Brookite (TiO2, $C21$) Structure: A2B_oP24_61_2c_c

Picture of Structure; Click for Big Picture
Prototype : TiO2
AFLOW prototype label : A2B_oP24_61_2c_c
Strukturbericht designation : $C21$
Pearson symbol : oP24
Space group number : 61
Space group symbol : $\mbox{Pbca}$
AFLOW prototype command : aflow --proto=A2B_oP24_61_2c_c
--params=
$a$,$b/a$,$c/a$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$


Other compounds with this structure

  • TeO2 tellurite

Simple Orthorhombic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \\ \mathbf{a}_2 & = & b \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}} \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&x_{1} \, \mathbf{a}_{1}+ y_{1} \, \mathbf{a}_{2}+ z_{1} \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{2} & =&\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}+ \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{3} & =&- x_{1} \, \mathbf{a}_{1}+ \left(\frac12 + y_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{4} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{1}\right) \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{1}\right) \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{5} & =&- x_{1} \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{6} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ y_{1} \, \mathbf{a}_{2}+ \left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{7} & =&x_{1} \, \mathbf{a}_{1}+ \left(\frac12 - y_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{8} & =&\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{1}\right) \, \mathbf{a}_{2}+ z_{1} \, \mathbf{a}_{3}& =&\left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{1}\right) \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O I} \\ \mathbf{B}_{9} & =&x_{2} \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{10} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{11} & =&- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{12} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{13} & =&- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{14} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{15} & =&x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{16} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{17} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{18} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{19} & =&- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{20} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{21} & =&- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{22} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{23} & =&x_{3} \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \mathbf{B}_{24} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Ti} \\ \end{array} \]

References

  • E. P. Meagher and G. A. Lager, Polyhedral thermal expansion in the TiO2 polymorphs; refinement of the crystal structures of rutile and brookite at high temperature, Can. Mineral. 17, 77–85 (1979).

Geometry files


Prototype Generator

aflow --proto=A2B_oP24_61_2c_c --params=

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