Rhombohedral CuTi2S4 Structure : AB4C2_hR28_166_2c_2c2h_abh

Picture of Structure; Click for Big Picture
Prototype : CuS4Ti2
AFLOW prototype label : AB4C2_hR28_166_2c_2c2h_abh
Strukturbericht designation : None
Pearson symbol : hR28
Space group number : 166
Space group symbol : $R\bar{3}m$
AFLOW prototype command : aflow --proto=AB4C2_hR28_166_2c_2c2h_abh
--params=
$a$,$c/a$,$x_{3}$,$x_{4}$,$x_{5}$,$x_{6}$,$x_{7}$,$z_{7}$,$x_{8}$,$z_{8}$,$x_{9}$,$z_{9}$


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} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \, \mathbf{\hat{x}} + 0 \, \mathbf{\hat{y}} + 0 \, \mathbf{\hat{z}} & \left(1a\right) & \mbox{Ti I} \\ \mathbf{B}_{2} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}c \, \mathbf{\hat{z}} & \left(1b\right) & \mbox{Ti II} \\ \mathbf{B}_{3} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & x_{3}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Cu I} \\ \mathbf{B}_{4} & = & -x_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & -x_{3}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Cu I} \\ \mathbf{B}_{5} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Cu II} \\ \mathbf{B}_{6} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Cu II} \\ \mathbf{B}_{7} & = & x_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & x_{5}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{S I} \\ \mathbf{B}_{8} & = & -x_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & -x_{5}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{S I} \\ \mathbf{B}_{9} & = & x_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & x_{6}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{S II} \\ \mathbf{B}_{10} & = & -x_{6} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2}-x_{6} \, \mathbf{a}_{3} & = & -x_{6}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{S II} \\ \mathbf{B}_{11} & = & x_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{12} & = & z_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{13} & = & x_{7} \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{14} & = & -z_{7} \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2}-x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{15} & = & -x_{7} \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{16} & = & -x_{7} \, \mathbf{a}_{1}-z_{7} \, \mathbf{a}_{2}-x_{7} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S III} \\ \mathbf{B}_{17} & = & x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{18} & = & z_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + x_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{19} & = & x_{8} \, \mathbf{a}_{1} + z_{8} \, \mathbf{a}_{2} + x_{8} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{20} & = & -z_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2}-x_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{21} & = & -x_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{22} & = & -x_{8} \, \mathbf{a}_{1}-z_{8} \, \mathbf{a}_{2}-x_{8} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{S IV} \\ \mathbf{B}_{23} & = & x_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \mathbf{B}_{24} & = & z_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \mathbf{B}_{25} & = & x_{9} \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2} + x_{9} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \mathbf{B}_{26} & = & -z_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2}-x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \mathbf{B}_{27} & = & -x_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \mathbf{B}_{28} & = & -x_{9} \, \mathbf{a}_{1}-z_{9} \, \mathbf{a}_{2}-x_{9} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{9}+\frac{1}{3}z_{9}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{Ti III} \\ \end{array} \]

References

  • N. Soheilnia, K. M. Kleinke, E. Dashjav, H. L. Cuthbert, J. E. Greedan, and H. Kleinke, Crystal Structure and Physical Properties of a New CuTi2S4 Modification in Comparison to the Thiospinel, Inorg. Chem. 43, 6473–6478 (2004), doi:10.1021/ic0495113.

Geometry files


Prototype Generator

aflow --proto=AB4C2_hR28_166_2c_2c2h_abh --params=

Species:

Running:

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