Zn22Zr Structure : A22B_cF184_227_cdfg_a

Picture of Structure; Click for Big Picture
Prototype : Zn22Zr
AFLOW prototype label : A22B_cF184_227_cdfg_a
Strukturbericht designation : None
Pearson symbol : cF184
Space group number : 227
Space group symbol : $Fd\bar{3}m$
AFLOW prototype command : aflow --proto=A22B_cF184_227_cdfg_a
--params=
$a$,$x_{4}$,$x_{5}$,$z_{5}$


Other compounds with this structure

  • Be22Mo and Be22W

  • (Samson, 1961) gives the atomic coordinates in terms of setting 1 of space group $F3m$ #227. We have shifted this to the standard setting 2, where the inversion site of the lattice is at the origin.
  • Samson also suggests that the Zn14Zr structure is created when zirconium atoms replace some of the zinc atoms on the ($16c$) site [the ($16d$) site in Samson's orientation].
  • This structure can be derived from the Mg3Cr2Al18 structure.

Face-centered Cubic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \mbox{Zr} \\ \mathbf{B}_{2} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \mbox{Zr} \\ \mathbf{B}_{3} & = & 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(16c\right) & \mbox{Zn I} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} & \left(16c\right) & \mbox{Zn I} \\ \mathbf{B}_{5} & = & \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Zn I} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{1} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Zn I} \\ \mathbf{B}_{7} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Zn II} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Zn II} \\ \mathbf{B}_{9} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Zn II} \\ \mathbf{B}_{10} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Zn II} \\ \mathbf{B}_{11} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{12} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{13} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{14} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{15} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{16} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{17} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{18} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{19} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{20} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{21} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{22} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{Zn III} \\ \mathbf{B}_{23} & = & z_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{24} & = & z_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{25} & = & \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{27} & = & \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{29} & = & z_{5} \, \mathbf{a}_{1} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{30} & = & z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{31} & = & z_{5} \, \mathbf{a}_{1} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{32} & = & z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{33} & = & \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{34} & = & \left(2x_{5}-z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{5}\right)a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{35} & = & -z_{5} \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{36} & = & -z_{5} \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{37} & = & \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{39} & = & \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{41} & = & -z_{5} \, \mathbf{a}_{1} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{42} & = & -z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{43} & = & -z_{5} \, \mathbf{a}_{1} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{44} & = & -z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{45} & = & \left(\frac{1}{2} +2x_{5} + z_{5}\right) \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \mathbf{B}_{46} & = & \left(-2x_{5}+z_{5}\right) \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96g\right) & \mbox{Zn IV} \\ \end{array} \]

References

Found in

  • T. B. Massalski, H. Okamoto, P. R. Subramanian, and L. Kacprzak, eds., Binary Alloy Phase Diagrams, vol. 3 (ASM International, Materials Park, Ohio, USA, 1990), 2nd edn. Hf–Re to Zn–Zr.

Geometry files


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aflow --proto=A22B_cF184_227_cdfg_a --params=

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