Phonons are quasi-particles that quantize the energy of lattice vibrations. The concept of phonon has been developed by Albert Einstein, Peter Joseph William Debye, Max Born, Theodore von Kármán, etc., and was formally named by Jacov Frenkel in 1932. The Debye model has played a pivotal role in history, but it is only applicable to linearly dispersive phonons.

In a study published in Nature Physics, a research team led by Prof. JIANG Minqiang from the Institute of Mechanics of the Chinese Academy of Sciences proposed a unified theory of phonon in solids to show when the Debye model breaks down as Van Hove singularities in crystals, the boson peaks in glasses, or their coexistence.

To explain why the Dulong-Petit law fails to describe the heat capacity of solids at very low temperatures, Einstein in 1907 applied Planck's quantum hypothesis to quantize the energy of lattice vibrations. However, due to the neglect of phonons' interaction, the Einstein model only qualitatively matched experimental observations. 

In 1912, Debye improved the Einstein theory by treating low-frequency phonons as continuum elastic waves, and derived the vibrational density of states (VDOS) as proportional to the squared frequency. The Debye model successfully predicted phononic contribution to the specific heat of solids in the continuum limit.

However, as the phonon wavenumber increases, the VDOS gradually deviates from the Debye prediction and eventually manifests as the Van Hove singularities (VHS) for crystals and a boson peak (BP) for glasses. These non-Debye anomalies have a profound influence on low-temperature properties of solids. In the past decades, there is much controversy about whether these two non-Debye anomalies, VHS and BP, are equivalent or not.

In this study, researchers developed a VDOS model universal for solids, either amorphous materials or ordered crystals. The model derived a phonon damping function that not only describes the low-wavenumber Rayleigh scattering but also captures the Mie damping at higher wavenumbers. This damping directly contributed to an extra softening of phonons in dispersion beyond the inherent softening near the (pseudo-) Brillouin zone boundary, which extends the Debye’s linear dispersion to nonlinear regimes.

In the competition between phonon propagation and damping, the model predicted a panoramic "phase diagram" of non-Debye phonon anomalies: VHS and BP. It demonstrated that the continuous softening of phonons induces either single VHS or single BP. The VHS mainly arose from phonon softening near the Brillouin zone boundary, leading to a piling-up of the VDOS. The BP came from a combined contribution from both extra and inherent softening. 

Accompanied by the occurrence of VHS or BP, the damping deviated from Rayleigh scattering and the acoustic behavior was lost. In this sense, the BP could be regarded as a broadened version of the VHS that moves to low frequencies due to extra softening. When the phonon damping/scattering exhibited a resonance peak, the extra softening in dispersion became significantly localized and thus separated from inherently global softening. In this scenario, the coexistence of VHS and BP emerged, demonstrating they are fundamentally distinct phenomena.

This unified theory, beyond the Debye model, is further supported by a comparison with experimental heat capacity data over a wide range of real solids, including 143 crystalline and glassy substances. The findings of this study settle a debate over the physical relationship between VHS and BP, and deepen the fundamental understanding of the continuum elasticity of real solids.