La$_{2}$O$_{3}$ ($D5_{2}$) Structure: A2B3_hP5_164_d_ad-001
| Prototype | La$_{2}$O$_{3}$ |
| AFLOW prototype label | A2B3_hP5_164_d_ad-001 |
| Strukturbericht designation | $D5_{2}$ |
| ICSD | 100204 |
| Pearson symbol | hP5 |
| Space group number | 164 |
| Space group symbol | $P\overline{3}m1$ |
| AFLOW prototype command |
aflow --proto=A2B3_hP5_164_d_ad-001
--params=$a, \allowbreak c/a, \allowbreak z_{2}, \allowbreak z_{3}$ |
Other compounds with this structure
$\alpha$-Bi$_{2}$Mg$_{3}$, $\alpha$-Sb$_{2}$Mg$_{3}$, Ac$_{2}$O$_{3}$, As$_{2}$Mg$_{3}$, Ce$_{2}$O$_{3}$, Gd$_{2}$O$_{3}$, Nd$_{2}$O$_{3}$, Pr$_{2}$O$_{3}$, Pu$_{2}$O$_{3}$, Th$_{2}$N$_{3}$, Th$_{2}$N$_{2}$O, U$_{2}$N$_{3}$
- This structure has proved rather controversial:
- (Zachariasen 1926, 1929) originally proposed that the crystal structure of La$_{2}$O$_{3}$ and other lanthanide series oxides belong to trigonal space group $P321$ #150.
- This was immediately criticized by (Pauling, 1929), who suggested the trigonal space group ($P\overline{3}m1$ #164) later confirmed by (Koehler, 1953).
- Much later, (Aldebert, 1979) stated that the structure was hexagonal, space group $P6_{3}/mmc$ #194. This would represent a doubling of the $P\overline{3}m1$ unit cell in the $z$-direction, however they give structural parameters, also used by us and by (Villars, 1991, 2016), which have a density consistent with the $P\overline{3}m1$ structure, and are in reasonable agreement with lattice parameters given in (Pearson, 1958) and (Koehler, 1953). We agree with (Villars, 1991, 2016) that these are close to the correct values for the structure.
- This structure is very similar to Al$_{3}$Ni$_{2}$ ($D5_{13}$, A3B2_hP5_164_ad_d). We follow (Pearson, 1958) and assign the intermetallics as $D5_{13}$, keeping $D5_{2}$ for oxides and related compounds.
- The ternary Ce$_{2}$O$_{2}$S structure, aka CaAl$_{2}$Si$_{2}$, is related to this structure, the sulfur atom replacing the oxygen on the (1a) site.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | O I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (2d) | La I |
| $\mathbf{B_{3}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{2} \,\mathbf{\hat{z}}$ | (2d) | La I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (2d) | O II |
| $\mathbf{B_{5}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{3} \,\mathbf{\hat{z}}$ | (2d) | O II |
References
- P. Aldebert and J. P. Traverse, Etude par diffraction neutronique des structures de haute temperature de La$_{2}$O$_{3}$ et Nd$_{2}$O$_{3}$, Mat. Res. Bull. 14, 303–323 (1979), doi:10.1016/0025-5408(79)90095-3.
- W. Zachariasen, Die Kristallstruktur der Α-Modifikation von den Sesquioxyden der seltenen Erdmetalle. (La$_{2}$O$_{3}$ Ce$_{2}$O$_{3}$, Pr$_{2}$O$_{3}$, Nd$_{2}$O$_{3}$), Z. phys. Chem. 123, 134 (1926), doi:10.1515/zpch-1926-12309.
- W. H. Zachariasen, Kürzere Originalmitteilungen und Notizen, Z. Kristallogr. 70, 187–188 (1929), doi:10.1524/zkri.1929.70.1.187.
- L. Pauling, The Crystal Structure of the A-Modification of the Rare Earth Sesquioxides, Z. Kristallogr. 69, 415 (1928), doi:10.1524/zkri.1929.69.1.415.
- W. C. Koehler and E. O. Wollan, Neutron-diffraction study of the structure of the A-form of the rare earth sesquioxides, Acta Cryst. 6, 741–742 (1953), doi:10.1107/S0365110X53002076.
- P. Villars, ed., PAULING FILE in: Inorganic Solid Phases (SpringerMaterials (online database), Heidelberg, 2016).
- W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, no. N.R.C. No. 4303 in International Series of Monographs on Metal Physics and Physical Metallurgy (Pergamon Press, Oxford, London, Edinburgh, New York, Paris, Frankfort, 1958), 1964 reprint with corrections edn.
Found in
- P. Villars and L. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases (ASM International, Materials Park, OK, 1991), vol. IV, chap. , p. 4189.
Prototype Generator
aflow --proto=A2B3_hP5_164_d_ad --params=$a,c/a,z_{2},z_{3}$