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

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.

💡 Research Summary

This paper tackles the long‑standing problem of linking the boson peak (BP) – an excess in the vibrational density of states (vDOS) over the Debye prediction – to experimentally measurable quantities, namely the dynamical structure factor S(q,ω) obtained from inelastic neutron and X‑ray scattering. The authors propose and compare two complementary routes for extracting the vDOS directly from S(q,ω).

The first route exploits the high‑wave‑vector regime (q > q_D, where q_D is the Debye wave number). In this region incoherent, single‑particle‑like scattering dominates, allowing the authors to approximate S_L(q,ω) ≈ (k_BT/2Nq²ω²) M⁻¹(ω) g(ω). Here M(ω) is a mode‑dependent effective mass that reduces to the particle mass for one‑component systems. By averaging over a suitable high‑q window, they recover a vDOS, g_inc(ω), that is mathematically equivalent to the traditional velocity‑autocorrelation‑function method used in molecular dynamics (MD).

The second, more informative route integrates S(q,ω) over the low‑q regime (q < q_D). The authors define a q‑resolved vDOS, g_low‑q(ω)=∫₀^{q_D}dq W(q,ω) S_L(q,ω), with the weight W(q,ω)=k_BT/(2Nq²ω²). Because the integration retains the q‑dependence, one can decompose the total vDOS into contributions from distinct wave‑number sectors, thereby gaining direct insight into the spatial character of the vibrational modes that build up the BP.

To test both approaches, the authors perform extensive MD simulations on three representative glass models: (i) a silica (SiO₂) network glass described by a BKS‑SHIK potential, (ii) a one‑component Lennard‑Jones (LJ) glass, and (iii) a binary soft‑sphere (SS) mixture. System sizes range from 1.5 × 10⁴ to over 10⁶ particles, allowing accurate normal‑mode analyses across a broad frequency window. From the eigenfrequencies and eigenvectors they compute the vDOS, the longitudinal and transverse dynamical structure factors, and the Debye parameters (sound speeds, q_D, ω_D).

Both extraction routes yield virtually identical g(ω) curves, confirming the reliability of the high‑q incoherent approximation. Crucially, the low‑q integration reveals that the BP originates from a nearly dispersionless excitation band in S(q,ω). In the frequency range of the BP, the intensity of S_L(q,ω) (and S_T(q,ω)) remains roughly constant over a wide span of q, forming a flat band that does not follow the usual acoustic dispersion ω ∝ q. When this flat band is integrated over q, it contributes an ω‑independent term to g(ω), which appears as the characteristic excess over the Debye ω² law.

The authors corroborate this picture with an effective‑medium theory (EMT) for random spring networks. EMT provides an analytical expression for the complex wave number k(ω)=q(ω)+i/ℓ(ω) and predicts that near the Ioffe–Regel crossover (ℓ q≈1) the phonon mean free path ℓ collapses, leading to a non‑propagating, dispersionless mode. The EMT‑derived S(q,ω) reproduces the flat band observed in the simulations, demonstrating that the mechanism is not model‑specific but a generic feature of disordered elastic media.

By comparing a covalently bonded network glass (silica) with packing‑dominated glasses (LJ and SS), the study shows that the dispersionless band and the associated BP are universal across both network‑type and packing‑type glasses. This unifies previously disparate interpretations of the BP—whether as a signature of strong acoustic damping, of localized “soft” modes, or of hybridized phonon–defect excitations—under a single, wave‑number‑resolved framework.

The paper concludes that the boson peak is best understood as the spectral fingerprint of a collective, non‑phononic excitation band that lacks conventional dispersion. This insight bridges the gap between experimental scattering data and the microscopic vibrational landscape of glasses, offering a practical route for experimentalists to isolate the BP by focusing on low‑q, dispersionless features in S(q,ω). Future work is suggested on temperature effects, anharmonicity, and multi‑component systems where the effective mass M(ω) becomes non‑trivial.