where 𝐶 𝜆 is the modal volumetric heat capacity, 𝒗 𝜆 ⊗ 𝒗 𝜆 is the outer product of the group velocity 𝒗 𝜆 , 𝜏 𝜆 is the lifetime, 𝑁 is the number of phonon wavevectors 𝐪 included in the summation, and 𝑉 0 is the volume of the unit cell. The product 𝒗 𝜆 𝜏 𝜆 is the phonon mean-free path 𝚲 𝜆 (MFP), which appears in an alternative form of Eq. 2.