Geometric Resonance Analysis of Superconductivity in CaC₆: Hexagonal and Rhombohedral Descriptions in the Roeser–Huber Framework

The superconducting transition temperature of CaC₆ is investigated within the Roeser–Huber (RH) formalism using both rhombohedral and hexagonal crystallographic representations. While these two descriptions are crystallographically equivalent, they differ in their geometric construction of superconducting paths and near-atom environments. In the rhombohedral representation, only translationally closed Ca–Ca vectors consistent with the primitive lattice are considered, yielding three symmetry-distinct RH paths. In the hexagonal representation, the same superconducting channels are expressed in an expanded conventional cell, where some paths appear as unfolded or symmetry-related sublattice connections. For each representation, the RH path lengths and effective near-atom counts are evaluated and used to compute the superconducting transition temperature. The rhombohedral description yields $T_c^{\rm(calc)} = 10.35$ K, while the hexagonal representation gives $T_c^{\rm(calc)} = 10.91$ K, both in good agreement with the experimental value $T_c^{\rm(exp)} = 11.5$ K. The difference between the calculat\( {The superconducting transition temperature of CaC$_6$ is investigated within the Roeser–Huber (RH) formalism using both rhombohedral and hexagonal crystallographic representations. While these two descriptions are crystallographically equivalent, they differ in their geometric construction of superconducting paths and near-atom environments. In the rhombohedral representation, only translationally closed Ca–Ca vectors consistent with the primitive lattice are considered, yielding three symmetry-distinct RH paths. In the hexagonal representation, the same superconducting channels are expressed in an expanded conventional cell, where some paths appear as unfolded or symmetry-related sublattice connections. For each representation, the RH path lengths and effective near-atom counts are evaluated and used to compute the superconducting transition temperature. The rhombohedral description yields $T_c^{\rm(calc)} = 10.35$ K, while the hexagonal representation gives $T_c^{\rm(calc)} = 10.91$ K, both in good agreement with the experimental value $T_c^{\rm(exp)} = 11.5$ K. The difference between the calculated values amounts to approximately 5.4\%. These results show that the underlying RH superconducting channels and their near-atom environments are representation independent, while minor quantitative differences in $T_c^{\rm(calc)}$ arise from metric redistribution of equivalent paths. This directly confirms that the RH formalism captures intrinsic structural features of superconductivity rather than artifacts of unit-cell representation. \)d values amounts to approximately 5.4\%. These results show that the underlying RH superconducting channels and their near-atom environments are representation independent, while minor quantitative differences in $T_c^{\rm(calc)}$ arise from metric redistribution of equivalent paths. This directly confirms that the RH formalism captures intrinsic structural features of superconductivity rather than artifacts of unit-cell representation.