The vibrational spectrum of vitreous silica: rigorous decomposition via recursive orthogonal splitting analysis

Abstract: Our understanding of vibrations in solids currently rests on concepts and techniques designed for crystals and explicitly relying on periodicity, hence inapplicable to amorphous materials. As a consequence, no established framework enables a systematic decomposition of the vibrational spectrum of amorphous solids into contributions associated with well-defined types of atomic motions. This methodological gap obscures the interpretation of various experimental probes of linear response, based on the measurements of acoustic, thermal, or optical properties. Here, we construct such a framework-Recursive Orthogonal Splitting Analysis (ROSA)-which decomposes the vibrational space by recursive applications of the projection formalism. Each step of the procedure exploits a dominant stiffness contrast to split a vibrational displacement subspace into two weakly coupled orthogonal complements. We illustrate ROSA by applying it to the archetypal covalent glass-vitreous silica. Successively separating bond-stretching, symmetric and antisymmetric oxygen motions, isotropic and deviatoric tetrahedral strains, and distinct classes of tetrahedral bending, reveals a hierarchical structure of the space of vibrational degrees of freedom, involving six mutually orthogonal subspaces. These subspaces selectively capture all salient spectral features, including the two-humped structure in the low-frequency range, the peak near 800 cm -1 , and the high-frequency doublet. In the low-frequency range, rigid-unit tetrahedral rotations do not constitute independent degrees of freedom but are kinematically enslaved to bending coordinates by no-stretch constraints. Because ROSA relies solely on the existence of contrasted stiffness scales associated with the point symmetry of local structural units, and on their separation by enforcing geometric constraints via the projection formalism, it is directly applicable to a broad class of covalent network glasses.