Cr5B3 ($D8_{l}$) Structure: A3B5_tI32_140_ah_cl

Picture of Structure; Click for Big Picture
Prototype : Cr5B3
AFLOW prototype label : A3B5_tI32_140_ah_cl
Strukturbericht designation : $D8_{l}$
Pearson symbol : tI32
Space group number : 140
Space group symbol : $I4/mcm$
AFLOW prototype command : aflow --proto=A3B5_tI32_140_ah_cl
--params=
$a$,$c/a$,$x_{3}$,$x_{4}$,$z_{4}$


Other compounds with this structure

  • Eu5Ge3, Pr5Si3, Ta5Ge3, Nb5Si3, Ta5Si3, EuIrGe2, Fe5PB2, Fe5SiB2, Fe4CoPB2, Fe4MnPB2, Fe4CoSiB2, Fe4MnSiB2, Mo5SiB2, Sr5In3, Gd5CoSi2, Ca5Si3, and Sr5Si3

  • We have been unable to obtain a copy of (Bertaut, 1953), so we use the data found online from (Downs, 2003). Although Cr5B3 is universally regarded as the prototype for $D8_{l}$, it appears that no determination of the internal parameters has ever been made. The values found in (Bertaut, 1953) seem to be resonable choices.

Body-centered Tetragonal primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} & = & \frac{1}{4}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{B I} \\ \mathbf{B}_{2} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} & = & \frac{3}{4}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{B I} \\ \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(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{5} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{y}} & \left(8h\right) & \mbox{B II} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{y}} & \left(8h\right) & \mbox{B II} \\ \mathbf{B}_{7} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} & \left(8h\right) & \mbox{B II} \\ \mathbf{B}_{8} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} & \left(8h\right) & \mbox{B II} \\ \mathbf{B}_{9} & = & \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{11} & = & \left(x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{12} & = & \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{13} & = & \left(x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{14} & = & \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{15} & = & \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - 2x_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16l\right) & \mbox{Cr II} \\ \end{array} \]

References

  • F. Bertaut and P. Blum, Étude des borures de chrome, C. R. Hebd. Séances Acad. Sci. 236, 1055–1056 (1953).

Found in

  • R. T. Downs and M. Hall–Wallace, The American Mineralogist Crystal Structure Database, Am. Mineral. 88, 247–250 (2003).

Geometry files


Prototype Generator

aflow --proto=A3B5_tI32_140_ah_cl --params=

Species:

Running:

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