Parallel computation of complex elementary functions using quaternary signed-digit arithmetic

The quantitative synthesis of four silicon and germanium compounds with the clathrate-II structure, Cs8Na16Si136 (1), Cs8Na16Ge136 (2), Rb8Na16Si136 (3), and Rb8Na16Ge36 (4), and their characterizations are reported. The corresponding Si–Si and Ge–Ge distances are determined with high accuracy from extensive single-crystal X-ray diffraction work. The compounds (cubic, space group Fd m, a=14.7560(4), 15.4805(6), 14.7400(4), and 15.4858(6) ? for 1, 2, 3, and 4, respectively) are stoichiometric, metallic, and remarkably stable. No evidence was found for vacancies in the silicon and germanium networks or partial occupancies of the alkali metal sites. The stoichiometry of these completely filled clathrates is consistent with the measured temperature-independent Pauli paramagnetism (=7.2×10-4, 6.5×10-4, 6.9×10-5, and 1.4×10-4 emu/mol for 1, 2, 3, and 4, respectively) and metallic resistivity (29310-5 -cm).