$\beta$–Mn ($A13$) Structure: A_cP20_213_cd

Picture of Structure; Click for Big Picture
Prototype : $\beta$–Mn
AFLOW prototype label : A_cP20_213_cd
Strukturbericht designation : $A13$
Pearson symbol : cP20
Space group number : 213
Space group symbol : $\mbox{P4}_{1}\mbox{32}$
AFLOW prototype command : aflow --proto=A_cP20_213_cd
--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{Mn I} \\ \mathbf{B}_{2} & = &\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - x_{1}\right) \, \, a \, \mathbf{\hat{x}}- x_{1} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{3} & = &- x_{1} \, \mathbf{a}_{1}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{1}\right) \, \mathbf{a}_{3}& = &- x_{1} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{1}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 - x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{4} & = &\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{1}\right) \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}& = &\left(\frac12 + x_{1}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{1}\right) \, \, a \, \mathbf{\hat{y}}- x_{1} \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{5} & = &\left(\frac34 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac14 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac14 - x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac34 + x_{1}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{1}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{6} & = &\left(\frac34 - x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac34 - x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac34 - x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac34 - x_{1}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac34 - x_{1}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac34 - x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{7} & = &\left(\frac14 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac14 - x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac34 + x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + x_{1}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{1}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac34 + x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{8} & = &\left(\frac14 - x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac34 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac14 + x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - x_{1}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac34 + x_{1}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{1}\right) \, \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Mn I} \\ \mathbf{B}_{9} & = &\frac18 \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ \left(\frac14 + y_{2}\right) \, \mathbf{a}_{3}& = &\frac18 \, \, a \, \mathbf{\hat{x}}+ y_{2} \, \, a \, \mathbf{\hat{y}}+ \left(\frac14 + y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{10} & = &\frac38 \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+ \left(\frac34 + y_{2}\right) \, \mathbf{a}_{3}& = &\frac38 \, \, a \, \mathbf{\hat{x}}- y_{2} \, \, a \, \mathbf{\hat{y}}+ \left(\frac34 + y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{11} & = &\frac78 \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac14 - y_{2}\right) \, \mathbf{a}_{3}& = &\frac78 \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac14 - y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{12} & = &\frac58 \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac34 - y_{2}\right) \, \mathbf{a}_{3}& = &\frac58 \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \left(\frac34 - y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{13} & = &\left(\frac14 + y_{2}\right) \, \mathbf{a}_{1}+ \frac18 \, \mathbf{a}_{2}+ y_{2} \, \mathbf{a}_{3}& = &\left(\frac14 + y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \frac18 \, \, a \, \mathbf{\hat{y}}+ y_{2} \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{14} & = &\left(\frac34 + y_{2}\right) \, \mathbf{a}_{1}+ \frac38 \, \mathbf{a}_{2}- y_{2} \, \mathbf{a}_{3}& = &\left(\frac34 + y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \frac38 \, \, a \, \mathbf{\hat{y}}- y_{2} \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{15} & = &\left(\frac14 - y_{2}\right) \, \mathbf{a}_{1}+ \frac78 \, \mathbf{a}_{2}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \frac78 \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{16} & = &\left(\frac34 - y_{2}\right) \, \mathbf{a}_{1}+ \frac58 \, \mathbf{a}_{2}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac34 - y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \frac58 \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 - y_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{17} & = &y_{2} \, \mathbf{a}_{1}+ \left(\frac14 + y_{2}\right) \, \mathbf{a}_{2}+ \frac18 \, \mathbf{a}_{3}& = &y_{2} \, \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \frac18 \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{18} & = &- y_{2} \, \mathbf{a}_{1}+ \left(\frac34 + y_{2}\right) \, \mathbf{a}_{2}+ \frac38 \, \mathbf{a}_{3}& = &- y_{2} \, \, a \, \mathbf{\hat{x}}+ \left(\frac34 + y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \frac38 \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{19} & = &\left(\frac12 + y_{2}\right) \, \mathbf{a}_{1}+ \left(\frac14 - y_{2}\right) \, \mathbf{a}_{2}+ \frac78 \, \mathbf{a}_{3}& = &\left(\frac12 + y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \frac78 \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \mathbf{B}_{20} & = &\left(\frac12 - y_{2}\right) \, \mathbf{a}_{1}+ \left(\frac34 - y_{2}\right) \, \mathbf{a}_{2}+ \frac58 \, \mathbf{a}_{3}& = &\left(\frac12 - y_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \left(\frac34 - y_{2}\right) \, \, a \, \mathbf{\hat{y}}+ \frac58 \, \, a \, \mathbf{\hat{z}}& \left(12d\right) & \mbox{Mn II} \\ \end{array} \]

References

  • C. Brink Shoemaker, D. P. Shoemaker, T. E. Hopkins, and S. Yindepit, Refinement of the structure of beta–manganese and of a related phase in the Mn–Ni–Si system, Acta Crystallogr. Sect. B Struct. Sci. 34, 3573–3576 (1978), doi:10.1107/S0567740878011620.

Geometry files


Prototype Generator

aflow --proto=A_cP20_213_cd --params=

Species:

Running:

Output: