Ferroelectric LiNbO3 Structure: ABC3_hR10_161_a_a_b

Picture of Structure; Click for Big Picture
Prototype : LiNbO3
AFLOW prototype label : ABC3_hR10_161_a_a_b
Strukturbericht designation : None
Pearson symbol : hR10
Space group number : 161
Space group symbol : $\mbox{R}3\mbox{c}$
AFLOW prototype command : aflow --proto=ABC3_hR10_161_a_a_b [--hex]
--params=
$a$,$c/a$,$x_{1}$,$x_{2}$,$x_{3}$,$y_{3}$,$z_{3}$


  • This is the ferroelectric phase of LiNbO3, which exists below 1430K. There is also a high-temperature paraelectric phase. This reduces to a double unit cell version of the cubic perovskite structure in the special case: {$c/a = \sqrt6$: This sets the angle between the rhombohedral primitive vectors to 60$^{o}$. Experimentally the value is about 56$^{o}$. {$z_{1} = 1/4$} {$z_{2} = 0$} {$x_{3} =1/2$} {$y_{3} = 0$} {$z_{3} = 0$

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac{1}{\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&x_{1} \, \mathbf{a}_{1}+ x_{1} \, \mathbf{a}_{2}+ x_{1} \, \mathbf{a}_{3}& =&x_{1} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Li} \\ \mathbf{B}_{2} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Li} \\ \mathbf{B}_{3} & =&x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& =&x_{2} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Nb} \\ \mathbf{B}_{4} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Nb} \\ \mathbf{B}_{5} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&\frac12 \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(x_{3} - 2 y_{3} + z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \mbox{O} \\ \mathbf{B}_{6} & =&z_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ y_{3} \, \mathbf{a}_{3}& =&\frac12 \left(z_{3} - y_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(z_{3} - 2 x_{3} + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \mbox{O} \\ \mathbf{B}_{7} & =&y_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac12 \left(y_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(y_{3} - 2 z_{3} + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \mbox{O} \\ \mathbf{B}_{8} & =&\left(\frac12 + y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(y_{3} - z_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(z_{3} - 2 x_{3} + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \mbox{O} \\ \mathbf{B}_{9} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(x_{3} - y_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(y_{3} - 2 z_{3} + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \mbox{O} \\ \mathbf{B}_{10} & =&\left(\frac12 + z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(x_{3} - 2 y_{3} + z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\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_161_a_a_b --params=

Species:

Running:

Output: