$\beta$–U ($A_{b}$) Structure: A_tP30_136_bf2ij

Picture of Structure; Click for Big Picture
Prototype : $\beta$–U
AFLOW prototype label : A_tP30_136_bf2ij
Strukturbericht designation : $A_{b}$
Pearson symbol : tP30
Space group number : 136
Space group symbol : $\mbox{P4}_{2}\mbox{/mnm}$
AFLOW prototype command : aflow --proto=A_tP30_136_bf2ij
--params=
$a$,$c/a$,$x_{2}$,$x_{3}$,$y_{3}$,$x_{4}$,$y_{4}$,$x_{5}$,$z_{5}$


  • According to (Donohue, 1982), there are three possible space groups which fit the diffraction data for $\beta$–U. This is the highest symmetry space group of the three. Except for a shift of the origin, this structure is crystallographically equivalent to $\sigma$–CrFe (D8b).

Simple Tetragonal primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&\frac12 \, \mathbf{a}_{3} & =&\frac12 c \, \mathbf{\hat{z}} & \left(2b\right) & \mbox{U I} \\ \mathbf{B}_{2} & =&\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac12 \, a \, \mathbf{\hat{y}}& \left(2b\right) & \mbox{U I} \\ \mathbf{B}_{3} & =&x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}& =&x_{2} \, a \, \mathbf{\hat{x}}+ x_{2} \, a \, \mathbf{\hat{y}}& \left(4f\right) & \mbox{U II} \\ \mathbf{B}_{4} & =&- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}& =&- x_{2} \, a \, \mathbf{\hat{x}}- x_{2} \, a \, \mathbf{\hat{y}}& \left(4f\right) & \mbox{U II} \\ \mathbf{B}_{5} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{U II} \\ \mathbf{B}_{6} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{U II} \\ \mathbf{B}_{7} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}& =&x_{3} \, a \, \mathbf{\hat{x}}+ y_{3} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{8} & =&- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{9} & =&\left(\frac12 - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 - y_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{10} & =&\left(\frac12 + y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 + y_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{11} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{12} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{13} & =&y_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}& =&y_{3} \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{14} & =&- y_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}& =&- y_{3} \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U III} \\ \mathbf{B}_{15} & =&x_{4} \, \mathbf{a}_{1}+ y_{4} \, \mathbf{a}_{2}& =&x_{4} \, a \, \mathbf{\hat{x}}+ y_{4} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{16} & =&- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}& =&- x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{17} & =&\left(\frac12 - y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{4}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{18} & =&\left(\frac12 + y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{4}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{19} & =&\left(\frac12 - x_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{20} & =&\left(\frac12 + x_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{21} & =&y_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}& =&y_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{22} & =&- y_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}& =&- y_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}& \left(8i\right) & \mbox{U IV} \\ \mathbf{B}_{23} & =&x_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ x_{5} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{24} & =&- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}- x_{5} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{25} & =&\left(\frac12 - x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{26} & =&\left(\frac12 + x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{27} & =&\left(\frac12 - x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{28} & =&\left(\frac12 + x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{29} & =&x_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ x_{5} \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \mathbf{B}_{30} & =&- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}- x_{5} \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(8j\right) & \mbox{U V} \\ \end{array} \]

References

  • C. W. Tucker, Jr., and P. Senio, An improved determination of the crystal structure of beta–uranium, Acta Cryst. 6, 753–760 (1953), doi:10.1107/S0365110X53002167.

Found in

  • J. Donohue, The Structure of the Elements (Robert E. Krieger Publishing Company, Malabar, Florida, 1982)., pp. 134-147.

Geometry files


Prototype Generator

aflow --proto=A_tP30_136_bf2ij --params=

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