Room Temperature Perovskite
CH$_{3}$NH$_{3}$PbBr$_{3}$ Structure: A3B12C144D12E_cP172_221_c_i_2k2n_i_b-001
| Prototype | Br$_{3}$CH$_{6}$NPb |
| AFLOW prototype label | A3B12C144D12E_cP172_221_c_i_2k2n_i_b-001 |
| Pearson symbol | cP172 |
| Space group number | 221 |
| Space group symbol | $Pm\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A3B12C144D12E_cP172_221_c_i_2k2n_i_b-001
--params=$a, \allowbreak y_{3}, \allowbreak y_{4}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}$ |
- CH$_{3}$NH$_{3}$PbBr$_{3}$ can be found in a variety of distorted perovskite ($E2_{1}$) structures, depending upon the temperature and pressure (Abia, 2022; Liang, 2023):
- A low temperature orthorhombic structure, stable below 145K.
- An intermediate temperature body-centered orthorhombic structure, stable from 145-157K.
- An intermediate temperature tetragonal structure, stable from 156-229K.
- This room-temperature cubic perovskite-like structure, stable above 229K.
- A cubic high-pressure Phase II structure, stable from 0.9-1.8 GPa.
- An orthorhombic Phase III structure, stable from 1.8-4.6 GPa.
- Another orthorhombic Phase IV structure was found by (Liang, 2023), but the atomic positions have yet to be calculated.
- We report the data taken at 298K.
- (Abia, 2022) report the carbon and nitrogen atoms to be on (12i) Wyckoff positions, coordinates (0,y,y), but this does not result in the perovskite-like structure the show in their Fig. 2. When we shift the atoms to (12j) sites (1/2,y,y), keeping y fixed, the problem is resolved. This view is consistent with (Evans, 2018) ICSD 120275 and to some extent (Xie, 2018) ICSD 130210, although those works do not locate the hydrogen atoms.
- The carbon, nitrogen and hydrogen sites are all only partially occupied: 8.333% (1/12) for C and H, 4.167% (1/24) for H. This puts one carbon, one nitrogen, and six hydrogen atoms in each unit cell, presumably arranged as CH$_{3}$ and NH$_{3}$ radicals.
- The AFLOW prototype label rules move the lead atom from the (1a) Wyckoff position to the (1b) Wyckoff position. This centers the CH$_{3}$NH$_{3}$ complex about the origin.
- A somewhat similar structure, including hydrogen atoms, appears in (Huang, 2022) with ICSD 48590. In that case the hydrogen atoms all seem to be associated with the carbon atom, which is bonded to the nitrogen. This seems rather implausible to us.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (1b) | Pb I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (3c) | Br I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (3c) | Br I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}$ | (3c) | Br I |
| $\mathbf{B_{5}}$ | = | $y_{3} \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{y}}+a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{6}}$ | = | $- y_{3} \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{y}}+a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{7}}$ | = | $y_{3} \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{y}}- a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{8}}$ | = | $- y_{3} \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{y}}- a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{9}}$ | = | $y_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{10}}$ | = | $y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{11}}$ | = | $- y_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{12}}$ | = | $- y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{z}}$ | (12i) | C I |
| $\mathbf{B_{13}}$ | = | $y_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}$ | = | $a y_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{y}}$ | (12i) | C I |
| $\mathbf{B_{14}}$ | = | $- y_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}$ | = | $- a y_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{y}}$ | (12i) | C I |
| $\mathbf{B_{15}}$ | = | $y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}$ | = | $a y_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{y}}$ | (12i) | C I |
| $\mathbf{B_{16}}$ | = | $- y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}$ | = | $- a y_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{y}}$ | (12i) | C I |
| $\mathbf{B_{17}}$ | = | $y_{4} \, \mathbf{a}_{2}+y_{4} \, \mathbf{a}_{3}$ | = | $a y_{4} \,\mathbf{\hat{y}}+a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{18}}$ | = | $- y_{4} \, \mathbf{a}_{2}+y_{4} \, \mathbf{a}_{3}$ | = | $- a y_{4} \,\mathbf{\hat{y}}+a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{19}}$ | = | $y_{4} \, \mathbf{a}_{2}- y_{4} \, \mathbf{a}_{3}$ | = | $a y_{4} \,\mathbf{\hat{y}}- a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{20}}$ | = | $- y_{4} \, \mathbf{a}_{2}- y_{4} \, \mathbf{a}_{3}$ | = | $- a y_{4} \,\mathbf{\hat{y}}- a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{21}}$ | = | $y_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{3}$ | = | $a y_{4} \,\mathbf{\hat{x}}+a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{22}}$ | = | $y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{3}$ | = | $a y_{4} \,\mathbf{\hat{x}}- a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{23}}$ | = | $- y_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{3}$ | = | $- a y_{4} \,\mathbf{\hat{x}}+a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{24}}$ | = | $- y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{3}$ | = | $- a y_{4} \,\mathbf{\hat{x}}- a y_{4} \,\mathbf{\hat{z}}$ | (12i) | N I |
| $\mathbf{B_{25}}$ | = | $y_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}$ | = | $a y_{4} \,\mathbf{\hat{x}}+a y_{4} \,\mathbf{\hat{y}}$ | (12i) | N I |
| $\mathbf{B_{26}}$ | = | $- y_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}$ | = | $- a y_{4} \,\mathbf{\hat{x}}+a y_{4} \,\mathbf{\hat{y}}$ | (12i) | N I |
| $\mathbf{B_{27}}$ | = | $y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}$ | = | $a y_{4} \,\mathbf{\hat{x}}- a y_{4} \,\mathbf{\hat{y}}$ | (12i) | N I |
| $\mathbf{B_{28}}$ | = | $- y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}$ | = | $- a y_{4} \,\mathbf{\hat{x}}- a y_{4} \,\mathbf{\hat{y}}$ | (12i) | N I |
| $\mathbf{B_{29}}$ | = | $y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{30}}$ | = | $- y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{31}}$ | = | $y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{32}}$ | = | $- y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{33}}$ | = | $z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{34}}$ | = | $z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{35}}$ | = | $- z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{36}}$ | = | $- z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{37}}$ | = | $y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}$ | = | $a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{38}}$ | = | $- y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}$ | = | $- a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{39}}$ | = | $y_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}$ | = | $a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{40}}$ | = | $- y_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}$ | = | $- a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{41}}$ | = | $y_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{42}}$ | = | $- y_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{43}}$ | = | $y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{44}}$ | = | $- y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{45}}$ | = | $z_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{y}}- a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{46}}$ | = | $z_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{y}}+a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{47}}$ | = | $- z_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{y}}- a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{48}}$ | = | $- z_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{y}}+a y_{5} \,\mathbf{\hat{z}}$ | (24k) | H I |
| $\mathbf{B_{49}}$ | = | $z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}$ | = | $a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{50}}$ | = | $z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}$ | = | $a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{51}}$ | = | $- z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}$ | = | $- a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{52}}$ | = | $- z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}$ | = | $- a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{y}}$ | (24k) | H I |
| $\mathbf{B_{53}}$ | = | $y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{54}}$ | = | $- y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{55}}$ | = | $y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{56}}$ | = | $- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{57}}$ | = | $z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{58}}$ | = | $z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{59}}$ | = | $- z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{60}}$ | = | $- z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{61}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}$ | = | $a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{62}}$ | = | $- y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}$ | = | $- a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{63}}$ | = | $y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}$ | = | $a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{64}}$ | = | $- y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}$ | = | $- a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{65}}$ | = | $y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{66}}$ | = | $- y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{67}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{68}}$ | = | $- y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{69}}$ | = | $z_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{y}}- a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{70}}$ | = | $z_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{y}}+a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{71}}$ | = | $- z_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{y}}- a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{72}}$ | = | $- z_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{y}}+a y_{6} \,\mathbf{\hat{z}}$ | (24k) | H II |
| $\mathbf{B_{73}}$ | = | $z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}$ | = | $a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{74}}$ | = | $z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}$ | = | $a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{75}}$ | = | $- z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}$ | = | $- a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{76}}$ | = | $- z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}$ | = | $- a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}$ | (24k) | H II |
| $\mathbf{B_{77}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{78}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{79}}$ | = | $- x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{80}}$ | = | $x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{81}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{82}}$ | = | $z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{83}}$ | = | $- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{84}}$ | = | $- z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{85}}$ | = | $y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{86}}$ | = | $- y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{87}}$ | = | $y_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{88}}$ | = | $- y_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{89}}$ | = | $y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{90}}$ | = | $- y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{91}}$ | = | $y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{92}}$ | = | $- y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{93}}$ | = | $x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{94}}$ | = | $- x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{95}}$ | = | $- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{96}}$ | = | $x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{97}}$ | = | $z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{98}}$ | = | $z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{99}}$ | = | $- z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{100}}$ | = | $- z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{101}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{102}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{103}}$ | = | $x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{104}}$ | = | $- x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{105}}$ | = | $- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{106}}$ | = | $- z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{107}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{108}}$ | = | $z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{109}}$ | = | $- y_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{110}}$ | = | $y_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{111}}$ | = | $- y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{112}}$ | = | $y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{113}}$ | = | $- y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{114}}$ | = | $y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{115}}$ | = | $- y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{116}}$ | = | $y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{117}}$ | = | $- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{118}}$ | = | $x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{119}}$ | = | $x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{120}}$ | = | $- x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a y_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{121}}$ | = | $- z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{122}}$ | = | $- z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{123}}$ | = | $z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{124}}$ | = | $z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (48n) | H III |
| $\mathbf{B_{125}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{126}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{127}}$ | = | $- x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{128}}$ | = | $x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{129}}$ | = | $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{130}}$ | = | $z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{131}}$ | = | $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{132}}$ | = | $- z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{133}}$ | = | $y_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{134}}$ | = | $- y_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{135}}$ | = | $y_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{136}}$ | = | $- y_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{137}}$ | = | $y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{138}}$ | = | $- y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{139}}$ | = | $y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{140}}$ | = | $- y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{141}}$ | = | $x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{142}}$ | = | $- x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{143}}$ | = | $- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{144}}$ | = | $x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{145}}$ | = | $z_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{146}}$ | = | $z_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{147}}$ | = | $- z_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{148}}$ | = | $- z_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{149}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{150}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{151}}$ | = | $x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{152}}$ | = | $- x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{153}}$ | = | $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{154}}$ | = | $- z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{155}}$ | = | $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{156}}$ | = | $z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{157}}$ | = | $- y_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{158}}$ | = | $y_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{159}}$ | = | $- y_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{160}}$ | = | $y_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{161}}$ | = | $- y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{162}}$ | = | $y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{163}}$ | = | $- y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{164}}$ | = | $y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{165}}$ | = | $- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{166}}$ | = | $x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{167}}$ | = | $x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{168}}$ | = | $- x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{169}}$ | = | $- z_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{170}}$ | = | $- z_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{171}}$ | = | $z_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
| $\mathbf{B_{172}}$ | = | $z_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48n) | H IV |
References
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- A. Liang, R. Turnbull, C. Popescu, I. Fernandez-Guillen, R. Abargues, P. P. Boix, and D. Errandonea, Pressure-Induced Phase Transition versus Amorphization in Hybrid Methylammonium Lead Bromide Perovskite, J. Phys. Chem. C 127, 12821–12826 (2023), doi:10.1021/acs.jpcc.3c03263.
- P. E. Evans, M. Pink, A. A. Zhumekenov, G. Hao, Y. Losovyj, O. M. Bakr, P. A. Dowben, and A. J. Yost, Rotationally Free and Rigid Sublattices of the Single Crystal Perovskite CH$_{3}$NH$_{3}$PbBr$_{3}$(001): The Case of the Lattice Polar Liquid, J. Phys. Chem. C 122, 25506–25514 (2018), doi:10.1021/acs.jpcc.8b08994.
- A. Xie, T. H. Nguyen, C. Hettiarachchi, M. E. Witkowski, W. Drozdowski, M. D. Birowosuto, H. Wang, and C. Dang, Thermal Quenching and Dose Studies of X‐ray Luminescence in Single Crystals of Halide Perovskites, J. Phys. Chem. C 122, 16265–16273 (2018), doi:10.1021/acs.jpcc.8b03622.
- X. Huang, X. Li, Y. Tao, S. Guo, J. Gu, H. Hong, Y. Yao, Y. Guan, Y. Gao, C. Li, X. Lü, and Y. Fu, Understanding Electron−Phonon Interactions in 3D Lead Halide Perovskites from the Stereochemical Expression of 6s$^2$ Lone Pairs, J. Am. Chem. Soc. 144, 12247–12260 (2022), doi:10.1021/jacs.2c03443.
First cited in
- N. Anderson, M. J. Mehl, H. Eckert, S. Divilov, X. Campilongo, S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 5. Submitted to Computational Materials Science (2026).
Geometry files
Prototype Generator
aflow --proto=A3B12C144D12E_cP172_221_c_i_2k2n_i_b --params=$a,y_{3},y_{4},y_{5},z_{5},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8}$