Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: A3B24C_hR28_160_b_2b3c_a

  • 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)

SbI3S24 Structure : A3B24C_hR28_160_b_2b3c_a

Picture of Structure; Click for Big Picture
Prototype : I3S24Sb
AFLOW prototype label : A3B24C_hR28_160_b_2b3c_a
Strukturbericht designation : None
Pearson symbol : hR28
Space group number : 160
Space group symbol : $R3m$
AFLOW prototype command : aflow --proto=A3B24C_hR28_160_b_2b3c_a
--params=
$a$,$c/a$,$x_{1}$,$x_{2}$,$z_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$


  • Since there are three S8 molecules in this structure, (Bjorvatten, 1963) refer to it as SbI3:3S8.
  • Space group $R3m$ #160 does not fix the zero of the $z$–axis. Here it is set to coincide with the plane of the iodine atoms.

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

References

Geometry files


Prototype Generator

aflow --proto=A3B24C_hR28_160_b_2b3c_a --params=

Species:

Running:

Output: