Bergman [Mg$_{32}$(Al,Zn)$_{49}$] Structure: AB32C48_cI162_204_a_2efg_2gh-001
| Prototype | AlMg$_{32}$Zn$_{48}$ |
| AFLOW prototype label | AB32C48_cI162_204_a_2efg_2gh-001 |
| Strukturbericht designation | $D8_{e}$ |
| Mineral name | bergman structure |
| ICSD | 57968 |
| Pearson symbol | cI162 |
| Space group number | 204 |
| Space group symbol | $Im\overline{3}$ |
| AFLOW prototype command |
aflow --proto=AB32C48_cI162_204_a_2efg_2gh-001
--params=$a, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}$ |
Other compounds with this structure
Li$_{20}$Mg$_{6}$Cu$_{13}$Al$_{42}$, Al$_{5}$CuLi$_{3}$
- Most of the sites in this lattice have random occupancy. In particular, according to (Bergman, 1957): The Al-I (2a) site is only occupied 80% of the time, the Zn-I (24g) site is occupied by Al 19% of the time, the Zn-II (24g) site is occupied by Al 43% of the time, and the Zn-III (48h) site is occupied by Al 36% of the time.
- The Li$_{20}$Mg$_{6}$Cu$_{13}$Al$_{42}$ structure found by (Pavlyuk, 2019) has all sites fully occupied.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- \frac{1}{2}a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (2a) | Al I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (12e) | Mg I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (12e) | Mg I |
| $\mathbf{B_{4}}$ | = | $x_{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+a x_{2} \,\mathbf{\hat{y}}$ | (12e) | Mg I |
| $\mathbf{B_{5}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- a x_{2} \,\mathbf{\hat{y}}$ | (12e) | Mg I |
| $\mathbf{B_{6}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+a x_{2} \,\mathbf{\hat{z}}$ | (12e) | Mg I |
| $\mathbf{B_{7}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}- a x_{2} \,\mathbf{\hat{z}}$ | (12e) | Mg I |
| $\mathbf{B_{8}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (12e) | Mg II |
| $\mathbf{B_{9}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (12e) | Mg II |
| $\mathbf{B_{10}}$ | = | $x_{3} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}$ | (12e) | Mg II |
| $\mathbf{B_{11}}$ | = | $- x_{3} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}$ | (12e) | Mg II |
| $\mathbf{B_{12}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (12e) | Mg II |
| $\mathbf{B_{13}}$ | = | $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (12e) | Mg II |
| $\mathbf{B_{14}}$ | = | $2 x_{4} \, \mathbf{a}_{1}+2 x_{4} \, \mathbf{a}_{2}+2 x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{15}}$ | = | $- 2 x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{16}}$ | = | $- 2 x_{4} \, \mathbf{a}_{2}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{17}}$ | = | $- 2 x_{4} \, \mathbf{a}_{1}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{18}}$ | = | $- 2 x_{4} \, \mathbf{a}_{1}- 2 x_{4} \, \mathbf{a}_{2}- 2 x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{19}}$ | = | $2 x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{20}}$ | = | $2 x_{4} \, \mathbf{a}_{2}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{21}}$ | = | $2 x_{4} \, \mathbf{a}_{1}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (16f) | Mg III |
| $\mathbf{B_{22}}$ | = | $\left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{23}}$ | = | $- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{24}}$ | = | $\left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{25}}$ | = | $- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{26}}$ | = | $y_{5} \, \mathbf{a}_{1}+\left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{27}}$ | = | $- y_{5} \, \mathbf{a}_{1}- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{28}}$ | = | $y_{5} \, \mathbf{a}_{1}+\left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{29}}$ | = | $- y_{5} \, \mathbf{a}_{1}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{z}}$ | (24g) | Mg IV |
| $\mathbf{B_{30}}$ | = | $z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+\left(y_{5} + z_{5}\right) \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}$ | (24g) | Mg IV |
| $\mathbf{B_{31}}$ | = | $z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}$ | (24g) | Mg IV |
| $\mathbf{B_{32}}$ | = | $- z_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+\left(y_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}$ | (24g) | Mg IV |
| $\mathbf{B_{33}}$ | = | $- z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a y_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}$ | (24g) | Mg IV |
| $\mathbf{B_{34}}$ | = | $\left(y_{6} + z_{6}\right) \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{35}}$ | = | $- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{36}}$ | = | $\left(y_{6} - z_{6}\right) \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{37}}$ | = | $- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{38}}$ | = | $y_{6} \, \mathbf{a}_{1}+\left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{39}}$ | = | $- y_{6} \, \mathbf{a}_{1}- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{40}}$ | = | $y_{6} \, \mathbf{a}_{1}+\left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{41}}$ | = | $- y_{6} \, \mathbf{a}_{1}- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- a z_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{z}}$ | (24g) | Zn I |
