Peculiarity of atomic displacements in garnets discovered through a single-crystal X-ray diffraction study of the Na-bearing vanadate garnet series

The structure refinements of the two Na-bearing vanadate garnet series Ca₂NaM₂V₃O₁₂ (M = Co, Ni) and Pb₂NaM₂V₃O₁₂ (M = Mg, Co, Ni) were conducted using single-crystal X-ray diffraction, together with the rerefinements of previously reported Ca₂NaM₂V₃O₁₂ (M = Mg, Mn, Zn, Cd). The crystal chemistry of both vanadate garnet series was discussed with special attention to atomic displacement behaviours, from the combination of the present results with those reported previously for various other garnet compounds. In the investigated compounds, except for Ca₂NaCd₂V₃O₁₂, the dodecahedral-dodecahedral and dodecahedral-octahedral shared edge lengths are unusually longer than or comparable to the corresponding unshared ones, contradicting Pauling's third rule. However, judging from the observation in the Ca₂NaM₂V₃O₁₂ series with a larger number of samples, the ratios of both shared edge lengths to the corresponding unshared ones decrease linearly with increasing octahedral-site cation size. Consequently, in Ca₂NaCd₂V₃O₁₂, with the largest octahedral cation (Cd²⁺) among those investigated here, the dodecahedral-dodecahedral and dodecahedral-octahedral shared edge lengths are significantly shortened compared with the corresponding unshared ones. These observations suggest that the relationships between both shared edge lengths and the corresponding unshared ones respond flexibly to structural geometric constraints depending on the octahedral-site cation size, rather than being subjected to Pauling's third rule. This flexibility results from the insignificant cation-cation repulsions across these shared edges, which are due to the lower mean valence (1.67+) and larger size of the dodecahedral cations than those of previously reported other garnets. In contrast, in all the investigated compounds, including Ca₂NaCd₂V₃O₁₂, the dodecahedral-tetrahedral shared edge lengths are sufficiently shorter than the tetrahedral unshared ones because of the strong repulsion between the dodecahedral and tetrahedral cations brought about by the larger valence of V⁵⁺ at the tetrahedral site. This strong repulsion should result in the smallest mean-square displacements of the tetrahedral cations in the direction of adjacent dodecahedral cations. However, in the Pb₂NaM₂V₃O₁₂ series, the displacements are largest in this direction. This peculiar property is generally observed when the dodecahedral sites are occupied by different cation species with different valences and largely different sizes, as in this series.