Bonding study of TiC and TiN. I. High-precision x-ray-diffraction determination of the valence-electron density distribution, Debye-Waller temperature factors, and atomic static displacements in ${\mathrm{TiC}}_{0.94}$ and ${\mathrm{TiN}}_{0.99}$

Single-crystal, high-precision, high-resolution x-ray-diffraction measurements of the substoichiometric refractory compounds TiC and TiN have been performed with AgKα radiation. Severe anisotropic general extinction affects the intense low-order reflections. Inhomogeneity in the mosaic spread and domain size produces small but significant differences between reflection and antireflection for the same plane of diffraction. These effects have been modeled and refined together with a scale factor, isotropic thermal parameters, a population parameter of the nonmetal site, the amplitude of metal-atom static displacements around nonmetal vacancies, and an atomic model which includes occupancy factors of the separate orbital contributions of the valence electrons combined with κ expansion-contraction parameters. At convergence, the ‘‘agreement indices’’ (or ‘‘reliability factors’’) were R=0.0025 for ${\mathrm{TiC}}_{0.94}$ and R=0.0023 for ${\mathrm{TiN}}_{0.99}$. The refined population parameters indicate a chemical composition of ${\mathrm{TiC}}_{0.939\left(9\right)\mathrm{}}$ and ${\mathrm{TiN}}_{0.99\left(2\right)\mathrm{}}$. The mean-square amplitudes of thermal vibrations, 〈$u^2$ ${〉}_{\mathrm{Ti}}$=0.002 38(2) A${\mathrm{̊}}^2$, 〈$u^2$ ${〉}_{\mathrm{C}}$=0.003 35(8) A${\mathrm{̊}}^2$, 〈$u^2$ ${〉}_{\mathrm{Ti}}$=0.002 94(1) A${\mathrm{̊}}^2$, and 〈$u^2$ ${〉}_{\mathrm{N}}$=0.003 08(12) A${\mathrm{̊}}^2$ are consistent with the respective atomic masses. 36% of the metal atoms in ${\mathrm{TiC}}_{0.94}$ are involved in a relaxation around the nonmetal vacancies, being displaced from their sublattice sites by 0.097(2) Å along [100]. No evidence for static displacements was found in ${\mathrm{TiN}}_{0.99}$. The valence-electron density distribution can be described satisfactorily in terms of deformed atoms. No buildup of charge density occurs between atomic sites. Our analysis, similar to a Mulliken partitioning, shows first that ionicity is important, with a charge transfer from the metal to the nonmetal of [2.1(4)]e in the carbide and [1.9(4)]e in the nitride, and secondly that the charge asphericity around the metal atoms is larger in the former than in the latter, while no departure from spherical symmetry is observed around the nonmetal atoms. The titanium 3d electrons can be split into a spherical shell that contains [1.27(6)]e plus an excess of [0.24(5)]e shared by two orbitals of $e_g$ symmetry in the carbide and conversely into a spherical shell that contains [0.88(11)]e plus an excess of [0.12(9)]e shared by three orbitals of $t_{2g}$ symmetry in the nitride. This suggests that the metal-to-metal bonding is similar in TiC and in TiN while the metal-to-nonmetal bonding is greater in TiC than in TiN.