TiBe12 (approximate, $D2_{a}$) Structure : A12B_hP13_191_cdei_a

Picture of Structure; Click for Big Picture
Prototype : Be12Ti
AFLOW prototype label : A12B_hP13_191_cdei_a
Strukturbericht designation : $D2_{a}$
Pearson symbol : hP13
Space group number : 191
Space group symbol : $P6/mmm$
AFLOW prototype command : aflow --proto=A12B_hP13_191_cdei_a
--params=
$a$,$c/a$,$z_{4}$,$z_{5}$


  • The structure of TiBe12 is not settled. (Raeuchle, 1952) described the structure as a somewhat disordered supercell containing 48 atoms with lattice constants $a = 29.44$ Å and $c = 7.33$ Å, but they stated that a ‘pseudo–cell’ existed with dimensions $a = 4.23$ Å and $c = 7.33$ Å. This pseudo–cell is described here, and was assigned the $D2_{a}$ Strukturbericht type by Smithells (Brandes, 1992). Raeuchle and Rundle suggested that the larger primitive cell was constructed from the multiple pseudo–cells, with the titanium atom alternating between the ($1a$) and ($1b$) $(0,0,1/2)$ Wyckoff positions.
  • Other experimenters have suggested that the actual structure of TiBe12 is tetragonal. (Jackson, 2016) presents first–principles calculations which suggest that the actual structure is of the tetragonal ThMn12 ($D2_{b}$) type.

Hexagonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{x}} - \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac{\sqrt3}2 \, a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & 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(1a\right) & \mbox{Ti} \\ \mathbf{B}_{2} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} & \left(2c\right) & \mbox{Be I} \\ \mathbf{B}_{3} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} & \left(2c\right) & \mbox{Be I} \\ \mathbf{B}_{4} & = & \frac{1}{3} \, \mathbf{a}_{1} + \frac{2}{3} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{Be II} \\ \mathbf{B}_{5} & = & \frac{2}{3} \, \mathbf{a}_{1} + \frac{1}{3} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{Be II} \\ \mathbf{B}_{6} & = & z_{4} \, \mathbf{a}_{3} & = & z_{4}c \, \mathbf{\hat{z}} & \left(2e\right) & \mbox{Be III} \\ \mathbf{B}_{7} & = & -z_{4} \, \mathbf{a}_{3} & = & -z_{4}c \, \mathbf{\hat{z}} & \left(2e\right) & \mbox{Be III} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \mathbf{B}_{9} & = & \frac{1}{2} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{4}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \mathbf{B}_{10} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \mathbf{B}_{11} & = & \frac{1}{2} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{4}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \mathbf{B}_{12} & = & \frac{1}{2} \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \mathbf{B}_{13} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}-z_{5}c \, \mathbf{\hat{z}} & \left(6i\right) & \mbox{Be IV} \\ \end{array} \]

References

  • E. A. Brandes and G. B. Brook, eds., Smithells Metals Reference Book (Butterworth Heinemann, Oxford, Auckland, Boston, Johannesburg, Melbourne, New Delhi, 1992), seventh edn. Strukturbericht designations start on page 6–36 (163 in PDF), see table 6.3 on page 6–63 (190).

Found in

  • M. L. Jackson, P. A. Burr, and R. W. Grimes, Resolving the structure of TiBe12, Acta Crystallogr. Sect. B Struct. Sci. 72, 277–280 (2016), doi:10.1107/S205252061600322X.

Geometry files


Prototype Generator

aflow --proto=A12B_hP13_191_cdei_a --params=

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