over the low-$q$ regime below $q_D$, which enables a decomposition of the vDOS into contributions from distinct wavenumber sectors and thereby provides direct access to the spatial character of vibrational modes. Focusing on the second route, we demonstrate that the BP in the vDOS emerges as the spectral consequence of a dispersionless excitation band in $S(q,ω)$. Our main results are obtained from molecular-dynamics simulations, and we further show that the same mechanism is captured by an effective-medium theory for random spring networks, providing a unified interpretation that connects the excess vDOS to the wavenumber-resolved structure of vibrational excitations in glasses.

Figures (6)

• Figure 1: Crossover wavenumber $q_T(\omega)$ between phononic and non-phononic excitations in silica glass. The transverse dynamical structure factor $S_T(q,\omega)/(k_B T)$ is plotted as a function of wavenumber $q$ for three representative frequencies: (a) $\omega=0.61~\mathrm{THz}\approx0.5\,\omega_{\mathrm{BP}}$, (b) $\omega=1.21~\mathrm{THz}=\omega_{\mathrm{BP}}$, and (c) $\omega=1.82~\mathrm{THz}\approx1.5\,\omega_{\mathrm{BP}}$. Orange circles indicate $q_T(\omega)$ for each frequency. Purple upward triangles and green downward triangles mark the wavenumbers $q_{\max}$ and $q_{\min}$ at which $S_T(q,\omega)$ takes its local maximum $S_{\max}$ and local minimum $S_{\min}$, respectively. The cyan squares indicate the wavenumber $q_{\max}+q_{\mathrm{half}}$, where $q_{\mathrm{half}}$ is defined as the half-width measured from the phonon-branch peak such that $S_T(q,\omega)$ has decreased from $S_{\max}$ by half the peak-to-trough amplitude, $(S_{\max}-S_{\min})/2$. The crossover wavenumber is then defined as $q_T(\omega)=q_{\max}+2\,q_{\mathrm{half}}$, and is shown by the orange circles.
• Figure 2: Dynamical structure factors. (a,b) Silica glass, (c,d) LJ glass, and (e,f) SS glass. The transverse component $S_T(q,\omega)/(k_B T)$ is shown as a function of $q$ and $\omega$ in upper panels (a,c,e), and the longitudinal component $S_L(q,\omega)/(k_B T)$ in lower panels (b,d,f). For silica glass, values are reported in units of $(\mathrm{eV}\,\mathrm{THz})^{-1}$. The horizontal solid and dashed lines indicate the BP frequency $\omega_\text{BP}$ and the Debye frequency $\omega_D$, respectively. The vertical solid line marks the Debye wavenumber $q_D$, while the vertical dashed lines in (b,d,f) mark $q_1$ and $q_2$ used to compute $g_\text{inc}(\omega)$ in Eq. (\ref{['eq_vdosexp']}).
• Figure 3: Vibrational density of states. (a,b) Silica glass, (c,d) LJ glass, and (e,f) SS glass. $g(\omega)$ and $g(\omega)/\omega^2$ (black) are shown as functions of frequency $\omega$, together with $g_\text{inc}(\omega)$ (purple), calculated from $S_L(q,\omega)$ over the high-wavenumber range $q_1 \le q \le q_2$ (with $q_1,\ q_2 > q_D$) using Eq. (\ref{['eq_vdosexp']}). Also shown are the transverse and longitudinal components $g_T(\omega)$ (cyan) and $g_L(\omega)$ (orange), obtained by integrating $S_T(q,\omega)$ and $S_L(q,\omega)$ over $q \le q_D$ as in Eq. (\ref{['eq_vdostl']}), and their sum $g_{T+L}(\omega)=g_T(\omega)+g_L(\omega)$ (green). The vertical and horizontal dotted lines in (b,d,f) indicate the BP frequency $\omega_\text{BP}$ and the Debye level $A_D$, respectively.
• Figure 4: Transverse dynamical structure factor at low wavenumbers and low frequencies. (a) Silica glass, (b) LJ glass, and (c) SS glass. $S_T(q,\omega)/(k_B T)$ is shown as a function of $q$ and $\omega$. For silica glass, values are reported in units of $(\mathrm{eV}\,\mathrm{THz})^{-1}$. The vertical line marks the Debye wavenumber $q_D$. The horizontal line indicates the BP frequency $\omega_\text{BP}$. The black curve shows the linear dispersion $\omega = c_T q$, where $c_T$ is the transverse sound speed, corresponding to phonon excitations. The green curve indicates $q_T(\omega)$, used to separate $g_T(\omega)$ into the phononic component $g_{\mathrm{Ph}}(\omega)$ and the non-phononic component $g_{\mathrm{Nph}}(\omega)$ in Eq. (\ref{['eq_vdos_decomp_T']}).
• Figure 5: Phononic and non-phononic components of the vibrational density of states. (a) Silica glass, (b) LJ glass, and (c) SS glass. The reduced vDOS $g(\omega)/\omega^2$ (black) is plotted as a function of the angular frequency $\omega$, together with the transverse contribution $g_T(\omega)/\omega^2$ (green), obtained by integrating $S_T(q,\omega)$ over $0\le q\le q_D$ according to Eq. (\ref{['eq_vdostl']}). We further decompose $g_T(\omega)/\omega^2$ into its phononic and non-phononic components, $g_{\mathrm{Ph}}(\omega)/\omega^2$ (cyan) and $g_{\mathrm{Nph}}(\omega)/\omega^2$ (orange), as defined in Eq. (\ref{['eq_vdos_decomp_T']}). The vertical and horizontal dotted lines mark the BP frequency $\omega_{\mathrm{BP}}$ and the Debye level $A_D$, respectively.