$\beta$–PdCl2 Structure: A2B_hR18_148_2f_f

Picture of Structure; Click for Big Picture
Prototype : PdCl2
AFLOW prototype label : A2B_hR18_148_2f_f
Strukturbericht designation : None
Pearson symbol : hR18
Space group number : 148
Space group symbol : $R\bar{3}$
AFLOW prototype command : aflow --proto=A2B_hR18_148_2f_f [--hex]
--params=
$a$,$c/a$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$


  • (Dell'Amico, 1996) did the original assessment of the crystal structure of $\beta$–PdCl2, but it is difficult to determine the Wyckoff positions from this paper. We relied on (Villars, 2010) for the Wyckoff positions.

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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Cl I} \\ \mathbf{B}_{4} & = & -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(6f\right) & \mbox{Cl I} \\ \mathbf{B}_{5} & = & -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(6f\right) & \mbox{Cl I} \\ \mathbf{B}_{6} & = & -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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Cl II} \\ \mathbf{B}_{10} & = & -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(6f\right) & \mbox{Cl II} \\ \mathbf{B}_{11} & = & -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(6f\right) & \mbox{Cl II} \\ \mathbf{B}_{12} & = & -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(6f\right) & \mbox{Cl 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(6f\right) & \mbox{Pd} \\ \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(6f\right) & \mbox{Pd} \\ \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(6f\right) & \mbox{Pd} \\ \mathbf{B}_{16} & = & -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(6f\right) & \mbox{Pd} \\ \mathbf{B}_{17} & = & -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(6f\right) & \mbox{Pd} \\ \mathbf{B}_{18} & = & -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(6f\right) & \mbox{Pd} \\ \end{array} \]

References

  • D. B. Dell'Amico, F. Calderazzo, F. Marchetti, and S. Ramello, Molecular Structure of [Pd6Cl12] in Single Crystals Chemically Grown at Room Temperature, Angew. Chem. Int. Ed. 35, 1331–1333 (1996), doi:10.1002/anie.199613311.
  • P. Villars and K. Cenzual, eds., Structure Types (Springer, Berlin, Heidelberg, 2010), Landolt–Börnstein – Group III Condensed Matter, vol. 43A8, chap. Part 8: Space Groups (156) P3m1 – (148) R–3, doi:10.1007/978-3-540-70892-6_423.

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