$\beta$–Alumina (Al2O3, $D5_{6}$) Structure : A2B3_hP60_194_3fk_cdef2k

Picture of Structure; Click for Big Picture
Prototype : Al2O3
AFLOW prototype label : A2B3_hP60_194_3fk_cdef2k
Strukturbericht designation : $D5_{6}$
Pearson symbol : hP60
Space group number : 194
Space group symbol : $P6_{3}/mmc$
AFLOW prototype command : aflow --proto=A2B3_hP60_194_3fk_cdef2k
--params=
$a$,$c/a$,$z_{3}$,$z_{4}$,$z_{5}$,$z_{6}$,$z_{7}$,$x_{8}$,$z_{8}$,$x_{9}$,$z_{9}$,$x_{10}$,$z_{10}$


  • (Hermann, 1937) assigned this the Strukturbericht designation $D5_{6}$, calling it $\beta$–corundum, and subtitles the section with small Na2O impurities. As noted by (Gottfried, 1937) and (Le Cars, 1975), the Na impurities replace Al atoms on the ($4f$) Wyckoff positions, along with some unspecified O sites. Charge neutrality requires that some of the displaced atoms remain in the lattice, and they are moved to ($2a$) Wyckoff positions at (0 0 0) and (0 0 1/2). Here we only list the ideal structure.

Hexagonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{x}} - \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{O I} \\ \mathbf{B}_{2} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{O I} \\ \mathbf{B}_{3} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{O II} \\ \mathbf{B}_{4} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{O II} \\ \mathbf{B}_{5} & = & z_{3} \, \mathbf{a}_{3} & = & z_{3}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{7} & = & -z_{3} \, \mathbf{a}_{3} & = & -z_{3}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{8} & = & \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{3}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{9} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al I} \\ \mathbf{B}_{10} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al I} \\ \mathbf{B}_{11} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al I} \\ \mathbf{B}_{12} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{4}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al I} \\ \mathbf{B}_{13} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al II} \\ \mathbf{B}_{14} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al II} \\ \mathbf{B}_{15} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al II} \\ \mathbf{B}_{16} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al II} \\ \mathbf{B}_{17} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al III} \\ \mathbf{B}_{18} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al III} \\ \mathbf{B}_{19} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al III} \\ \mathbf{B}_{20} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{Al III} \\ \mathbf{B}_{21} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{O IV} \\ \mathbf{B}_{22} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{O IV} \\ \mathbf{B}_{23} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{O IV} \\ \mathbf{B}_{24} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(4f\right) & \mbox{O IV} \\ \mathbf{B}_{25} & = & x_{8} \, \mathbf{a}_{1} + 2x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{8}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{26} & = & -2x_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{8}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{27} & = & x_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & -\sqrt{3}x_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{28} & = & -x_{8} \, \mathbf{a}_{1}-2x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{8}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{29} & = & 2x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{8}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{30} & = & -x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \sqrt{3}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{31} & = & 2x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{8}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{32} & = & -x_{8} \, \mathbf{a}_{1}-2x_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{8}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{33} & = & -x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \sqrt{3}x_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{34} & = & -2x_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{8}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{35} & = & x_{8} \, \mathbf{a}_{1} + 2x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{8}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{36} & = & x_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & -\sqrt{3}x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{Al IV} \\ \mathbf{B}_{37} & = & x_{9} \, \mathbf{a}_{1} + 2x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{9}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{38} & = & -2x_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{9}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{39} & = & x_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & -\sqrt{3}x_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{40} & = & -x_{9} \, \mathbf{a}_{1}-2x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{9}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{41} & = & 2x_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{9}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{42} & = & -x_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \sqrt{3}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{43} & = & 2x_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{9}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{44} & = & -x_{9} \, \mathbf{a}_{1}-2x_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{9}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{45} & = & -x_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \sqrt{3}x_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{46} & = & -2x_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{9}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{47} & = & x_{9} \, \mathbf{a}_{1} + 2x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{9}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{48} & = & x_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & -\sqrt{3}x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O V} \\ \mathbf{B}_{49} & = & x_{10} \, \mathbf{a}_{1} + 2x_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{10}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{50} & = & -2x_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{10}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{51} & = & x_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & -\sqrt{3}x_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{52} & = & -x_{10} \, \mathbf{a}_{1}-2x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{10}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{53} & = & 2x_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{10}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{54} & = & -x_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \sqrt{3}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{55} & = & 2x_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{10}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{56} & = & -x_{10} \, \mathbf{a}_{1}-2x_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{10}a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{57} & = & -x_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & \sqrt{3}x_{10}a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{58} & = & -2x_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{10}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{59} & = & x_{10} \, \mathbf{a}_{1} + 2x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \frac{3}{2}x_{10}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \mathbf{B}_{60} & = & x_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & -\sqrt{3}x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(12k\right) & \mbox{O VI} \\ \end{array} \]

References

  • W. L. Bragg, C. Gottfried, and J. West, The Structure of $\beta$ Alumina, Zeitschrift für Kristallographie – Crystalline Materials 77, 255–274 (1931), doi:10.1524/zkri.1931.77.1.255.
  • Y. Le Cars, D. Gratias, R. Portier, and J. Théry, Planar defects in $\beta$–alumina, J. Solid State Chem. 15, 218–222 (1975), doi:10.1016/0022-4596(75)90205-4.

Found in

  • C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928–1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).

Geometry files


Prototype Generator

aflow --proto=A2B3_hP60_194_3fk_cdef2k --params=

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