Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: ABC3_hR10_167_a_b_e-001

This structure originally had the label ABC3_hR10_167_a_b_e.LiNbO3. Calls to that address will be redirected here.

If you are using this page, please cite:
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)

Links to this page

https://aflow.org/p/TXPE
or https://aflow.org/p/ABC3_hR10_167_a_b_e-001
or PDF Version

Paraelectric LiNbO$_{3}$ Structure: ABC3_hR10_167_a_b_e-001

Picture of Structure; Click for Big Picture
Prototype LiONbO$_{3}$
AFLOW prototype label ABC3_hR10_167_a_b_e-001
ICSD 81240
Pearson symbol hR10
Space group number 167
Space group symbol $R\overline{3}c$
AFLOW prototype command aflow --proto=ABC3_hR10_167_a_b_e-001
--params=$a, \allowbreak c/a, \allowbreak x_{3}$

  • This is the paraelectric phase, stable above 1430K. There is also a ferroelectric phase.
  • Paraelectric LiNbO$_{3}$ and calcite CaCO$_{3}$ have the same AFLOW prototype label, ABC3_hR10_167_a_b_e. They are generated by the same symmetry operations with different sets of parameters (--params) specified in their corresponding CIF files.
  • Hexagonal settings rhombohedral structures can be obtained with the option --hex.

\[ \begin{array}{ccc} \mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}} \end{array}\]

Basis vectors

Lattice coordinates Cartesian coordinates Wyckoff position Atom type
$\mathbf{B_{1}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{1}{4}c \,\mathbf{\hat{z}}$ (2a) Li I
$\mathbf{B_{2}}$ = $\frac{3}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $\frac{3}{4}c \,\mathbf{\hat{z}}$ (2a) Li I
$\mathbf{B_{3}}$ = $0$ = $0$ (2b) Nb I
$\mathbf{B_{4}}$ = $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ = $\frac{1}{2}c \,\mathbf{\hat{z}}$ (2b) Nb I
$\mathbf{B_{5}}$ = $x_{3} \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{1}{8}a \left(4 x_{3} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{8}a \left(4 x_{3} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ (6e) O I
$\mathbf{B_{6}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $\frac{1}{8}a \left(4 x_{3} - 1\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{8}a \left(4 x_{3} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ (6e) O I
$\mathbf{B_{7}}$ = $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ = $- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ (6e) O I
$\mathbf{B_{8}}$ = $- x_{3} \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $- \frac{1}{8}a \left(4 x_{3} + 3\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{24}a \left(12 x_{3} + 1\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ (6e) O I
$\mathbf{B_{9}}$ = $\frac{3}{4} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- \frac{1}{8}a \left(4 x_{3} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{24}a \left(12 x_{3} + 5\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ (6e) O I
$\mathbf{B_{10}}$ = $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ = $a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ (6e) O I

References

  • H. Boysen and F. Altorfer, A neutron powder investigation of the high-temperature structure and phase transition in LiNbO$_3$, Acta Crystallogr. Sect. B 50, 405–414 (1994), doi:10.1107/S0108768193012820.

Prototype Generator

aflow --proto=ABC3_hR10_167_a_b_e --params=$a,c/a,x_{3}$

Species:

Running:

Output: