$\alpha$–BaB2O4 (Low–Temperature) Structure : A2BC4_hR42_161_2b_b_4b

Picture of Structure; Click for Big Picture
Prototype : B2BaO4
AFLOW prototype label : A2BC4_hR42_161_2b_b_4b
Strukturbericht designation : None
Pearson symbol : hR42
Space group number : 161
Space group symbol : $R3c$
AFLOW prototype command : aflow --proto=A2BC4_hR42_161_2b_b_4b
--params=
$a$,$c/a$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$


  • This is the low–temperature structure. Heating to temperatures between 100–400 °C transforms it into $\beta$–BaB2O4. The principle difference between the two forms is the lack of inversion symmetry in the low–temperature structure.

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac{1}{\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

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} & = & \frac{1}{2}\left(x_{1}-z_{1}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{1}+\frac{1}{\sqrt{3}}y_{1}-\frac{1}{2\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{1}+y_{1}+z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{2} & = & z_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + y_{1} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{1}+z_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{1}-\frac{1}{2\sqrt{3}}y_{1}-\frac{1}{2\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{1}+y_{1}+z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{3} & = & y_{1} \, \mathbf{a}_{1} + z_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{1}-\frac{1}{2\sqrt{3}}y_{1}+\frac{1}{\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{1}+y_{1}+z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{1}+z_{1}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{1}+\frac{1}{\sqrt{3}}y_{1}-\frac{1}{2\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{1} + \frac{1}{3}y_{1} + \frac{1}{3}z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{5} & = & \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{1}-z_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{1}-\frac{1}{2\sqrt{3}}y_{1}-\frac{1}{2\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{1} + \frac{1}{3}y_{1} + \frac{1}{3}z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{1}-y_{1}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{1}-\frac{1}{2\sqrt{3}}y_{1}+\frac{1}{\sqrt{3}}z_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{1} + \frac{1}{3}y_{1} + \frac{1}{3}z_{1}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B I} \\ \mathbf{B}_{7} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{2}-z_{2}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{2}+\frac{1}{\sqrt{3}}y_{2}-\frac{1}{2\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{2}+y_{2}+z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{8} & = & z_{2} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + y_{2} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{2}+z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{2}-\frac{1}{2\sqrt{3}}y_{2}-\frac{1}{2\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{2}+y_{2}+z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{9} & = & y_{2} \, \mathbf{a}_{1} + z_{2} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{2}-\frac{1}{2\sqrt{3}}y_{2}+\frac{1}{\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{2}+y_{2}+z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{2}+z_{2}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{2}+\frac{1}{\sqrt{3}}y_{2}-\frac{1}{2\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{2} + \frac{1}{3}y_{2} + \frac{1}{3}z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{11} & = & \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{2}-z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{2}-\frac{1}{2\sqrt{3}}y_{2}-\frac{1}{2\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{2} + \frac{1}{3}y_{2} + \frac{1}{3}z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{2}-y_{2}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{2}-\frac{1}{2\sqrt{3}}y_{2}+\frac{1}{\sqrt{3}}z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{2} + \frac{1}{3}y_{2} + \frac{1}{3}z_{2}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{B II} \\ \mathbf{B}_{13} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{3}+\frac{1}{\sqrt{3}}y_{3}-\frac{1}{2\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{3}+y_{3}+z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{14} & = & z_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + y_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{3}-\frac{1}{2\sqrt{3}}y_{3}-\frac{1}{2\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{3}+y_{3}+z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{15} & = & y_{3} \, \mathbf{a}_{1} + z_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{3}-\frac{1}{2\sqrt{3}}y_{3}+\frac{1}{\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{3}+y_{3}+z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{3}+\frac{1}{\sqrt{3}}y_{3}-\frac{1}{2\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{3} + \frac{1}{3}y_{3} + \frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{17} & = & \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{3}-\frac{1}{2\sqrt{3}}y_{3}-\frac{1}{2\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{3} + \frac{1}{3}y_{3} + \frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{3}-\frac{1}{2\sqrt{3}}y_{3}+\frac{1}{\sqrt{3}}z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{3} + \frac{1}{3}y_{3} + \frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{Ba} \\ \mathbf{B}_{19} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{4}+\frac{1}{\sqrt{3}}y_{4}-\frac{1}{2\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{4}+y_{4}+z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{20} & = & z_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + y_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{4}+z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{4}-\frac{1}{2\sqrt{3}}y_{4}-\frac{1}{2\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{4}+y_{4}+z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{21} & = & y_{4} \, \mathbf{a}_{1} + z_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{4}-\frac{1}{2\sqrt{3}}y_{4}+\frac{1}{\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{4}+y_{4}+z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{4}+\frac{1}{\sqrt{3}}y_{4}-\frac{1}{2\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{4} + \frac{1}{3}y_{4} + \frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{23} & = & \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{4}-\frac{1}{2\sqrt{3}}y_{4}-\frac{1}{2\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{4} + \frac{1}{3}y_{4} + \frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{4}-\frac{1}{2\sqrt{3}}y_{4}+\frac{1}{\sqrt{3}}z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{4} + \frac{1}{3}y_{4} + \frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O I} \\ \mathbf{B}_{25} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{26} & = & z_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{27} & = & y_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{29} & = & \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O II} \\ \mathbf{B}_{31} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{32} & = & z_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + y_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{33} & = & y_{6} \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O III} \\ \mathbf{B}_{37} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \mathbf{B}_{38} & = & z_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + y_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \mathbf{B}_{39} & = & y_{7} \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+y_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \mathbf{B}_{41} & = & \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6b\right) & \mbox{O IV} \\ \end{array} \]

References

  • R. Fröhlich, Crystal Structure of the low–temperature form of BaB2O4, Zeitschrift für Kristallographie – Crystalline Materials 168, 109–112 (1984), doi:10.1524/zkri.1984.168.14.109.

Geometry files


Prototype Generator

aflow --proto=A2BC4_hR42_161_2b_b_4b --params=

Species:

Running:

Output: