https://phys.org/news/2018-10-rationalizing-phonon-dispersion-efficient-precise.html

This results in a significant deviation of Debye dispersion for periodic crystalline materials when phonons with wave vectors are close to the Brillouin boundaries (high frequency phonons). When these phonons are involved for phonon transport (i.e. at not extremely low temperatures), Debye dispersion leads to an overestimation of lattice thermal conductivity due to the overestimation of group velocity for these high-frequency phonons, as observed in materials with hundreds of known measured lattice thermal conductivity and necessary details for a time- and cost-effective model prediction to our best knowledge (Fig. 2g and h showing a mean absolute deviation of ~+40%). In addition, Debye dispersion overestimates the theoretically available lower bound of lattice thermal conductivity as well, leading the violations of the measured lattice thermal conductivity to be even lower than the current theoretical minimum predicted (based on the Debye-Cahill model) as observed in tens of materials.

This work takes into account the BvK boundary condition, and reveals that the product of acoustic and optical dispersions yields a sine function. In the case of which the mass (or the force constant) contrast between atoms is large, the acoustic dispersion tends to be a sine-function. This sine type dispersion indeed exists in both the simplest and the most complex materials. Approximating the acoustic dispersion to be sine, the BvK boundary condition subsequently reduces the remaining optical branches to be a series of localized modes with a series of constant frequencies. While first-principles calculations enable a more detailed phonon dispersion, a development of rationalized phonon dispersion for a time- and cost-effective prediction of phonon transport is significant due to the time-consuming and computationally expensive for first-principles calculations.