Paraelectric LiNbO3 Structure: ABC3_hR10_167_a_b_e

Picture of Structure; Click for Big Picture
Prototype : LiNbO3
AFLOW prototype label : ABC3_hR10_167_a_b_e
Strukturbericht designation : None
Pearson symbol : hR10
Space group number : 167
Space group symbol : $\mbox{R}\bar{3}\mbox{c}$
AFLOW prototype command : aflow --proto=ABC3_hR10_167_a_b_e [--hex]
--params=
$a$,$c/a$,$x_{3}$


  • This is the paraelectric phase, which exists above 1430K. There is also a ferroelectric phase. Note that paraelectric LiNbO_3 (ABC3_hR10_167_a_b_e, LiNbO3) and calcite (ABC3_hR10_167_a_b_e, CaCO3) have the same AFLOW prototype label. They are generated by the same symmetry operations with different sets of parameters (––params) specified in their corresponding CIF files. Hexagonal settings of this structure can be obtained with the option ––hex.

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac1{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac1{\sqrt3} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac1{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =& \frac14 \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =& \frac14 c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Li} \\ \mathbf{B}_{2} & =& \frac34 \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =& \frac34 c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Li} \\ \mathbf{B}_{3} & =&0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & =&0 \mathbf{\hat{x}} + 0 \mathbf{\hat{y}} + 0 \mathbf{\hat{z}} & \left(2b\right) & \mbox{Nb} \\ \mathbf{B}_{4} & =& \frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =& \frac12 c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Nb} \\ \mathbf{B}_{5} & =&x_{3} \, \mathbf{a}_{1}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&- \frac18 \left(1 - 4 x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac{\sqrt{3}}{8} \left(1 - 4 x_{3}\right) \, a \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{6} & =&\frac14 \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{3}& =&- \frac18 \left(1 - 4 x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac{\sqrt{3}}{8} \left(1 - 4 x_{3}\right) \, a \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{7} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac14 \left(1 - 4 x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{8} & =&- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&- \frac18 \left(3 + 4 x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{8\sqrt3} \left(1 + 12 x_{3}\right) \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{9} & =&\frac34 \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& =&\frac18 \left(1 - 4 x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{8\sqrt3} \left(5 + 12 x_{3}\right) \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{10} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& =&\frac14 \left(1 + 4 x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \end{array} \]

References

  • H. Boysen and F. Altorfer, A neutron powder investigation of the high–temperature structure and phase transition in LiNbO3, Acta Crystallogr. Sect. B Struct. Sci. 50, 405–414 (1994), doi:10.1107/S0108768193012820.

Geometry files


Prototype Generator

aflow --proto=ABC3_hR10_167_a_b_e --params=

Species:

Running:

Output: