$\beta$–Potassium Nitrate (KNO3) Structure : ABC6_hR8_166_a_b_h

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Prototype : KNO3
AFLOW prototype label : ABC6_hR8_166_a_b_h
Strukturbericht designation : None
Pearson symbol : hR8
Space group number : 166
Space group symbol : $R\bar{3}m$
AFLOW prototype command : aflow --proto=ABC6_hR8_166_a_b_h
--params=
$a$,$c/a$,$x_{3}$,$z_{3}$


  • On heating, $\alpha$–KNO3 (either Structure I or Structure II) transforms into $\beta$–KNO3 at 128 °C. When heated above 200 °C and then cooled, the $\beta$ phase transforms into the metastable ferroelectric $\gamma$–KNO3 phase, which can remain down to room temperature.
  • In the $\beta$–phase the oxygen (6h) sites are only 50% filled. Presumably only the $+z$ or $–z$ sites will be occupied around a given nitrogen atom.
  • (Nimmo, 1976) give the data for $\beta$–KNO3 taken at 151 °C.

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} & = & 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(1a\right) & \mbox{K} \\ \mathbf{B}_{2} & = & \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}} & \left(1b\right) & \mbox{N} \\ \mathbf{B}_{3} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \mathbf{B}_{4} & = & z_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \mathbf{B}_{5} & = & x_{3} \, \mathbf{a}_{1} + z_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \mathbf{B}_{6} & = & -z_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \mathbf{B}_{7} & = & -x_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \mathbf{B}_{8} & = & -x_{3} \, \mathbf{a}_{1}-z_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}}-c\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right) \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{O} \\ \end{array} \]

References

  • J. K. Nimmo and B. W. Lucas, The crystal structures of $\gamma$–and $\beta$–KNO3 and the $\alpha$–$\beta$–$\gamma$ phase transformations, Acta Crystallogr. Sect. B Struct. Sci. 32, 1968–1971 (1976), doi:10.1107/S0567740876006894.

Found in

  • R. T. Downs and M. Hall–Wallace, The American Mineralogist Crystal Structure Database, Am. Mineral. 88, 247–250 (2003).

Geometry files


Prototype Generator

aflow --proto=ABC6_hR8_166_a_b_h --params=

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