Mg3Ru2 Structure : A3B2_cP20_213_d_c

Picture of Structure; Click for Big Picture
Prototype : Mg3Ru2
AFLOW prototype label : A3B2_cP20_213_d_c
Strukturbericht designation : None
Pearson symbol : cP20
Space group number : 213
Space group symbol : $P4_{1}32$
AFLOW prototype command : aflow --proto=A3B2_cP20_213_d_c
--params=
$a$,$x_{1}$,$y_{2}$



Simple Cubic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \\ \mathbf{a}_2 & = & a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & a \, \mathbf{\hat{z}} \\ \end{array} \]

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} + x_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}} + x_{1}a \, \mathbf{\hat{y}} + x_{1}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{1}\right)a \, \mathbf{\hat{x}}-x_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{3} & = & -x_{1} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{2}-x_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{1}\right)a \, \mathbf{\hat{y}}-x_{1}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{5} & = & \left(\frac{3}{4} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{6} & = & \left(\frac{3}{4} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{7} & = & \left(\frac{1}{4} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{8} & = & \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Ru} \\ \mathbf{B}_{9} & = & \frac{1}{8} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{4} +y_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{10} & = & \frac{3}{8} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{3}{4} +y_{2}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{11} & = & \frac{7}{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - y_{2}\right) \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{12} & = & \frac{5}{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - y_{2}\right) \, \mathbf{a}_{3} & = & \frac{5}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{13} & = & \left(\frac{1}{4} +y_{2}\right) \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + y_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + y_{2}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{14} & = & \left(\frac{3}{4} +y_{2}\right) \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2}-y_{2} \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}}-y_{2}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{15} & = & \left(\frac{1}{4} - y_{2}\right) \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{2}\right)a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{16} & = & \left(\frac{3}{4} - y_{2}\right) \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{2}\right)a \, \mathbf{\hat{x}} + \frac{5}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{2}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{17} & = & y_{2} \, \mathbf{a}_{1} + \left(\frac{1}{4} +y_{2}\right) \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & y_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{18} & = & -y_{2} \, \mathbf{a}_{1} + \left(\frac{3}{4} +y_{2}\right) \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & -y_{2}a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +y_{2}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{19} & = & \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - y_{2}\right) \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{2}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - y_{2}\right) \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-y_{2}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Mg} \\ \end{array} \]

References

  • R. Pöttgen, V. Hlukhyy, A. Baranov, and Y. Grin, Crystal Structure and Chemical Bonding of Mg3Ru2, Inorg. Chem. 47, 6051–6055 (2008), doi:10.1021/ic800387a.

Geometry files


Prototype Generator

aflow --proto=A3B2_cP20_213_d_c --params=

Species:

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