Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: AB4C_hP72_184_d_4d_d

  • 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)

Al[PO4] (Framework type AFI) Structure: AB4C_hP72_184_d_4d_d

Picture of Structure; Click for Big Picture
Prototype : Al[PO4]
AFLOW prototype label : AB4C_hP72_184_d_4d_d
Strukturbericht designation : None
Pearson symbol : hP72
Space group number : 184
Space group symbol : $P6cc$
AFLOW prototype command : aflow --proto=AB4C_hP72_184_d_4d_d
--params=
$a$,$c/a$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$



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} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & x_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{1}+y_{1}\right)a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{2} & = & -y_{1} \, \mathbf{a}_{1} + \left(x_{1}-y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{1}-y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{3} & = & \left(-x_{1}+y_{1}\right) \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(-x_{1}+\frac{1}{2}y_{1}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{1}a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{4} & = & -x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{1}-y_{1}\right)a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{5} & = & y_{1} \, \mathbf{a}_{1} + \left(-x_{1}+y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{1}+y_{1}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{6} & = & \left(x_{1}-y_{1}\right) \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(x_{1}-\frac{1}{2}y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{1}a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{7} & = & -y_{1} \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{1}+y_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{8} & = & \left(-x_{1}+y_{1}\right) \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{9} & = & x_{1} \, \mathbf{a}_{1} + \left(x_{1}-y_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(x_{1}-\frac{1}{2}y_{1}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{10} & = & y_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{1}+y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{1}-y_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{11} & = & \left(x_{1}-y_{1}\right) \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{1}-y_{1}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{12} & = & -x_{1} \, \mathbf{a}_{1} + \left(-x_{1}+y_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(-x_{1}+\frac{1}{2}y_{1}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{Al} \\ \mathbf{B}_{13} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{2}+y_{2}\right)a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{14} & = & -y_{2} \, \mathbf{a}_{1} + \left(x_{2}-y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{2}-y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{15} & = & \left(-x_{2}+y_{2}\right) \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(-x_{2}+\frac{1}{2}y_{2}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{16} & = & -x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{2}-y_{2}\right)a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{17} & = & y_{2} \, \mathbf{a}_{1} + \left(-x_{2}+y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{2}+y_{2}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{18} & = & \left(x_{2}-y_{2}\right) \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(x_{2}-\frac{1}{2}y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{19} & = & -y_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{2}+y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{20} & = & \left(-x_{2}+y_{2}\right) \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{21} & = & x_{2} \, \mathbf{a}_{1} + \left(x_{2}-y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(x_{2}-\frac{1}{2}y_{2}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{22} & = & y_{2} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{2}+y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{2}-y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{23} & = & \left(x_{2}-y_{2}\right) \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{2}-y_{2}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{24} & = & -x_{2} \, \mathbf{a}_{1} + \left(-x_{2}+y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(-x_{2}+\frac{1}{2}y_{2}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O I} \\ \mathbf{B}_{25} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{3}+y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{26} & = & -y_{3} \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{3}-y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{27} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(-x_{3}+\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{28} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{3}-y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{29} & = & y_{3} \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{3}+y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{30} & = & \left(x_{3}-y_{3}\right) \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(x_{3}-\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{31} & = & -y_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{3}+y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{32} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{33} & = & x_{3} \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(x_{3}-\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{34} & = & y_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{3}-y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{35} & = & \left(x_{3}-y_{3}\right) \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{3}-y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{36} & = & -x_{3} \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(-x_{3}+\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O II} \\ \mathbf{B}_{37} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{4}+y_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{38} & = & -y_{4} \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{39} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(-x_{4}+\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{40} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{4}-y_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{41} & = & y_{4} \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{4}+y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{42} & = & \left(x_{4}-y_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(x_{4}-\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{43} & = & -y_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{4}+y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{44} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{45} & = & x_{4} \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(x_{4}-\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{46} & = & y_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{4}-y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{47} & = & \left(x_{4}-y_{4}\right) \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{4}-y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{48} & = & -x_{4} \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(-x_{4}+\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O III} \\ \mathbf{B}_{49} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{50} & = & -y_{5} \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{5}-y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{51} & = & \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(-x_{5}+\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{52} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{5}-y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{53} & = & y_{5} \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{5}+y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{54} & = & \left(x_{5}-y_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(x_{5}-\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{55} & = & -y_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{56} & = & \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{57} & = & x_{5} \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(x_{5}-\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{58} & = & y_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{5}-y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{59} & = & \left(x_{5}-y_{5}\right) \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{5}-y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{60} & = & -x_{5} \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(-x_{5}+\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{O IV} \\ \mathbf{B}_{61} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{6}+y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{62} & = & -y_{6} \, \mathbf{a}_{1} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{6}-y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{63} & = & \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(-x_{6}+\frac{1}{2}y_{6}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{64} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{6}-y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{65} & = & y_{6} \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{6}+y_{6}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{66} & = & \left(x_{6}-y_{6}\right) \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(x_{6}-\frac{1}{2}y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{67} & = & -y_{6} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{6}+y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{68} & = & \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{69} & = & x_{6} \, \mathbf{a}_{1} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(x_{6}-\frac{1}{2}y_{6}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{70} & = & y_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(x_{6}-y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{71} & = & \left(x_{6}-y_{6}\right) \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{6}-y_{6}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \mathbf{B}_{72} & = & -x_{6} \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(-x_{6}+\frac{1}{2}y_{6}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(12d\right) & \text{P} \\ \end{array} \]

References

  • G. J. Klap, H. van Koningsveld, H. Graafsma, and A. M. M. Schreurs, Absolute configuration and domain structure of AlPO4–5 studied by single crystal X–ray diffraction, Microporous Mesoporous Mater. 38, 403–412 (2000), doi:10.1016/S1387-1811(00)00161-X.

Found in

  • P. Villars and K. Cenzual, Pearson's Crystal Data – Crystal Structure Database for Inorganic Compounds, ASM International (2013).

Geometry files


Prototype Generator

aflow --proto=AB4C_hP72_184_d_4d_d --params=

Species:

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Output: