TaTi (BCC SQS–16) Structure : AB_aP16_2_4i_4i

Picture of Structure; Click for Big Picture
Prototype : TaTi
AFLOW prototype label : AB_aP16_2_4i_4i
Strukturbericht designation : None
Pearson symbol : aP16
Space group number : 2
Space group symbol : $P\bar{1}$
AFLOW prototype command : aflow --proto=AB_aP16_2_4i_4i
--params=
$a$,$b/a$,$c/a$,$\alpha$,$\beta$,$\gamma$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$


  • This is a special quasirandom structure with 16 atoms per unit cell (SQS–16) for a bcc binary substitutional alloy $AxB1–x (Jiang, 2004). This prototype is the equicompositional structure ($x=0.5)$. The $x=0.25$ and $x=0.75$ structures are given by AB3_mC32_8_4a_12a.

Triclinic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & a \mathbf{\hat{x}} \\ \mathbf{a}_2 & = & b \cos\gamma \, \mathbf{\hat{x}} + b \sin\gamma \,\mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c_x \mathbf{\hat{x}} + c_y \, \mathbf{\hat{y}} + c_z \, \mathbf{\hat{z}}\\\\ c_x & = & c \, \cos\beta \\ c_y & = & c \, (\cos\alpha -\cos\beta \cos\gamma)/\sin\gamma \\ c_z & = & \sqrt{c^2-c_x^2-c_y^2} \end{array} \]

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} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(x_{1}a+y_{1}b\cos\gamma+z_{1}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{1}b\sin\gamma+z_{1}c_{y}\right) \, \mathbf{\hat{y}} + z_{1}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta I} \\ \mathbf{B}_{2} & = & -x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & \left(-x_{1}a-y_{1}b\cos\gamma-z_{1}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{1}b\sin\gamma-z_{1}c_{y}\right) \, \mathbf{\hat{y}}-z_{1}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta I} \\ \mathbf{B}_{3} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(x_{2}a+y_{2}b\cos\gamma+z_{2}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{2}b\sin\gamma+z_{2}c_{y}\right) \, \mathbf{\hat{y}} + z_{2}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta II} \\ \mathbf{B}_{4} & = & -x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & \left(-x_{2}a-y_{2}b\cos\gamma-z_{2}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{2}b\sin\gamma-z_{2}c_{y}\right) \, \mathbf{\hat{y}}-z_{2}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta II} \\ \mathbf{B}_{5} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(x_{3}a+y_{3}b\cos\gamma+z_{3}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{3}b\sin\gamma+z_{3}c_{y}\right) \, \mathbf{\hat{y}} + z_{3}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta III} \\ \mathbf{B}_{6} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(-x_{3}a-y_{3}b\cos\gamma-z_{3}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{3}b\sin\gamma-z_{3}c_{y}\right) \, \mathbf{\hat{y}}-z_{3}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta III} \\ \mathbf{B}_{7} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(x_{4}a+y_{4}b\cos\gamma+z_{4}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{4}b\sin\gamma+z_{4}c_{y}\right) \, \mathbf{\hat{y}} + z_{4}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta IV} \\ \mathbf{B}_{8} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(-x_{4}a-y_{4}b\cos\gamma-z_{4}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{4}b\sin\gamma-z_{4}c_{y}\right) \, \mathbf{\hat{y}}-z_{4}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ta IV} \\ \mathbf{B}_{9} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(x_{5}a+y_{5}b\cos\gamma+z_{5}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{5}b\sin\gamma+z_{5}c_{y}\right) \, \mathbf{\hat{y}} + z_{5}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti I} \\ \mathbf{B}_{10} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(-x_{5}a-y_{5}b\cos\gamma-z_{5}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{5}b\sin\gamma-z_{5}c_{y}\right) \, \mathbf{\hat{y}}-z_{5}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti I} \\ \mathbf{B}_{11} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(x_{6}a+y_{6}b\cos\gamma+z_{6}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{6}b\sin\gamma+z_{6}c_{y}\right) \, \mathbf{\hat{y}} + z_{6}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti II} \\ \mathbf{B}_{12} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(-x_{6}a-y_{6}b\cos\gamma-z_{6}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{6}b\sin\gamma-z_{6}c_{y}\right) \, \mathbf{\hat{y}}-z_{6}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti II} \\ \mathbf{B}_{13} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \left(x_{7}a+y_{7}b\cos\gamma+z_{7}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{7}b\sin\gamma+z_{7}c_{y}\right) \, \mathbf{\hat{y}} + z_{7}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti III} \\ \mathbf{B}_{14} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \left(-x_{7}a-y_{7}b\cos\gamma-z_{7}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{7}b\sin\gamma-z_{7}c_{y}\right) \, \mathbf{\hat{y}}-z_{7}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti III} \\ \mathbf{B}_{15} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \left(x_{8}a+y_{8}b\cos\gamma+z_{8}c_{x}\right) \, \mathbf{\hat{x}} + \left(y_{8}b\sin\gamma+z_{8}c_{y}\right) \, \mathbf{\hat{y}} + z_{8}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti IV} \\ \mathbf{B}_{16} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \left(-x_{8}a-y_{8}b\cos\gamma-z_{8}c_{x}\right) \, \mathbf{\hat{x}} + \left(-y_{8}b\sin\gamma-z_{8}c_{y}\right) \, \mathbf{\hat{y}}-z_{8}c_{z} \, \mathbf{\hat{z}} & \left(2i\right) & \mbox{Ti IV} \\ \end{array} \]

References

  • C. Jiang, C. Wolverton, J. Sofo, L.–Q. Chen, and Z.–K. Liu, First–principles study of binary bcc alloys using special quasirandom structures, Phys. Rev. B 69, 214202 (2004), doi:10.1103/PhysRevB.69.214202.

Geometry files


Prototype Generator

aflow --proto=AB_aP16_2_4i_4i --params=

Species:

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