$B30$ (MgZn?) Structure (Problematic): AB_oI48_44_6c_abc2de-001
| Prototype | MgZn |
| AFLOW prototype label | AB_oI48_44_6c_abc2de-001 |
| Strukturbericht designation | $B30$ |
| ICSD | 151402 |
| Pearson symbol | oI48 |
| Space group number | 44 |
| Space group symbol | $Imm2$ |
| AFLOW prototype command |
aflow --proto=AB_oI48_44_6c_abc2de-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak z_{1}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak z_{9}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}$ |
- It is rather a mystery why (Hermann, 1937) gave this the Strukturbericht designation $B30$, as the structure presented in the literature contradicts itself. (Tarschish, 1933) derived this structure from the hexagonal Laves structure MgZn$_{2}$ (C14) by doubling the unit cell in all directions to obtain a 96 atom unit cell, replacing 16 of the zinc atoms in this structure by magnesium, and shifting the $z$-coordinates of these atoms by $\pm c/16$. He then states that the space group remains $P6_{3}/mmc$ #194.
- (McKeehan, 1935) pointed out that this is impossible, as the converted Mg atoms only have a two-fold rotation axis about the $z$-axis. He assigned the structure to space group $Pmm2$ #25.
- (Hermann, 1937) referenced both papers, giving the space group as $P6_{3}/mmc$ but listing the atomic coordinates enumerated by McKeehan.
- In fact, the McKeehan structure has space group $Imm2$ #44, with 48 atoms in the conventional cell, half of the original, and 24 atoms in the primitive cell. This was noted, without reference, by (Parthé, 1993), which is the only comprehensive list of Strukturbericht symbols to include the $B30$ structure. We have reproduced this $Imm2$ structure from McKeehan's data.
- The true structure of MgZn is unclear. It is seen in the Mg-Zn binary phase diagram (Massalski, 1990) over a small range of compositions, but a complete crystallographic study has never been published. It is possible that the actual structure is off-stoichiometry. There is some evidence of an Mg$_{12}$Zn$_{13}$ structure (Mezbahul-Islam, 2014), and Mg$_{21}$Zn$_{25}$ has been determined (Cerný, 2002) to have the Zr$_{21}$Re$_{25}$ structure.
- There are similar problems with the $D2_{2}$ MgZn$_{5}$ structure, which we discuss on that page.
- The ICSD entry is from (Tarschish, 1933). It gives the atomic positions in space group $P1$ #1, but AFLOW finds that the structure is in space group $Imm2$ #44, as found from our analysis of (McKeehan, 1935). Unsurprisingly, this structure does not agree with our interpretation of the data.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}- \frac{1}{2}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $z_{1} \, \mathbf{a}_{1}+z_{1} \, \mathbf{a}_{2}$ | = | $c z_{1} \,\mathbf{\hat{z}}$ | (2a) | Zn I |
| $\mathbf{B_{2}}$ | = | $\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}b \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (2b) | Zn II |
| $\mathbf{B_{3}}$ | = | $z_{3} \, \mathbf{a}_{1}+\left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+c z_{3} \,\mathbf{\hat{z}}$ | (4c) | Mg I |
| $\mathbf{B_{4}}$ | = | $z_{3} \, \mathbf{a}_{1}- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+c z_{3} \,\mathbf{\hat{z}}$ | (4c) | Mg I |
| $\mathbf{B_{5}}$ | = | $z_{4} \, \mathbf{a}_{1}+\left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+c z_{4} \,\mathbf{\hat{z}}$ | (4c) | Mg II |
| $\mathbf{B_{6}}$ | = | $z_{4} \, \mathbf{a}_{1}- \left(x_{4} - z_{4}\right) \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+c z_{4} \,\mathbf{\hat{z}}$ | (4c) | Mg II |
| $\mathbf{B_{7}}$ | = | $z_{5} \, \mathbf{a}_{1}+\left(x_{5} + z_{5}\right) \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+c z_{5} \,\mathbf{\hat{z}}$ | (4c) | Mg III |
| $\mathbf{B_{8}}$ | = | $z_{5} \, \mathbf{a}_{1}- \left(x_{5} - z_{5}\right) \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+c z_{5} \,\mathbf{\hat{z}}$ | (4c) | Mg III |
| $\mathbf{B_{9}}$ | = | $z_{6} \, \mathbf{a}_{1}+\left(x_{6} + z_{6}\right) \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+c z_{6} \,\mathbf{\hat{z}}$ | (4c) | Mg IV |
| $\mathbf{B_{10}}$ | = | $z_{6} \, \mathbf{a}_{1}- \left(x_{6} - z_{6}\right) \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}+c z_{6} \,\mathbf{\hat{z}}$ | (4c) | Mg IV |
