$\alpha$–Mn ($A12$) Structure: A_cI58_217_ac2g

Picture of Structure; Click for Big Picture
Prototype : $\alpha$–Mn
AFLOW prototype label : A_cI58_217_ac2g
Strukturbericht designation : $A12$
Pearson symbol : cI58
Space group number : 217
Space group symbol : $\mbox{I}\bar{4}\mbox{3m}$
AFLOW prototype command : aflow --proto=A_cI58_217_ac2g
--params=
$a$,$x_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$


Body-centered Cubic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = &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(2a\right) & \mbox{Mn I} \\ \mathbf{B}_{2} & = &2 x_{2} \, \mathbf{a}_{1}+ 2 x_{2} \, \mathbf{a}_{2}+ 2 x_{2} \, \mathbf{a}_{3}& = &x_{2} \, \, a \, \mathbf{\hat{x}}+ x_{2} \, \, a \, \mathbf{\hat{y}}+ x_{2} \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn II} \\ \mathbf{B}_{3} & = &- 2 x_{2} \, \mathbf{a}_{3}& = &- x_{2} \, \, a \, \mathbf{\hat{x}}- x_{2} \, \, a \, \mathbf{\hat{y}}+ x_{2} \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn II} \\ \mathbf{B}_{4} & = &- 2 x_{2} \, \mathbf{a}_{2}& = &- x_{2} \, \, a \, \mathbf{\hat{x}}+ x_{2} \, \, a \, \mathbf{\hat{y}}- x_{2} \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn II} \\ \mathbf{B}_{5} & = &- 2 x_{2} \, \mathbf{a}_{1}& = &x_{2} \, \, a \, \mathbf{\hat{x}}- x_{2} \, \, a \, \mathbf{\hat{y}}- x_{2} \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn II} \\ \mathbf{B}_{6} & = &\left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ 2 x_{3} \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}+ z_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{7} & = &\left(z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{2}- 2 x_{3} \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}+ z_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{8} & = &\left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}- z_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{9} & = &- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}& = &x_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}- z_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{10} & = &2 x_{3} \, \mathbf{a}_{1}+ \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &z_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{11} & = &- 2 x_{3} \, \mathbf{a}_{1}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &z_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{12} & = &\left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &- z_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{13} & = &- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &- z_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{14} & = &\left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ 2 x_{3} \, \mathbf{a}_{2}+ \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}+ z_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{15} & = &\left(z_{3} - x_{3}\right) \, \mathbf{a}_{1}- 2 x_{3} \, \mathbf{a}_{2}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}+ z_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{16} & = &- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}- z_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{17} & = &\left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}- z_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn III} \\ \mathbf{B}_{18} & = &\left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ 2 x_{4} \, \mathbf{a}_{3}& = &x_{4} \, \, a \, \mathbf{\hat{x}}+ x_{4} \, \, a \, \mathbf{\hat{y}}+ z_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{19} & = &\left(z_{4} - x_{4}\right) \, \mathbf{a}_{1}+ \left(z_{4} - x_{4}\right) \, \mathbf{a}_{2}- 2 x_{4} \, \mathbf{a}_{3}& = &- x_{4} \, \, a \, \mathbf{\hat{x}}- x_{4} \, \, a \, \mathbf{\hat{y}}+ z_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{20} & = &\left(x_{4} - z_{4}\right) \, \mathbf{a}_{1}- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}& = &- x_{4} \, \, a \, \mathbf{\hat{x}}+ x_{4} \, \, a \, \mathbf{\hat{y}}- z_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{21} & = &- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} - z_{4}\right) \, \mathbf{a}_{2}& = &x_{4} \, \, a \, \mathbf{\hat{x}}- x_{4} \, \, a \, \mathbf{\hat{y}}- z_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{22} & = &2 x_{4} \, \mathbf{a}_{1}+ \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} + z_{4}\right) \, \mathbf{a}_{3}& = &z_{4} \, \, a \, \mathbf{\hat{x}}+ x_{4} \, \, a \, \mathbf{\hat{y}}+ x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{23} & = &- 2 x_{4} \, \mathbf{a}_{1}+ \left(z_{4} - x_{4}\right) \, \mathbf{a}_{2}+ \left(z_{4} - x_{4}\right) \, \mathbf{a}_{3}& = &z_{4} \, \, a \, \mathbf{\hat{x}}- x_{4} \, \, a \, \mathbf{\hat{y}}- x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{24} & = &\left(x_{4} - z_{4}\right) \, \mathbf{a}_{2}- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{3}& = &- z_{4} \, \, a \, \mathbf{\hat{x}}- x_{4} \, \, a \, \mathbf{\hat{y}}+ x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{25} & = &- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} - z_{4}\right) \, \mathbf{a}_{3}& = &- z_{4} \, \, a \, \mathbf{\hat{x}}+ x_{4} \, \, a \, \mathbf{\hat{y}}- x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{26} & = &\left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ 2 x_{4} \, \mathbf{a}_{2}+ \left(x_{4} + z_{4}\right) \, \mathbf{a}_{3}& = &x_{4} \, \, a \, \mathbf{\hat{x}}+ z_{4} \, \, a \, \mathbf{\hat{y}}+ x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{27} & = &\left(z_{4} - x_{4}\right) \, \mathbf{a}_{1}- 2 x_{4} \, \mathbf{a}_{2}+ \left(z_{4} - x_{4}\right) \, \mathbf{a}_{3}& = &- x_{4} \, \, a \, \mathbf{\hat{x}}+ z_{4} \, \, a \, \mathbf{\hat{y}}- x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{28} & = &- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} - z_{4}\right) \, \mathbf{a}_{3}& = &x_{4} \, \, a \, \mathbf{\hat{x}}- z_{4} \, \, a \, \mathbf{\hat{y}}- x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \mathbf{B}_{29} & = &\left(x_{4} - z_{4}\right) \, \mathbf{a}_{1}- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{3}& = &- x_{4} \, \, a \, \mathbf{\hat{x}}- z_{4} \, \, a \, \mathbf{\hat{y}}+ x_{4} \, \, a \, \mathbf{\hat{z}}& \left(24g\right) & \mbox{Mn IV} \\ \end{array} \]

References

  • J. A. Oberteuffer and J. A. Ibers, A refinement of the atomic and thermal parameters of alpha–manganese from a single crystal, Acta Crystallogr. Sect. B Struct. Sci. 26, 1499–1504 (1970), doi:10.1107/S0567740870004399.

Found in

  • J. Donohue, The Structure of the Elements (Robert E. Krieger Publishing Company, Malabar, Florida, 1982)., pp. 191-196.

Geometry files


Prototype Generator

aflow --proto=A_cI58_217_ac2g --params=

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