$\beta'$–AuCd ($B19$) Structure: AB_oP4_51_e_f

Picture of Structure; Click for Big Picture
Prototype : $\beta'$–AuCd
AFLOW prototype label : AB_oP4_51_e_f
Strukturbericht designation : $B19$
Pearson symbol : oP4
Space group number : 51
Space group symbol : $\mbox{Pmma}$
AFLOW prototype command : aflow --proto=AB_oP4_51_e_f
--params=
$a$,$b/a$,$c/a$,$z_{1}$,$z_{2}$


  • When $a = b = c$, $z_{1} = 1/4$, and $z_{2} = 3/4$ the atoms are on the sites of a face-centered cubic lattice. When $a = c$, $z_{1} = 1/4$, and $z_{2} = 3/4$ the system reduces to the L10 (AuCu) structure. When $a/b = (8/3)^{2/3}$, $c/b = 3^{1/2}$, $z_{1} = 1/3$, and $z_{2} = 5/6$, the atoms are on the sites of the hcp structure. Finally, when $z_{2} = 1/2 + z_{1}$ the atoms are at the positions of the $\alpha$–U structure.

Simple Orthorhombic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =& \frac14 \, \mathbf{a}_{1} + z_{1} \, \mathbf{a}_{3}& =& \frac14 \, a \, \mathbf{\hat{x}} + z_{1} \, c \, \mathbf{\hat{z}}& \left(2e\right) & \mbox{Au} \\ \mathbf{B}_{2} & =& \frac34 \, \mathbf{a}_{1} - z_{1} \, \mathbf{a}_{3}& =& \frac34 \, a \, \mathbf{\hat{x}} - z_{1} \, c \, \mathbf{\hat{z}}& \left(2e\right) & \mbox{Au} \\ \mathbf{B}_{3} & =& \frac14 \, \mathbf{a}_{1} + \frac12 \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3}& =& \frac14 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} + z_{2} \, c \, \mathbf{\hat{z}}& \left(2f\right) & \mbox{Cd} \\ \mathbf{B}_{4} & =& \frac34 \, \mathbf{a}_{1} + \frac12 \, \mathbf{a}_{2} - z_{2} \, \mathbf{a}_{3}& =& \frac34 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} - z_{2} \, c \, \mathbf{\hat{z}}& \left(2f\right) & \mbox{Cd} \\ \end{array} \]

References

  • L.–C. Chang, Atomic displacements and crystallographic mechanisms in diffusionless transformation of gold–cadium single crystals containing 47.5 atomic percent cadmium, Acta Cryst. 4, 320–324 (1951), doi:10.1107/S0365110X51001057.

Found in

  • W. B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys (Wiley– Interscience, New York, London, Sydney, Toronto, 1972)., pp. 313-314.

Geometry files


Prototype Generator

aflow --proto=AB_oP4_51_e_f --params=

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