| $\mathbf{B_{11}}$ | = | $z_{7} \, \mathbf{a}_{1}+\left(x_{7} + z_{7}\right) \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+c z_{7} \,\mathbf{\hat{z}}$ | (4c) | Mg V |
| $\mathbf{B_{12}}$ | = | $z_{7} \, \mathbf{a}_{1}- \left(x_{7} - z_{7}\right) \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+c z_{7} \,\mathbf{\hat{z}}$ | (4c) | Mg V |
| $\mathbf{B_{13}}$ | = | $z_{8} \, \mathbf{a}_{1}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+c z_{8} \,\mathbf{\hat{z}}$ | (4c) | Mg VI |
| $\mathbf{B_{14}}$ | = | $z_{8} \, \mathbf{a}_{1}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+c z_{8} \,\mathbf{\hat{z}}$ | (4c) | Mg VI |
| $\mathbf{B_{15}}$ | = | $z_{9} \, \mathbf{a}_{1}+\left(x_{9} + z_{9}\right) \, \mathbf{a}_{2}+x_{9} \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+c z_{9} \,\mathbf{\hat{z}}$ | (4c) | Zn III |
| $\mathbf{B_{16}}$ | = | $z_{9} \, \mathbf{a}_{1}- \left(x_{9} - z_{9}\right) \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}+c z_{9} \,\mathbf{\hat{z}}$ | (4c) | Zn III |
| $\mathbf{B_{17}}$ | = | $\left(y_{10} + z_{10}\right) \, \mathbf{a}_{1}+z_{10} \, \mathbf{a}_{2}+y_{10} \, \mathbf{a}_{3}$ | = | $b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (4d) | Zn IV |
| $\mathbf{B_{18}}$ | = | $- \left(y_{10} - z_{10}\right) \, \mathbf{a}_{1}+z_{10} \, \mathbf{a}_{2}- y_{10} \, \mathbf{a}_{3}$ | = | $- b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (4d) | Zn IV |
| $\mathbf{B_{19}}$ | = | $\left(y_{11} + z_{11}\right) \, \mathbf{a}_{1}+z_{11} \, \mathbf{a}_{2}+y_{11} \, \mathbf{a}_{3}$ | = | $b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (4d) | Zn V |
| $\mathbf{B_{20}}$ | = | $- \left(y_{11} - z_{11}\right) \, \mathbf{a}_{1}+z_{11} \, \mathbf{a}_{2}- y_{11} \, \mathbf{a}_{3}$ | = | $- b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (4d) | Zn V |
| $\mathbf{B_{21}}$ | = | $\left(y_{12} + z_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} + z_{12}\right) \, \mathbf{a}_{2}+\left(x_{12} + y_{12}\right) \, \mathbf{a}_{3}$ | = | $a x_{12} \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8e) | Zn VI |
| $\mathbf{B_{22}}$ | = | $- \left(y_{12} - z_{12}\right) \, \mathbf{a}_{1}- \left(x_{12} - z_{12}\right) \, \mathbf{a}_{2}- \left(x_{12} + y_{12}\right) \, \mathbf{a}_{3}$ | = | $- a x_{12} \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8e) | Zn VI |
| $\mathbf{B_{23}}$ | = | $- \left(y_{12} - z_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} + z_{12}\right) \, \mathbf{a}_{2}+\left(x_{12} - y_{12}\right) \, \mathbf{a}_{3}$ | = | $a x_{12} \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8e) | Zn VI |
| $\mathbf{B_{24}}$ | = | $\left(y_{12} + z_{12}\right) \, \mathbf{a}_{1}- \left(x_{12} - z_{12}\right) \, \mathbf{a}_{2}- \left(x_{12} - y_{12}\right) \, \mathbf{a}_{3}$ | = | $- a x_{12} \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8e) | Zn VI |
References
- C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928-1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
- L. Tarschisch, Röntgenographische Untersuchung der Verbindungen MgZn und MgZn$_{5}$, Z. Kristallogr. 86, 423–438 (1933), doi:10.1524/zkri.1933.86.1.423.
- L. W. McKeehan, Note on MgZn and MgZn$_{5}$}, Z. Kristallogr. 91, 501–503 (1935), doi:10.1524/zkri.1935.91.1.501. \bibitem{parthe93:TYPIXE. Parthé, L. Gelato, B. Chabot, M. Penso, K. Cenzula, and R. Gladyshevskii, Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types, Gmelin Handbook of Inorganic and Organometallic Chemistry}, vol. 2 (Springer-Verlag, Berlin, Heidelberg, 1993), 8 edn., doi:10.1007/978-3-662-02909-1_3.\bibAnnoteFile{parthe93:TYPIX
- T. B. Massalski, H. Okamoto, P. R. Subramanian, and L. Kacprzak, eds., Binary Alloy Phase Diagrams}, vol. 3 (ASM International, Materials Park, Ohio, USA, 1990), 2$^{nd$ edn. Hf-Re to Zn-Zr.
- M. Mezbahul-Islam, A. O. Mostafa, and M. Medraj, Essential Magnesium Alloys Binary Phase Diagrams and Their Thermochemical Data, J. Mater. 2014, 704283 (2014), doi:10.1155/2014/704283.
- R. Cerný and G. Renaudin, The intermetallic compound Mg$_{21}$Zn$_{25}$, Acta Crystallogr. Sect. C 58, i154–i155 (2002), doi:10.1107/S0108270102018103.
Prototype Generator
aflow --proto=AB_oI48_44_6c_abc2de --params=$a,b/a,c/a,z_{1},z_{2},x_{3},z_{3},x_{4},z_{4},x_{5},z_{5},x_{6},z_{6},x_{7},z_{7},x_{8},z_{8},x_{9},z_{9},y_{10},z_{10},y_{11},z_{11},x_{12},y_{12},z_{12}$