The predominant hydrogen-containing species in Ti${\mathrm{O}}_2$ (rutile) are O${\mathrm{H}}^{-}$ (hydroxyl) ions, in which the oxygen occupies a regular oxygen ion site, and the O-H bond is perpendicular to the $c$ axis. It is proposed that diffusion of hydrogen parallel to the $c$ axis proceeds by a proton jump from one ${\mathrm{O}}^{2-}$ ion to another along the channel as represented by O${\mathrm{H}}^{-}$···${\mathrm{O}}^{2-}$ → ${\mathrm{O}}^{2-}$···${\mathrm{H}}^{+}$···${\mathrm{O}}^{2-}$ → ${\mathrm{O}}^{2-}$···H${\mathrm{O}}^{-}$. It is also proposed that diffusion perpendicular to the $c$ axis proceeds by a rotation of the O${\mathrm{H}}^{-}$ bond to move the proton from one channel to an adjacent channel, followed by a proton jump to another ${\mathrm{O}}^{2-}$ ion in the same channel. From a potential-energy model, which includes a Morse function to represent the O${\mathrm{H}}^{-}$ bond, as well as electrostatic and repulsive terms, the activation energies for hydrogen and tritium diffusion parallel to the $c$ axis were calculated to be (including a zero-point energy correction) 0.60 and 0.69 eV, respectively, in good agreement with the respective experimental values of 0.59 and 0.75 eV. The calculated activation energy for diffusion perpendicular to the $c$ axis was 1.23 eV (no zero-point energy correction), as compared to the experimental values of 1.28 and 1.11 eV, respectively, for hydrogen and tritium. The calculated equilibrium orientation of the O${\mathrm{H}}^{-}$ ion in Ti${\mathrm{O}}_2$ and the calculated stretching frequency of this species were also in good agreement with the respective experimental results.

• Received 22 September 1978

DOI:https://doi.org/10.1103/PhysRevB.19.4130