β-BaB$_{2}$O$_{4}$ (High Temperature) Structure: A2BC4_hR42_167_f_ac

Basis vectors

		Lattice coordinates		Cartesian coordinates	Wyckoff position	Atom type
$B_{1}$	=	$\frac{1}{4} a_{1} + \frac{1}{4} a_{2} + \frac{1}{4} a_{3}$	=	$\frac{1}{4} c \hat{z}$	(2a)	Ba I
$B_{2}$	=	$\frac{3}{4} a_{1} + \frac{3}{4} a_{2} + \frac{3}{4} a_{3}$	=	$\frac{3}{4} c \hat{z}$	(2a)	Ba I
$B_{3}$	=	$x_{2} a_{1} + x_{2} a_{2} + x_{2} a_{3}$	=	$c x_{2} \hat{z}$	(4c)	Ba II
$B_{4}$	=	$- (x_{2} - \frac{1}{2}) a_{1} - (x_{2} - \frac{1}{2}) a_{2} - (x_{2} - \frac{1}{2}) a_{3}$	=	$- c (x_{2} - \frac{1}{2}) \hat{z}$	(4c)	Ba II
$B_{5}$	=	$- x_{2} a_{1} - x_{2} a_{2} - x_{2} a_{3}$	=	$- c x_{2} \hat{z}$	(4c)	Ba II
$B_{6}$	=	$(x_{2} + \frac{1}{2}) a_{1} + (x_{2} + \frac{1}{2}) a_{2} + (x_{2} + \frac{1}{2}) a_{3}$	=	$c (x_{2} + \frac{1}{2}) \hat{z}$	(4c)	Ba II
$B_{7}$	=	$x_{3} a_{1} + y_{3} a_{2} + z_{3} a_{3}$	=	$\frac{1}{2} a (x_{3} - z_{3}) \hat{x} - \frac{3}{6} a (x_{3} - 2 y_{3} + z_{3}) \hat{y} + \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{8}$	=	$z_{3} a_{1} + x_{3} a_{2} + y_{3} a_{3}$	=	$- \frac{1}{2} a (y_{3} - z_{3}) \hat{x} + \frac{3}{6} a (2 x_{3} - y_{3} - z_{3}) \hat{y} + \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{9}$	=	$y_{3} a_{1} + z_{3} a_{2} + x_{3} a_{3}$	=	$- \frac{1}{2} a (x_{3} - y_{3}) \hat{x} - \frac{3}{6} a (x_{3} + y_{3} - 2 z_{3}) \hat{y} + \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{10}$	=	$- (z_{3} - \frac{1}{2}) a_{1} - (y_{3} - \frac{1}{2}) a_{2} - (x_{3} - \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{3} - z_{3}) \hat{x} + \frac{3}{6} a (x_{3} - 2 y_{3} + z_{3}) \hat{y} - \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} - 3) \hat{z}$	(12f)	B I
$B_{11}$	=	$- (y_{3} - \frac{1}{2}) a_{1} - (x_{3} - \frac{1}{2}) a_{2} - (z_{3} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (y_{3} - z_{3}) \hat{x} - \frac{3}{6} a (2 x_{3} - y_{3} - z_{3}) \hat{y} - \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} - 3) \hat{z}$	(12f)	B I
$B_{12}$	=	$- (x_{3} - \frac{1}{2}) a_{1} - (z_{3} - \frac{1}{2}) a_{2} - (y_{3} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{3} - y_{3}) \hat{x} + \frac{3}{6} a (x_{3} + y_{3} - 2 z_{3}) \hat{y} - \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} - 3) \hat{z}$	(12f)	B I
$B_{13}$	=	$- x_{3} a_{1} - y_{3} a_{2} - z_{3} a_{3}$	=	$- \frac{1}{2} a (x_{3} - z_{3}) \hat{x} + \frac{3}{6} a (x_{3} - 2 y_{3} + z_{3}) \hat{y} - \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{14}$	=	$- z_{3} a_{1} - x_{3} a_{2} - y_{3} a_{3}$	=	$\frac{1}{2} a (y_{3} - z_{3}) \hat{x} - \frac{3}{6} a (2 x_{3} - y_{3} - z_{3}) \hat{y} - \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{15}$	=	$- y_{3} a_{1} - z_{3} a_{2} - x_{3} a_{3}$	=	$\frac{1}{2} a (x_{3} - y_{3}) \hat{x} + \frac{3}{6} a (x_{3} + y_{3} - 2 z_{3}) \hat{y} - \frac{1}{3} c (x_{3} + y_{3} + z_{3}) \hat{z}$	(12f)	B I
$B_{16}$	=	$(z_{3} + \frac{1}{2}) a_{1} + (y_{3} + \frac{1}{2}) a_{2} + (x_{3} + \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{3} - z_{3}) \hat{x} - \frac{3}{6} a (x_{3} - 2 y_{3} + z_{3}) \hat{y} + \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} + 3) \hat{z}$	(12f)	B I
$B_{17}$	=	$(y_{3} + \frac{1}{2}) a_{1} + (x_{3} + \frac{1}{2}) a_{2} + (z_{3} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (y_{3} - z_{3}) \hat{x} + \frac{3}{6} a (2 x_{3} - y_{3} - z_{3}) \hat{y} + \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} + 3) \hat{z}$	(12f)	B I
$B_{18}$	=	$(x_{3} + \frac{1}{2}) a_{1} + (z_{3} + \frac{1}{2}) a_{2} + (y_{3} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{3} - y_{3}) \hat{x} - \frac{3}{6} a (x_{3} + y_{3} - 2 z_{3}) \hat{y} + \frac{1}{6} c (2 x_{3} + 2 y_{3} + 2 z_{3} + 3) \hat{z}$	(12f)	B I
$B_{19}$	=	$x_{4} a_{1} + y_{4} a_{2} + z_{4} a_{3}$	=	$\frac{1}{2} a (x_{4} - z_{4}) \hat{x} - \frac{3}{6} a (x_{4} - 2 y_{4} + z_{4}) \hat{y} + \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{20}$	=	$z_{4} a_{1} + x_{4} a_{2} + y_{4} a_{3}$	=	$- \frac{1}{2} a (y_{4} - z_{4}) \hat{x} + \frac{3}{6} a (2 x_{4} - y_{4} - z_{4}) \hat{y} + \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{21}$	=	$y_{4} a_{1} + z_{4} a_{2} + x_{4} a_{3}$	=	$- \frac{1}{2} a (x_{4} - y_{4}) \hat{x} - \frac{3}{6} a (x_{4} + y_{4} - 2 z_{4}) \hat{y} + \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{22}$	=	$- (z_{4} - \frac{1}{2}) a_{1} - (y_{4} - \frac{1}{2}) a_{2} - (x_{4} - \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{4} - z_{4}) \hat{x} + \frac{3}{6} a (x_{4} - 2 y_{4} + z_{4}) \hat{y} - \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} - 3) \hat{z}$	(12f)	O I
$B_{23}$	=	$- (y_{4} - \frac{1}{2}) a_{1} - (x_{4} - \frac{1}{2}) a_{2} - (z_{4} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (y_{4} - z_{4}) \hat{x} - \frac{3}{6} a (2 x_{4} - y_{4} - z_{4}) \hat{y} - \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} - 3) \hat{z}$	(12f)	O I
$B_{24}$	=	$- (x_{4} - \frac{1}{2}) a_{1} - (z_{4} - \frac{1}{2}) a_{2} - (y_{4} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{4} - y_{4}) \hat{x} + \frac{3}{6} a (x_{4} + y_{4} - 2 z_{4}) \hat{y} - \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} - 3) \hat{z}$	(12f)	O I
$B_{25}$	=	$- x_{4} a_{1} - y_{4} a_{2} - z_{4} a_{3}$	=	$- \frac{1}{2} a (x_{4} - z_{4}) \hat{x} + \frac{3}{6} a (x_{4} - 2 y_{4} + z_{4}) \hat{y} - \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{26}$	=	$- z_{4} a_{1} - x_{4} a_{2} - y_{4} a_{3}$	=	$\frac{1}{2} a (y_{4} - z_{4}) \hat{x} - \frac{3}{6} a (2 x_{4} - y_{4} - z_{4}) \hat{y} - \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{27}$	=	$- y_{4} a_{1} - z_{4} a_{2} - x_{4} a_{3}$	=	$\frac{1}{2} a (x_{4} - y_{4}) \hat{x} + \frac{3}{6} a (x_{4} + y_{4} - 2 z_{4}) \hat{y} - \frac{1}{3} c (x_{4} + y_{4} + z_{4}) \hat{z}$	(12f)	O I
$B_{28}$	=	$(z_{4} + \frac{1}{2}) a_{1} + (y_{4} + \frac{1}{2}) a_{2} + (x_{4} + \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{4} - z_{4}) \hat{x} - \frac{3}{6} a (x_{4} - 2 y_{4} + z_{4}) \hat{y} + \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} + 3) \hat{z}$	(12f)	O I
$B_{29}$	=	$(y_{4} + \frac{1}{2}) a_{1} + (x_{4} + \frac{1}{2}) a_{2} + (z_{4} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (y_{4} - z_{4}) \hat{x} + \frac{3}{6} a (2 x_{4} - y_{4} - z_{4}) \hat{y} + \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} + 3) \hat{z}$	(12f)	O I
$B_{30}$	=	$(x_{4} + \frac{1}{2}) a_{1} + (z_{4} + \frac{1}{2}) a_{2} + (y_{4} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{4} - y_{4}) \hat{x} - \frac{3}{6} a (x_{4} + y_{4} - 2 z_{4}) \hat{y} + \frac{1}{6} c (2 x_{4} + 2 y_{4} + 2 z_{4} + 3) \hat{z}$	(12f)	O I
$B_{31}$	=	$x_{5} a_{1} + y_{5} a_{2} + z_{5} a_{3}$	=	$\frac{1}{2} a (x_{5} - z_{5}) \hat{x} - \frac{3}{6} a (x_{5} - 2 y_{5} + z_{5}) \hat{y} + \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{32}$	=	$z_{5} a_{1} + x_{5} a_{2} + y_{5} a_{3}$	=	$- \frac{1}{2} a (y_{5} - z_{5}) \hat{x} + \frac{3}{6} a (2 x_{5} - y_{5} - z_{5}) \hat{y} + \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{33}$	=	$y_{5} a_{1} + z_{5} a_{2} + x_{5} a_{3}$	=	$- \frac{1}{2} a (x_{5} - y_{5}) \hat{x} - \frac{3}{6} a (x_{5} + y_{5} - 2 z_{5}) \hat{y} + \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{34}$	=	$- (z_{5} - \frac{1}{2}) a_{1} - (y_{5} - \frac{1}{2}) a_{2} - (x_{5} - \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{5} - z_{5}) \hat{x} + \frac{3}{6} a (x_{5} - 2 y_{5} + z_{5}) \hat{y} - \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} - 3) \hat{z}$	(12f)	O II
$B_{35}$	=	$- (y_{5} - \frac{1}{2}) a_{1} - (x_{5} - \frac{1}{2}) a_{2} - (z_{5} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (y_{5} - z_{5}) \hat{x} - \frac{3}{6} a (2 x_{5} - y_{5} - z_{5}) \hat{y} - \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} - 3) \hat{z}$	(12f)	O II
$B_{36}$	=	$- (x_{5} - \frac{1}{2}) a_{1} - (z_{5} - \frac{1}{2}) a_{2} - (y_{5} - \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{5} - y_{5}) \hat{x} + \frac{3}{6} a (x_{5} + y_{5} - 2 z_{5}) \hat{y} - \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} - 3) \hat{z}$	(12f)	O II
$B_{37}$	=	$- x_{5} a_{1} - y_{5} a_{2} - z_{5} a_{3}$	=	$- \frac{1}{2} a (x_{5} - z_{5}) \hat{x} + \frac{3}{6} a (x_{5} - 2 y_{5} + z_{5}) \hat{y} - \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{38}$	=	$- z_{5} a_{1} - x_{5} a_{2} - y_{5} a_{3}$	=	$\frac{1}{2} a (y_{5} - z_{5}) \hat{x} - \frac{3}{6} a (2 x_{5} - y_{5} - z_{5}) \hat{y} - \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{39}$	=	$- y_{5} a_{1} - z_{5} a_{2} - x_{5} a_{3}$	=	$\frac{1}{2} a (x_{5} - y_{5}) \hat{x} + \frac{3}{6} a (x_{5} + y_{5} - 2 z_{5}) \hat{y} - \frac{1}{3} c (x_{5} + y_{5} + z_{5}) \hat{z}$	(12f)	O II
$B_{40}$	=	$(z_{5} + \frac{1}{2}) a_{1} + (y_{5} + \frac{1}{2}) a_{2} + (x_{5} + \frac{1}{2}) a_{3}$	=	$- \frac{1}{2} a (x_{5} - z_{5}) \hat{x} - \frac{3}{6} a (x_{5} - 2 y_{5} + z_{5}) \hat{y} + \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} + 3) \hat{z}$	(12f)	O II
$B_{41}$	=	$(y_{5} + \frac{1}{2}) a_{1} + (x_{5} + \frac{1}{2}) a_{2} + (z_{5} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (y_{5} - z_{5}) \hat{x} + \frac{3}{6} a (2 x_{5} - y_{5} - z_{5}) \hat{y} + \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} + 3) \hat{z}$	(12f)	O II
$B_{42}$	=	$(x_{5} + \frac{1}{2}) a_{1} + (z_{5} + \frac{1}{2}) a_{2} + (y_{5} + \frac{1}{2}) a_{3}$	=	$\frac{1}{2} a (x_{5} - y_{5}) \hat{x} - \frac{3}{6} a (x_{5} + y_{5} - 2 z_{5}) \hat{y} + \frac{1}{6} c (2 x_{5} + 2 y_{5} + 2 z_{5} + 3) \hat{z}$	(12f)	O II

Prototype	B $_{2}$ BaO $_{4}$
AFLOW prototype label	A2BC4_hR42_167_f_ac_2f-001
ICSD	14376
Pearson symbol	hR42
Space group number	167
Space group symbol	$R \overline{3} c$
AFLOW prototype command	aflow --proto=A2BC4_hR42_167_f_ac_2f-001 --params= $a, c / a, x_{2}, x_{3}, y_{3}, z_{3}, x_{4}, y_{4}, z_{4}, x_{5}, y_{5}, z_{5}$

Encyclopedia of Crystallographic Prototypes

β-BaB $_{2}$ O $_{4}$ (High Temperature) Structure: A2BC4_hR42_167_f_ac_2f-001

Rhombohedral primitive vectors

References

Found in

Running:

Output:

Encyclopedia of Crystallographic Prototypes

β-BaB2​O4​ (High Temperature) Structure: A2BC4_hR42_167_f_ac_2f-001

Rhombohedral primitive vectors

References

Found in

Running:

Output:

β-BaB $_{2}$ O $_{4}$ (High Temperature) Structure: A2BC4_hR42_167_f_ac_2f-001