MgNi2 Hexagonal Laves ($C36$) Structure: AB2_hP24_194_ef_fgh

Picture of Structure; Click for Big Picture
Prototype : MgNi2
AFLOW prototype label : AB2_hP24_194_ef_fgh
Strukturbericht designation : $C36$
Pearson symbol : hP24
Space group number : 194
Space group symbol : $\mbox{P6}_{3}\mbox{/mmc}$
AFLOW prototype command : aflow --proto=AB2_hP24_194_ef_fgh
--params=
$a$,$c/a$,$z_{1}$,$z_{2}$,$z_{3}$,$x_{5}$


Other compounds with this structure

  • NbZn2, ScFe2, ThMg2, HfCr2, UPt2

  • An AFLOW user, Mehdi Noori, identified an inconsistency with this entry and the original reference: the $z$-value for Mg (I) $4e$ should have been 0.09400, not 0.04598. This entry has been corrected (March 19, 2021).

Hexagonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{x}} - \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}}\\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1}& = &z_{1} \, \mathbf{a}_{3}& = &z_{1} \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Mg I} \\ \mathbf{B}_{2}& = &- z_{1} \, \mathbf{a}_{3}& = &- z_{1} \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Mg I} \\ \mathbf{B}_{3}& = &\left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Mg I} \\ \mathbf{B}_{4}& = &\left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Mg I} \\ \mathbf{B}_{5}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+\frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Mg II} \\ \mathbf{B}_{6}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Mg II} \\ \mathbf{B}_{7}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Mg II} \\ \mathbf{B}_{8}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Mg II} \\ \mathbf{B}_{9}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+\frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ni I} \\ \mathbf{B}_{10}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ni I} \\ \mathbf{B}_{11}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ni I} \\ \mathbf{B}_{12}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ni I} \\ \mathbf{B}_{13}& = &\frac12 \, \mathbf{a}_{1}& = &\frac14 \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}4 \, a \, \mathbf{\hat{y}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{14}& = &\frac12 \, \mathbf{a}_{2}& = &\frac14 \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}4 \, a \, \mathbf{\hat{y}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{15}& = &\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}& = &\frac12 \, a \, \mathbf{\hat{x}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{16}& = &\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{3}& = &\frac14 \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}4 \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{17}& = &\frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& = &\frac14 \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}4 \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{18}& = &\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(6g\right) & \mbox{Ni II} \\ \mathbf{B}_{19}& = &x_{5} \, \mathbf{a}_{1}+ 2 \, x_{5} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &\frac32 \, x_{5} \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}{2} \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \mathbf{B}_{20}& = &- 2 \, x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &- \frac32 \, x_{5} \, a \, \mathbf{\hat{x}}+ \frac{\sqrt3}{2} \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \mathbf{B}_{21}& = &x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &- \sqrt3 \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \mathbf{B}_{22}& = &- x_{5} \, \mathbf{a}_{1}- 2 \, x_{5} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &- \frac32 \, x_{5} \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}{2} \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \mathbf{B}_{23}& = &2 \, x_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &\frac32 \, x_{5} \, a \, \mathbf{\hat{x}}- \frac{\sqrt3}{2} \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \mathbf{B}_{24}& = &- x_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &+ \sqrt3 \, x_{5} \, a \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{Ni III} \\ \end{array} \]

References

  • Y. Komura and K. Tokunaga, Structural studies of stacking variants in Mg–base Friauf–Laves phases, Acta Crystallogr. Sect. B Struct. Sci. 36, 1548–1554 (1980), doi:10.1107/S0567740880006565.

Geometry files


Prototype Generator

aflow --proto=AB2_hP24_194_ef_fgh --params=

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