## 3 Results
An evaluation on the electronic thermal conductivity can be facilitated by combining the resistivity \(\rho\) measurements and the Wiedemann-Franz relation (\(\kappa_{\rm e}=L_{0}T\rho\), where \(L_{0}\) = 2.45 \(\times\) 10\({}^{-8}\) W \(\Omega\) K\({}^{-2}\)) [31]. A negligible \(\kappa_{\rm e}\) is normally characterized by a quite large resistivity whose value remains beyond the scope of 10 m\(\Omega\)\(\cdot\)cm [28; 29]. Due to similar reasons, the lattice thermal conductivity of ternary clathrate (Cs,Rb)\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) is given essential importance here. Inspired by the previously suggested \"rattling\" model [32], which states guest impurity resonantly vibrates with the cage constituent, one becomes familiar with the minimization of lattice thermal conductivity. This suppression is furnished when the glass-like phonons, which carry thermal energy, are scattered by a large scale of loosely bound \"rattlers\". Accordingly, our estimated lattice vibration contribution to the thermal conductivity are given from a simple kinetic equation \(\kappa_{\rm L}=(1/3)C_{\rm V}v_{\rm s}\lambda\). Here, \(\lambda\) is also assumed that such neighboring guests act as almost localized \"scattering centers\". Consequently, we predict the lattice thermal conductivity (see Table 2) of the ternary compound Cs\({}_{8}\)Na\({}_{16}\)Si\({}_{136}\), whose value remains lower than the experimental data [23]. To account for this increased \(\kappa_{\rm L}\) arising from the observed measurement, we anticipate that the lattice thermal conductibility is strongly affected by the avoided-crossing effect caused by flat localized phonons rather than guest atoms that serve as scattering centers. In other words, the mean free path of acoustic phonons (transverse acoustic (TA) and longitudinal acoustic (LA)) near the avoided-crossing point might be recalculated due to the predominant three-phonon processes suggested by Ref. [33; 34]. The previously used separation of rattlers (Cs-Cs distance \(\sim\) 6.34 A) shows an underestimation on the mean free path. It is noted that the relation \(\lambda_{\rm q}=v_{\rm q}\tau_{\rm q}\) helps to find the mean free path \(\lambda_{\rm q}\) of acoustic phonons in the presence of a computed phonon lifetime \(\tau_{\rm q}\) while considering cubic anharmonicity originating from three-phonon interaction [34]. Specifically, the phonon lifetime \(\tau_{\rm q}\) due to anharmonic phonon scattering is determined by the inverse of imaginary part of bubble self-energy, which remains roughly proportional to summation corresponding to the strength of the cubic coupling. Previously, the computation of \(\kappa_{\rm L}\) based on the phonon lifetime of the TA, LA mode due to the \"avoided-crossing\" effect leads to a quantitatively good agreement with the experimental data [34; 35; 36; 37]. Figure 4 shows the \"avoided-crossing\" effect because of guest-host coupling with respect to Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) along the \(\Gamma\)-X line. The dotted lines describing the dispersive LA and TA phonons of Al\({}_{8}\)Si\({}_{128}\) are also given for comparison. It is shown that those crossings due to the \"intersection\" of the flat localized Cs modes appear roughly at 40-41 cm\({}^{-1}\). Specifically, crossings of TA phonons and LA phonons occur at approximately 3/5 and 3/10 along the \(\Gamma\)-X line. The consequence of these \"avoided-crossings\" is attributable to strong interactions between the acoustic phonons and rattling modes. Near the crossing point, which is below the optic band of Cs, the group velocity of TA and LA phonons is facing substantial reduction (\(v_{\rm TA(LA)}\)\(\rightarrow\)0). This indicates that the phonon-rattler coupling is dominantly modeled by three-phonon processes at crossings. The suggested \(\lambda_{\rm q}=v_{\rm q}\tau_{\rm q}\) can be used to evaluate the mean free path in a more reasonable way, resulting in a good agreement with the experimentally determined \(\kappa_{\rm L}\).