X-ray diffuse scattering from alkali, silver, and europiumβ-alumina

Crystals of varying compensation mechanism (extra oxygen bridges or ${\mathrm{Co}}^{2+}$ substitution) and phase ($\beta$ or ${\beta }^{\prime \prime }$) have been grown by adding cobalt oxide to a melt containing NaC${\mathrm{O}}_3$, ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$, and ${\mathrm{Bi}}_2$ ${\mathrm{O}}_3$ in the proportion 2.9:4.4:100. In $\mathrm{}(1+x)\mathrm{}{\mathrm{Na}}_2\mathrm{O}·11{\mathrm{Al}}_2{\mathrm{O}}_3$ the value of $x$ is ∼0.16(3) for crystals grown from the melt and ∼0.38(3) for crystals grown from the flux. Diffuse x-ray scattering measurements confirm the formation of Al Frenkel defects which lock the extra oxygen ions in the diffusing plane. Rods of diffuse scattering are observed in $\beta$-alumina grown from the flux with Ag, K, Rb, and Eu substituted for Na in the diffusing plane. The rods are at ($h+\frac{1}{3},k+\frac{1}{3}$) and ($h+\frac{2}{3},k+\frac{2}{3}$) but not at integral ($\mathrm{hk}$). A two-dimensional short-range-order model consisting of three types of cells: (i) containing one diffusing ion, (ii) two ions, and (iii) 2 ions plus an extra bridge [cells (ii) and (iii) have no diffusing ions in the Eu case] agrees semiquantitatively with the observed data. The conditional probability for doubly occupied cells or defect cells being nearest neighbors is $P_{22}\mathrm{}\left(1\right)=P_{33}\mathrm{}\left(1\right)\mathrm{}\sim 0.05$ (random value is 1/6) and the correlations are assumed to fall off as $\frac{e^{\frac{-R}{\xi }}}{R^{\frac{1}{2}}}$ with the signs of the coefficients given by a superlattice which isolates cell (ii) and (iii). The correlation lengths in the plane and the activation energy for ionic conductivity both vary linearly with the concentration of extra bridges for K $\beta$-alumina. The correlation lengths in units of $a_0$ are 0.9(1), 1.2(1), 2.5(1), and 6.1(5) for Na, Ag, K, and Rb from the flux and 1.0(1), 1.2(1), 1.3(1) for Ag, K, and Rb from melt-grown crystals.