PdAl Structure: AB_hR26_148_b2f_a2f

Picture of Structure; Click for Big Picture
Prototype : PdAl
AFLOW prototype label : AB_hR26_148_b2f_a2f
Strukturbericht designation : None
Pearson symbol : hR26
Space group number : 148
Space group symbol : $\mbox{R}\bar{3}$
AFLOW prototype command : aflow --proto=AB_hR26_148_b2f_a2f [--hex]
--params=
$a$,$c/a$,$x_3$,$y_3$,$z_3$,$x_4$,$y_4$,$z_4$,$x_5$,$y_5$,$z_5$,$x_6$,$y_6$,$z_6$


  • Hexagonal settings of this structure can be obtained with the option ––hex.

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{Pd I} \\ \mathbf{B_2} & =& \frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =& \frac12 \, c \, \mathbf{\hat{z}}& \left(1b\right) & \mbox{Al I} \\ \mathbf{B_3} & =& x_3 \, \mathbf{a}_{1}+ y_3 \, \mathbf{a}_{2}+ z_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_3 - z_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 y_3 - z_3 - x_3\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_4} & =& z_3 \, \mathbf{a}_{1}+ x_3 \, \mathbf{a}_{2}+ y_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_3 - y_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 x_3 - y_3 - z_3\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_5} & =& y_3 \, \mathbf{a}_{1}+ z_3 \, \mathbf{a}_{2}+ x_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_3 - x_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 z_3 - x_3 - y_3\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_6} & =& - x_3 \, \mathbf{a}_{1}- y_3 \, \mathbf{a}_{2}- z_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_3 - x_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(z_3 + x_3 - 2 y_3\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_7} & =& - z_3 \, \mathbf{a}_{1}- x_3 \, \mathbf{a}_{2}- y_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_3 - z_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(y_3 + z_3 - 2 x_3\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_8} & =& - y_3 \, \mathbf{a}_{1}- z_3 \, \mathbf{a}_{2}- x_3 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_3 - y_3\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(x_3 + y_3 - 2 z_3\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_3 + y_3 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al II} \\ \mathbf{B_9} & =& x_4 \, \mathbf{a}_{1}+ y_4 \, \mathbf{a}_{2}+ z_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_4 - z_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 y_4 - z_4 - x_4\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{10} & =& z_4 \, \mathbf{a}_{1}+ x_4 \, \mathbf{a}_{2}+ y_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_4 - y_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 x_4 - y_4 - z_4\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{11} & =& y_4 \, \mathbf{a}_{1}+ z_4 \, \mathbf{a}_{2}+ x_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_4 - x_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 z_4 - x_4 - y_4\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{12} & =& - x_4 \, \mathbf{a}_{1}- y_4 \, \mathbf{a}_{2}- z_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_4 - x_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(z_4 + x_4 - 2 y_4\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{13} & =& - z_4 \, \mathbf{a}_{1}- x_4 \, \mathbf{a}_{2}- y_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_4 - z_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(y_4 + z_4 - 2 x_4\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{14} & =& - y_4 \, \mathbf{a}_{1}- z_4 \, \mathbf{a}_{2}- x_4 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_4 - y_4\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(x_4 + y_4 - 2 z_4\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_4 + y_4 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Al III} \\ \mathbf{B}_{15} & =& x_5 \, \mathbf{a}_{1}+ y_5 \, \mathbf{a}_{2}+ z_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_5 - z_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 y_5 - z_5 - x_5\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{16} & =& z_5 \, \mathbf{a}_{1}+ x_5 \, \mathbf{a}_{2}+ y_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_5 - y_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 x_5 - y_5 - z_5\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{17} & =& y_5 \, \mathbf{a}_{1}+ z_5 \, \mathbf{a}_{2}+ x_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_5 - x_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 z_5 - x_5 - y_5\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{18} & =& - x_5 \, \mathbf{a}_{1}- y_5 \, \mathbf{a}_{2}- z_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_5 - x_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(z_5 + x_5 - 2 y_5\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{19} & =& - z_5 \, \mathbf{a}_{1}- x_5 \, \mathbf{a}_{2}- y_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_5 - z_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(y_5 + z_5 - 2 x_5\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{20} & =& - y_5 \, \mathbf{a}_{1}- z_5 \, \mathbf{a}_{2}- x_5 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_5 - y_5\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(x_5 + y_5 - 2 z_5\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_5 + y_5 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd II} \\ \mathbf{B}_{21} & =& x_6 \, \mathbf{a}_{1}+ y_6 \, \mathbf{a}_{2}+ z_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_6 - z_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 y_6 - z_6 - x_6\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \mathbf{B}_{22} & =& z_6 \, \mathbf{a}_{1}+ x_6 \, \mathbf{a}_{2}+ y_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_6 - y_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 x_6 - y_6 - z_6\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \mathbf{B}_{23} & =& y_6 \, \mathbf{a}_{1}+ z_6 \, \mathbf{a}_{2}+ x_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_6 - x_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(2 z_6 - x_6 - y_6\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \mathbf{B}_{24} & =& - x_6 \, \mathbf{a}_{1}- y_6 \, \mathbf{a}_{2}- z_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(z_6 - x_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(z_6 + x_6 - 2 y_6\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \mathbf{B}_{25} & =& - z_6 \, \mathbf{a}_{1}- x_6 \, \mathbf{a}_{2}- y_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(y_6 - z_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(y_6 + z_6 - 2 x_6\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \mathbf{B}_{26} & =& - y_6 \, \mathbf{a}_{1}- z_6 \, \mathbf{a}_{2}- x_6 \, \mathbf{a}_{3}& =& \frac12 \, \left(x_6 - y_6\right) \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, \left(x_6 + y_6 - 2 z_6\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_6 + y_6 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(6f\right) & \mbox{Pd III} \\ \end{array} \]

References

Found in

  • P. Villars, K. Cenzual, J. Daams, R. Gladyshevskii, O. Shcherban, V. Dubenskyy, V. Kuprysyuk, and I. Savesyuk, Landolt–Börnstein – Group III Condensed Matter (Springer–Verlag GmbH, Heidelberg, 2010). Accessed through the Springer Materials site.

Geometry files


Prototype Generator

aflow --proto=AB_hR26_148_b2f_a2f --params=

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