## 6 Chemical Fingerprints in the Physical Properties
Thus, QCs containing heavier TM atoms, such as AlPdRe and AlCuOs, possess higher resistivity and R ratio values than those observed in AlPdMn and AlCu(Fe,Ru) phases, the i-AlPdRe being the most resistive QC discovered so far, exhibiting maximum \(\varrho\)(4 K) and R values two orders of magnitude larger than those of the isostructural i-AlPdMn phase. Such a resistivity behavior was interpreted as signaling an enhancement of covalent bonding between Al and TM atoms in AlPdRe i-QCs [68] (see Section 7). By inspecting the two last rows of Table 4, one would expect remarkably low electrical conductivity values in QCs belonging to the quaternary system i-AlPdOsRe. However, a room temperature resistivity \(\varrho\) = 1.2 \(\times\) 10\({}^{4}\)\(\upmu\Omega\)cm and a resistivity ratio R = 5 was reported for i-Al\({}_{71.5}\)Pd\({}_{19}\)Os\({}_{5,5}\)Re\({}_{4}\)[69]. These figures are well below those listed for i-AlPdRe phases in Table 4, indicating the peculiar role of fhenium atoms in icosahedral phases. \begin{table}
\begin{tabular}{c c c c c} \hline
**Compound** & \(\mathbf{\Theta_{D}}\) **(K)** & \(\mathbf{T_{m}}\) **(K)** & **Cluster-Type** & **References** \\ \hline Al\({}_{70.5}\)Pd\({}_{21}\)Res\({}_{5.5}\) & 425 & 1298 & Mackay & [57] \\ \hline Al\({}_{63}\)Cu\({}_{25}\)Fe\({}_{12}\) & 509 & 1169 & Mackay & [58] \\ Al\({}_{63}\)Cu\({}_{24}\)Fe\({}_{13}\) & 536 & 1145 & Mackay & [59] \\ Al\({}_{65}\)Cu\({}_{20}\)Fe\({}_{15}\) & 360–560 & 1135–1163 & Mackay & [56; 58; 60] \\ Al\({}_{72}\)Pd\({}_{20}\)Mn\({}_{8}\) & 362 & 1140 & [PERSON] & [61] \\ Ti\({}_{40}\)Zr\({}_{40}\)Ni\({}_{20}\) & 325 & 1083 & [PERSON] & [62] \\ Cd\({}_{85.1}\)Yb\({}_{14.9}\) & 140 & 909 & [PERSON] & [63] \\ Al\({}_{60}\)Li\({}_{30}\)Cu\({}_{10}\) & 465 & 895 & [PERSON] & [64] \\ Au\({}_{44.2}\)In\({}_{41.7}\)Ca\({}_{14.1}\) & & 848 & [PERSON] & [37] \\ Zn\({}_{88}\)Sc\({}_{12}\) & & 778 & [PERSON] & [65] \\ Zn\({}_{43}\)Mg\({}_{37}\)Ga\({}_{20}\) & 240 & 680 & [PERSON] & [8] \\ \hline \end{tabular}
\end{table} 
Table 3: Debye (\(\mathbf{\Theta_{D}}\)) and melting (\(\mathbf{T_{m}}\)) temperatures of i-QCs belonging to different alloy systems listed by their decreasing melting temperature value. The electrical resistivity trends illustrated in Table 4 may arise from the relativistic contraction of the s and p states relative to the d and f ones, which lowers the orbital energies of the former and screens the nucleus effective charge, so that the outer d electrons experience a lesser binding, hence displaying a larger spatial extent. Accordingly, relativistic effects lead to a lowering of the energy of the s and p bands, along with a simultaneous raising of the energy of the d bands. This enhances sp-d hybridization among these bands, ultimately leading to an increase of cohesive energy and a localization of electrons involved in the formation of covalent bonds. In Figure 9a we plot the room temperature electrical resistivity for samples belonging to the i-AlCuRu system with a constant aluminum content but different Cu and Ru concentrations. By inspecting this figure we readily appreciate that the \(\varrho\)(300 K) value systematically increases as we increase the number of electrons per atom. A similar enhancement of the resistivity with increasing concentration of TM atoms was reported in the i-AlPdRe system as well [68]. This is a quite unexpected behavior if we think of it in terms of the free electron model, but it can be better understood if we consider that most electrons are mainly employed in chemical bonding instead, hence becoming essentially localized, rather than extended in character. Indeed, from the crossing among the various spectral electronic distribution curves shown in Figure 9b, one realizes that a substantial electronic overlapping exists between Al 3p and Ru 4d states close to the Fermi level, as well as between Cu 3d and Al 3s,d states in the middle of the band. In particular, the Al-Ru interaction shifts the Al states far from E\({}_{\text{F}}\), thereby deepening the pseudogap [78; 79; 80].