Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: A2B_tP36_96_3b_ab

  • 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)

Keatite (SiO2) Structure: A2B_tP36_96_3b_ab

Picture of Structure; Click for Big Picture
Prototype : SiO2
AFLOW prototype label : A2B_tP36_96_3b_ab
Strukturbericht designation : None
Pearson symbol : tP36
Space group number : 96
Space group symbol : $\text{P4}_{3}\text{2}_{1}\text{2}$
AFLOW prototype command : aflow --proto=A2B_tP36_96_3b_ab
--params=
$a$,$c/a$,$x_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$


  • All references, including (Wyckoff, 1963), (Shropshire, 1959) and (Demuth, 1999) note that keatite can exist in both space group P41212–D44 (#92) and its enantiomorph P43212–D48 (#96). Wyckoff uses the coordinates proposed by Shropshire and assumes the space group is P41212. He then notes that one of the Si–O bonds in this structure is very long (3.69 Å), and is so improbable that there is something wrong either with the parameters as stated or the structure itself. If we use space group P43212 while retaining Shropshire's coordinates we obtain a much more convincing structure, one that looks much like the structure in Shropshire's Fig. 3. For this reason we place this structure in P43212.

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} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & =&x_{1} \, \mathbf{a}_{1}+ x_{1} \, \mathbf{a}_{2}& =&x_{1} \, a \, \mathbf{\hat{x}}+ x_{1} \, a \, \mathbf{\hat{y}}& \left(4a\right) & \text{Si I} \\ \mathbf{B}_{2} & =&- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}- x_{1} \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(4a\right) & \text{Si I} \\ \mathbf{B}_{3} & =&\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4a\right) & \text{Si I} \\ \mathbf{B}_{4} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{1}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4a\right) & \text{Si I} \\ \mathbf{B}_{5} & =&x_{2} \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ y_{2} \, a \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{6} & =&- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}- y_{2} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{7} & =&\left(\frac12 - y_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac34 + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - y_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{8} & =&\left(\frac12 + y_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac14 + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + y_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{9} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac34 - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{10} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac14 - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{11} & =&y_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& =&y_{2} \, a \, \mathbf{\hat{x}}+ x_{2} \, a \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{12} & =&- y_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&- y_{2} \, a \, \mathbf{\hat{x}}- x_{2} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O I} \\ \mathbf{B}_{13} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ y_{3} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{14} & =&- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{15} & =&\left(\frac12 - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac34 + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - y_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{16} & =&\left(\frac12 + y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac14 + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + y_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{17} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac34 - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{18} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac14 - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{19} & =&y_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&y_{3} \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{20} & =&- y_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&- y_{3} \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O II} \\ \mathbf{B}_{21} & =&x_{4} \, \mathbf{a}_{1}+ y_{4} \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ y_{4} \, a \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{22} & =&- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+ \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{23} & =&\left(\frac12 - y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{4}\right) \, \mathbf{a}_{2}+ \left(\frac34 + z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{24} & =&\left(\frac12 + y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{4}\right) \, \mathbf{a}_{2}+ \left(\frac14 + z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{25} & =&\left(\frac12 - x_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2}+ \left(\frac34 - z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{26} & =&\left(\frac12 + x_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2}+ \left(\frac14 - z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{27} & =&y_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& =&y_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{28} & =&- y_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+ \left(\frac12 - z_{4}\right) \, \mathbf{a}_{3}& =&- y_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{O III} \\ \mathbf{B}_{29} & =&x_{5} \, \mathbf{a}_{1}+ y_{5} \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{30} & =&- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{31} & =&\left(\frac12 - y_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac34 + z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{32} & =&\left(\frac12 + y_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac14 + z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{33} & =&\left(\frac12 - x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{5}\right) \, \mathbf{a}_{2}+ \left(\frac34 - z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{34} & =&\left(\frac12 + x_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{5}\right) \, \mathbf{a}_{2}+ \left(\frac14 - z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{35} & =&y_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& =&y_{5} \, a \, \mathbf{\hat{x}}+ x_{5} \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \mathbf{B}_{36} & =&- y_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+ \left(\frac12 - z_{5}\right) \, \mathbf{a}_{3}& =&- y_{5} \, a \, \mathbf{\hat{x}}- x_{5} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8b\right) & \text{Si II} \\ \end{array} \]

References

  • J. Shropshire, P. P. Keat, and P. A. Vaughan, The crystal structure of keatite, a new form of silica, Zeitschrift für Kristallographie 112, 409–413 (1959), doi:10.1524/zkri.1959.112.1-6.409.
  • R. W. G. Wyckoff, Crystal Structures Vol. 1 (Wiley, 1963), 2nd edn.

Found in

  • T. Demuth, Y. Jeanvoine, J. Hafner, and J. G. Ángyán, Polymorphism in silica studied in the local density and generalized–gradient approximations, J. Phys. Condens. Matter 11, 3833–3874 (1999), doi:10.1088/0953-8984/11/19/306.

Geometry files


Prototype Generator

aflow --proto=A2B_tP36_96_3b_ab --params=

Species:

Running:

Output: