# Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: AB_oI48_44_6d_ab2cde

• M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
• D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
• D. Hicks, M.J. Mehl, M. Esters, C. Oses, O. Levy, G.L.W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

## $B30$ (MgZn?) Structure : AB_oI48_44_6d_ab2cde

 Prototype : MgZn AFLOW prototype label : AB_oI48_44_6d_ab2cde Strukturbericht designation : $B30$ Pearson symbol : oI48 Space group number : 44 Space group symbol : $Imm2$ AFLOW prototype command : aflow --proto=AB_oI48_44_6d_ab2cde --params=$a$,$b/a$,$c/a$,$z_{1}$,$z_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$,$y_{5}$,$z_{5}$,$y_{6}$,$z_{6}$,$y_{7}$,$z_{7}$,$y_{8}$,$z_{8}$,$y_{9}$,$z_{9}$,$y_{10}$,$z_{10}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$

• 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 MgZn2 ($C14$) 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, although it is seen in the Mg–Zn binary phase diagram (Massalski, 1990) over a small range of compositions, a complete crystallographic study has never been published. It is possible that the actual structure is off–stoichiometry. There is some evidence of a Mg12Zn13 structure (Mezbahul–Islam, 2014), and Mg21Zn25 has been determined (Cerný, 2002) to have the Zr21Re25 structure. There are similar problems with the $D2_{2}$ MgZn5 structure, which we discuss on that page.

### Body-centered Orthorhombic primitive vectors:

$\begin{array}{ccc} \mathbf{a}_1 & = & - \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} + \frac12 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac12 \, b \, \mathbf{\hat{y}} + \frac12 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} - \frac12 \, c \, \mathbf{\hat{z}} \\ \end{array}$

Basis vectors:

$\begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & z_{1} \, \mathbf{a}_{1} + z_{1} \, \mathbf{a}_{2} & = & z_{1}c \, \mathbf{\hat{z}} & \left(2a\right) & \text{Zn I} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{1} + z_{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(2b\right) & \text{Zn II} \\ \mathbf{B}_{3} & = & z_{3} \, \mathbf{a}_{1} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + z_{3}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Zn III} \\ \mathbf{B}_{4} & = & z_{3} \, \mathbf{a}_{1} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + z_{3}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Zn III} \\ \mathbf{B}_{5} & = & z_{4} \, \mathbf{a}_{1} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + z_{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Zn IV} \\ \mathbf{B}_{6} & = & z_{4} \, \mathbf{a}_{1} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + z_{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \text{Zn IV} \\ \mathbf{B}_{7} & = & \left(y_{5}+z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg I} \\ \mathbf{B}_{8} & = & \left(-y_{5}+z_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2}-y_{5} \, \mathbf{a}_{3} & = & -y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg I} \\ \mathbf{B}_{9} & = & \left(y_{6}+z_{6}\right) \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2} + y_{6} \, \mathbf{a}_{3} & = & y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg II} \\ \mathbf{B}_{10} & = & \left(-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2}-y_{6} \, \mathbf{a}_{3} & = & -y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg II} \\ \mathbf{B}_{11} & = & \left(y_{7}+z_{7}\right) \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2} + y_{7} \, \mathbf{a}_{3} & = & y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg III} \\ \mathbf{B}_{12} & = & \left(-y_{7}+z_{7}\right) \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2}-y_{7} \, \mathbf{a}_{3} & = & -y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg III} \\ \mathbf{B}_{13} & = & \left(y_{8}+z_{8}\right) \, \mathbf{a}_{1} + z_{8} \, \mathbf{a}_{2} + y_{8} \, \mathbf{a}_{3} & = & y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg IV} \\ \mathbf{B}_{14} & = & \left(-y_{8}+z_{8}\right) \, \mathbf{a}_{1} + z_{8} \, \mathbf{a}_{2}-y_{8} \, \mathbf{a}_{3} & = & -y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg IV} \\ \mathbf{B}_{15} & = & \left(y_{9}+z_{9}\right) \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2} + y_{9} \, \mathbf{a}_{3} & = & y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg V} \\ \mathbf{B}_{16} & = & \left(-y_{9}+z_{9}\right) \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2}-y_{9} \, \mathbf{a}_{3} & = & -y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg V} \\ \mathbf{B}_{17} & = & \left(y_{10}+z_{10}\right) \, \mathbf{a}_{1} + z_{10} \, \mathbf{a}_{2} + y_{10} \, \mathbf{a}_{3} & = & y_{10}b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg VI} \\ \mathbf{B}_{18} & = & \left(-y_{10}+z_{10}\right) \, \mathbf{a}_{1} + z_{10} \, \mathbf{a}_{2}-y_{10} \, \mathbf{a}_{3} & = & -y_{10}b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Mg VI} \\ \mathbf{B}_{19} & = & \left(y_{11}+z_{11}\right) \, \mathbf{a}_{1} + z_{11} \, \mathbf{a}_{2} + y_{11} \, \mathbf{a}_{3} & = & y_{11}b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{Zn V} \\ \mathbf{B}_{20} & = & \left(-y_{11}+z_{11}\right) \, \mathbf{a}_{1} + z_{11} \, \mathbf{a}_{2}-y_{11} \, \mathbf{a}_{3} & = & -y_{11}b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(4d\right) & \text{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} & = & x_{12}a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8e\right) & \text{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} & = & -x_{12}a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8e\right) & \text{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} & = & x_{12}a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8e\right) & \text{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} & = & -x_{12}a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8e\right) & \text{Zn VI} \\ \end{array}$

### 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 MgZn5, Zeitschrift für Kristallographie – Crystalline Materials 86, 423–438 (1933), doi:10.1524/zkri.1933.86.1.423.
• L. W. McKeehan, Note on MgZn and MgZn5, Zeitschrift für Kristallographie – Crystalline Materials 91, 501–503 (1935), doi:10.1524/zkri.1935.91.1.501.
• E. Parthé, L. Gelato, B. Chabot, M. Penso, K. Cenzual, and R. Gladyshevskii, in Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types (Springer–Verlag, Berlin, Heidelberg, 1993), Gmelin Handbook of Inorganic and Organometallic Chemistry, vol. 2, chap. Crystal Chemical Characterization of Inorganic Structure Types, 8 edn., doi:10.1007/978-3-662-02909-1_3.
• 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), 2nd 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.

### Prototype Generator

aflow --proto=AB_oI48_44_6d_ab2cde --params=