| $\mathbf{B_{42}}$ | = | $z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+\left(y_{6} + z_{6}\right) \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}$ | (24g) | Zn I |
| $\mathbf{B_{43}}$ | = | $z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}$ | (24g) | Zn I |
| $\mathbf{B_{44}}$ | = | $- z_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+\left(y_{6} - z_{6}\right) \, \mathbf{a}_{3}$ | = | $a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}$ | (24g) | Zn I |
| $\mathbf{B_{45}}$ | = | $- z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{3}$ | = | $- a y_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}$ | (24g) | Zn I |
| $\mathbf{B_{46}}$ | = | $\left(y_{7} + z_{7}\right) \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{47}}$ | = | $- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{48}}$ | = | $\left(y_{7} - z_{7}\right) \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{49}}$ | = | $- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{50}}$ | = | $y_{7} \, \mathbf{a}_{1}+\left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{51}}$ | = | $- y_{7} \, \mathbf{a}_{1}- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{52}}$ | = | $y_{7} \, \mathbf{a}_{1}+\left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{53}}$ | = | $- y_{7} \, \mathbf{a}_{1}- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{z}}$ | (24g) | Zn II |
| $\mathbf{B_{54}}$ | = | $z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+\left(y_{7} + z_{7}\right) \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}$ | (24g) | Zn II |
| $\mathbf{B_{55}}$ | = | $z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}$ | (24g) | Zn II |
| $\mathbf{B_{56}}$ | = | $- z_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+\left(y_{7} - z_{7}\right) \, \mathbf{a}_{3}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}$ | (24g) | Zn II |
| $\mathbf{B_{57}}$ | = | $- z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{3}$ | = | $- a y_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}$ | (24g) | Zn II |
| $\mathbf{B_{58}}$ | = | $\left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{59}}$ | = | $- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{60}}$ | = | $\left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{61}}$ | = | $- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{62}}$ | = | $\left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{63}}$ | = | $- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{64}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{65}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{66}}$ | = | $\left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{67}}$ | = | $- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{68}}$ | = | $- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{69}}$ | = | $\left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{70}}$ | = | $- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{71}}$ | = | $\left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{72}}$ | = | $- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{73}}$ | = | $\left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{74}}$ | = | $- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{75}}$ | = | $\left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{76}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{77}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a y_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{78}}$ | = | $- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{79}}$ | = | $\left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{80}}$ | = | $\left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
| $\mathbf{B_{81}}$ | = | $- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $a y_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (48h) | Zn III |
References
- G. Bergman, J. L. T. Waugh, and L. Pauling, The crystal structure of the metallic phase Mg$_{32}$(Al,Zn)$_{49}$ 10, 254–9 (1957), doi:10.1107/S0365110X57000808.
- N. Pavlyuk, G. Dmytriv, V. Pavlyuk, and H. Ehrenberg, Li$_{20}$Mg$_{6}$Cu$_{13}$Al$_{42}$: a new ordered quaternary superstructure to the icosahedral T-Mg$_{32}$(Zn,Al)$_{49}$ phase with fullerene‐like Al$_{60}$ cluster, Acta Crystallogr. Sect. B 75, 168–174 (2019), doi:10.1107/S2052520619000349.
Prototype Generator
aflow --proto=AB32C48_cI162_204_a_2efg_2gh --params=$a,x_{2},x_{3},x_{4},y_{5},z_{5},y_{6},z_{6},y_{7},z_{7},x_{8},y_{8},z_{8}$