Encyclopedia of Crystallographic Prototypes

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)

Al5C3N ($E9_{4}$) Structure: A5B3C_hP18_186_2a3b_2ab_b

 Prototype : Al5C3N AFLOW prototype label : A5B3C_hP18_186_2a3b_2ab_b Strukturbericht designation : $E9_{4}$ Pearson symbol : hP18 Space group number : 186 Space group symbol : $\mbox{P6}_{3}\mbox{mc}$ AFLOW prototype command : aflow --proto=A5B3C_hP18_186_2a3b_2ab_b --params=$a$,$c/a$,$z_1$,$z_2$,$z_3$,$z_4$,$z_5$,$z_6$,$z_7$,$z_8$,$z_9$

• Since space group #186 has no $z = 0$ mirror plane, we are free to uniformly shift the $z$ coordinates of the atoms. We have done this so that the first carbon atom is at the origin.

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} & = & z_1 \, \mathbf{a}_{3} & = & z_1 \, \, c \, \mathbf{\hat{z}} &\left(2a\right) & \mbox{Al I} \\ \mathbf{B_2} & = & \left(\frac12 + z_1\right) \, \mathbf{a}_{3} & = & \left(\frac12 +z_1\right) \, \, c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Al I} \\ \mathbf{B_3} & = & z_2 \, \mathbf{a}_{3} & = & z_2 \, \, c \, \mathbf{\hat{z}} &\left(2a\right) & \mbox{Al II} \\ \mathbf{B_4} & = & \left(\frac12 + z_2\right) \, \mathbf{a}_{3} & = & \left(\frac12 +z_2\right) \, \, c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Al II} \\ \mathbf{B_5} & = & z_3 \, \mathbf{a}_{3} & = & z_3 \, \, c \, \mathbf{\hat{z}} &\left(2a\right) & \mbox{C I} \\ \mathbf{B_6} & = & \left(\frac12 + z_3\right) \, \mathbf{a}_{3} & = & \left(\frac12 +z_3\right) \, \, c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{C I} \\ \mathbf{B_7} & = & z_4 \, \mathbf{a}_{3} & = & z_4 \, \, c \, \mathbf{\hat{z}} &\left(2a\right) & \mbox{C II} \\ \mathbf{B_8} & = & \left(\frac12 + z_4\right) \, \mathbf{a}_{3} & = & \left(\frac12 +z_4\right) \, \, c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{C II} \\ \mathbf{B_9} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_5 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_5 \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al III} \\ \mathbf{B}_{10} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_5\right) \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al III} \\ \mathbf{B}_{11} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_6 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_6 \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al IV} \\ \mathbf{B}_{12} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_6\right) \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_6\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al IV} \\ \mathbf{B}_{13} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_7 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_7 \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al V} \\ \mathbf{B}_{14} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_7\right) \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_7\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Al V} \\ \mathbf{B}_{15} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_8 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_8 \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{C III} \\ \mathbf{B}_{16} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_8\right) \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_8\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{C III} \\ \mathbf{B}_{17} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_9 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ z_9 \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{N} \\ \mathbf{B}_{18} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_9\right) \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_9\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{N} \\ \end{array}$

Found in

• P. Villars, K. Cenzual, J. Daams, R. Gladyshevskii, O. Shcherban, V. Dubenskyy, N. Melnichenko–Koblyuk, O. Pavlyuk, I. Savesyuk, S. Stoiko, and L. Sysa, Landolt–Börnstein – Group III Condensed Matter (Springer–Verlag Berlin Heidelberg, 2006). Accessed through the Springer Materials site.

Prototype Generator

aflow --proto=A5B3C_hP18_186_2a3b_2ab_b --params=