Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: A_cF136_227_aeg

  • M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
  • D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
  • D. Hicks, M.J. Mehl, M. Esters, C. Oses, O. Levy, G.L.W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

Si34 Clathrate Structure: A_cF136_227_aeg

Picture of Structure; Click for Big Picture
Prototype : Si
AFLOW prototype label : A_cF136_227_aeg
Strukturbericht designation : None
Pearson symbol : cF136
Space group number : 227
Space group symbol : $\text{Fd}\bar{3}\text{m}$
AFLOW prototype command : aflow --proto=A_cF136_227_aeg
--params=
$a$,$x_{2}$,$x_{3}$,$z_{3}$


  • Silicon clathrates are open structures of pentagonal dodecahedra connected so that all of the silicon atoms have sp$^{3}$ bonding. In nature these structures are stabilized by alkali impurity atoms. This structure and the Si46 structure are proposed pure silicon clathrate structures. For more information about these structures and their possible stability, see (Adams, 1994). See (Gryko, 2000) for a possible experimental realization of this structure (Si34Nax, were x is very small). We have used the fact that all vectors of the form $\left(0, \pm \, a/2, \pm \, a/2\right)$, $\left(\pm \, a/2, 0, \pm \, a/2 \right)$, and $\left( \pm \, a/2, \pm \, a/2, 0 \right)$ are primitive vectors of the face-centered cubic lattice to simplify the positions of some atoms in both lattice and Cartesian coordinates.

Face-centered Cubic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = &\frac18 \, \, \mathbf{a}_{1}+ \frac18 \, \, \mathbf{a}_{2}+ \frac18 \, \, \mathbf{a}_{3}& = &\frac18 \, a \, \mathbf{\hat{x}}+ \frac18 \, a \, \mathbf{\hat{y}}+ \frac18 \, a \, \mathbf{\hat{z}}& \left(8a\right) & \text{Si I} \\ \mathbf{B}_{2} & = &\frac78 \, \, \mathbf{a}_{1}+ \frac78 \, \, \mathbf{a}_{2}+ \frac78 \, \, \mathbf{a}_{3}& = &\frac78 \, a \, \mathbf{\hat{x}}+ \frac78 \, a \, \mathbf{\hat{y}}+ \frac78 \, a \, \mathbf{\hat{z}}& \left(8a\right) & \text{Si I} \\ \mathbf{B}_{3} & = &x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &x_{2} \, a \, \mathbf{\hat{x}}+ x_{2} \, a \, \mathbf{\hat{y}}+ x_{2} \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{4} & = &x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ \left(\frac12 - 3 \, x_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{y}}+ x_{2} \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{5} & = &x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - 3 \, x_{2}\right) \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &\left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ x_{2} \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{6} & = &\left(\frac12 - 3 \, x_{2}\right) \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{7} & = &- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+ \left(\frac12 + 3 \, x_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{y}}- x_{2} \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{8} & = &- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &- x_{2} \, a \, \mathbf{\hat{x}}- x_{2} \, a \, \mathbf{\hat{y}}- x_{2} \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{9} & = &- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + 3 \, x_{2}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &\left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{x}}- x_{2} \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{10} & = &\left(\frac12 + 3 \, x_{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(32e\right) & \text{Si II} \\ \mathbf{B}_{11} & = &z_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}+ z_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{12} & = &z_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ z_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{13} & = &\left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{14} & = &\left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{15} & = &\left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &z_{3} \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}+ x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{16} & = &\left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &z_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{17} & = &z_{3} \, \mathbf{a}_{1}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{18} & = &z_{3} \, \mathbf{a}_{1}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{19} & = &z_{3} \, \mathbf{a}_{1}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}+ z_{3} \, a \, \mathbf{\hat{y}}+ x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{20} & = &z_{3} \, \mathbf{a}_{1}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ z_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{21} & = &\left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ \left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{22} & = &\left(2 x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ \left(\frac12 - 2 x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - z_{3}\right) \, a \, \mathbf{\hat{y}}+ x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{23} & = &- z_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{y}}- z_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{24} & = &- z_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}- z_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{25} & = &\left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{26} & = &\left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{27} & = &\left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{y}}- x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{28} & = &\left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{29} & = &- z_{3} \, \mathbf{a}_{1}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}- z_{3} \, a \, \mathbf{\hat{y}}- x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{30} & = &- z_{3} \, \mathbf{a}_{1}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}- z_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{31} & = &- z_{3} \, \mathbf{a}_{1}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{y}}- x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{32} & = &- z_{3} \, \mathbf{a}_{1}+ \left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac14 + z_{3}\right) \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{33} & = &\left(\frac12 + 2 x_{3} + z_{3}\right) \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- z_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \mathbf{B}_{34} & = &\left(z_{3} - 2 x_{3}\right) \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- z_{3} \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}- x_{3} \, a \, \mathbf{\hat{z}}& \left(96g\right) & \text{Si III} \\ \end{array} \]

References

  • G. B. Adams, M. O'Keeffe, A. A. Demkov, O. F. Sankey, and Y.–M. Huang, Wide–band–gap Si in open fourfold–coordinated clathrate structures, Phys. Rev. B 49, 8048–8053 (1994), doi:10.1103/PhysRevB.49.8048.
  • J. Gryko, P. F. McMillan, R. F. Marzke, G. K. Ramachandran, D. Patton, S. K. Deb, and O. F. Sankey, Low–density framework form of crystalline silicon with a wide optical band gap, Phys. Rev. B 62, R7707–7710 (2000), doi:10.1103/PhysRevB.62.R7707.

Geometry files


Prototype Generator

aflow --proto=A_cF136_227_aeg --params=

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