Ionic order and defect conductivity in the one-dimensional superionic conductor hollandite

The effects of impurities and stoichiometry on the ionic ordering proposed by Beyeler for hollandite (${\mathrm{K}}_{1.54}$ ${\mathrm{Mg}}_{0.77}$ ${\mathrm{Ti}}_{7.23}$ ${\mathrm{O}}_{16}$) are studied. Even relatively strong impurities are found to have little qualitative effect on the predicted x-ray diffraction peaks for the ground state of this model. Deviation of the potassium concentration from the perfect stoichiometric concentration of 0.75 ions per available site introduces new peaks in the x-ray scattering by enlarging the size of the unit cell from 4 to 13 lattice sites. The activation energy for the conductivity due to thermally activated defects in the ordered phase is an order of magnitude smaller than that observed in actual conductivity measurements.