Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: A2BC4_hR42_167_f_ac_2f

  • M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
  • D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
  • D. Hicks, M.J. Mehl, M. Esters, C. Oses, O. Levy, G.L.W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

$\beta$–BaB2O4 (High–Temperature) Structure : A2BC4_hR42_167_f_ac_2f

Picture of Structure; Click for Big Picture
Prototype : B2BaO4
AFLOW prototype label : A2BC4_hR42_167_f_ac_2f
Strukturbericht designation : None
Pearson symbol : hR42
Space group number : 167
Space group symbol : $R\bar{3}c$
AFLOW prototype command : aflow --proto=A2BC4_hR42_167_f_ac_2f
--params=
$a$,$c/a$,$x_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$


  • This is the high–temperature structure of BaB2O4. Cooling to temperatures between 100–400 °C, it transforms into $\alpha$–BaB2O4. The principle difference between the two forms is the lack of inversion symmetry in the low–temperature structure.
  • This structure was determined by cooling liquid BaB2O4 to room temperature.

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} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}c \, \mathbf{\hat{z}} & \left(2a\right) & \text{Ba I} \\ \mathbf{B}_{2} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{3}{4}c \, \mathbf{\hat{z}} & \left(2a\right) & \text{Ba I} \\ \mathbf{B}_{3} & = & x_{2} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & x_{2}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Ba II} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{2}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Ba II} \\ \mathbf{B}_{5} & = & -x_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & -x_{2}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Ba II} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Ba II} \\ \mathbf{B}_{7} & = & 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(12f\right) & \text{B} \\ \mathbf{B}_{8} & = & 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(12f\right) & \text{B} \\ \mathbf{B}_{9} & = & 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(12f\right) & \text{B} \\ \mathbf{B}_{10} & = & \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(12f\right) & \text{B} \\ \mathbf{B}_{11} & = & \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(12f\right) & \text{B} \\ \mathbf{B}_{12} & = & \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{B} \\ \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(12f\right) & \text{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(12f\right) & \text{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(12f\right) & \text{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(12f\right) & \text{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(12f\right) & \text{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(12f\right) & \text{O I} \\ \mathbf{B}_{25} & = & -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(12f\right) & \text{O I} \\ \mathbf{B}_{26} & = & -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(12f\right) & \text{O I} \\ \mathbf{B}_{27} & = & -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(12f\right) & \text{O I} \\ \mathbf{B}_{28} & = & \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(12f\right) & \text{O I} \\ \mathbf{B}_{29} & = & \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(12f\right) & \text{O I} \\ \mathbf{B}_{30} & = & \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(12f\right) & \text{O I} \\ \mathbf{B}_{31} & = & 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(12f\right) & \text{O II} \\ \mathbf{B}_{32} & = & 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(12f\right) & \text{O II} \\ \mathbf{B}_{33} & = & 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(12f\right) & \text{O II} \\ \mathbf{B}_{34} & = & \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(12f\right) & \text{O II} \\ \mathbf{B}_{35} & = & \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(12f\right) & \text{O II} \\ \mathbf{B}_{36} & = & \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(12f\right) & \text{O II} \\ \mathbf{B}_{37} & = & -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(12f\right) & \text{O II} \\ \mathbf{B}_{38} & = & -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(12f\right) & \text{O II} \\ \mathbf{B}_{39} & = & -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(12f\right) & \text{O II} \\ \mathbf{B}_{40} & = & \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(12f\right) & \text{O II} \\ \mathbf{B}_{41} & = & \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(12f\right) & \text{O II} \\ \mathbf{B}_{42} & = & \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(12f\right) & \text{O II} \\ \end{array} \]

References

  • A. D. Mighell, A. Perloff, and S. Block, The crystal structure of the high temperature form of barium borate, BaO·B2O3, Acta Cryst. 20, 819–823 (1966), doi:10.1107/S0365110X66001920.

Found in

  • 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_167_f_ac_2f --params=

Species:

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Output: