Similar analyses to Fig. were performed for the five filled materials (Figs. S2.19-S2.23), and the corresponding 𝑃 ̃ at 300 and 600 K are listed in Table and Table .3. Taking into account the prefactor of 1 (3𝑛 𝑎 ) 2 ⁄ in Eq. 12 gives three possible scenarios: (1) if there are no interactions between the filler and framework modes -i.e. the corresponding Φ 𝜆𝜆 ′ 𝜆 ′′ are zero -the 𝑃 𝜆 would be scaled by 32 2 33 2 ⁄ = 0.94; (2) if the interactions between the filler and framework modes are of comparable strength to those between the framework modes, the 𝑃 𝜆 would be unchanged; and (2) if the filler and framework modes interact more strongly than the framework modes, the 𝑃 𝜆 would increase. For HeCo12Sb24, the 𝑃 ̃ are increased by ~1.3 % compared to pristine CoSb3, indicating a strong interaction between the He and framework modes. For Ne-Xe the values are reduced by 0.5-2 %, indicating that the interactions between the filler and framework modes are not negligible, but are weaker than the interactions among framework modes. This contrast provides a natural explanation for why filling with He leads to a larger reduction in the 𝜏 CRTA than the other four noble gases. The weaker -Page 21 -interaction strengths for the heavier fillers is also consistent with modelling studies on YbFe4Sb12, which found that the third-order (anharmonic) interactions involving Yb were relatively weak. We also examined the effect of the filler atoms on the numbers of energy-conserving scattering channels by comparing the 𝑁 2 (𝐪, 𝜔) defined in Eqs. 14-16 averaged over wavevectors 𝐪, i.e.: