High Pressure Phase III CH$_{3}$NH$_{3}$PbBr$_{3}$ Structure: A3BC12DE_oP144_31_4a4b_4a_24b_4a_2b-001
| Prototype | Br$_{3}$CH$_{6}$NPb |
| AFLOW prototype label | A3BC12DE_oP144_31_4a4b_4a_24b_4a_2b-001 |
| ICSD | 63763 |
| CCDC | 2194530 |
| Pearson symbol | mP138 |
| Space group number | 4 |
| Space group symbol | $P2_1$ |
| AFLOW prototype command |
aflow --proto=A3BC12DE_oP144_31_4a4b_4a_24b_4a_2b-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \beta, \allowbreak x_{1}, \allowbreak y_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}, \allowbreak x_{20}, \allowbreak y_{20}, \allowbreak z_{20}, \allowbreak x_{21}, \allowbreak y_{21}, \allowbreak z_{21}, \allowbreak x_{22}, \allowbreak y_{22}, \allowbreak z_{22}, \allowbreak x_{23}, \allowbreak y_{23}, \allowbreak z_{23}, \allowbreak x_{24}, \allowbreak y_{24}, \allowbreak z_{24}, \allowbreak x_{25}, \allowbreak y_{25}, \allowbreak z_{25}, \allowbreak x_{26}, \allowbreak y_{26}, \allowbreak z_{26}, \allowbreak x_{27}, \allowbreak y_{27}, \allowbreak z_{27}, \allowbreak x_{28}, \allowbreak y_{28}, \allowbreak z_{28}, \allowbreak x_{29}, \allowbreak y_{29}, \allowbreak z_{29}, \allowbreak x_{30}, \allowbreak y_{30}, \allowbreak z_{30}, \allowbreak x_{31}, \allowbreak y_{31}, \allowbreak z_{31}, \allowbreak x_{32}, \allowbreak y_{32}, \allowbreak z_{32}, \allowbreak x_{33}, \allowbreak y_{33}, \allowbreak z_{33}, \allowbreak x_{34}, \allowbreak y_{34}, \allowbreak z_{34}, \allowbreak x_{35}, \allowbreak y_{35}, \allowbreak z_{35}, \allowbreak x_{36}, \allowbreak y_{36}, \allowbreak z_{36}, \allowbreak x_{37}, \allowbreak y_{37}, \allowbreak z_{37}, \allowbreak x_{38}, \allowbreak y_{38}, \allowbreak z_{38}, \allowbreak x_{39}, \allowbreak y_{39}, \allowbreak z_{39}, \allowbreak x_{40}, \allowbreak y_{40}, \allowbreak z_{40}, \allowbreak x_{41}, \allowbreak y_{41}, \allowbreak z_{41}, \allowbreak x_{42}, \allowbreak y_{42}, \allowbreak z_{42}, \allowbreak x_{43}, \allowbreak y_{43}, \allowbreak z_{43}, \allowbreak x_{44}, \allowbreak y_{44}, \allowbreak z_{44}, \allowbreak x_{45}, \allowbreak y_{45}, \allowbreak z_{45}, \allowbreak x_{46}, \allowbreak y_{46}, \allowbreak z_{46}, \allowbreak x_{47}, \allowbreak y_{47}, \allowbreak z_{47}, \allowbreak x_{48}, \allowbreak y_{48}, \allowbreak z_{48}, \allowbreak x_{49}, \allowbreak y_{49}, \allowbreak z_{49}, \allowbreak x_{50}, \allowbreak y_{50}, \allowbreak z_{50}, \allowbreak x_{51}, \allowbreak y_{51}, \allowbreak z_{51}, \allowbreak x_{52}, \allowbreak y_{52}, \allowbreak z_{52}, \allowbreak x_{53}, \allowbreak y_{53}, \allowbreak z_{53}, \allowbreak x_{54}, \allowbreak y_{54}, \allowbreak z_{54}, \allowbreak x_{55}, \allowbreak y_{55}, \allowbreak z_{55}, \allowbreak x_{56}, \allowbreak y_{56}, \allowbreak z_{56}, \allowbreak x_{57}, \allowbreak y_{57}, \allowbreak z_{57}, \allowbreak x_{58}, \allowbreak y_{58}, \allowbreak z_{58}, \allowbreak x_{59}, \allowbreak y_{59}, \allowbreak z_{59}, \allowbreak x_{60}, \allowbreak y_{60}, \allowbreak z_{60}, \allowbreak x_{61}, \allowbreak y_{61}, \allowbreak z_{61}, \allowbreak x_{62}, \allowbreak y_{62}, \allowbreak z_{62}, \allowbreak x_{63}, \allowbreak y_{63}, \allowbreak z_{63}, \allowbreak x_{64}, \allowbreak y_{64}, \allowbreak z_{64}, \allowbreak x_{65}, \allowbreak y_{65}, \allowbreak z_{65}, \allowbreak x_{66}, \allowbreak y_{66}, \allowbreak z_{66}, \allowbreak x_{67}, \allowbreak y_{67}, \allowbreak z_{67}, \allowbreak x_{68}, \allowbreak y_{68}, \allowbreak z_{68}, \allowbreak x_{69}, \allowbreak y_{69}, \allowbreak z_{69}$ |
- 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.
- The room-temperature cubic perovskite-like structure, stable above 229K.
- A cubic high-pressure Phase II structure, stable from 0.9-1.8 GPa.
- This 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 measurements taken by (Liang, 2022) at 2.3GPa.
- The hydrogen sites are all half-filled.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \cos{\beta} \,\mathbf{\hat{x}}+c \sin{\beta} \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\left(a x_{1} + c z_{1} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{1} \,\mathbf{\hat{y}}+c z_{1} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br I |
| $\mathbf{B_{2}}$ | = | $- x_{1} \, \mathbf{a}_{1}+\left(y_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}$ | = | $- \left(a x_{1} + c z_{1} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{1} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{1} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br I |
| $\mathbf{B_{3}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\left(a x_{2} + c z_{2} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+c z_{2} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br II |
| $\mathbf{B_{4}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $- \left(a x_{2} + c z_{2} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{2} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br II |
| $\mathbf{B_{5}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br III |
| $\mathbf{B_{6}}$ | = | $- x_{3} \, \mathbf{a}_{1}+\left(y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $- \left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br III |
| $\mathbf{B_{7}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br IV |
| $\mathbf{B_{8}}$ | = | $- x_{4} \, \mathbf{a}_{1}+\left(y_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $- \left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br IV |
| $\mathbf{B_{9}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br V |
| $\mathbf{B_{10}}$ | = | $- x_{5} \, \mathbf{a}_{1}+\left(y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br V |
| $\mathbf{B_{11}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VI |
| $\mathbf{B_{12}}$ | = | $- x_{6} \, \mathbf{a}_{1}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VI |
| $\mathbf{B_{13}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VII |
| $\mathbf{B_{14}}$ | = | $- x_{7} \, \mathbf{a}_{1}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VII |
| $\mathbf{B_{15}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VIII |
| $\mathbf{B_{16}}$ | = | $- x_{8} \, \mathbf{a}_{1}+\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br VIII |
| $\mathbf{B_{17}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br IX |
| $\mathbf{B_{18}}$ | = | $- x_{9} \, \mathbf{a}_{1}+\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{9} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br IX |
| $\mathbf{B_{19}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br X |
| $\mathbf{B_{20}}$ | = | $- x_{10} \, \mathbf{a}_{1}+\left(y_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $- \left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{10} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br X |
| $\mathbf{B_{21}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br XI |
| $\mathbf{B_{22}}$ | = | $- x_{11} \, \mathbf{a}_{1}+\left(y_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $- \left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{11} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br XI |
| $\mathbf{B_{23}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br XII |
| $\mathbf{B_{24}}$ | = | $- x_{12} \, \mathbf{a}_{1}+\left(y_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $- \left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{12} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Br XII |
| $\mathbf{B_{25}}$ | = | $x_{13} \, \mathbf{a}_{1}+y_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $\left(a x_{13} + c z_{13} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C I |
| $\mathbf{B_{26}}$ | = | $- x_{13} \, \mathbf{a}_{1}+\left(y_{13} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $- \left(a x_{13} + c z_{13} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{13} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{13} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C I |
| $\mathbf{B_{27}}$ | = | $x_{14} \, \mathbf{a}_{1}+y_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $\left(a x_{14} + c z_{14} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C II |
| $\mathbf{B_{28}}$ | = | $- x_{14} \, \mathbf{a}_{1}+\left(y_{14} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $- \left(a x_{14} + c z_{14} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{14} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{14} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C II |
| $\mathbf{B_{29}}$ | = | $x_{15} \, \mathbf{a}_{1}+y_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $\left(a x_{15} + c z_{15} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C III |
| $\mathbf{B_{30}}$ | = | $- x_{15} \, \mathbf{a}_{1}+\left(y_{15} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $- \left(a x_{15} + c z_{15} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{15} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{15} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C III |
| $\mathbf{B_{31}}$ | = | $x_{16} \, \mathbf{a}_{1}+y_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $\left(a x_{16} + c z_{16} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c z_{16} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C IV |
| $\mathbf{B_{32}}$ | = | $- x_{16} \, \mathbf{a}_{1}+\left(y_{16} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $- \left(a x_{16} + c z_{16} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{16} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{16} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | C IV |
| $\mathbf{B_{33}}$ | = | $x_{17} \, \mathbf{a}_{1}+y_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $\left(a x_{17} + c z_{17} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c z_{17} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H I |
| $\mathbf{B_{34}}$ | = | $- x_{17} \, \mathbf{a}_{1}+\left(y_{17} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $- \left(a x_{17} + c z_{17} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{17} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{17} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H I |
| $\mathbf{B_{35}}$ | = | $x_{18} \, \mathbf{a}_{1}+y_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $\left(a x_{18} + c z_{18} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c z_{18} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H II |
| $\mathbf{B_{36}}$ | = | $- x_{18} \, \mathbf{a}_{1}+\left(y_{18} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $- \left(a x_{18} + c z_{18} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{18} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{18} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H II |
| $\mathbf{B_{37}}$ | = | $x_{19} \, \mathbf{a}_{1}+y_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $\left(a x_{19} + c z_{19} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c z_{19} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H III |
| $\mathbf{B_{38}}$ | = | $- x_{19} \, \mathbf{a}_{1}+\left(y_{19} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $- \left(a x_{19} + c z_{19} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{19} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{19} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H III |
| $\mathbf{B_{39}}$ | = | $x_{20} \, \mathbf{a}_{1}+y_{20} \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $\left(a x_{20} + c z_{20} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c z_{20} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H IV |
| $\mathbf{B_{40}}$ | = | $- x_{20} \, \mathbf{a}_{1}+\left(y_{20} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{20} \, \mathbf{a}_{3}$ | = | $- \left(a x_{20} + c z_{20} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{20} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{20} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H IV |
| $\mathbf{B_{41}}$ | = | $x_{21} \, \mathbf{a}_{1}+y_{21} \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $\left(a x_{21} + c z_{21} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c z_{21} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H V |
| $\mathbf{B_{42}}$ | = | $- x_{21} \, \mathbf{a}_{1}+\left(y_{21} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{21} \, \mathbf{a}_{3}$ | = | $- \left(a x_{21} + c z_{21} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{21} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{21} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H V |
| $\mathbf{B_{43}}$ | = | $x_{22} \, \mathbf{a}_{1}+y_{22} \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $\left(a x_{22} + c z_{22} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c z_{22} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VI |
| $\mathbf{B_{44}}$ | = | $- x_{22} \, \mathbf{a}_{1}+\left(y_{22} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{22} \, \mathbf{a}_{3}$ | = | $- \left(a x_{22} + c z_{22} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{22} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{22} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VI |
| $\mathbf{B_{45}}$ | = | $x_{23} \, \mathbf{a}_{1}+y_{23} \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $\left(a x_{23} + c z_{23} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c z_{23} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VII |
| $\mathbf{B_{46}}$ | = | $- x_{23} \, \mathbf{a}_{1}+\left(y_{23} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{23} \, \mathbf{a}_{3}$ | = | $- \left(a x_{23} + c z_{23} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{23} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{23} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VII |
| $\mathbf{B_{47}}$ | = | $x_{24} \, \mathbf{a}_{1}+y_{24} \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $\left(a x_{24} + c z_{24} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c z_{24} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VIII |
| $\mathbf{B_{48}}$ | = | $- x_{24} \, \mathbf{a}_{1}+\left(y_{24} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{24} \, \mathbf{a}_{3}$ | = | $- \left(a x_{24} + c z_{24} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{24} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{24} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H VIII |
| $\mathbf{B_{49}}$ | = | $x_{25} \, \mathbf{a}_{1}+y_{25} \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $\left(a x_{25} + c z_{25} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c z_{25} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H IX |
| $\mathbf{B_{50}}$ | = | $- x_{25} \, \mathbf{a}_{1}+\left(y_{25} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{25} \, \mathbf{a}_{3}$ | = | $- \left(a x_{25} + c z_{25} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{25} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{25} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H IX |
| $\mathbf{B_{51}}$ | = | $x_{26} \, \mathbf{a}_{1}+y_{26} \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $\left(a x_{26} + c z_{26} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c z_{26} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H X |
| $\mathbf{B_{52}}$ | = | $- x_{26} \, \mathbf{a}_{1}+\left(y_{26} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{26} \, \mathbf{a}_{3}$ | = | $- \left(a x_{26} + c z_{26} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{26} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{26} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H X |
| $\mathbf{B_{53}}$ | = | $x_{27} \, \mathbf{a}_{1}+y_{27} \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $\left(a x_{27} + c z_{27} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c z_{27} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XI |
| $\mathbf{B_{54}}$ | = | $- x_{27} \, \mathbf{a}_{1}+\left(y_{27} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{27} \, \mathbf{a}_{3}$ | = | $- \left(a x_{27} + c z_{27} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{27} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{27} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XI |
| $\mathbf{B_{55}}$ | = | $x_{28} \, \mathbf{a}_{1}+y_{28} \, \mathbf{a}_{2}+z_{28} \, \mathbf{a}_{3}$ | = | $\left(a x_{28} + c z_{28} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{28} \,\mathbf{\hat{y}}+c z_{28} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XII |
| $\mathbf{B_{56}}$ | = | $- x_{28} \, \mathbf{a}_{1}+\left(y_{28} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{28} \, \mathbf{a}_{3}$ | = | $- \left(a x_{28} + c z_{28} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{28} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{28} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XII |
| $\mathbf{B_{57}}$ | = | $x_{29} \, \mathbf{a}_{1}+y_{29} \, \mathbf{a}_{2}+z_{29} \, \mathbf{a}_{3}$ | = | $\left(a x_{29} + c z_{29} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{29} \,\mathbf{\hat{y}}+c z_{29} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIII |
| $\mathbf{B_{58}}$ | = | $- x_{29} \, \mathbf{a}_{1}+\left(y_{29} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{29} \, \mathbf{a}_{3}$ | = | $- \left(a x_{29} + c z_{29} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{29} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{29} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIII |
| $\mathbf{B_{59}}$ | = | $x_{30} \, \mathbf{a}_{1}+y_{30} \, \mathbf{a}_{2}+z_{30} \, \mathbf{a}_{3}$ | = | $\left(a x_{30} + c z_{30} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{30} \,\mathbf{\hat{y}}+c z_{30} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIV |
| $\mathbf{B_{60}}$ | = | $- x_{30} \, \mathbf{a}_{1}+\left(y_{30} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{30} \, \mathbf{a}_{3}$ | = | $- \left(a x_{30} + c z_{30} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{30} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{30} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIV |
| $\mathbf{B_{61}}$ | = | $x_{31} \, \mathbf{a}_{1}+y_{31} \, \mathbf{a}_{2}+z_{31} \, \mathbf{a}_{3}$ | = | $\left(a x_{31} + c z_{31} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{31} \,\mathbf{\hat{y}}+c z_{31} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XV |
| $\mathbf{B_{62}}$ | = | $- x_{31} \, \mathbf{a}_{1}+\left(y_{31} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{31} \, \mathbf{a}_{3}$ | = | $- \left(a x_{31} + c z_{31} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{31} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{31} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XV |
| $\mathbf{B_{63}}$ | = | $x_{32} \, \mathbf{a}_{1}+y_{32} \, \mathbf{a}_{2}+z_{32} \, \mathbf{a}_{3}$ | = | $\left(a x_{32} + c z_{32} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{32} \,\mathbf{\hat{y}}+c z_{32} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVI |
| $\mathbf{B_{64}}$ | = | $- x_{32} \, \mathbf{a}_{1}+\left(y_{32} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{32} \, \mathbf{a}_{3}$ | = | $- \left(a x_{32} + c z_{32} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{32} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{32} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVI |
| $\mathbf{B_{65}}$ | = | $x_{33} \, \mathbf{a}_{1}+y_{33} \, \mathbf{a}_{2}+z_{33} \, \mathbf{a}_{3}$ | = | $\left(a x_{33} + c z_{33} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{33} \,\mathbf{\hat{y}}+c z_{33} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVII |
| $\mathbf{B_{66}}$ | = | $- x_{33} \, \mathbf{a}_{1}+\left(y_{33} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{33} \, \mathbf{a}_{3}$ | = | $- \left(a x_{33} + c z_{33} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{33} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{33} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVII |
| $\mathbf{B_{67}}$ | = | $x_{34} \, \mathbf{a}_{1}+y_{34} \, \mathbf{a}_{2}+z_{34} \, \mathbf{a}_{3}$ | = | $\left(a x_{34} + c z_{34} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{34} \,\mathbf{\hat{y}}+c z_{34} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVIII |
| $\mathbf{B_{68}}$ | = | $- x_{34} \, \mathbf{a}_{1}+\left(y_{34} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{34} \, \mathbf{a}_{3}$ | = | $- \left(a x_{34} + c z_{34} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{34} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{34} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XVIII |
| $\mathbf{B_{69}}$ | = | $x_{35} \, \mathbf{a}_{1}+y_{35} \, \mathbf{a}_{2}+z_{35} \, \mathbf{a}_{3}$ | = | $\left(a x_{35} + c z_{35} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{35} \,\mathbf{\hat{y}}+c z_{35} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIX |
| $\mathbf{B_{70}}$ | = | $- x_{35} \, \mathbf{a}_{1}+\left(y_{35} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{35} \, \mathbf{a}_{3}$ | = | $- \left(a x_{35} + c z_{35} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{35} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{35} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XIX |
| $\mathbf{B_{71}}$ | = | $x_{36} \, \mathbf{a}_{1}+y_{36} \, \mathbf{a}_{2}+z_{36} \, \mathbf{a}_{3}$ | = | $\left(a x_{36} + c z_{36} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{36} \,\mathbf{\hat{y}}+c z_{36} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XX |
| $\mathbf{B_{72}}$ | = | $- x_{36} \, \mathbf{a}_{1}+\left(y_{36} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{36} \, \mathbf{a}_{3}$ | = | $- \left(a x_{36} + c z_{36} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{36} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{36} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XX |
| $\mathbf{B_{73}}$ | = | $x_{37} \, \mathbf{a}_{1}+y_{37} \, \mathbf{a}_{2}+z_{37} \, \mathbf{a}_{3}$ | = | $\left(a x_{37} + c z_{37} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{37} \,\mathbf{\hat{y}}+c z_{37} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXI |
| $\mathbf{B_{74}}$ | = | $- x_{37} \, \mathbf{a}_{1}+\left(y_{37} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{37} \, \mathbf{a}_{3}$ | = | $- \left(a x_{37} + c z_{37} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{37} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{37} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXI |
| $\mathbf{B_{75}}$ | = | $x_{38} \, \mathbf{a}_{1}+y_{38} \, \mathbf{a}_{2}+z_{38} \, \mathbf{a}_{3}$ | = | $\left(a x_{38} + c z_{38} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{38} \,\mathbf{\hat{y}}+c z_{38} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXII |
| $\mathbf{B_{76}}$ | = | $- x_{38} \, \mathbf{a}_{1}+\left(y_{38} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{38} \, \mathbf{a}_{3}$ | = | $- \left(a x_{38} + c z_{38} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{38} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{38} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXII |
| $\mathbf{B_{77}}$ | = | $x_{39} \, \mathbf{a}_{1}+y_{39} \, \mathbf{a}_{2}+z_{39} \, \mathbf{a}_{3}$ | = | $\left(a x_{39} + c z_{39} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{39} \,\mathbf{\hat{y}}+c z_{39} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIII |
| $\mathbf{B_{78}}$ | = | $- x_{39} \, \mathbf{a}_{1}+\left(y_{39} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{39} \, \mathbf{a}_{3}$ | = | $- \left(a x_{39} + c z_{39} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{39} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{39} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIII |
| $\mathbf{B_{79}}$ | = | $x_{40} \, \mathbf{a}_{1}+y_{40} \, \mathbf{a}_{2}+z_{40} \, \mathbf{a}_{3}$ | = | $\left(a x_{40} + c z_{40} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{40} \,\mathbf{\hat{y}}+c z_{40} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIV |
| $\mathbf{B_{80}}$ | = | $- x_{40} \, \mathbf{a}_{1}+\left(y_{40} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{40} \, \mathbf{a}_{3}$ | = | $- \left(a x_{40} + c z_{40} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{40} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{40} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIV |
| $\mathbf{B_{81}}$ | = | $x_{41} \, \mathbf{a}_{1}+y_{41} \, \mathbf{a}_{2}+z_{41} \, \mathbf{a}_{3}$ | = | $\left(a x_{41} + c z_{41} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{41} \,\mathbf{\hat{y}}+c z_{41} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXV |
| $\mathbf{B_{82}}$ | = | $- x_{41} \, \mathbf{a}_{1}+\left(y_{41} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{41} \, \mathbf{a}_{3}$ | = | $- \left(a x_{41} + c z_{41} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{41} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{41} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXV |
| $\mathbf{B_{83}}$ | = | $x_{42} \, \mathbf{a}_{1}+y_{42} \, \mathbf{a}_{2}+z_{42} \, \mathbf{a}_{3}$ | = | $\left(a x_{42} + c z_{42} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{42} \,\mathbf{\hat{y}}+c z_{42} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVI |
| $\mathbf{B_{84}}$ | = | $- x_{42} \, \mathbf{a}_{1}+\left(y_{42} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{42} \, \mathbf{a}_{3}$ | = | $- \left(a x_{42} + c z_{42} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{42} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{42} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVI |
| $\mathbf{B_{85}}$ | = | $x_{43} \, \mathbf{a}_{1}+y_{43} \, \mathbf{a}_{2}+z_{43} \, \mathbf{a}_{3}$ | = | $\left(a x_{43} + c z_{43} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{43} \,\mathbf{\hat{y}}+c z_{43} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVII |
| $\mathbf{B_{86}}$ | = | $- x_{43} \, \mathbf{a}_{1}+\left(y_{43} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{43} \, \mathbf{a}_{3}$ | = | $- \left(a x_{43} + c z_{43} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{43} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{43} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVII |
| $\mathbf{B_{87}}$ | = | $x_{44} \, \mathbf{a}_{1}+y_{44} \, \mathbf{a}_{2}+z_{44} \, \mathbf{a}_{3}$ | = | $\left(a x_{44} + c z_{44} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{44} \,\mathbf{\hat{y}}+c z_{44} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVIII |
| $\mathbf{B_{88}}$ | = | $- x_{44} \, \mathbf{a}_{1}+\left(y_{44} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{44} \, \mathbf{a}_{3}$ | = | $- \left(a x_{44} + c z_{44} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{44} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{44} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXVIII |
| $\mathbf{B_{89}}$ | = | $x_{45} \, \mathbf{a}_{1}+y_{45} \, \mathbf{a}_{2}+z_{45} \, \mathbf{a}_{3}$ | = | $\left(a x_{45} + c z_{45} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{45} \,\mathbf{\hat{y}}+c z_{45} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIX |
| $\mathbf{B_{90}}$ | = | $- x_{45} \, \mathbf{a}_{1}+\left(y_{45} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{45} \, \mathbf{a}_{3}$ | = | $- \left(a x_{45} + c z_{45} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{45} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{45} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXIX |
| $\mathbf{B_{91}}$ | = | $x_{46} \, \mathbf{a}_{1}+y_{46} \, \mathbf{a}_{2}+z_{46} \, \mathbf{a}_{3}$ | = | $\left(a x_{46} + c z_{46} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{46} \,\mathbf{\hat{y}}+c z_{46} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXX |
| $\mathbf{B_{92}}$ | = | $- x_{46} \, \mathbf{a}_{1}+\left(y_{46} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{46} \, \mathbf{a}_{3}$ | = | $- \left(a x_{46} + c z_{46} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{46} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{46} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXX |
| $\mathbf{B_{93}}$ | = | $x_{47} \, \mathbf{a}_{1}+y_{47} \, \mathbf{a}_{2}+z_{47} \, \mathbf{a}_{3}$ | = | $\left(a x_{47} + c z_{47} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{47} \,\mathbf{\hat{y}}+c z_{47} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXI |
| $\mathbf{B_{94}}$ | = | $- x_{47} \, \mathbf{a}_{1}+\left(y_{47} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{47} \, \mathbf{a}_{3}$ | = | $- \left(a x_{47} + c z_{47} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{47} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{47} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXI |
| $\mathbf{B_{95}}$ | = | $x_{48} \, \mathbf{a}_{1}+y_{48} \, \mathbf{a}_{2}+z_{48} \, \mathbf{a}_{3}$ | = | $\left(a x_{48} + c z_{48} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{48} \,\mathbf{\hat{y}}+c z_{48} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXII |
| $\mathbf{B_{96}}$ | = | $- x_{48} \, \mathbf{a}_{1}+\left(y_{48} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{48} \, \mathbf{a}_{3}$ | = | $- \left(a x_{48} + c z_{48} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{48} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{48} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXII |
| $\mathbf{B_{97}}$ | = | $x_{49} \, \mathbf{a}_{1}+y_{49} \, \mathbf{a}_{2}+z_{49} \, \mathbf{a}_{3}$ | = | $\left(a x_{49} + c z_{49} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{49} \,\mathbf{\hat{y}}+c z_{49} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIII |
| $\mathbf{B_{98}}$ | = | $- x_{49} \, \mathbf{a}_{1}+\left(y_{49} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{49} \, \mathbf{a}_{3}$ | = | $- \left(a x_{49} + c z_{49} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{49} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{49} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIII |
| $\mathbf{B_{99}}$ | = | $x_{50} \, \mathbf{a}_{1}+y_{50} \, \mathbf{a}_{2}+z_{50} \, \mathbf{a}_{3}$ | = | $\left(a x_{50} + c z_{50} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{50} \,\mathbf{\hat{y}}+c z_{50} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIV |
| $\mathbf{B_{100}}$ | = | $- x_{50} \, \mathbf{a}_{1}+\left(y_{50} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{50} \, \mathbf{a}_{3}$ | = | $- \left(a x_{50} + c z_{50} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{50} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{50} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIV |
| $\mathbf{B_{101}}$ | = | $x_{51} \, \mathbf{a}_{1}+y_{51} \, \mathbf{a}_{2}+z_{51} \, \mathbf{a}_{3}$ | = | $\left(a x_{51} + c z_{51} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{51} \,\mathbf{\hat{y}}+c z_{51} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXV |
| $\mathbf{B_{102}}$ | = | $- x_{51} \, \mathbf{a}_{1}+\left(y_{51} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{51} \, \mathbf{a}_{3}$ | = | $- \left(a x_{51} + c z_{51} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{51} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{51} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXV |
| $\mathbf{B_{103}}$ | = | $x_{52} \, \mathbf{a}_{1}+y_{52} \, \mathbf{a}_{2}+z_{52} \, \mathbf{a}_{3}$ | = | $\left(a x_{52} + c z_{52} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{52} \,\mathbf{\hat{y}}+c z_{52} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVI |
| $\mathbf{B_{104}}$ | = | $- x_{52} \, \mathbf{a}_{1}+\left(y_{52} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{52} \, \mathbf{a}_{3}$ | = | $- \left(a x_{52} + c z_{52} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{52} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{52} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVI |
| $\mathbf{B_{105}}$ | = | $x_{53} \, \mathbf{a}_{1}+y_{53} \, \mathbf{a}_{2}+z_{53} \, \mathbf{a}_{3}$ | = | $\left(a x_{53} + c z_{53} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{53} \,\mathbf{\hat{y}}+c z_{53} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVII |
| $\mathbf{B_{106}}$ | = | $- x_{53} \, \mathbf{a}_{1}+\left(y_{53} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{53} \, \mathbf{a}_{3}$ | = | $- \left(a x_{53} + c z_{53} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{53} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{53} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVII |
| $\mathbf{B_{107}}$ | = | $x_{54} \, \mathbf{a}_{1}+y_{54} \, \mathbf{a}_{2}+z_{54} \, \mathbf{a}_{3}$ | = | $\left(a x_{54} + c z_{54} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{54} \,\mathbf{\hat{y}}+c z_{54} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVIII |
| $\mathbf{B_{108}}$ | = | $- x_{54} \, \mathbf{a}_{1}+\left(y_{54} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{54} \, \mathbf{a}_{3}$ | = | $- \left(a x_{54} + c z_{54} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{54} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{54} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXVIII |
| $\mathbf{B_{109}}$ | = | $x_{55} \, \mathbf{a}_{1}+y_{55} \, \mathbf{a}_{2}+z_{55} \, \mathbf{a}_{3}$ | = | $\left(a x_{55} + c z_{55} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{55} \,\mathbf{\hat{y}}+c z_{55} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIX |
| $\mathbf{B_{110}}$ | = | $- x_{55} \, \mathbf{a}_{1}+\left(y_{55} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{55} \, \mathbf{a}_{3}$ | = | $- \left(a x_{55} + c z_{55} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{55} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{55} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XXXIX |
| $\mathbf{B_{111}}$ | = | $x_{56} \, \mathbf{a}_{1}+y_{56} \, \mathbf{a}_{2}+z_{56} \, \mathbf{a}_{3}$ | = | $\left(a x_{56} + c z_{56} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{56} \,\mathbf{\hat{y}}+c z_{56} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XL |
| $\mathbf{B_{112}}$ | = | $- x_{56} \, \mathbf{a}_{1}+\left(y_{56} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{56} \, \mathbf{a}_{3}$ | = | $- \left(a x_{56} + c z_{56} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{56} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{56} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XL |
| $\mathbf{B_{113}}$ | = | $x_{57} \, \mathbf{a}_{1}+y_{57} \, \mathbf{a}_{2}+z_{57} \, \mathbf{a}_{3}$ | = | $\left(a x_{57} + c z_{57} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{57} \,\mathbf{\hat{y}}+c z_{57} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLI |
| $\mathbf{B_{114}}$ | = | $- x_{57} \, \mathbf{a}_{1}+\left(y_{57} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{57} \, \mathbf{a}_{3}$ | = | $- \left(a x_{57} + c z_{57} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{57} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{57} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLI |
| $\mathbf{B_{115}}$ | = | $x_{58} \, \mathbf{a}_{1}+y_{58} \, \mathbf{a}_{2}+z_{58} \, \mathbf{a}_{3}$ | = | $\left(a x_{58} + c z_{58} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{58} \,\mathbf{\hat{y}}+c z_{58} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLII |
| $\mathbf{B_{116}}$ | = | $- x_{58} \, \mathbf{a}_{1}+\left(y_{58} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{58} \, \mathbf{a}_{3}$ | = | $- \left(a x_{58} + c z_{58} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{58} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{58} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLII |
| $\mathbf{B_{117}}$ | = | $x_{59} \, \mathbf{a}_{1}+y_{59} \, \mathbf{a}_{2}+z_{59} \, \mathbf{a}_{3}$ | = | $\left(a x_{59} + c z_{59} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{59} \,\mathbf{\hat{y}}+c z_{59} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLIII |
| $\mathbf{B_{118}}$ | = | $- x_{59} \, \mathbf{a}_{1}+\left(y_{59} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{59} \, \mathbf{a}_{3}$ | = | $- \left(a x_{59} + c z_{59} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{59} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{59} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLIII |
| $\mathbf{B_{119}}$ | = | $x_{60} \, \mathbf{a}_{1}+y_{60} \, \mathbf{a}_{2}+z_{60} \, \mathbf{a}_{3}$ | = | $\left(a x_{60} + c z_{60} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{60} \,\mathbf{\hat{y}}+c z_{60} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLIV |
| $\mathbf{B_{120}}$ | = | $- x_{60} \, \mathbf{a}_{1}+\left(y_{60} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{60} \, \mathbf{a}_{3}$ | = | $- \left(a x_{60} + c z_{60} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{60} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{60} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLIV |
| $\mathbf{B_{121}}$ | = | $x_{61} \, \mathbf{a}_{1}+y_{61} \, \mathbf{a}_{2}+z_{61} \, \mathbf{a}_{3}$ | = | $\left(a x_{61} + c z_{61} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{61} \,\mathbf{\hat{y}}+c z_{61} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLV |
| $\mathbf{B_{122}}$ | = | $- x_{61} \, \mathbf{a}_{1}+\left(y_{61} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{61} \, \mathbf{a}_{3}$ | = | $- \left(a x_{61} + c z_{61} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{61} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{61} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | H XLV |
| $\mathbf{B_{123}}$ | = | $x_{62} \, \mathbf{a}_{1}+y_{62} \, \mathbf{a}_{2}+z_{62} \, \mathbf{a}_{3}$ | = | $\left(a x_{62} + c z_{62} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{62} \,\mathbf{\hat{y}}+c z_{62} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N I |
| $\mathbf{B_{124}}$ | = | $- x_{62} \, \mathbf{a}_{1}+\left(y_{62} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{62} \, \mathbf{a}_{3}$ | = | $- \left(a x_{62} + c z_{62} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{62} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{62} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N I |
| $\mathbf{B_{125}}$ | = | $x_{63} \, \mathbf{a}_{1}+y_{63} \, \mathbf{a}_{2}+z_{63} \, \mathbf{a}_{3}$ | = | $\left(a x_{63} + c z_{63} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{63} \,\mathbf{\hat{y}}+c z_{63} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N II |
| $\mathbf{B_{126}}$ | = | $- x_{63} \, \mathbf{a}_{1}+\left(y_{63} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{63} \, \mathbf{a}_{3}$ | = | $- \left(a x_{63} + c z_{63} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{63} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{63} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N II |
| $\mathbf{B_{127}}$ | = | $x_{64} \, \mathbf{a}_{1}+y_{64} \, \mathbf{a}_{2}+z_{64} \, \mathbf{a}_{3}$ | = | $\left(a x_{64} + c z_{64} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{64} \,\mathbf{\hat{y}}+c z_{64} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N III |
| $\mathbf{B_{128}}$ | = | $- x_{64} \, \mathbf{a}_{1}+\left(y_{64} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{64} \, \mathbf{a}_{3}$ | = | $- \left(a x_{64} + c z_{64} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{64} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{64} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N III |
| $\mathbf{B_{129}}$ | = | $x_{65} \, \mathbf{a}_{1}+y_{65} \, \mathbf{a}_{2}+z_{65} \, \mathbf{a}_{3}$ | = | $\left(a x_{65} + c z_{65} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{65} \,\mathbf{\hat{y}}+c z_{65} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N IV |
| $\mathbf{B_{130}}$ | = | $- x_{65} \, \mathbf{a}_{1}+\left(y_{65} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{65} \, \mathbf{a}_{3}$ | = | $- \left(a x_{65} + c z_{65} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{65} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{65} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | N IV |
| $\mathbf{B_{131}}$ | = | $x_{66} \, \mathbf{a}_{1}+y_{66} \, \mathbf{a}_{2}+z_{66} \, \mathbf{a}_{3}$ | = | $\left(a x_{66} + c z_{66} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{66} \,\mathbf{\hat{y}}+c z_{66} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb I |
| $\mathbf{B_{132}}$ | = | $- x_{66} \, \mathbf{a}_{1}+\left(y_{66} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{66} \, \mathbf{a}_{3}$ | = | $- \left(a x_{66} + c z_{66} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{66} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{66} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb I |
| $\mathbf{B_{133}}$ | = | $x_{67} \, \mathbf{a}_{1}+y_{67} \, \mathbf{a}_{2}+z_{67} \, \mathbf{a}_{3}$ | = | $\left(a x_{67} + c z_{67} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{67} \,\mathbf{\hat{y}}+c z_{67} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb II |
| $\mathbf{B_{134}}$ | = | $- x_{67} \, \mathbf{a}_{1}+\left(y_{67} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{67} \, \mathbf{a}_{3}$ | = | $- \left(a x_{67} + c z_{67} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{67} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{67} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb II |
| $\mathbf{B_{135}}$ | = | $x_{68} \, \mathbf{a}_{1}+y_{68} \, \mathbf{a}_{2}+z_{68} \, \mathbf{a}_{3}$ | = | $\left(a x_{68} + c z_{68} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{68} \,\mathbf{\hat{y}}+c z_{68} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb III |
| $\mathbf{B_{136}}$ | = | $- x_{68} \, \mathbf{a}_{1}+\left(y_{68} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{68} \, \mathbf{a}_{3}$ | = | $- \left(a x_{68} + c z_{68} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{68} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{68} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb III |
| $\mathbf{B_{137}}$ | = | $x_{69} \, \mathbf{a}_{1}+y_{69} \, \mathbf{a}_{2}+z_{69} \, \mathbf{a}_{3}$ | = | $\left(a x_{69} + c z_{69} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{69} \,\mathbf{\hat{y}}+c z_{69} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb IV |
| $\mathbf{B_{138}}$ | = | $- x_{69} \, \mathbf{a}_{1}+\left(y_{69} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{69} \, \mathbf{a}_{3}$ | = | $- \left(a x_{69} + c z_{69} \cos{\beta}\right) \,\mathbf{\hat{x}}+b \left(y_{69} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{69} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Pb IV |
References
- A. Liang, J. Gonzalez-Platas, R. Turnbull, C. Popescu, I. Fernandez-Guillen, R. Abargues, P. P. Boix, L.-T. Shi, and D. Errandonea, Reassigning the Pressure-Induced Phase Transitions of Methylammonium Lead Bromide Perovskite}, J. Am. Chem. Soc. 144, 20099–20108 (2022), doi:10.1021/jacs.2c09457.
- C. Abia, C. A. López, L. C. {nadillas Delgado, M. T. Fernández-Diaz, and J. A. Alonso, Crystal structure thermal evolution and novel orthorhombic phase of methylammonium lead bromide, CH$_{3}$NH$_{3}$PbBr$_{3}$, Sci. Rep. 12, 18647 (2022), doi:10.1038/s41598-022-21544-2.
- 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.
Found in
- 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.
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=A12B4C45D4E4_mP138_4_12a_4a_45a_4a_4a --params=$a,b/a,c/a,\beta,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19},x_{20},y_{20},z_{20},x_{21},y_{21},z_{21},x_{22},y_{22},z_{22},x_{23},y_{23},z_{23},x_{24},y_{24},z_{24},x_{25},y_{25},z_{25},x_{26},y_{26},z_{26},x_{27},y_{27},z_{27},x_{28},y_{28},z_{28},x_{29},y_{29},z_{29},x_{30},y_{30},z_{30},x_{31},y_{31},z_{31},x_{32},y_{32},z_{32},x_{33},y_{33},z_{33},x_{34},y_{34},z_{34},x_{35},y_{35},z_{35},x_{36},y_{36},z_{36},x_{37},y_{37},z_{37},x_{38},y_{38},z_{38},x_{39},y_{39},z_{39},x_{40},y_{40},z_{40},x_{41},y_{41},z_{41},x_{42},y_{42},z_{42},x_{43},y_{43},z_{43},x_{44},y_{44},z_{44},x_{45},y_{45},z_{45},x_{46},y_{46},z_{46},x_{47},y_{47},z_{47},x_{48},y_{48},z_{48},x_{49},y_{49},z_{49},x_{50},y_{50},z_{50},x_{51},y_{51},z_{51},x_{52},y_{52},z_{52},x_{53},y_{53},z_{53},x_{54},y_{54},z_{54},x_{55},y_{55},z_{55},x_{56},y_{56},z_{56},x_{57},y_{57},z_{57},x_{58},y_{58},z_{58},x_{59},y_{59},z_{59},x_{60},y_{60},z_{60},x_{61},y_{61},z_{61},x_{62},y_{62},z_{62},x_{63},y_{63},z_{63},x_{64},y_{64},z_{64},x_{65},y_{65},z_{65},x_{66},y_{66},z_{66},x_{67},y_{67},z_{67},x_{68},y_{68},z_{68},x_{69},y_{69},z_{69}$